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Investigations of Mie resonance-mediated all dielectric functional metastructures as component-less on-chip classical and quantum optical circuits
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Investigations of Mie resonance-mediated all dielectric functional metastructures as component-less on-chip classical and quantum optical circuits
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Content
Investigations of Mie Resonance-Mediated All Dielectric
Functional Metastructures as Component-less On-Chip
Classical and Quantum Optical Circuits
by
Swarnabha Chattaraj
A Dissertation Presented to the
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHEN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(ELECTRICAL ENGINEERING)
August 2021
Copyright 2021 Swarnabha Chattaraj
ii
Dedication
To my parents
iii
Acknowledgements
First and foremost, I thank my professor, Prof. Anupam Madhukar, who has
been a teacher, a scientific and moral beacon. Far beyond his enormous contribution in
my scientific growth, he has set up an extraordinary example of zeal for understanding
the world simply, and an impeccable example of values and integrity. Through
numerous examples, he has reminded of the words of Tagore everyday-- “Where tireless
striving stretches its arms towards perfection .. ”. I will be lucky to have imparted with
a fraction of his scientific and personal teachings in my life.
I have also been fortunate to have known and worked with Dr. Siyuan Lu. His
professional help to this dissertation work is immense without which this dissertation
will not exist. I will always aspire to the scientific spirit and excellence he represents.
His personal help and guidance throughout these years have been equally important for
which I am even more grateful.
As a part of the NMDL family I have been lucky to get to know many wonderful
scientists in person and through their work. I thank all of them for setting up such a rich
heritage and great standards. I have been fortunate to have worked in a daily basis with
friends like Dr. Jiefei Zhang, Lucas Jordao and Qi Huang, and always take pride in the
solidarity, honesty, and camaraderie that exist between us. I have also been lucky to
have known and worked with Dr. Zach Lingley in my initial year.
iv
Especially I would like to express my gratitude to Dr. Jiefei Zhang who has been
not only a friend and colleague but also a mentor throughout my journey and helped me
countless times in professional and personal fronts. Also, this dissertation work was
inspired by her dissertation work on the mesa-top single quantum dot single photon
sources and would not exist without it.
I am thankful to Prof. Anupam Madhukar, Prof. Hossein Hashemi, Prof. Alan
Willner and Prof. Paolo Zanardi for kindly serving on my Ph.D. Dissertation
Committee.
Finally, I thank Army Research Office and Air Force Office of Scientific
Research for providing financial support for this work and making this dissertation work
possible. I also thank the Viterbi Graduate Fellowship that supported me in the initial
phase of my studies.
v
Table of Contents
Dedication ii
Acknowledgements iii
List of Figures viii
List of Tables xxi
Abbreviations xxii
Abstract xxiii
Chapter 1. Introduction 1
§1.1 Central Theme and Motivation 1
§1.2 Concept of Mie Resonance 12
§1.3 Organization of the dissertation 16
§1.4 References 18
Chapter 2. Nanoantenna-- Emission Rate Enhancement and Directionality 24
§2.1 Introduction and Motivation 24
§2.1.1 SPS-Optical Circuit Integration: Status and Requirements 25
§2.2 Fundamental Concepts of SPS- DBB Array Coupling 36
§2.2.1 Principle of Nanoantenna- Purcell Enhancement 38
§2.2.2 EM Green-Function to Density of States 39
§2.2.3 Mie Resonances of Spherical DBBs: 43
§2.2.4 Principle of Nanoantenna-Directionality 48
§2.3 The Primitive Nanoantenna: A dipole coupled to a single DBB 50
§2.3.1 Mie Resonance of DBBs of non-spherical shape-
Multipole Decomposition 50
§2.3.2 Design of Nanoantenna 55
§2.3.3 Effect of Dielectric Membrane 62
§2.4 Yagi-Uda Nanoantenna 68
§2.4.1 Principles of Yagi-Uda Nanoantenna 70
§2.4.2 Design of the Nanoantenna Structure: 73
§2.4.3 Effect of Membrane: 76
§2.4.4 Why use a Nanoantenna? 80
§2.5 Summary and Conclusion 82
§2.6 References 85
Chapter 3. From Nanoantenna towards Optical Circuits--
Nanoantenna-Waveguide 90
vi
§3.1 Motivation and Background 90
§3.1.1 Key Figures of Merits of an Optical Circuit 91
§3.1.2 Current Approaches in the Literature: 94
§3.1.3 Our approach 96
§3.2 The Nature of the Propagating Mode of a DBB Array 98
§3.2.1 Existing Approaches to On-Chip Waveguiding 98
§3.2.2 Dispersion Characteristics of DBB Arrays 100
§3.2.3 Light Propagation via Array of DBBs 110
§3.3 Experimental Studies: Silicon DBB Linear Array Based Waveguiding 121
§3.3.1 Fabrication and Structural Characterization 122
§3.3.2. Optical Measurement Setup and Results 128
§3.3.2.a Vertical Optical Access via Objective Lens: 129
§3.3.2.b Horizontal Excitation-Detection Measurement using a Lensed Fiber 136
§3.4. Nanoantenna-Waveguide 142
§3.4.1 Transition from Nanoantenna to Nanoantenna-Waveguide 143
§3.4.2 Local Density of States of Photon: 152
§3.4.3. Response of the Nanoantenna-Waveguide: 154
§3.5 Beam-Splitting and Beam-Combining 157
§3.5.1 Conventional Approach to On-Chip Beam Splitter: Directional Coupler 158
§3.5.2 Alternative Architecture of
Nanoantenna / Waveguide / Beam-Splitter/ Beam-Combiner System: 161
§3.5.3 Beamsplitter-Combiner and Photon Interference: 164
§3.6 Conclusion and Next Steps. 167
§3.7 References 170
Chapter 4. SPS-SPS Coupling and Super Radiance
in Mie-Resonance Based Optical Circuits: Classical Approach 175
§4.1 Introduction and Motivation 175
§4.1.1 Importance of Coherence of the On-Chip SPSs 177
§4.1.2 Synchronization in Collective Emission- Background 180
§4.1.3 Our Approach 182
§4.2 Classical Formulation of Super Radiance using EM Green Function 184
§4.2.1 Green Function for Multiple SPS Emitters 185
§4.2.2 Super-radiant and Sub-radiant States 187
§4.3 Design of SPS-SPS Coupling Metastructure 190
§4.4 Summary and Conclusions 196
§4.5 References 197
Chapter 5. Emergence of SPS-SPS Coherence- Quantum Approach 201
§5.1 Introduction and Motivation 201
§5.2 Formulation of Coherent Emission: Quantum Approach 205
§5.2.1 SPS-SPS Energy Transfer 224
§5.2.2. SPS-SPS Collective Emission 227
vii
§5.3 Effect of Measurements—Inclusion of the Detector States 229
§5.3.1 One Detector 229
§5.3.2 HBT Measurement Outcome- Two Detectors 235
§5.4 Summary and Conclusions 239
§5.5 References 242
Chapter 6. Conclusions and Outlook 245
§6.1 Mie Resonant Metastructures Established as Viable Approach
to On-Chip Light Manipulating Elements and Units 248
§6.2 Mie Resonance Established as a Viable Approach
towards Collective Emission and Entanglement of Multiple SPSs 251
§6.3 Outlook 254
§6.3.1 Extending the Quantum Approach to More Than 2 SPSs 254
§6.3.2 General Quantum Approach to Mie Scattering 255
§6.3.3 Topological Photonics and Applicability of Mie Resonance 255
§6.3.4 Integration of Nanoantenna with MTSQD SPSs 256
§6.4 References 258
Bibliography 260
Appendix A: Mie Theory of Spherical DBBs 270
Appendix B: Green function and Local Density of States 281
Appendix C: Instrumentation: Optical Measurement on DBB Array 297
viii
List of Figures
Figure. 1.1. A generic schematic depiction of on-chip quantum optical
circuit based on integrated on-chip single photon sources and optical
circuit comprising the key functional elements of cavity/ nanoantenna,
waveguiding, splitting and combining for manipulation of the emitted
photons in the on-chip architecture.
2
Figure 1.2. (a) SEM image of MTSQD array, (b) Single photon emission
behavior of the MTSQD array shown by the g
(2)
plot, (c) Planarized
MTSQD array as a platform for integration with light manipulating
circuits. (d)-(e) Overview of the existing approaches to on-chip light
manipulation. (f) The new approach introduced and studied in this
dissertation. 7
Figure 1.3. Schematic showing our approach of on-chip DBB array (blue
blocks) based multifunctional optical circuit embedded with on-chip SPSs
(Pyramids) and array of detectors (purple blocks) that exploits on-chip
photon interference to create path entanglement. 12
Figure. 1.4. Schematic representation of the E-field on the interface of a
high index DBB. The boundary condition enforces the E-field inside the
interface to be dominantly tangential, thus resulting in dominant magnetic
resonances. 13
Figure 1.5. (a) Schematic representation of interference between electric
and magnetic resonance resulting in directionality without strong field
localization. (b) Schematic representation of mode-mode coupling
resulting in guiding. 15
Figure 2.1. Overall vision of On-chip quantum optical circuits based SPS
array (such as the MTSQDs) codesigned and integrated with dielectric
building block (DBB) metastructure based multifunctional quantum
optical circuits. The nanoantenna component is indicated with a red box. 24
ix
Figure 2.2. A schematic drawing of the typical excitation and photon
emission in a QD SPS- indicating the key processes that result in
dephasing of the emitted photon. 26
Figure 2.3 (a) High resolution PL of the MTSQD indicating a dephasing
time ~100ps. (b) time resolved PL measurement on the MTSQD
indicating radiative decay lifetime of ~1ns [2.7]. 28
Figure 2.4. (a) Schematic drawing of MTSQD on GaAs substrate and (b)
the angular distribution of the Poynting vector on a spherical enclosing
surface of radius 3𝜇𝑚 for a radiating transition dipole of 1 Debye strength
at 930nm indicating the angular distribution of the emitted photons and
pointing to the fact that most of the emitted photons are lost into the
substrate. (c) The MTSQD on the GaAs mesa on a 65.5nmGaAs/78.5nm
AlAs DBR in pillar structure of 2um height with (d) showing the
corresponding Poynting vector angular distribution. 30
Figure 2.5. (a) Schematic drawing of MTSQD on GaAs substrate and (b)
the angular distribution of the Poynting vector on a spherical enclosing
surface of radius 3𝜇𝑚 for a radiating transition dipole of 1 Debye strength
at 930nm indicating the angular distribution of the emitted photons and
pointing to the fact that most of the emitted photons are lost into the
substrate. (c) MTSQD mesa- now placed on a 1𝜇𝑚 thick membrane of
dielectric of refractive index ~1.5 supported by GaAs substrate and (d)
corresponding angular distribution of photon flux indicating a significant
reduction in number of photons lost into the substrate. 33
Figure 2.6 (a) A general schematic representing a single photon source
coupled to a DBB metastructure antenna. (b) schematically shows the
transition between the ground level electron and hole state of the SPS
approximated as a two level system populating a photon of energy 𝜔 𝑆𝑃𝑆 -
with a spontaneous emission rate proportional to the local density of states
of photons at that particular position of that particular energy. 37
Figure. 2.7 (a) A single isolated spherical DBB of radius 130nm and
refractive index 3.5. (b) Photon local density of states plotted along line
AB in (a). x represents the position from the center of the DBB. (c)
Schematic capturing the physics of Magnetic resonance- the reason why
the LDOS at the surface is enhanced. 42
Figure 2.8. (a) and (b) Mie resonance scattering cross section spectra of a
single spherical DBB of radius 130nm and refractive index 3.5. (c)
Scattering cross section where the wavelength has been normalized to the
x
diameter of the DBB- indicating that the electric and magnetic dipole
modes exist when the diameter of the DBB ~ wavelength of light in
dielectric. 47
Figure. 2.9. A schematic showing the notion of radiation pattern by a
nanoantenna structure. 49
Figure 2.10. (a) and (b) Mie resonance scattering cross section spectra of
a single cubic DBB of size 220nm and refractive index 3.5. (c) Scattering
cross section where the wavelength has been normalized to the diameter of
the DBB- indicating that the electric and magnetic dipole modes exist
when the diameter of the DBB ~ wavelength of light in dielectric. 52
Figure 2.11. The Electric and Magnetic dipole mode peak wavelengths in
a single rectangular DBB as a function of (a) size in the Z direction
(parallel to the incident magnetic field) and (b) size in the Y direction
(parallel to the incident E-field). 54
Figure. 2.12. (a) E-field distribution within the DBB associated with the
Magnetic dipole mode with dipole moment along Z (MDZ). (b) Radiation
pattern corresponding to the MDZ mode. (c) and (d) panels show the
corresponding plots for the EDY mode (Electric dipole mode along Y
direction) 56
Figure 2.13. A generic schematic of the primitive nanoantenna structure. 57
Figure 2.14. Multipole decomposition (amplitude and phase) of a
primitive nanoantenna comprising a dipole source embedded in a cubic
DBB of size 220nm and refractive index 3.5. As the position of the dipole
is shifted from the center of the DBB, the amplitude of the MDZ mode
goes up and consequently attains higher directionality. 59
Figure 2.15. Directivity spectrum for primitive nanoantenna for four
different cases of different positions of the source dipole with respect to
the center of the DBB. 60
Figure 2.16. (a) The geometry of the single DBB with embedded SPS
transition dipole and (b) the spatial distribution of local density of states at
980nm on an XY plane passing through the center of the DBB. (c) The
spectrum of the density of state for different fixed positions of the
transition dipole and (d) resultant Purcell enhancement spectrum for the
primitive nanoantenna structure. 61
xi
Figure 2.17. Comparative Poynting vector angular distribution (Panel
(b),(d),(f), (h)) corresponding to the schematics shown in panel (a), (c), (e),
(g) correspondingly where the thickness of a low-index (1.5) dielectric
membrane under the SPS in a DBB is increased gradually. It is evident
that 1𝜇𝑚 thickness of the dielectric membrane is enough to cut-off the
effect of the underlying GaAs substrate completely. 64
Figure 2.18. Primitive nanoantenna on membrane- Plot of |EY| distribution
on YZ cross-section plane on the front and back face of the DBB as well
as ~270nm away from the surface of the DBB for (a) DBB on bulk GaAs
(light blue here) (b) DBB on a 500nm thick membrane of refractive index
1.5 (light green) (c) 1𝜇𝑚 thick membrane and (d) the extreme case of
semi-infinite membrane. The top of the DBB is assumed to be of
refractive index 1.5 representing a protective polymer. 65
Figure 2.19. Primitive nanoantenna on membrane- Plot of |HZ| distribution
on YZ cross-section plane on the front and back face of the DBB as well
as ~270nm away from the surface of the DBB for (a) DBB on bulk GaAs
(light blue here) (b) DBB on a 500nm thick membrane of refractive index
1.5 (light green) (c) 1𝜇𝑚 thick membrane and (d) the extreme case of
semi-infinite membrane. The top of the DBB is assumed to be of
refractive index 1.5 representing a protective polymer. 66
Figure 2.20. Primitive nanoantenna on membrane- Plot of |SX| (Poynting
vector/energy flow towards X) distribution on YZ cross-section plane on
the front and back face of the DBB as well as ~270nm away from the
surface of the DBB for (a) DBB on bulk GaAs (light blue here) (b) DBB
on a 500nm thick membrane of refractive index 1.5 (light green) (c) 1𝜇𝑚
thick membrane and (d) the extreme case of semi-infinite membrane. The
top of the DBB is assumed to be of refractive index 1.5 representing a
protective polymer.. 67
Figure 2.21 (a) A microwave Yagi-Uda antenna used in day-to-day
communication. (b) Nanoscale DBB metastructure based Yagi-Uda
antenna. (c) The basic principle of the Yagi-Uda antenna architecture. 69
Figure 2.22. (a) Radiation pattern of a dipole embedded in a single DBB ,
compared to dipole embedded in (b) yagi-Uda antenna. (c) Electric field
distribution to the +X and -X direction, shown as a superposition of the E-
fields generated by the reflector, feed and director- indicating that the
directionality is resulted by constructive interference in +X and destructive
interference in -X direction. (d) Photon local density of state distribution at
980nm on the XY plane passing through the center of the DBBs. (e)
xii
Directivity spectrum of the Yagi-Uda antenna (blue) compared with the
primitive single DBB antenna (dashed black). (f) Purcell enhancement
spectrum of the Yagi-Uda and primitive antenna. 72
Figure 2.23. Effect of the size of reflector in the Y direction on (a)
Directivity and (b) Photon LDOS. 74
Figure 2.24. Effect of the pitch of the reflector (P1) and director (P2) on
directivity and Green function (photon LDOS) 75
Figure 2.25. Response surface of the Directivity and Purcell Factor as a
function of P1 and P2, the pitch of the reflector-feed and director-feed.. 75
Figure 2.26. Yagi-Uda nanoantenna on membrane- Plot of |EY| distribution
on YZ cross-section plane on the front and back face of the DBB as well
as on the surface of the reflector and director DBBs for (a) underlying
bulk GaAs (light blue) (b) on a 500nm thick membrane of refractive index
1.5 (light green) (c) 1𝜇𝑚 thick membrane and (d) the extreme case of
semi-infinite membrane. The top of the DBB is assumed to be of
refractive index 1.5 representing a protective polymer. 77
Figure 2.27. Yagi-Uda nanoantenna on membrane- Plot of |HZ| distribution
on YZ cross-section plane on the front and back face of the DBB as well
as on the surface of the reflector and director DBBs for (a) underlying
bulk GaAs (light blue) (b) on a 500nm thick membrane of refractive index
1.5 (light green) (c) 1𝜇𝑚 thick membrane and (d) the extreme case of
semi-infinite membrane. The top of the DBB is assumed to be of
refractive index 1.5 representing a protective polymer. 78
Figure 2.28. Yagi-Uda nanoantenna on membrane- Plot of distribution of
the x directional Poynting vector ( SX ) on YZ cross-section plane on the
front and back face of the DBB as well as on the surface of the reflector
and director DBBs for (a) underlying bulk GaAs (light blue) (b) on a
500nm thick membrane of refractive index 1.5 (light green) (c) 1𝜇𝑚 thick
membrane and (d) the extreme case of semi-infinite membrane. The top of
the DBB is assumed to be of refractive index 1.5 representing a protective
polymer. 79
Figure 2.29. Schematic drawing indicating photon collection from the
nanoantenna in the horizontal architecture. 80
xiii
Figure 2.30. Comparison of the spectral width of the response of (a) a
typical photonic crystal L3 cavity [2.6] and (b) DBB metastructure based
Yagi-Uda nanoantenna 81
Figure 2.31. The nanoantenna structure enables realization of optical
circuits by coupling it to continuous waveguides or DBB array
waveguides – discussed in the next chapter. 84
Figure 3.1. A schematic of the paradigm of on-chip optical circuits based
on DBB metastructures showing the different needed components. 90
Figure 3.2. Our approach to on-chip optical circuit providing all the
needed functions via a single collective Mie resonance. 97
Figure 3.3. Schematic showing an infinite array of spherical DBBs for
which we solve the photonic band-structure in the following to gain
understanding on the dispersion characteristics of the propagating
collective Mie mode. 101
Figure 3.4. Band-structure of an array of spherical DBBs of radius 130nm
and refractive index of 3.5 for different values of the center-center spacing.
Panel (a), (b), and (c) shows the band structure for p=275nm, 300nm and
375nm in the 𝛽 − 𝑘 𝐵 plane. The dashed line indicates the light cone-𝛽 =
𝑘 𝐵 line. Panel (d)-(f) shows the corresponding band structure where the
vertical axis displays wavelength.. 108
Figure. 3.5. Phase-space of the photon band-structure showing the
wavelength of the collective Mie mode as a function of its Bloch wave-
vector and the pitch (center-center spacing of the DBBs). The DBBs are
chosen to be 130nm in radius and refractive index ~3.5. 109
Figure 3.6.(a) Light propagation via an array of 40 DBBs of spherical
shape. (b) response in the phase space of varying wavelength of light and
pitch of the array. 111
Figure 3.7. The 𝜆 − 𝑃 phase space response of array of spherical DBBs
(same plot as Figure 3.6(b))- indicating the Fabry-Perot resonant fringes. 112
Figure 3.8. FEM simulation of propagation of a plane incident wave
through an array of 20 cubic DBBs of size 220nmx220nmx220nm and
refractive index 3.5. Panel (b) shows the 𝜆 − 𝑃 phase space of the MD
xiv
mode amplitude of the last DBB of the array- a plot like fig. 3.7 for
spherical DBBs. 114
Figure 3.9. Distribution of the Ey field on a XY plane passing the center of
the DBB array. The array here comprises of 20 DBBs of cubic size of
220nm and pitch of 275nm. The Fabry-Perot oscillation along the length
of the array are identified with red dashed curves. The color scale of all
the plots are the same and only shown once. 116
Figure 3.10. Distribution of the HZ field on a XY plane passing the center
of the DBB array. The array here comprises of 20 DBBs of cubic size of
220nm and pitch of 250nm. The Fabry-Perot oscillation along the length
of the array are identified using red dashed lines. The color scale of all the
plots are the same and only shown once. 117
Figure 3.11. Cross-sectional distribution of the (a) EY and (b) HZ fields
and (c) Poynting vector along x direction, for the linear DBB array of
20DBB length for a plane incident wave of wavelength 980nm. The mode
profile indicate that the EY and HZ field attains maximum at the center of
the waveguide cross-section- consistent with the propagating mode being
the EDY and MDZ type. 118
Figure 3.12. Partial dispersion characteristics of the DBB array for three
different value of pitch – where the Bloch wave-vector is inferred from the
spatial distribution of the E- and H-fields as indicated in Fig. 3.9-3.10. 119
Figure 3.13. Fabricated DBB array on SOI wafer. (a) Top-view of the
HSQ mesas on SOI used as etching masks. (b) Tilted SEM image of the
fabricated DBB arrays comprising DBB of size ~200nmx200nmx200nm
and separation of ~50nm. Panel (c) and (d) shows top-view and side-view
of the constituent BDBs of the array. 127
Figure 3.14. Two experimental measurement geometry studied for
characterization of the DBB array in this dissertation work. 128
Figure. 3.15. (a) Simulation geometry of array of 40DBBs of size 200nm
cube and refractive index 3.8 mimicking Si- with pitch of the array 250nm
and surrounding medium 1.5 representing the SiO2 membrane and
surrounding microscope oil for the measurement. (b) The MD and ED
mode amplitudes along the DBB array at 900nm. (c) The expected
spectrum of collected photons in the vertical-vertical measurement
geometry. This shows good resemblance with the experimentally acquired
spectrum. 130
xv
Figure 3.16.- (a) Experimental measurement geometry and (b) optical
setup. 132
Figure 3.17.-(a) Measurement geometry of the angle-integrated photon
collection. (b), (c) and (d) Result from three distinct DBB arrays of
experimental measurement of far-field angle-dependent scattering
spectroscopy with vertical incident and vertical detection geometry (e)
The three spectra plotted on the same graph- showing the existence of the
Fabry-Perot fringes in all the three spectra. The existence of these fringes
indicates lossless/ near-lossless on-chip propagation. 134
Figure 3.18 (a) Angle dependent photon collection- simulation and
experimental result (b) Key physics governing the photon directionality. 135
Figure 3.19. Experimental geometry for measurement of transmission
efficiency via DBB array in horizontal geometry using Objective lens for
excitation and lensed optical fiber for detection. 137
Figure 3.20. (a) Simulation geometry mimicking the horizontal
excitation/detection measurement on DBB array waveguides shown in Fig.
3.14(b). (b) Magnetic field distribution on the XY plane passing through
the center of the DBBs indicating the collective magnetic dipole mode of
the DBB array at ~930nm. (c) Estimated overall transmission spectrum
including effect of insertion efficiency and collection efficiency of the
photons into the optical fiber. 139
Figure 3.21. (a) Measured transmission via three DBB arrays of length
24𝜇𝑚 , 41𝜇𝑚 and 60𝜇𝑚 - showing the collective Mie resonance around
800nm and ~1% overall transmission efficiency. (b)The peak transmission
as a function of the length of the DBB array 12 different DBB arrays of
different lengths in the range of 24𝜇𝑚 and 60𝜇𝑚 . The dashed straight line
indicates ~70 to 100dB/mm propagation loss. 140
Figure 3.22.(a) Nanoantenna-Waveguide structure- the Yagi-Uda
nanoantenna structure discussed in Chapter 2 now integrated with a
waveguide segment of N DBBs- (b)-(f) shows the E-field distribution for a
1debye transition dipole emitting at 980nm for different numbers of DBBs
in the waveguide segment- 1,2,3,5,and 10 respectively. E-field symmetry
indicates that the same collective Mie mode based on Magnetic dipole and
electric dipole mode is now extended for the whole “nanoantenna-
waveguide” unit. 144
xvi
Figure 3.23. Distribution of the EY field on YZ cross-sectional planes for
the nanoantenna-waveguide structure with varying number of DBBs in the
waveguide section shown in panel (a): only Yagi-Uda nanoantenna (b)
nanoantenna+1DBB (c) Nanoantenna+2DBBs (d) Nanoantenna+5DBBs
and (e) Nanoantenna+10DBBs for a transition dipole of 1 debye strength
radiating at 980nm representing the SPS. 145
Figure 3.24. Distribution of the HZ field on YZ cross-sectional planes for
the nanoantenna-waveguide structure with varying number of DBBs in the
waveguide section shown in panel (a): only Yagi-Uda nanoantenna (b)
nanoantenna+1DBB (c) Nanoantenna+2DBBs (d) Nanoantenna+5DBBs
and (e) Nanoantenna+10DBBs for a transition dipole of 1 debye strength
radiating at 980nm representing the SPS. 146
Figure 3.25. Distribution of the x-directional Poynting vector on YZ cross-
sectional planes for the nanoantenna-waveguide structure with varying
number of DBBs in the waveguide section shown in panel (a): only Yagi-
Uda nanoantenna (b) nanoantenna+1DBB (c) Nanoantenna+2DBBs (d)
Nanoantenna+5DBBs and (e) Nanoantenna+10DBBs for a transition
dipole of 1 debye strength radiating at 980nm representing the SPS. 147
Figure 3.26.- Evolution of Purcell enhancement as the nanoantenna
structure is extended gradually to nanoantenna waveguide. 148
Figure 3.27. - Evolution of coupling efficiency (𝛽 ) as the nanoantenna
structure is extended gradually to nanoantenna waveguide. 149
Figure. 3.28.- Peak Purcell enhancement and coupling efficiency (𝛽 ) as a
function of number of DBBs in the array extended from the nanoantenna.
Both the figures of merit saturate over ~5 DBBs (~1.5-2 𝜇𝑚 ) distance. 151
Figure 3.29. Spatial distribution of the Local density of photon states at
fixed wavelength of 980nm for the (a) nanoantenna-waveguide structure
where the Yagi-Uda nanoantenna is combined with array of 10DBBs as
waveguide. The nanoantenna comprises of reflector of size
220nmx250nmx220nm. The Feed DBB, director, and waveguide DBBs
are of cubic shape and size 220nm. The surface-to-surface separation
between the DBBs are 55nm. Note- in the feed DBB- the DBB bearing the
QD in our design, the density of states obtains a maximum ~50nm from
the center of the DBB- at which point the QD is placed. 153
xvii
Figure 3.30. Spectrum of local photon density of states at the location of
the SPS transition dipole for different numbers of DBBs in the DBB array
integrated with the nanoantenna. Vertical division for each of the plots is
12× 10
30
states/m
3
/ J. 154
Figure 3.31. (a) Yagi-Uda nanoantenna-waveguide structure. The SPS
transition dipole is embedded in a cubic DBB of size 220nm. The reflector
DBB of the nanoantenna is of size 220nm×250nm×220nm, where as the
director DBB and the waveguide DBBs are of cubic shape of size 220nm.
(b) Angular distribution of the photon flux on a spherical surface (radius ~
500nm) surrounding the nanoantenna for a radiating dipole of strength 1-
debye at 980nm. The asymmetry in this angular distribution indicates the
nanoantenna effect. (c) Finite element method-based calculated spatial
distribution in the nanoantenna-waveguide of the electric field for a 1-
debye point oscillating electric dipole source representing the SPS
emitting at 980nm. (d) Purcell enhancement spectrum indicating
broadband response with Purcell enhancement ~7. 156
Figure 3.32. Conventional directional coupler structure using DBB array.
The nanoantenna and DBB array waveguide has same design as the
structure presented in Fig. 3.26. The coupling region is ~1𝜇𝑚 in length.
The surface-surface separation between the two parallel sections of the
DBB arrays in the coupled region (𝑑 𝐺𝑎𝑝
) is varied and the resultant E-field
distribution is shown in the different panels. 159
Figure 3.33. The ratio of photon flux (integrated Poynting vector over the
waveguide cross-section) at the two output branches of the structure
shown in Fig. 3.31- when only one of the SPSs is emitting photon- plotted
as a function of the surface-surface separation of the two parallel DBB
array waveguides in the coupling section. 160
Figure 3.34 (a) Schematic showing the nanoantenna-waveguide-
beamsplitter-combiner circuit. The junction comprises of a DBB of
cylindrical shape of diameter 230nm. The rectangular DBB size and pitch
are identical to the nanoantenna-waveguide structure discussed before. (b)
E-field distribution of the collective Mie mode of the whole circuit when
only SPS1 transition dipole is emitting. (b) E-field distribution of the
collective Mie mode in the spatial region of the beam-splitter that
participates in the equal splitting of the photons into the two branches. (c)
Cross sectional Poynting vector distribution of the input waveguide
section and the two output branches of the beamsplitter. (e) E-field
distribution of the collective Mie mode in the spatial region of the beam-
combiner. (c) Cross sectional Poynting vector distribution of the two input
waveguide sections and the output branch for the beamcombiner part. 163
xviii
Figure 3.35. (a) Nanoantenna-waveguide-beamsplitting-combining circuit
based on rectangular DBBs. (b) Distribution of Ey around the SPS1 when
only SPS1-transition dipole is emitting. (c) Purcell enhancement spectrum
when only SPS1 is emitting. (d) The Ey distribution of the collective Mie
resonance of the whole unit when the only SPS1 is emitting: showing the
splitting of the photon at the Y-junction. (e) The Ey distribution of the
collective Mie resonance of the unit when the both the SPSs are
emitting—showing recombining of the photons in the common branch. 166
Figure 4.1. The nanoantenna-waveguide-splitter-combiner metastructure
towards DBB based optical circuits. 175
Figure 4.2. A schematic drawing of the typical excitation and photon
emission in a QD SPS- indicating the key processes that result in
dephasing of the emitted photon. 178
Figure 4.3. Our Approach of MTSQD-MTSQD Coupled structure via a
collective Mie mode of the DBB array. 183
Figure 4.4. Representing the second-order Green function when the both
the emitters interact with the emitted photon. 187
Figure 4.5. complex poles corresponding to the super-radiant and sub-
radiant emission of the two coupled QDs. 189
Figure 4.6. (a) schematic of the back-to-back nanoantenna waveguide
structure. E-field distribution of when only (b) SPS1 and (c) SPS2 is
emitting a photon. (b) and (c) thus represents the classically computed
zeroth order Green function as discussed in the previous section. (d)
Spectrum of the Green function component 𝐺 11
(𝜔 ) = 𝑝 ̂ 1
⋅ 𝐺 ̿ (𝑟 ̅ 1
, 𝑟 ̅ 1
, 𝜔 ) ⋅ 𝑝 ̂ 1
- at the location of the emitter QD (e) Spectrum of the Green function
component 𝐺 12
(𝜔 ) = 𝑝 ̂ 1
⋅ 𝐺 ̿ (𝑟 ̅ 1
, 𝑟 ̅ 2
, 𝜔 ) ⋅ 𝑝 ̂ 2
– representing the propagator
from one QD to the other. 192
Figure 4.7. The emission rate of the sub-radiant and super-radiant states
for the system shown in Fig. 4.6(a). 193
Figure 4.8. Dependence of the Green function on the distance between the
two emitters. (a) and (b) Back-to-back nanoantenna waveguide structure
where the number of DBBs of the waveguide section is varied to change
the emitter-emitter distance. For reference, (c) and (d) two emitters
coupled via a continuous waveguide; and (e) and (f) via free-space
propagation in a uniform medium. 195
xix
Figure 5.1. (a) A generic on-chip SPS-DBB integrated quantum optical
circuit (b) The specific simple two-SPS coupled back-to-back nanoantenna
waveguide structure investigated in this chapter.. 201
Figure 5.2. Flowchart capturing the content and interconnection of the
different chapters to the overall theme of this dissertation: on-chip
quantum optical circuits 203
Figure 5.3. A single SPS with the nanoantenna-waveguide from the
quantum mechanical point of view. 205
Figure 5.4. Contour for integrating the expression for Green function in
the 𝜔 -sum in equation (5.11). 209
Figure 5.5. (a) and (b) represents the DBB array nanoantenna waveguide
structure (same as in Chapter 3) and the conventional ridge waveguide
structure of 220nmx220nm cross section coupled to a single SPS. (c) and
(d) shows the cross-sectional Poynting vector distribution for the
Nanoantenna waveguide and the conventional waveguide for the forward
direction and reverse direction- demonstrating the nanoantenna effect. (e)
and (f) shows the E-field distribution on the XY plane passing though the
center of the DBBs- indicating a higher E-field enhancement and higher
coupling to the guided mode in the case of the nanoantenna waveguide
compared to the standard ridge waveguide. Panel (e) and (f) share the
same color-scale to the right. 213
Figure 5.6. System of two SPSs coupled via the collective Mie mode, but
also independently decaying into radiation continuum. 215
Figure 5.7. Decay pathways for the two-SPS coupled system. 218
Figure 5.8. Decay pathways of the Hamiltonian and their connection to the
multiple scattering Green function as discussed in the last section. 220
Figure 5.9. Exchange of photon between two SPSs mediated by the Mie
resonance—simulation using the master equation approach when at t=0,
only SPS1 is in the excited state. (b) shows the overall decay process-
involving energy exchange between the two SPSs. This is seen in the
occupation probability calculated from the Master equation shown in (c)
and (d)- in the single SPS basis and the super-radiant-and sub-radiant basis
respectively.
226
xx
Figure 5.10. (a) SPS-SPS coupled system in the back-to-back nanoantenna
waveguide structure with (b) and (c) indicating the decay process when
the initial state is | 𝑒 1
𝑒 2
⟩. (d) and (e) show the occupation probability of
the basis states in the single-SPS basis and the super-and sub-radiant basis
respectively. 228
Figure 5.11. A schematic indicating the detector state included into the
Hilbert space of the system. (a) including the state of a single detector
represented by a 2 level system and (b) two 2-level systems representing
two detector arranged in HBT set-up. 230
Figure 5.12. Structure of the density matrix when the two emitters are
joined by a single detector, also approximated as a two level system. 231
Figure 5.13. (a) Two SPSs coupled via the back-to-back nanoantenna
waveguide structure in the presence of a single detector (b) decay
pathways when the system starts with both SPSs excited. (c) The
probability of the super-radiant and the sub-radiant state once a single
photon has been detected. (d) The Von-Neumann entropy of the two SPSs,
and a single SPS, and their difference indicating “shared information”
between the two SPSs. 234
Figure 5.14. (a) SPS-SPS coupled system and two detectors in HBT set-up.
(b)The transitions in the 16x16 dimensional density matrix that correspond
to the triggering of the start and the stop detectors in an HBT measurement.
237
Figure 5.15. Expected coincident count rate (normalized per excitation
pulse) for 2Γ = 5 /ns , Γ
+
~1.7 Γ, Γ
+
~0.3 Γ. Shown in (a) linear scale and
(b) log-scale- clearly showing the super-radiant and sub-radiant decay
channels. 238
Figure 6.1. A representative schematic of the MTSQD-DBB on-chip
quantum optical circuit that has been investigated in this thesis.. 246
Figure 6.2. The Back-to-back nanoantenna-waveguide structure studied in
this dissertation towards SPS-SPS coupling and entanglement via Mie
mode. 252
Figure 6.3. Planarized MTSQD SPSs towards integration with DBBs. 257
xxi
List of Tables
Table 3.1. Current approaches to efficient extraction/ harvesting of
photons to an on-chip optical circuit. 94
Table 3.2: Existing approaches to light guiding 99
Table 3.3. Fabrication Steps for creating the DBB array 122
Table 4.1. Current status in the literature of super-radiance/ emitter-emitter
coupling in on-chip optical circuits 181
Table 5.1. Basis transformation, from single to collective, and its impact
on the Hamiltonian 217
xxii
Abbreviations
DBB: Dielectric Building Block
DOS: Density Of States
EM: Electromagnetic
ED: Electric Dipole
HBT: Hanbury-Brown-Twiss
HOM: Hong Ou Mandel
LDOS: Local Density of States
LME: Light Manipulating Elements
LMU: Light Manipulating Unit
MD: Magnetic Dipole
MTSQD: Mesa Top Single Quantum Dot
PhC: Photonic Crystal
PLDOS: Partial/ Projected Local Density of States
QD: Quantum Dot
QIP: Quantum Information Processing
SEM: Scanning Electron Microscope
SPS: Single Photon Source
SQD: Single Quantum Dot
xxiii
Abstract
This dissertation proposes and explores a new and novel class of on-chip optical
circuits aimed at optical quantum information processing (QIP). Realization of on-
chip optical QIP systems demands conceiving, fabricating, and examining material
structures that exploit designed modulation in the refractive index (in the wavelength
regime of interest) to manipulate on-chip generated photons from their source
through their pathway to the detector where they are finally detected having
performed the many desired information carrying and transfer functions on the way.
Such light manipulating functional metastructures made of appropriate material
combinations laid out in planar architectures to mediate photon interactions with
other photons and material entities (typically electrons) are dubbed optical circuits
(OCs). Such OCs are to embed on-chip single photon sources (SPSs) for
manipulation of the emitted photons to enable on-chip photon interference and
establish communication between distinct SPSs for quantum entanglement as a
resource for quantum information processing. Generically the optical circuits are
required to provide such functions as (1) enhancement of emission rate of the SPS
(2) enhancement of emission directionality, (3) state-preserving propagation of
emitted photons, (4) splitting and (5) combining to enable interference between
photons emitted from pre-determined distinct on-chip SPSs. In the conventional
approaches to on-chip photon manipulation, these functions are implemented using
distinct functional structures, categorized as discrete components such as cavity,
waveguide, beam splitter etc.- using existing platforms such as 2D photonic crystal
and ridge waveguides and couplers. However, this modular approach also demands
xxiv
mode-matching between the distinct functional components such as cavity,
waveguide, etc.. This has always been an obstacle towards scalability. As a result, an
optical circuit that provides all the above-mentioned five functions eludes realization
to this day.
In contrast, we envision and pursue an approach fundamentally different from
this conventional way of thinking of an optical circuit as a collection of mode-
matched “components”. The physics we exploit is also new—the collective Mie
resonance of arrays (metastrucutres) of subwavelength scale dielectric building
blocks (DBBs) in contrast to Bragg scattering in photonic crystal. Engineering the
collective Mie resonance of a DBB based optical metastructure allows simultaneous
control over the E-field spatial distribution and dispersion characteristics of the
collective mode, which, in turn, allows to implement all the above noted needed five
light manipulating functions using the same collective Mie mode. Thus, on-chip
integration of SPS arrays with DBB metastructures opens a new paradigm for
quantum information processing applications. The conceptualization, theoretical
(classical and quantum) modelling, and numerical examination of this new paradigm
constitutes the core of this dissertation.
Our conceptualization and thus modelling of the DBB metastructure optical
circuits and judicious classical and quantum theoretical studies are guided by the
recently established mesa-top single quantum dots (MTSQDs) in spatially regular
arrays as on-chip scalable SPSs with considerable potential for providing sufficiently
spectrally uniform emission (< 2nm standard deviation) from arrays distributed over
1000µm
2
areas to enable proven local tuning methodologies to bring to resonance
xxv
photons emitted from different known emitters in the array to enable controlled
interference and entanglement. Thus the simulation studies undertaken are of DBB
units built around each MTSQD SPS of the array to provide-- using the same
collective Mie-mode of the interconnected units forming the metastructure dubbed
the optical circuit-- all the needed functions: enhancement of SPS emission rate and
directionality, propagation, splitting, and recombining aimed at photon interference
and entanglement. As all light manipulating functions are provided by the nature of
the electric and magnetic field distribution of the same collective Mie mode in
different spatial regions of the unit as part of the larger metastructure, there is no
concept of “components” and therefore no issue of “impedance matching” between
them. In this new paradigm based upon collective Mie resonance of the whole
system (i.e. the metastructure that is the optical circuit itself) the same mode of the
whole system provides the different needed “functions” in pre-specified spatial
regions as part of the co-design of the optical circuit (i.e. the metastructure) to
include even the DBB that contains inside the effectively point-like photon emitting
source.
To expose the fundamental physics of the nature of the Mie resonances in high
refractive index dielectric blocks of subwavelength size for optical wavelengths, we
have carried out analytical and numerical (finite element method) studies for
spherical and cubic DBBs respectively, exploiting the Mie theory to establish the
nature of the Mie resonance of the DBBs. The various magnetic and electric Mie
resonance are identified by studying the different symmetries associated with the
oscillation of displacement current within an individual dielectric block and it is
xxvi
revealed that the interference between these oscillation of different symmetries
provide the fundamental basis behind light manipulation functions. The dominant
magnetic resonance of the dielectric was identified to result in enhanced density of
photon states and, for cubic DBBs of linear dimension 200nm with nearest wall-to-
wall separation ~50nm, provide Purcell enhancement of ~5 to 10 in the spontaneous
emission rate of the source modelled as a dipole-driven two-level system emitting at
980nm. The interference between the magnetic and electric dipole modes controls
the degree of directionality in photon emission with ~0.5 coupling efficiency of the
emitted photon to the collective Mie resonance. Supporting the extensive theoretical
studies, we also pursued experimental work with limited scope of fabrication array of
DBB array in the Silicon on Insulator platform and via far-field scattering
spectroscopy measurement, demonstrating the propagating collective Mie resonance
of such DBB array validating our approach.
Complementing the above examined phenomena of single photon generation to
interference between photons, the direct coupling between two or more SPSs
mediated by the Mie mode is of significance to QIP. To examine this coupling, we
exploited a Dyson sequence of the classical Green dyadic description to account for
the radiative rate enhancement of coupled SPSs. We found that the collective Mie
resonance of the DBB array is capable of inducing on-chip coupling between distinct
SPSs over large on-chip separations of ~10 to 100 𝜇𝑚 . Moreover, theoretically a
super-radiant emission rate, approximately 1.7 times the single emitter is
demonstrated. However, as a classical description cannot account for two photon
states, to account for the coherent and incoherent decay processes that lead to such
xxvii
super-radiant coupled SPS state, we exploit Von-Neumann Lindblad approach and
provide a detailed quantum field theoretic study of the evolution of the density
matrix of SPSs coupled via the collective Mie resonance to show that the collective
Mie resonance is responsible for the emergence of coherence and entanglement
between the coupled SPSs over time. The reported study lays the foundation for
further exploration and study on this new paradigm of “component-less” optical
circuits based on common Mie resonance of DBB metastructures providing spatial
location dependent photon manipulation functions for optical classical and quantum
information processing.
1
Chapter 1. Introduction
§1.1. Central Theme and Motivation:
This dissertation comprises original contribution to the emerging field of on-
chip scalable nanophotonic systems aimed at manipulating on-chip generated photons
in the optical (~1000nm) wavelength regime to realize controlled interference and
entanglement between two or more photons-- the two basic functions underpinning
quantum information processing [1.1, 1.2] via quantum optical “circuits” as
schematically depicted in Fig.1. The yellow pyramidal structures in Fig.1 depict on-
demand single photon sources arranged in spatially-regular array with the
metastructure shown in black and blue providing the indicated basic functions needed
for the control and manipulation of the photons generated on-chip in the designed
network of interconnected single photon sources, collectively the whole nanophotonic
system dubbed “quantum optical circuit”. The original ideas, their analysis, and
findings that constitute the major part of this dissertation relate to the introduction here
of a new paradigm [1.3] for the implementation of the five indicated spatially-
dependent circuit functions of (1) local enhancement of the photon emission rate
(Purcell effect), (2) introducing highly directional emission into (3) a state-preserving
2
Figure. 1.1. A generic schematic depiction of on-chip quantum optical circuit based on
integrated on-chip single photon sources and optical circuit comprising the key
functional elements of cavity/ nanoantenna, waveguiding, splitting and combining for
manipulation of the emitted photons in the on-chip architecture.
propagation region (traditionally called wave guiding) to (4) “beam-split” and then (5)
combine the photons are implemented exploiting a collective magnetic and electric
Mie resonance [1.3, 1.4] of the entire metastructure made of subwavelength sized
dielectric building blocks (DBBs), co-designed with a bandwidth to incorporate the
emission bandwidth of the photon emitter array.
This new on-chip photon manipulation paradigm, though applicable to any
type of single photon emitters in spatially regular arrays, is motivated by the first
realization of spatially regular arrays of on-demand single photon emitters in the form
of mesa-top single quantum dots (MTSQDs) by my fellow graduate student, Jiefei
Zhang, as part of her dissertation and post-doctoral work [1.5-1.10]. As such, most of
the specific choice of parameters of the metastructures analyzed and simulated are
guided by the parallel experimental efforts on the MTSQDs.
3
In on-chip optical circuits such as Fig. 1, controlled photon interference, and
emitter-photon and photon-photon entanglement finds promising applications towards
several areas [1.2] such as quantum key distribution [1.11, 1.12], quantum metrology
[1.13], quantum computation [1.11] and quantum simulation [1.14]. Amongst the
competing platforms of trapped ion [1.15], superconducting transmon qubits [1.16],
topological [1.17] atomic and nuclear spin based systems, photons have the distinct
advantage of (a) very weak interaction with the environment- thus good prospect for
long decoherence time, and (b) fast propagation, hence carrying information over long
distances [1.2]. Owing to this relative immunity to environment, photons, from the
early days have been instrumental to any form of wireless long-distance reliable
communication. It is also for the same reason secure information transfer using
photons as a potent carrier of quantum information [1.2] via its spin (polarization),
Intensity (number), or angular momentum (spatial field-distribution) degrees of
freedom [1.2] has been the earliest proposition and demonstration amongst the various
areas of application of quantum optical systems -- the first proposition by Bennett in
1983[1.12]- followed by experimental demonstration in 1992 of secure
communication over a free space distance of 32cm [1.18], setting the paradigm. Since
then, numerous experimental and theoretical investigations have established such
paradigm– establishing beyond any doubt, photons as a well-established carrier of
quantum information over hundreds of Kms [1.19].
While only on-demand single photon states are required for quantum key
distribution, coherent super-position of many-photon states find application in
4
quantum metrology [1.13], quantum information processing [1.11]. Specifically, an N-
photon coherent state shows an effective de-Broglie wavelength reduced by a factor of
N compared to the wavelength of a single photon. Exploiting this effect, quantum
imaging surpassing the diffraction-limited classical resolution limit in measurements
of distance by a factor of N [1.20] and has been recently demonstrated using photon
pairs generated in Spontaneous Parametric Down Conversion (SPDC) sources [1.13,
1.21] and entangled two-photon emission from biexciton-exciton cascade in
QDs[1.22]. Beyond metrology, a breakthrough in understanding the role of entangled
photons in quantum information processing was made in 2001 by two seminal papers:
(1) Knill, Laflamme and Milbourne [1.23] proposed a scheme of universal quantum
computation using linear optical circuits. and (2) Russendorf and Briegel [1.24, 1.25]
proposed an alternate cluster state quantum computation architecture. Both of these
schemes utilize interference of distinct mutually coherent photons in beam-splitter
structures. Recently many of these ideas have been demonstrated using on-chip
waveguides and directional couplers as beam-splitters in silicon photonics. Photon
interference in such on-chip circuits has been used to provide proof-of-concept
demonstrations such as (1) implementing primitive quantum computation algorithms
such as Shor’s factorization [1.26], (2) arbitrary high-fidelity quantum information
processing logic gates up to two logical qubits [1.27, 1.28], and many others [1.2].
So far, the demonstrations of optical quantum information processing as discussed
above, however, have been based on weak coherent laser with SPDC (Spontaneous
Parametric Down Conversion) and SFWM (Spontaneous Four Wave Mixing)
5
processes [1.2, 1.28] to create state that mimic single and entangled photons and thus
ultimately fail in scaling to large systems needed for practical applications. Realization
of distinct components for the needed on-chip quantum optical circuits including on-
chip single photon sources as shown in Fig. 1 has been attempted with self-assembled
island quantum dots as on-chip on-demand single photon emitters coupled to typically
ridge waveguide based optical circuits and super-conducting nanowire based on-chip
single photon detectors. Recently there has been demonstrations of on-chip single
photon generation, propagation, and detection [1.29] and also on-chip Hanbury Brown
and Twiss measurements [1.30] involving splitting of a single photon via the on-chip
beam-splitter structures. However, more relevant goal of on-chip HOM interference,
essential for realization of quantum entanglement needed for any form of optical
quantum information processing or metrology, has been out of reach mostly because
of the huge scale of uncertainty and randomness in the co-design of the photon-source
and light manipulating structures. A significant part causing these uncertainties lies in
spectral and spatial irregularity of the source, typically the island QDs that have been
for the last two decade the dominant approach to realize the on-chip SPSs. The island
QDs were first demonstrated as single photon sources in early 2000’s [1.31], and from
then used in several demonstrations of integration with ridge waveguides [1.32, 1.33]
or photonic crystal cavity/waveguides [1.33-1.41]. However, the spectral and spatial
irregular nature of the island QDs precludes scalable integration. Moreover, the typical
excitonic decay lifetime of a typical semiconductor QD is ~1ns and the dephasing time
is typically ~100 to 200ps [1.31]. Thus, it is imperative that the emission rate be
6
enhanced (lifetime shortened) by a factor of ~10 to maintain coherence—essential to
Quantum Information Processing [1.42, 1.43].
SPS Arrays:
To this end, the novel approach to realize spectrally uniform and spatially regular
array of integrable Mesa top single Quantum Dots as SPSs has recently been
demonstrated [1.5-1.10]- by controlled MBE growth on lithographically patterned
mesa allowing forming a single quantum dot of known size and shape on the apex of
the nanomesas in the array as shown in the SEM image of Fig. 1.2 (a). These
spectrally uniform arrays have also been demonstrated to be efficient single photon
sources with greater than 99% single photon purity (Fig. 1.2(b)). With the advent of
this new class of MTSQDs as SPSs, the need for nanometer-level precision of the
position in all three directions, as well as the need to have on-chip distinct QDs
sufficiently spectrally close is satisfied. Furthermore, the MTSQD array can be
overgrown and planarized, resulting in a substrate with buried deterministic SPS array
[1.10] (Fig. 1.2(c)) enabling deterministic monolithic integration with light
manipulating elements providing the functions needed for on-chip photon interference
needed for entanglement. Towards realization of on-chip optical circuits as indicated
in Fig 1.1, the most general set of these needed light manipulating functions can be
listed as (1) enhancement of emission rate of the QD SPS (2) enhancement of photon
directionality, (3) State-preserving photon propagation (4) Splitting and (5)
Recombining photons from distinct on-chip sources [1.3].
7
Figure 1.2. (a) SEM image of MTSQD array, (b) Single photon emission behavior of
the MTSQD array shown by the g
(2)
plot, (c) Planarized MTSQD array as a platform
for integration with light manipulating circuits. (d)-(e) Overview of the existing
approaches to on-chip light manipulation. (f) The new approach introduced and
studied in this dissertation.
8
So far, the major approaches to implementing these needed light manipulating
functions have been based on either continuous ridge waveguide structures (Fig.
1.2.(d)), or photonic crystal-based architecture (Fig. 1.2.(e)). By contrast, this
dissertation introduces and examines a new class of light manipulating circuits (Fig.
1.2.(f)) that exploit collective Mie resonance of suitably designed array
(metastructure) of subwavelength sized dielectric building blocks that provides all the
needed light manipulating functions. In all the above noted approaches a prime
consideration is to reduce the loss of photons into the underlying Si or III-V substrate
and thus requires incorporation of mirror/ membrane structure to prevent photon loss.
Particularly, since functions of the on-chip optical circuits are dominantly for photons
with k-vector parallel to the substrate, the underlying mirror is required to be
broadband. This is remedied by exploiting total internal reflection by the interface to
an underlying low refractive index substrate—a concept that is exploited in photonic
crystals as air-suspended membrane structures [1.33, 1.34] and in conventional Si
ridge waveguide approach with high-quality underlying native SiO2 layer [1.44]. For
III-V platform, achieving the same underlying membrane is more challenging.
However recently there has been progress on epitaxially integrating low-index
dielectrics such as BaF2 and CaF2 with GaAs [1.45, 1.46], or by exploiting flip-chip
bonding to place III-V active layer on high quality SiO2 membrane [1.47]. Such
established techniques pave the way to realize the optical structures based on our new
approach of DBB array based metastructures where structural support is of paramount
importance alongside with preventing photon loss into substrate. In the design of the
9
DBB metastructures presented in this thesis, we have thus assumed a low index
(n~1.5) membrane under the DBB arrays.
To put Mie resonance approach in proper perspective, next we provide a brief
recapitulation of the status of the dominant approaches of ridge waveguide structures
and photonic crystal waveguide and cavity.
Waveguides:
Historically the main contender for light manipulation on-chip has been ridge
waveguide based structures [1.48] allowing near-lossless on-chip propagation and
guiding [below 0.2dB/cm [1.44, 1.47] for on-chip device applications- a feature that
has been well exploited in now established technologies of classical optical
information processing in Si photonics platform. Efforts to integrate the QD SPSs with
ridge waveguide [1.25, 1.26] has been limited by lack of essential functions of
enhancement of the emission rate and emission directionality of the SPS- critical for
the needed quantum information processing applications.
2D Photonic Crystal Platform:
Alternatively, it has been established that by introducing suitable absence of
holes in an otherwise periodic array of holes characteristic of a photonic crystal [1.49,
1.50], one may exploit the coupling of the SPS to a localized photon state created in
this region by virtue of the absence of Bragg scattering in the medium surrounding the
10
SPS to achieve the effect of a local cavity. This has proven to be an efficient means to
create high Q cavities with Q~10
5
[1.51], and also waveguiding with loss of the order
of ~10dB/cm [1.52]—still suited for optical circuit designs. Although promising, the
photonic crystal based approach typically results in a very narrow (typically ~100 𝜇 eV)
width of the cavity mode requiring high degree of fabrication control to tune the QD
SPS emission and the cavity mode to be exploited. Furthermore, to achieve
directionality in photon emission, it is essential to couple the cavity and waveguides
together, typically resulting in ~50-60% collection efficiency of the emitted photons
into the waveguide modes [1.35, 1.39].
Mie Resonance Metastructures
An alternate way to achieve emission directionality and Purcell enhancement is
by constructing controlled interference of the emitted photon with itself in one
particular direction without the need for strong field localization in open resonator
structures usually referred to as nanoantenna [1.53] as it functionally mimics the
antennae used heavily in all microwave and RF communications worldwide.
Compared to high-Q cavity typical in the photonic crystal-based implementations, the
nanoantennae provide a broadband response of emission directionality and Purcell
enhancement, alleviating the need for spectral matching with the emitter. However,
such nanoantennae are only explored as individual elements coupling to the free-space
electromagnetic modes- unsuited for design of scalable quantum optical circuits, and a
more holistic approach is required.
11
Investigations reported in this dissertation on the new class of optical circuits
aim to overcome many of the limitations of the different approaches of ridge
waveguide, photonic crystals and standard nanoantenna structures as mentioned above
and holistically addresses all the needs of on-chip integration. We demonstrate that by
arranging judiciously a co-designed array of subwavelength scale dielectric building
blocks around a single photon emitter one can exploit the collective Mie resonance of
the array to provide not only the nanoantenna functionalities of enhanced emission
rate and directed emission but also all the rest needed functions such as beam splitting
beam combining, and phase delay control in a completely mode-matched way. The
basic physics of Mie-like resonances [1.54] exploited in this new approach is quite
different from that underlying either the ridge waveguide or photonic crystals- as
outlined in Fig. 1.2(f). Our approach is in a regime where the resonance of each block
spectrally coincides with the photonic band and also the emission frequency of the
single photon source. This allows us to control, simultaneously, the propagation of the
photon (dispersion characteristic- by controlling the periodicity of the array) and the
local E-field distribution (Mie resonance of the individual blocks-- by controlling the
size and shape of the individual dielectric building blocks- hereby referred to as
DBBs)- thus allowing realization of all the needed light manipulating functions, both
cavity-like, such as enhancement of emission rate of the SPS, enhancement of photon
directionality; and waveguide-like, such as propagation, splitting and recombining
towards on-chip entanglement between coupled single photon emitters for quantum
information processing. Unlike the conventional approaches, this allows realization of
optical networks that focus holistically on all the needed functions rather than specific
12
cavity or waveguides. A schematic of such SPS-DBB optical network is shown in Fig.
1.3 that represents these five essential light manipulating functions i.e. (1)
Enhancement of the emission rate of the SPS, (2) Enhancement of the directionality of
emission, (3) On-chip propagation, (4) Beamsplitting and (5) Beam combining- all
exploiting the same Mie resonance and thus implemented with no mode-mishatch
between the different network components.
Figure 1.3. Schematic showing our approach of on-chip DBB array (blue blocks)
based multifunctional optical circuit embedded with on-chip SPSs (Pyramids) and
array of detectors (purple blocks) that exploits on-chip photon interference to create
path entanglement.
Next, we provide an overview of the phenomenon of Mie resonance that stands as
the key physical process exploited in our approach.
§1.2. Concept of Mie Resonance
The phenomenon of Mie scattering of light by scatterers whose size is comparable to
the incident light has been well-known for a century [1.55, 1.56]. Controlling and
probing such Mie resonances in sub-micron scale dielectric particles, however, has
13
been undertaken very recently [1.54, 1.57]. The major interest in these phenomena has
been the fact that the high index dielectric particles show dominant magnetic
resonance along with the typical electrical counterpart [1.58].
The physical origin of this unique magnetic response is explained in the
schematic drawing in Fig. 1.4.
Figure. 1.4. Schematic representation of the E-field on the interface of a high index
DBB. The boundary condition enforces the E-field inside the interface to be
dominantly tangential, thus resulting in dominant magnetic resonances.
Let us consider an interface between a dielectric material (with relative
permittivity 𝜖 𝑟 ) and air. Under any arbitrary excitation, the boundary condition
enforces:
𝐸 ̅
1 , 𝑡 = 𝐸 ̅
2 , 𝑡 ( 1 . 1 )
and,
𝜖 0
𝐸 ̅
1 , 𝑛 = 𝜖 0
𝜖 𝑟 𝐸 ̅
2 , 𝑛 ( 1 . 2 )
where the subscript 𝑡 and 𝑛 represents the component tangential and normal to the
interface, respectively. Thus, as we move from the medium-2 (the outside medium)
14
into inside the dielectric, the tangential component of the electric field remains same,
but the normal component of the electric field is reduced by a factor of 𝜖 𝑟 . The effect
of these two boundary conditions can be observed as the total electric field (red arrow
in Fig. 1.4) is dominantly tangential inside the interface in the dielectric medium. For
high values of 𝜖 𝑟 , the dielectric-to-air interface thus acts as a perfect magnetic
conductor boundary, suppressing the normal component of the E-field and
consequently the tangential component of the H-field inside the interface.
If, now the volume of the dielectric material is taken to be finite, such as in the
left panel of Fig. 1.4 the dominant tangential E-field vector inside the interface now
has a chance to close onto itself, resulting in a circulating E-field (displacement
current) inside the DBB near the interface. Such circulating E-field in the dielectric
results in an enhanced magnetic field in the center. This is the origin of the strong
magnetic resonance of the high-index dielectric building blocks. For specific size of
the DBBs, the circulating E-field vector near the dielectric surface will combine onto
itself with a constructive interference owing to a phase shift multiple of 2 𝜋 . Very
roughly speaking, these constructive interferences are the so-called magnetic Mie
resonances. Note, that the same physical principle of the circulating displacement
current also results in enhanced photon density of states close to the dielectric
interface- an aspect that we have exploited towards achieving SPS-Mie mode coupling
in this work.
15
Two application-oriented principles emerge from this conceptual foundation of
electric and magnetic Mie resonances in a dielectric that carry importance towards
practical applications of light manipulation.
Figure 1.5. (a) Schematic representation of interference between electric and magnetic
resonance resulting in directionality without strong field localization. (b) Schematic
representation of mode-mode coupling resulting in guiding.
First, typically these Mie resonances are spectrally broad in nature [1.54]
allowing co-existence of electric and magnetic resonance at the same frequency. This
results in interference of the E- and H-field generated by modes of different
symmetries resulting in directionality, as schematically depicted in Fig. 1.5 (a). Unlike
standard photonic crystal cavity structures, this directionality is caused by interference
and not reflection and thus directionality in photon propagation in Mie resonance-
based structures can be achieved without any strong field localization. Such
interference is the main operating principle of nanoantenna structures- providing the
key functions of emission directionality and emission rate enhancement of the SPS.
16
Second, the Mie resonances of interacting DBBs can be coupled to each other,
as schematically depicted in Fig. 1.5.(b), resulting in a propagating collective mode.
Light propagation via such emergent collective mode emerges in an interacting array
was first conceptually demonstrated in [1.59] and since then has been demonstrated as
a platform for applications towards waveguiding [1.60, 1.61].
Notably, the presented approach in this dissertation combines the two above-
noted features in the same system and thus allows implementing all needed functions
simultaneously for holistic applications towards on-chip quantum information
processing.
§1.3. Organization of the dissertation
The work presented in this dissertation is organized as follows:
Chapter 2 presents study of the nanoantenna structure exploiting the unique
approach of collective Mie resonance of subwavelength DBB arrays to explore the
functionality depicted in Fig. 1.5(a) and discussed above. We present study of the
nature of the electric and magnetic Mie resonances for single DBB using two
approaches- (1) An analytical approach based on spherical vector harmonics for
spherically symmetric DBBs and (2) A finite element method based numerical
approach towards rectangular DBBs that are fabricable and integrable with the on-chip
SPS array such as the MTSQDs. As figures of merits, both the photon density of states
17
derived from the classical Green function resulting in enhancement of SPS emission
rate, and directionality of the emitted single photons based on the notion of classical
Poynting vector is studied. Finally, the principle and studies on the Yagi-Uda
nanoantenna structure is presented in both analytical and numerical pathways.
Chapter 3 extends the nanoantenna structure presented in chapter 2 to a more
holistic system that includes also the guiding of light via mode-mode coupling in
arrays of DBBs- the key principle depicted in Fig. 1.5(b). The resultant nanoantenna-
waveguide structure is then extended to beamsplitting and beamcombining
demonstrating all the needed functions towards photon interference and entanglement
in an on-chip optical circuit. In this section, we also present experimental studies of
fabrication of the DBB arrays using SOI platform and far-field spectroscopic
measurement on such arrays providing the experimental verification to some of the
ideas presented via simulation studies.
Chapter 4 builds upon the conceptual and engineering approach to Mie
resonance metastructure built in Chapter 2 and 3 to addressing the important aspect of
achieving SPS-SPS coupling mediated by the collective Mie resonance of the DBB
metastructures. To address the emission and manipulation of a single photon by and in
the presence of multiple emitters, we invoke a Dyson series on the classical Green
function- allowing study of the radiative decay rate of a single photon from a
collective excited SPS-SPS coupled states.
In Chapter 5, we finally extend the studies of Chapter 4 on SPS-SPS coupling
mediated by the collective Mie resonance to phenomena involving emission of
18
multiple photons from coupled SPSs via the DBB array metastructures. To do this, we
formulate a quantum approach based on the Von-Neumann Lindblad formulation to
address coherent and incoherent processes of state evolution and demonstrate
emergence of super-radiance and entanglement in MTSQD SPSs coupled via
collective Mie mode of the co-designed DBB array.
Chapter 6 concludes based on the results presented in chapters 2 to 5 and
suggests some directions for future research.
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24
Chapter 2. Nanoantenna-- Emission Rate Enhancement and
Directionality
§2.1. Introduction and Motivation
As discussed in the preceding chapter, in this dissertation we investigate a new
class of light manipulating elements based on the paradigm of collective Mie
resonance of array of subwavelength scale dielectric building blocks. This new
paradigm exploits the Mie resonance to simultaneously implement the cavity-like and
waveguide-like functions towards a holistic design of the optical circuits [2.1, 2.2].
One example of such holistic design was depicted by Fig. 1.3- reiterated here as Fig.
2.1- an example where the same collective Mie mode of the whole unit representing
Figure 2.1. Overall vision of On-chip quantum optical circuits based SPS array (such
as the MTSQDs) codesigned and integrated with dielectric building block (DBB)
metastructure based multifunctional quantum optical circuits. The nanoantenna
component is indicated with a red box.
the five essential functions of (1) enhancement of the emission rate of the SPS, (2)
enhancement of the directionality of emission, (3) on-chip propagation, (4) beam
splitting and (5) beam combining. In this chapter we discuss the design of the elements
associated with the first two functions- the nanoantenna structure (indicated in Fig.
2.1). The nanoantenna serves the critical function of coupling the emitted photons to
25
the optical circuit, and thus, is required to enhance (1) the emission rate of the SPS and
(2) the emission directionality towards integration of the SPSs with the on-chip DBB
metastructure based optical circuits as in Fig. 2.1.
§2.1.1. SPS-Optical Circuit Integration: Status and Requirements
Over the past nearly two decades, integration of single photon sources on-chip
with light manipulating elements to achieve on-chip manipulation of a single photon
state towards optical quantum communication and information processing has been a
major goal [2.3, 2.4]. The dominant platform that has been explored is the class of
self-assembled island quantum dots as on-chip single photon source integrated with
cavity or waveguide structures involving photonic crystal or ridge waveguide
structures [2.5]. However, the spatial and spectral randomness of the self-assembled
QDs have hindered deterministic integration with cavity/waveguide structures as it has
prevented deterministic mode-matching, both spatially and spectrally, of the QD SPS
with the mode of the light manipulating unit [2.5, 2.6]. On other platforms such as Si-
V and N-V center in diamond [2.7]- while the spectral uniformity problem is not
present, the issue of high degree of defects in the ion implantation process results in
low quantum efficiency. Contrary to these approaches, as discussed in Chapter 1,
recently a new class of QD SPSs- the Mesa-top Single Quantum Dots (MTSQDs) [2.2,
2.8-2.11 ] has been able to circumvent this barrier by realizing on-chip single quantum
dot single photon source in regular arrays and in high spectral uniformity of
inhomogeneous broadening ~2nm [2.9, 2.10] across a regular array distributed over
26
1000 µm
2
area. Such MTSQD array can further be overgrown and planarized [2.9] –
resulting in planar substrate with buried SPSs that are in known locations – thus
enabling on-chip fabrication of optical circuits around them.
This new class of MTSQD SPSs provide a viable paradigm for achieving
deterministic integration to realize quantum optical circuit such as in Fig. 2.1 to enable
on-chip photon interference. To assess the required light manipulating functions, it is
however important to take a close look at the nature of the QD as a single photon
emitter and the nature of the emitted photon state.
Figure 2.2. A schematic drawing of the typical excitation and photon emission in a QD
SPS- indicating the key processes that result in dephasing of the emitted photon.
Figure 2.2 illustrates the concept of the different uncertainty associated with
the emitted photon [2.12]- expressed as an exponentially decaying wave packet using
the Wegner-Weiskopf approach. The intensity decays with a timescale of T1- the
radiative lifetime- typically ~1ns for the semiconductor QDs [2.10, 2.11]. Owing to
interaction with phonons and trapped charges in the surrounding medium, the phase of
the emitted photon wave packet is gradually randomized over time- becoming
27
completely uncertain after a time T2
*
- also known as the dephasing time. The
combined effect of the amplitude decay rate and dephasing rate results in the overall
decoherence of the photon wave packet- denoted typically using the symbol T2, where
1
𝑇 2
=
2
𝑇 1
+
1
𝑇 2
∗
. For semiconductor QDs, the decoherence time is found to be typically
~100-200ps [2.14, 2.15]. Decoherence of the photon wave packet results in inherent
loss in contrast in the interferometer output- also known as visibility [2.16] that
typically has a lower bound of [2.17]
𝐼 𝑚𝑎𝑥
− 𝐼 𝑚𝑖𝑛 𝐼 𝑚𝑎𝑥
~ (
𝑇 2
2𝑇 1
) (2.1)
It is thus important to reduce the spontaneous decay lifetime 𝑇 1
using the enhancement
of local density of photon states at the location of the SPS which is referred to as
Purcell enhancement. As an illustrative example, in Fig. 2.3 we show the measured
response of the MTSQD SPS [2.8]. Fig. 2.3(a) shows the PL spectrum of a
representative MTSQD indicating a ~10ueV spectral width [2.9, 2.10]. Such spectral
width, when translated into timescale, corresponds to a T2~60-100ps. On the other
hand, the Fig. 2.3(b) indicates the time-resolved PL measurement on the PL intensity
indicating a radiative lifetime of ~800ps [2.8, 2.10]. Thus, towards photon interference
as a tool towards quantum information processing, we require that
𝑇 2
𝑇 1
be enhanced by a
factor of ~5-10. However, T2, being determined by electron-phonon scattering and
stark effect owing to trapped charges [2.17], is difficult to control. Thus, needed a
reduction of T1 is needed.
28
Figure 2.3 (a) High resolution PL of the MTSQD indicating a dephasing time ~100ps.
(b) time resolved PL measurement on the MTSQD indicating radiative decay lifetime
of ~1ns [2.7].
Additionally, the relaxation time (𝜏 𝑅 ) and the inherent time uncertainty in the
spontaneous emission process results in a global phase to the wave packet of the
photon- shown using the 𝑒 𝑖𝜙
phase factor. Typically, this delay is only accounted for
in Lab environment by adjusting the relative delay between two photons by using
continuous wave excitations [2.13, 2.14]. A way to reduce this uncertainty is to induce
coupling between two SPSs- which we discuss in Chapter 4 and Chapter 5.
Requirement on efficient photon extraction
In addition to the requirement of Purcell enhancement, another important function the
light manipulating structures around the SPS must provide is to enable efficient
photon extraction into the “waveguiding” function in the horizontal on-chip geometry.
For III-V platform based photonics, such as the MTSQDs, this is a pressing issue as
the high refractive index of the substrate and its high contrast with the medium above
(typically air) causes most of the emitted photons to go into the substrate. [2.18]. A
29
simple analysis leads to an estimate of ~1/2n
2
fraction of the photons that escape the
substrate, n being the refractive index of the substrate. For GaAs (n~3.5) therefore this
fraction emitted into air is typically no more than 5%.
As an illustration, for the mesa-top single quantum dots (MTSQDs) as
integrable single photon sources mentioned above, we undertook FEM based
simulations of the angular distribution of the photons coming out of a representative
MTSQD on GaAs substrate in the configuration shown in Fig. 2.4(a) [2.8, 2.10].
Figure 2.4(b) shows the angular distribution of the calculated Poynting vector at a
distance 3𝜇𝑚 from the QD which represents the angular distribution of the flux of the
emitted photons, here for a QD dipole strength normalized to 1 Debye. Typical
GaAs/InGaAs quantum dot is characterized by a dipole of ~50 Debye [2.19]. By
integrating the calculated Poynting vector over the lower hemisphere, we find that
~95% of the emitted photons go into the substrate and therefore are lost. This loss
must be eliminated and not simply by introducing a means of reflecting most the
photons upwards but, to be useful for on-chip optical circuits, guiding the emitted
photons into a horizontally propagating mode thereby enabling efficient extraction of
the photons into optical circuits such as the one shown in Fig. 2.1.
30
Figure 2.4. (a) Schematic drawing of MTSQD on GaAs substrate and (b) the angular
distribution of the Poynting vector on a spherical enclosing surface of radius 3𝜇𝑚 for
a radiating transition dipole of 1 Debye strength at 930nm indicating the angular
distribution of the emitted photons and pointing to the fact that most of the emitted
photons are lost into the substrate. (c) The MTSQD on the GaAs mesa on a
65.5nmGaAs/78.5nm AlAs DBR in pillar structure of 2um height with (d) showing
the corresponding Poynting vector angular distribution.
Approach of DBR:
A popular approach to prevent photon loss has been distributed Bragg
reflectors [2.20] as they can be readily grown underneath the QDs using techniques
such as MBE and MOCVD exploiting the refractive index contrast between lattice-
matched GaAs (n~3.5) and AlAs (n~2.95). An illustration is shown in Fig. 2.4(c) that
31
represents the MTSQD placed on a DBR that comprises of 8 period GaAs/AlAs pillar
structure. The DBR layer thicknesses are designed to accommodate the lateral
confinement effect of the photon mode of the pillar structure so that the criterion of
quarter wavelength layer thickness [2.20] is met. The FEM simulation of the angular
distribution of the Poynting vector, shown in Fig. 2.4(d) indicate significant
improvement in preventing the loss of photons into the substrate. The fraction of
emitted photons being lost into the substrate is reduced to ~50% compared to the case
when the DBR was not present. The fraction collected into vacuum has increased from
5% for no mirror to 50% with the mirror. However, one of the major limitations of the
DBR structure is that the Bragg scattering is very sensitive to the wave-vector of the
photon perpendicular to the layers of the DBR. Typically, DBR mirrors designed at
Optical and NIR wavelengths (~900-1000nm) with GaAs/AlAs layers show a
bandwidth ~100nm. This bandwidth, in turn, results in decrease in the reflectivity
when the angle of incidence is not normal- that changes the effective wave-vector of
the photon perpendicular to the mirror. Thus, the DBR can only be effective as long as
the angle of incidence of the photons does not vary more than ~0.1 radians (~6
0
) from
the surface normal. This is clearly not compatible with the horizontal architecture of
on-chip optical circuits. A means of directing the emitted photons horizontally is
needed while preventing loss into the substrate even more effectively.
32
Approach of Dielectric Membrane Structure:
A more effective approach to prevent photon loss into substrate for optical
circuits in horizontal on-chip architecture is, as discussed in Chapter 1, using a low-
index (n~1.5) dielectric membrane underneath the SPSs [2.21]. Incorporation of such
membrane structures can be achieved by either lattice matched epitaxial growth of
low-index dielectrics such as CaF2 and BaF2 on GaAs [2.22, 2.23], or by flip chip
bonding of GaAs on SiO2 [2.24]. Specifically, with the later, recently very low loss
ridge waveguide structure in III-V has been demonstrated that opens possibility of
realization of the DBB metastructure based optical circuits on underlying oxide
membrane for preventing photon loss, as well as maintaining structural support.
To understand the effect of such dielectric membrane structures underneath the
GaAs comprising the QD SPSs, we undertook simulations of the emitted photon
distribution with n=1.5 of varying thickness immediately below as shown in Fig.2.5
(c). An illustrative finite element method-based simulation of the angular distribution
of the Poynting vector is shown in Fig. 2.5(d). Figures 2.5(a) and (b) are repeat of Fig.
2.4 (a) and (b) to help draw visual comparison more effectively. As we place the same
MTSQD mesa on a dielectric membrane of thickness 1𝜇𝑚 as indicated in Fig. 2.5(c),
the radiation pattern (Fig. 2.5(d)) is significantly modified- suppressing the loss of
photons into the substrate to 80%. With membrane thickness higher than 1𝜇𝑚 , this
fraction remains unaffected- indicating that it is contribution from the shape of the
mesa rather than the substrate. The directionality can be thus further enhanced by
controlling the shape of the SQD bearing DBB and, indeed, embedding such a DBB in
33
a nanoantenna geometry to direct the photon emission and propagation horizontally as
discussed in Sec.2.3.
Figure 2.5. (a) Schematic drawing of MTSQD on GaAs substrate and (b) the angular
distribution of the Poynting vector on a spherical enclosing surface of radius 3𝜇𝑚 for
a radiating transition dipole of 1 Debye strength at 930nm indicating the angular
distribution of the emitted photons and pointing to the fact that most of the emitted
photons are lost into the substrate. (c) MTSQD mesa- now placed on a 1𝜇𝑚 thick
membrane of dielectric of refractive index ~1.5 supported by GaAs substrate and (d)
corresponding angular distribution of photon flux indicating a significant reduction in
number of photons lost into the substrate.
Conventionally the collection of the emitted photon to cavity and waveguide
structure has been optimized by blocking all but a very specific set of propagation
directions using Bragg scattering in the photonic crystal approach. Enhancement of the
34
emission rate, on the other hand requires photonic crystal cavity structure that exploits
the same principle of Bragg scattering block all the propagation directions, forming a
cavity [2.3, 2.5, 2.6, 2.14 2.25, 2.26] that creates spatially confined electromagnetic
field. In fact, in photonic crystal platform high degree of spatial confinement of the
photon mode has allowed realization of strong coupling in the platform of photonic
crystal cavities [2.27]. However, the act of confining the photon from all directions
also results in loss in directionality of the photon emission. Thus, physically there is an
inherent tradeoff between the cavity function and the waveguide function which is
typically addressed by coupling a photonic crystal cavity with a waveguide structure.
As mentioned in Chapter 1, however, this approach has two major limitations- (1)
such spatially confined modes are typically very narrow in spectral width [2.5, 2.6],
and thus suffers from problem of spectral matching with the emitter. (2) The notion of
simultaneous realization of both the enhanced emission rate and directionality requires
spectral and spatial mode-matching between a cavity and waveguide which is also no
easy task [2.5, 2.6, 2.25].
As we discussed in Chapter 1, our approach of Mie resonant metastructure
employs the principle of interference of modes of different symmetries (such as the
magnetic dipole and the electric dipole) to provide directionality in photon emission.
This is fundamentally different principle from Bragg scattering in photonic crystal
platform. This principle of antenna has been mostly applied the field of microwave
and radio wave [2.28] and has recently been applied in the field of photonics in a few
cases [2.29, 2.30, 2.31, 2.32]. Unlike the photonic cavity structures that exploit spatial
35
confinement of E-field by preventing propagation in certain areas exploiting Bragg
scattering, the nanoantenna structure functions by exploiting interference of certain
specific modes of the structure in an engineered way. Thus, unlike photonic crystals
that typically results in a high Q (~10
5
) cavity [2.5], the nanoantenna structures acts as
open resonators- with no specifically defined concept of mode volume or quality
factor [2.32]. Therefore. antennas are broadband and fast [2.32] (few fs) in response
compared to ~few hundred picosecond timescale of photonic crystal cavities.
Particular importance is the broadband response that allows nanoantenna to extract
photon efficiently from a SPS without much concern of spectral matching, and
efficiently transfer the extracted photons to different classes of on-chip optical circuits
for light manipulation, as will be seen later. This also facilitates design of the overall
optical circuit from a holistic point. In this chapter we study how the Mie resonance of
subwavelength scale dielectric building blocks (DBBs) can be exploited to gain this
nanoantenna effect on an integrated on-chip SPS. The physics, design and relative
merits and demerits of such DBB array based nanoantenna structure towards the
overall goal of on-chip quantum optical circuits is the main topic addressed in this
chapter.
36
§2.2. Fundamental Concepts of SPS- DBB Array Coupling
The conceptual picture of the phenomenon we are interested in is captured in Fig. 2.6.
Panel (a) of Fig. 2.6 indicates a single SPS integrated with an array of DBBs-
indicated by the blue blocks. The emitted single photon is shown with red wavy
arrows. Two competing processes are indicated- i.e. photon emitted into the Mie
resonance of the DBB array, and photon lost to lossy radiation modes that are not a
part of the Mie resonance. We will show studies in this chapter that the Mie mode
results in an enhanced emission direction to the +X direction in Fig. 2.6(a). The Flux
density of the emitted photons in that direction is indicated by 𝑆 𝑋 and is higher
compared to a spherically symmetric flux distribution and usually referred to as
directivity [2.33].
The physics behind the effect of the Mie resonance on the spontaneous
emission of the SPS is captured in Fig. 2.6(b). The SPS here is approximated as a 2-
level system. Typically, radiative transition rate of a 2-level system can be defined as
[2.34, 2.35]
Γ =
𝜋𝜔
3ℏ𝜖 𝑝 2
𝜌 𝐿𝐷𝑂𝑆 (𝑟 ̅
𝑆𝑃𝑆 , 𝜔 𝑆𝑃𝑆 ) (2.2)
Here 𝑝 represents the transition dipole moment of the SPS- denoted by 𝑝 ̅ =<
𝑔 |−𝑒 𝑟 ̅ |𝑒 >. 𝜌 𝐿𝐷𝑂𝑆 (𝑟 ̅
𝑆𝑃𝑆 , 𝜔 𝑆𝑃𝑆 ) represents the local photon density of states projected
along the direction of the transition dipole [2.36]. Typically, the size of the
37
Figure 2.6 (a) A general schematic representing a single photon source coupled to a
DBB metastructure antenna. (b) schematically shows the transition between the
ground level electron and hole state of the SPS approximated as a two level system
populating a photon of energy 𝜔 𝑆𝑃𝑆 - with a spontaneous emission rate proportional to
the local density of states of photons at that particular position of that particular
energy.
SPS (~10nm) is much smaller compared to the wavelength of light (~300nm in GaAs).
Thus, in most cases, the SPS can be replaced by a point electric dipole of the net
dipole moment 𝑝 ̅ . In cases where the mesoscopic size of the SPS needs to be taken
into account, the point dipole moment can be often replaced by a distributed dipole
moment [2.37, 2.38]. Note, this is also equivalent to taking into account higher order
multipoles (such as electric quadrupole) for the point-like emitter. In this work,
however, we will stick to the point electric dipole approximation of the SPS emitter.
38
§2.2.1. Principle of Nanoantenna- Purcell Enhancement
Historically, the concept of Purcell enhancement was first introduced by EM
Purcell in [2.39]. and since then the concept of Purcell enhancement has been referred
to numerously in the context of emission of a point-line source in the presence of a
cavity. Typically, for a mode volume V and quality factor Q, such Purcell
enhancement can be expressed as [2.35, 2.38]
𝐹 𝑝 =
3
4𝜋 2
(
𝜆 𝑛 𝑖 )
2
(
𝑄 𝑉 ) (2.3)
For example, in a typical photonic crystal L3 cavity [2.5, 2.25] the mode volume ~
results in a theoretical Purcell enhancement ~Q/10- although in practice this is hardly
true owing to high effect of fabrication disorders.
The notion of Purcell enhancement as denoted in equation (2.3) only applies
for cavities where the mode volume can be defined, such as photonic crystal cavities,
and starts to break down when we deal with open resonators, such as the Mie modes
that are exploited in our approach. It is thus important to discuss the concept of Purcell
enhancement in its most fundamental form. This physical insight comes from the
Fermi’s golden rule in equation (2.2) that captures the time-dynamics of a transition
rate of a 2-level state in the presence of an external electric field belonging to photon
modes with certain density of states ρ
LDOS
- weakly coupled to the 2-level state. As
seen in equation (2.2), the emission rate of the SPS is proportional to the local photon
density of states at the location of the emitter. This is the key to enhanced emission
decay rate of an emitter. Any kind of radiative decay is not the property of the emitter,
39
but the property of its environment and thus depends on the electromagnetic Green
function. Thus, we employ here the general concept of electromagnetic propagator, or
Green function, to quantify the notion of density of states of photons. In contrast to
equation (2.3), the notion of the radiative rate captured in equation (2.2) and its
dependence on Green function is a general concept, and thus are equally applicable to
photonic crystals and Mie resonances. This is discussed next.
§2.2.2. EM Green-Function to Density of States
Electromagnetic Green function is an effective classical approach to calculate the local
photon density of states [2.36, 2.40-2.44]. The detailed derivation of the
electromagnetic Green function can be found in Appendix B. here we note the key
results.
The electromagnetic Green function is defined by
∇
̅
× ∇
̅
× 𝐺 ̿
(𝑟 ̅ , 𝑟 ̅ ′, 𝜔 ) − 𝛽 2
𝐺 ̿
(𝑟 ̅ , 𝑟 ̅ ′, 𝜔 ) =
𝛽 0
2
𝜖 0
𝐼 ̿
𝛿 (𝑟 ̅ − 𝑟 ̅
′
) (2.4)
where 𝐼 ̿
is the unit dyadic and can be expressed as 𝑥 ̂𝑥 ̂ + 𝑦 ̂𝑦 ̂ + 𝑧 ̂ 𝑧 ̂ and 𝛽 2
= 𝜔 2
𝜇𝜖 is
the wave-vector at as a function of position. The Green dyadic 𝐺 ̿
(𝑟 ̅ , 𝑟 ̅ ′, 𝜔 ) captures the
complete description of the dielectric environment excluding the SPS, and completely
determines the propagation of the single photon once its emitted. However, there is
also a probability of the emitted photon interacting back with the SPS which is not
present in 𝐺 ̿
(𝑟 ̅ , 𝑟 ̅ ′, 𝜔 ). To account for such scattering event, we approximate the SPS
40
as a point scatterer with a susceptibility 𝛼 ̿(𝜔 ) = 𝑝 ̂ 𝑉 (𝜔 )𝑝 ̂ where 𝑉 (𝜔 ) =
𝑝 ℏ
2𝜔 𝑆𝑃 𝑆 𝜔 2
−𝜔 𝑆𝑃𝑆
2
(Details in Appendix B).
To account for the influence of the SPS, the Green dyadic is modified by
exploiting a Dyson sequence including the multiple scattering resulting into a
modified first order Green function given by
𝐺 ̿
(1)
(𝑟 ̅ , 𝑟 ̅
𝑆𝑃𝑆 , 𝜔 ) =
(𝜔 2
− 𝜔 𝑆𝑃𝑆 2
)𝐺 ̿
(𝑟 ̅ , 𝑟 ̅
𝑆𝑃𝑆 , 𝜔 )
𝜔 2
− 𝜔 𝑆𝑃𝑆 2
− 2 𝜔 𝑆𝑃𝑆 𝑝 2
ℏ
𝑝 ̂ ⋅ 𝐺 ̿
(𝑟 ̅
𝑆𝑃𝑆 , 𝑟 ̅
𝑆𝑃𝑆 , 𝜔 ) ⋅ 𝑝 ̂
(2.5)
The imaginary component of the pole of this Green function determines the radiative
decay rate. In the weak coupling limit, we have
Γ =
2𝑝 2
ℏ
𝐼𝑚 (𝑝 ̂ ⋅ 𝐺 ̿
(𝑟 ̅
𝑆𝑃𝑆 , 𝑟 ̅
𝑆𝑃𝑆 , 𝜔 ) ⋅ 𝑝 ̂ ) (2.6)
Comparing this expression for Γ with the Fermi golden rule in equation (2.2), we get
𝜌 𝐿𝐷𝑂𝑆
(𝑟 , 𝜔 𝑆𝑃𝑆 ) =
6𝜖 0
𝜋𝜔
𝐼𝑚 (𝑝 ̂ ⋅ 𝐺 ̿
(𝑟 ̅
𝑆𝑃𝑆 , 𝑟 ̅
𝑆𝑃𝑆 , 𝜔 ) ⋅ 𝑝 ̂ ) (2.7)
The unique thing about equation (2.7) is that the right-hand side is a completely
classically computed quantity from Maxwell equation, whereas the left-hand side
denotes the photon density of states which is quantum mechanical in nature. This is
the key to understanding the effect of any arbitrary dielectric interface on a quantum
emitter and has been exploited in many platforms in the general field of nanophotonics
[2.35, 2.38].
41
As an illustration to the effect of dielectric interface on the photon density of
states, in Fig. 2.7 we show the density of states of photons for a single spherical DBB.
To find out the density of states as a function of position, we exploit the expansion of
the Green function in the basis state of the Maxwell equation [2.41] that are in
spherical geometry analytically described by the spherical vector harmonics [Ref
Appendix A and B].
In particular, as discussed in Chapter 1, the unique property of the Mie
resonances is the enhanced magnetic resonance that results in an enhanced E-field and
thus enhanced density of states at the boundary of the DBB. This is shown Fig. 2.7-
where we plot the photon local density of states as a function of position in space for a
single DBB of spherical shape of radius 130nm and refractive index 3.5 sitting in a
uniform medium of refractive index 1.5. We specifically show the density of states
along the line shown in black and denoted as AB in panel Fig. 2.7(b). Note that the
density of states is enhanced at the dielectric interface. This is central to the Mie
resonance and is caused by the circular displacement current density at the interface
resulting the unique magnetic Mie resonance of the DBB, as captured in Chapter 1 and
repeated here as Fig. 2.7(c).
42
Figure. 2.7 (a) A single isolated spherical DBB of radius 130nm and refractive index
3.5. (b) Photon local density of states plotted along line AB in (a). x represents the
position from the center of the DBB. (c) Schematic capturing the physics of Magnetic
resonance- the reason why the LDOS at the surface is enhanced.
Purcell enhancement from Green function
From the photon density of states, we can also arrive at a more operational
definition of Purcell factor, which, contrary to equation (2.3), is applicable for
arbitrary dielectric environment. The Purcell factor, i.e. the enhancement of the
radiative decay rate of the QD dipole in the nanoantenna compared to the reference
case where the QD dipole is in infinite homogeneous dielectric can be defined as:
𝐹 𝑝 =
Γ
Γ
𝐻𝑜𝑚 =
𝐼𝑚 (𝑝 ̂ ⋅ 𝐺 ̿
(𝑟 ̅
𝑆𝑃𝑆 , 𝑟 ̅
𝑆𝑃𝑆 , 𝜔 ) ⋅ 𝑝 ̂ )
𝐼𝑚 (𝑝 ̂ ⋅ 𝐺 ̿
𝐻𝑜𝑚 (𝑟 ̅
𝑆𝑃𝑆 , 𝑟 ̅
𝑆𝑃𝑆 , 𝜔 ) ⋅ 𝑝 ̂ )
(2.8)
43
By plugging in the well-known analytical form of the Green function in homogeneous
medium- i.e. 𝑝 ̂ ⋅ 𝐺 ̿
𝐻𝑜𝑚 (𝑟 ̅
𝑆𝑃𝑆 , 𝑟 ̅
𝑆𝑃𝑆 , 𝜔 ) ⋅ 𝑝 ̂=
𝑖 𝑛 𝑖 6𝜋 𝜖 0
(
𝜔 𝑐 )
3
, we finally get,
𝐹 𝑝 =
6𝜋 𝜖 0
𝑐 3
𝑛 𝑖 𝜔 𝑆𝑃𝑆 3
𝐼𝑚 (𝑝 ̂ ⋅ 𝐺 ̿
(𝑟 ̅
𝑆𝑃𝑆 , 𝑟 ̅
𝑆𝑃𝑆 , 𝜔 ) ⋅ 𝑝 ̂ ) (2.9)
We note that the notion of the density of states and Purcell enhancement as
derived in equation (2.8) and (2.9) is in its most general form as long as the weak
coupling limit applies [2.44]- and is equally applicable to spectrally narrow cavity
modes such as in photonic crystals, or, spectrally broad modes as in Mie resonance
metastructures. In the following, we thus apply these key results to determine the
figures of merit of the nanoantenna structure under design in this chapter. We further
note that, the concept of Multiple scattering in Green function as exploited in the
derivation above goes beyond the case of single emitter and can be extended to
understand the time-dynamics of an ensemble of multiple emitters coupled via a
spectrally broad photon mode. This is extensively used in Chapter 4 and Chapter 5 to
determine the evolution of entanglement of systems of coupled emitters.
§2.2.3. Mie Resonances of Spherical DBBs:
Scattering of light by particles smaller/ comparable in size to the wavelength of
light has been a problem that has intrigued physicists from the early days of advent of
electromagnetism. A formal solution of such scattering process by spherical particles
was first given by Mie in 1908- and has been known since as Mie resonances [2.45].
44
However, over a large part of the 20
th
century, these Mie resonances remained a tool
to understanding scattering by a large ensemble of particles, and found its applications
in understanding light/ microwave propagation in atmosphere, colored glasses,
colloidal solutions etc. [2.46]. Only recently in the last few decades we have acquired
the capability controlling the size and position of individual particles in the scale of 1
𝜇 m. With this, the long-existing concepts of Mie resonance were rejuvenated [2.31],
only this time focused on the modes of individual particles. Some early examples of
such experimental studies have been on Silicon nanoparticles fabricated using laser
ablation techniques [2.47, 2.48], or even non-spherical shaped particles made using
lithographic techniques.
Importantly all these studies reveal a unique property of Mie resonance of
dielectric particles. Typically, the different modes that exists have strong magnetic
properties, i.e. they enhance the magnetic field within the dielectric, mimicking a
magnetic multipole moment. This has been the cornerstone to application of Mie
resonance towards photonics platforms. In the next, we review such Mie resonances
for DBBs of spherical and non-spherical shapes.
As shown in the last section in Fig. 2.3, the Mie resonance results in an
enhanced local density of photon states near the boundary of the dielectric. In this
section we explore the nature of these Mie resonances. The Mie solutions [2.45] exists
as the spherical vector harmonics – closed form solution to Maxwell equations under
spherically symmetric boundary [2.49, 2.50, 2.51]. [Analytical expressions for these
45
spherical vector harmonics can be found in the Appendix A] The vector spherical
harmonics are defined as
𝑀 ̅
𝑛 ,𝑚 = ∇
̅
× (𝑟 ̅ 𝜓 𝑛 ,𝑚 (𝑟 , 𝜃 , 𝜙 )) (2.10)
and
𝑁 ̅
𝑛 ,𝑚 = ∇
̅
× ∇
̅
× (𝑟 ̅ 𝜓 𝑛 ,𝑚 (𝑟 , 𝜃 , 𝜙 )) (2.11)
where 𝜓 𝑛 ,𝑚 (𝑟 , 𝜃 , 𝜙 ) is the scalar spherical harmonic defined as
𝜓 𝑛 ,𝑚 (𝑟 , 𝜃 , 𝜙 ) = 𝑧 𝑛 (𝛽𝑟 )𝑌 𝑛 ,𝑚 (𝜃 , 𝜙 )(𝑛 (𝑛 + 1))
−
1
2
(2.12)
With these basis functions, the E-field associated with the magnetic resonances
(TEn,m) and the electric resonances (TMn,m) are defined as
𝐸 ̅
𝑇𝐸𝑛 ,𝑚 (𝑟 ̅ ) = 𝑀 ̅
𝑛 ,𝑚 (𝑟 ̅ ); 𝐸 ̅
𝑇𝑀𝑛 ,𝑚 (𝑟 ̅ ) = 𝑁 ̅
𝑛 ,𝑚 (𝑟 ̅ ) (2.13)
We note that,
n
z represents spherical Bessel function of order n. The type of this
Bessel function is chosen according to the nature of the wave [Appendix A].
In general, the overall scattered E-field is expressed as a superposition of these
multipole Mie modes, as
𝐸 ̅
𝑆𝑐 𝑎𝑡𝑡𝑒𝑟𝑒𝑑 (𝑟 ̅ ) = ∑ 𝑎 𝑛 ,𝑚 𝐸 ̅
𝑇𝐸𝑛 ,𝑚 (𝑟 ̅ ) + 𝑏 𝑛 ,𝑚 𝐸 ̅
𝑇𝑀𝑛 ,𝑚 (𝑟 ̅ )
𝑛 ,𝑚 (2.14)
46
Here 𝑎 𝑛 ,𝑚 and 𝑏 𝑛 ,𝑚 are complex numbers representing the multipole mode
coefficients and thus provide insight on the symmetries of the displace current
distribution in the dielectric [2.51].
Furthermore, the spherical vector harmonics in our analysis is normalized in
such a way that the total scattered power over the whole 4𝜋 solid angle by a particular
TE or TM mode is given by [2.50]
𝑃 𝑅𝑎𝑑 =
1
2𝛽 2
𝜂 |𝑎 𝑛 ,𝑚 |
2
, 𝑓𝑜𝑟 𝑇 𝐸 𝑛 ,𝑚 𝑚𝑜𝑑𝑒𝑠
=
1
2𝛽 2
𝜂 |𝑏 𝑛 ,𝑚 |
2
, 𝑓𝑜𝑟 𝑇 𝑀 𝑛 ,𝑚 𝑚𝑜𝑑𝑒𝑠 (2.15)
This normalization is exploited to estimate the scattering cross sections of the Mie
resonances of the single DBB as discussed next.
To probe the nature of the Mie resonances we present here analytical
calculation [2.49, 2.52, Appendix A] of scattering cross section of the Mie resonances
for a spherical DBB of refractive index 3.5, mimicking GaAs, surrounded by an
infinite medium of uniform refractive index of 1.5 (Fig. 2.8(a)). The scattering cross
section spectrum of the individual Mie resonances for a specific size of the DBB of
diameter 260nm is presented in Fig. 2.8(b). Notably, the scattering cross section is
found to be almost an order of magnitude larger than the physical cross section of the
DBB, ~0.05𝜇 𝑚 2
- thus indicating presence of strong resonance. Furthermore, from the
total cross section spectrum shown using the black line, we observe that the strongest
47
contribution of the peaks of the scattering cross section spectrum arise from the
magnetic dipole, quadrupole and higher order multipole resonances.
Figure 2.8. (a) and (b) Mie resonance scattering cross section spectra of a single
spherical DBB of radius 130nm and refractive index 3.5. (c) Scattering cross section
where the wavelength has been normalized to the diameter of the DBB- indicating that
the electric and magnetic dipole modes exist when the diameter of the DBB ~
wavelength of light in dielectric.
48
To get insight into the physical process, we also plot in Fig. 2.8(c) the
scattering cross section spectrum as a function of the wavelength normalized to the
diameter of the DBB (𝜆 →
𝜆 𝑛 𝐷𝐵𝐵 ×2𝑅 𝐷𝐵𝐵
). We observe that the spectrum of the electric
dipole mode and the magnetic dipole mode both coincide in spectral response when
the wavelength is comparable to the diameter of the DBB. This is not surprising, as,
after all, the Mie resonances are resonances of the oscillating fields of displacement
currents in the dielectric medium, as was indicated in Chapter 1.
§2.2.4. Principle of Nanoantenna-Directionality:
The approach in the design of nanoantenna to enhance directionality is quite
different from other approaches such as photonic crystals. In photonic crystals,
directionality is implemented by blocking the emitted photons in particular directions
with the help of Bragg scattering, and by creating absence of Bragg scattering in one
particular direction. However, in nanoantenna, directionality in photon propagation is
implemented by enabling interference between multipole modes that are spectrally
coincident and spatially of different symmetry. This has been the key principle
exploited in all nanoantenna structures historically- such as phase array antenna [2.33],
patch antenna, Yagi-Uda antenna etc. [2.33].
49
Measures of Directionality:
The directionality for nanoantenna structures is typically expressed in terms of
Directivity. We discuss the concept of directivity of a nanoantenna with the help of the
general schematic shown in Fig. 2.9. Here, we denote an infinitesimal surface
extending a solid angle 𝑑 Ω to an angular direction given by the polar coordinates 𝜃
and 𝜙 , and residing on a sphere of radius R. We can define the angular distribution of
the Poynting vector on that spherical surface, as
𝑆 𝑅 (𝜃 , 𝜙 ) = 𝑆 ̅
(𝑟 ̅ = 𝑅 𝑟 ̂ ) ⋅ 𝑟 ̂ (2.16)
Figure. 2.9. A schematic showing the notion of radiation pattern by a nanoantenna
structure.
50
Directivity towards a particular direction is defined as the ratio of the Poynting
vector in that direction to the average pointing vector over the whole 4𝜋 solid angle of
radiation, i.e.
𝐷 (𝜃 , 𝜙 ) =
𝑆 𝑅 (𝜃 , 𝜙 )
1
4𝜋 ∫𝑑 Ω𝑆 𝑅 (𝜃 , 𝜙 )
(2.17)
Importantly, the quantity 𝐷 (𝜃 , 𝜙 ) turns out to be independent R- the distance from the
antenna phase-center as long as R is large. This threshold-- the distance after which
the angular distribution of the radiation flux becomes independent of distance is
known as the Far-field or Fraunhofer region [2.33]. For a typical antenna of an overall
maximum dimension of D, and radiating at a wavelength of 𝜆 , this far-field region can
be defined as 𝑅 > 2 𝐷 2
/ 𝜆 [2.33]. For our case, the maximum size of the nanoantenna
structure that we investigate here is ~600nm and the wavelength of light in the
medium surrounding the nanoantenna is ~ 1𝜇𝑚 /1.5 , or ~666nm. This yields a
minimum distance for far-field domain ~1𝜇𝑚 . This is the criteria used to calculate the
directivity of the designed nanoantenna structures presented in this chapter.
§2.3. The Primitive Nanoantenna: A dipole coupled to a single DBB
§2.3.1. Mie Resonance of DBBs of non-spherical shape- Multipole Decomposition
On Section 2.2.3, we presented the Mie resonance E-field distributions for
spherically symmetric DBBs. When the spherical symmetry of the DBB is broken, the
spherical vector harmonics individually are no longer able to satisfy the boundary
51
condition at the dielectric interface, and therefore are no-longer valid solutions of
Maxwell equations. However, the same spherical vector harmonics still serve as a
viable basis to understand the nature of the symmetry of the oscillation of the charges
in the dielectric within the DBB.
In the available numerical techniques to solve for the electric field response of
rectangular DBBs such as finite element method and finite difference time domain
method, the information on the individual multipolar excitations of the DBBs are
inherently lost. To remedy, in this work we have appended the finite-element-method
based calculation done in the platform of COMSOL Multiphysics with our own
subroutines of decomposition of the total electric field into the individual multipole
mode components. To this end we adapt the formulation presented in [2.53]. In the
following we outline the main approach to this.
The multipole decomposition allows expanding any arbitrary distribution of
Polarization of a finite dielectric medium as a superposition of the multipole modes
that are defined as the vector spherical harmonics as in equation (2.10) and (2.11). We
start with the total E-field distribution inside the DBB at a particular frequency 𝜔 ,
denoted as, 𝐸 ̅
(𝑟 ̅ , 𝜔 ). From these, the polarization of the dielectric medium as a
function of position is found as
𝑃 ̅
(𝑟 ̅ , 𝜔 ) = 𝜖 0
(𝑛 𝐷𝐵𝐵 2
− 𝑛 𝑜𝑢𝑡 2
)𝐸 ̅
(𝑟 ̅ , 𝜔 ) (2.18)
The polarization of the dielectric medium can be transformed to effective
displacement current density as
52
𝐽 ̅
(𝑟 ̅ , 𝜔 ) = −𝑖𝜔 𝑃 ̅
(𝑟 ̅ , 𝜔 ) = −𝑖𝜔 𝜖 0
(𝑛 𝐷𝐵𝐵 2
− 𝑛 𝑜𝑢𝑡 2
)𝐸 ̅
(𝑟 ̅ , 𝜔 ) (2.19)
Finally, the displacement current density is now exploited to expand into the basis of
the multipole modes. Following the formulation in [2.51] we have,
Figure 2.10. (a) and (b) Mie resonance scattering cross section spectra of a single
cubic DBB of size 220nm and refractive index 3.5. (c) Scattering cross section where
the wavelength has been normalized to the diameter of the DBB- indicating that the
electric and magnetic dipole modes exist when the diameter of the DBB ~ wavelength
of light in dielectric.
53
TE (magnetic Mie resonance) coefficients:
𝑎 𝑛 ,𝑚 = 𝑛 𝑜𝑢𝑡 𝛽 0
2
𝜂 0
∫ 𝑑 𝑟 ̅
𝑟 ̅ 𝑖𝑛𝑠𝑖𝑑𝑒 𝐷𝐵𝐵 𝑀 ̅
𝑛 ,𝑚 ∗
(𝑟 ̅ , 𝜔 ) ⋅ 𝐽 ̅
(𝑟 ̅ , 𝜔 ) (2.20)
TM (electric Mie resonance) coefficients:
𝑏 𝑛 ,𝑚 = 𝑛 𝑜𝑢𝑡 𝛽 0
2
𝜂 0
∫ 𝑑 𝑟 ̅
𝑟 ̅ 𝑖𝑛𝑠𝑖𝑑𝑒 𝐷𝐵𝐵 𝑁 ̅
𝑛 ,𝑚 ∗
(𝑟 ̅ , 𝜔 ) ⋅ 𝐽 ̅
(𝑟 ̅ , 𝜔 ) (2.21)
where 𝛽 0
denotes free-space wave-vector and 𝜂 0
denotes free-space impedance,
approximately 120 𝜋 Ω.
Exploiting the multipole decomposition approach as noted above, in Figure
2.10, we therefore present the results on scattering of a plane incident wave by a single
DBB of cubic shape, as shown in Fig. 2.10(a). Figure 2.10(b) represents the scattering
cross section spectrum for the specific case when the size of the DBB is 220nm cube.
This is supplemented with panel 2.10(c) where we show the spectral response of the
scattering cross section as a function of the wavelength normalized to the size of the
DBB. Similar to what we discovered for the spherical shaped DBBs in Fig. 2.8, we
observe that for the cubic DBBs, both the magnetic and electric dipole modes are
spectrally located when the wavelength of light in the dielectric is ~1.2 times the size
of the cubic DBB. Why this value is larger than what we found out for the spherical
DBB (~1) can be explained readily. Since the Mie resonance is dominantly dependent
on the perimeter of the DBB as it is the perimeter that governs resonance of circulating
displacement current in the dielectric. However, for the same size of cubic DBB and
spherical DBB, cubic DBB has a perimeter (~4×Size) -- ~20% larger compared to a
54
spherical DBB (~𝜋 × Diameter). This confirms the validity of the simplified physical
picture drawn in Chapter 1, Fig. 1.4.
Figure 2.11. The Electric and Magnetic dipole mode peak wavelengths in a single
rectangular DBB as a function of (a) size in the Z direction (parallel to the incident
magnetic field) and (b) size in the Y direction (parallel to the incident E-field).
Further, we investigate the variation of the spectral position of the magnetic
dipole and electric dipole as a function of the aspect ratio of the DBB when the shape
is deviated from the cubic shape. The results are shown in Fig. 2.11. The two panels
(a) and (b) represents when the size of the DBB is varied in the Z direction, the
55
direction of the incident magnetic field and in the Y direction, the direction of the
incident E-field respectively. The results indicate that both the magnetic and electric
Mie resonance is more sensitive to the dimension of the DBB in the direction of the
incident E-field.
§2.3.2. Design of Nanoantenna
In the last section, we explored the spectrum of the various electric and
magnetic dipole mode Mie resonances of single DBBs under excitation of a plane
wave. We also observed that, irrespective of the shape of the DBB, the electric and
magnetic dipole mode spectrally co-exists at a wavelength compared to the size of the
DBB. This spectral co-existence of the electric and magnetic dipole mode is the key to
achieving the nanoantenna effect. To understand, we need to now shift our attention to
not just the spectral response, but the spatial distribution of the E -fields corresponding
to the magnetic dipole and electric dipole modes.
This is show in Figure 2.12. Panel (a) and (c) shows the typical electric field
distribution within the dielectric of the DBB corresponding to the magnetic dipole
mode (in this specific case, magnetic dipole with the dipole moment along Z, or, MD-
Z) and the electric dipole mode (in this specific case, the electric dipole mode with the
dipole moment along Y, or, the EDY). The E-field distribution within the DBB is
critical to achieving and optimizing the coupling of a point dipole source representing
the SPS transition- as will be seen later.
56
Figure. 2.12. (a) E-field distribution within the DBB associated with the Magnetic
dipole mode with dipole moment along Z (MDZ). (b) Radiation pattern corresponding
to the MDZ mode. (c) and (d) panels show the corresponding plots for the EDY mode
(Electric dipole mode along Y direction)
Also important is, however, the distribution of the E-field at a distance from
the DBB as it is this that determines the radiation pattern and directionality. This is
57
shown in Figure 2.12(b) and (d) as the angular distribution of the Poynting vector. For
both magnetic dipole and electric dipole, we see that the donut-shaped radiation
pattern is present, but in different orientations. Thus, super-position of the E-fields
generated by these two modes breaks the symmetry and results directional photon
propagation. This is depicted in Figure 2.12(e). We note that, it is this interference
between the electric and magnetic dipole modes of the same DBB that is the key
working principle behind the single-DBB primitive nanoantenna.
In Figure 2.13 we show the antenna structure where a single point electric
dipole representing the SPS transition is embedded in a single dielectric building
block, here of cubic shape of size 220nm and refractive index 3.5 mimicking GaAs.
The surrounding medium is assumed to be a uniform medium of refractive index 1.5
that represents a protective polymer layer on the top and membrane structure on the
bottom.
Figure 2.13. A generic schematic of the primitive nanoantenna structure.
We note from Figure 2.12(e), that directional propagation works by the
constructive interference of the E-field of the electric dipole and magnetic dipole to
58
the +X direction and destructive interference to the -X direction. This requires that
both the MDZ and EDY modes are excited with comparable amplitude. Furthermore, to
enable constructive interference to the +X direction is enabled by a magnetic dipole
moment towards Z and an electric dipole moment towards -Y. This demands a 𝜋 phase
shift between the MDZ and EDY modes.
The position of the SPS dipole plays an important role in determining the two
above noted effects. To investigate we here present simulation of the primitive
nanoantenna structure shown in Fig. 2.13 for different positions of the SPS dipole with
respect to the center of the DBB. The results are presented in Fig. 2.14.
Figure 2.14 above shows simulation results of the primitive nanoantenna
structure for 4 different position of the source dipole with respect to the center of the
DBB (Δ𝐿 =0nm, 25nm, 50nm, 75nm), as shown in the left insets. Panel (a), (d), (g) and
(j) shows the spectrum of the amplitude of the MD and ED modes- represented by the
𝑎 1,1
and the 𝑏 1,1
coefficients as defined in equation (2.20) and (2.21). Next to it, panel
(b), (e), (h) and (k) shows the corresponding phase. We notice when the source dipole
is at the center, the magnetic dipole excitation is zero. This is consistent from the fact
that at the center of the DBB, the MD mode has zero electric field. However, as the
source dipole is shifted towards the edge of the DBB, the relative amplitude of the MD
mode with respect to the ED mode goes up. The crossover between the dominant MD
and dominant ED excitations happen at Δ𝐿 = 25𝑛𝑚 − 50𝑛𝑚 range- resulting in
highest directionality of the emitted photons to the +X direction.
59
Figure 2.14. Multipole decomposition (amplitude and phase) of a primitive
nanoantenna comprising a dipole source embedded in a cubic DBB of size 220nm and
refractive index 3.5. As the position of the dipole is shifted from the center of the
DBB, the amplitude of the MDZ mode goes up and consequently attains higher
directionality.
This is further illustrated in Figure 2.15, which shows the spectrum of the
directivity towards +X direction for different values of Δ𝐿 . Note, for the range of Δ𝐿
where ED and MD amplitudes crosses over, the directivity is highest ~2.1.
60
Figure 2.15. Directivity spectrum for primitive nanoantenna for four different cases of
different positions of the source dipole with respect to the center of the DBB.
Finally, in Figure 2.16, we show the density of states as a function of spatial
location for the single DBB shown in Fig. 2.16(a) of cubic shape and size 220nm. To
compute the density of state at any arbitrary location in space, we assume a point
electrical source dipole at that location and calculate the Green function of the E-field
at the same point in space- i.e. 𝐺 ̿
(𝑟 ̅
𝑆𝑃𝑆 , 𝑟 ̅
𝑆𝑃𝑆 , 𝜔 ). To make things more practical, we
only compute the 𝑦 ̂ ⋅ 𝑦 ̂ component of the Green dyadic- which represent the density of
states that is seen by a Y-directional transition dipole of the SPS. This quantity is also
commonly referred to as the Partial photon density of states in the literature. The
partial density of states for a y-dipole is calculated from the Green function using
equation (2.7)- 𝜌 𝐿𝐷𝑂𝑆
(𝑟 , 𝜔 𝑆𝑃𝑆 ) =
1
𝜋 𝐼𝑚 (𝑦 ̂ ⋅ 𝐺 ̿
(𝑟 ̅
𝑆𝑃𝑆 , 𝑟 ̅
𝑆𝑃𝑆 , 𝜔 ) ⋅ 𝑦 ̂). In Fig. 2.16(b) we
plot the local density of states at 980nm on the XY plane passing through the center of
the DBB. Note- that the symmetry of the density of states is much similar to the
symmetry of the magnetic dipole mode shown in Fig. 2.10. This indicate that the
Magnetic dipole mode is the dominant mode controlling the density of photons states-
61
i.e. spontaneous decay enhancement whereas the electric dipole mode helps to
enhance the directionality via interference as shown before. On Fig. 2.16(c) the
spectrum of the Local density of states as a function of wavelength is shown for
Figure 2.16. (a) The geometry of the single DBB with embedded SPS transition dipole
and (b) the spatial distribution of local density of states at 980nm on an XY plane
passing through the center of the DBB. (c) The spectrum of the density of state for
different fixed positions of the transition dipole and (d) resultant Purcell enhancement
spectrum for the primitive nanoantenna structure.
62
different locations of the transition dipole denoted by the quantity Δ𝐿 shown in Fig.
2.16(a). The dashed line shows the density of state of homogeneous medium of
refractive index 3.5 (GaAs). Note the enhancement of the density of states for the
DBB compared to the case of uniform medium. This enhancement ratio results in
enhancement of the radiative decay rate of the SPS or Purcell enhancement following
eqn. (2.9) which is plotted in Fig. 2.16(d).
§2.3.3. Effect of Dielectric Membrane:
As discussed earlier in this chapter, realization of such DBB metastructure based
antenna structure will require incorporation of a dielectric membrane structure
underneath the DBB to prevent photon loss into the substrate. Here we study the effect
of the thickness of such membrane structure. This is shown in Fig. 2.17- by comparing
cases where the dielectric block bearing the SPS is on (1) semi-infinite GaAs bulk
(Fig. 2.17 (a)), (2) 500nm thick dielectric membrane on GaAs substrate (Fig. 2.17 (c)),
(3) 1𝜇 𝑚 thick dielectric membrane on GaAs (Fig. 2.17 (e)) and (4) Semi-infinite
dielectric (Fig. 2.17 (g)). It is clear from the corresponding angular distribution of the
Poynting vector shown in Panel (b), (d), (f), (h) respectively. To substantiate the
angular distribution of Poynting vector shown in Fig. 2.17, we also show the cross-
sectional distribution of the most dominant Field quantities for the Y-directional
transition dipole source that are the E-field along the Y direction and the magnetic
field along the Z direction- resulting in a Poynting vector along the X direction that
represent photon propagation in X direction-the direction chosen in our design for
63
enhancement of the photon directionality. Fig. 2.18, 2.19, and 2.20 show the
distribution of the EY Field, HZ Field and SX (Poynting vector along X) on YZ cross-
sectional planes on the DBB’s front and back surface and also ~275nm away from the
DBB in both +X and -X directions. Particularly, panel (a), (b), (c), and (d) respectively
represents the DBB on bulk GaAs (shown in light blue), on 500nm thick membrane
(shown in light green), 1000nm thick membrane and finally, the other extreme case of
the dielectric building block on semi-infinite membrane. Fig. 2.17-2.20 confirm that
when the DBB is directly on bulk GaAs there is significant loss into the substrate- but
~1𝜇𝑚 thickness membrane underneath the DBB is enough so that the SPS does not
feel the presence of the underlying GaAs substrate at all. We note that the effect of the
dielectric membrane remains same for DBB metastructures containing higher number
of DBBs- such as the Yagi-Uda nanoantenna- discussed in the next section. The effect
of the membrane thickness for such Yagi-Uda nanoantenna is shown in Fig. 2.26-2.28
in the next section. Incorporation of such underlying low index oxide layer thus
provides a way towards design of light manipulating structures that now can enhance
the directionality of photon emission in a specific horizontal direction for optimal
photon extraction into a horizontal architecture optical circuit.
64
Figure 2.17. Comparative Poynting vector angular distribution (Panel (b),(d),(f), (h))
corresponding to the schematics shown in panel (a), (c), (e), (g) correspondingly
where the thickness of a low-index (1.5) dielectric membrane under the SPS in a DBB
is increased gradually. It is evident that 1𝜇𝑚 thickness of the dielectric membrane is
enough to cut-off the effect of the underlying GaAs substrate completely.
65
Figure 2.18. Primitive nanoantenna on membrane- Plot of |EY| distribution on YZ
cross-section plane on the front and back face of the DBB as well as ~270nm away
from the surface of the DBB for (a) DBB on bulk GaAs (light blue here) (b) DBB on a
500nm thick membrane of refractive index 1.5 (light green) (c) 1𝜇𝑚 thick membrane
and (d) the extreme case of semi-infinite membrane. The top of the DBB is assumed to
be of refractive index 1.5 representing a protective polymer.
66
Figure 2.19. Primitive nanoantenna on membrane- Plot of |HZ| distribution on YZ
cross-section plane on the front and back face of the DBB as well as ~270nm away
from the surface of the DBB for (a) DBB on bulk GaAs (light blue here) (b) DBB on a
500nm thick membrane of refractive index 1.5 (light green) (c) 1𝜇𝑚 thick membrane
and (d) the extreme case of semi-infinite membrane. The top of the DBB is assumed to
be of refractive index 1.5 representing a protective polymer.
67
Figure 2.20. Primitive nanoantenna on membrane- Plot of |SX| (Poynting vector/energy
flow towards X) distribution on YZ cross-section plane on the front and back face of
the DBB as well as ~270nm away from the surface of the DBB for (a) DBB on bulk
GaAs (light blue here) (b) DBB on a 500nm thick membrane of refractive index 1.5
(light green) (c) 1𝜇𝑚 thick membrane and (d) the extreme case of semi-infinite
membrane. The top of the DBB is assumed to be of refractive index 1.5 representing a
protective polymer.
68
§2.4. Yagi-Uda Nanoantenna
The primitive nanoantenna as discussed in the previous section is readily
extended to the higher level Yagi-Uda nanostructure. The Yagi-Uda nanoantenna is an
elegant design for propagating wave antennas that was first proposed by S. Uda, and
later by H. Yagi in 1928 [2.54, 2.55]. Since then, Yagi-Uda antenna has been a major
component in radio communications to this day [2.56]. Additionally, owing to the
linear array-structure of the design of a Yagi-Uda antenna, it also provides a platform
that can be readily fabricated in nanoscale. This has also led to investigation with the
Yagi-Uda architecture in the optical wavelength. One of the first examples was set by
Curto et. al [2.57] that has been followed by other experimental and theoretical work
in metallic and dielectric platform [2.32, 2.57].
The basic structure of a conventional Yagi-Uda antenna is represented in Fig.
2.21(a). The excitation, conventionally in MHz-GHz range is fed to a driven metallic
element named feed. On one side, an array of metallic rods, named directors, are
arranged in specific positions, and on the other side, a single metallic rod, known as
the reflector is placed. In general, the lengths of the reflector and the director(s) are so
chosen that the currents induced in the reflector and director elements are
progressively phase shifted – resulting in directionality to the forward direction with a
specific phase velocity [2.56].
Conventionally in such Yagi-Uda antenna structures, the operating principle
relies on induced currents in straight metallic elements- which has only electric dipole
response. In our case, however, (Fig. 2.21 panel (b)), we explore the Yagi-Uda
69
architecture made of DBBs of near-cubic shape- allowing co-existence of the electric
and magnetic dipole modes as shown previously. Thus, the Yagi-Uda nanoantenna can
exploit both (1) directional interference of electric and magnetic dipole modes of the
same DBB and (2) directional constructive interference of E-field generated by
different DBBs. This idea is schematically shown in Fig. 2.21(c). This combined
effect allows improvement of the directivity of such nanoantenna structures as will be
seen next.
Figure 2.21 (a) A microwave Yagi-Uda antenna used in day-to-day communication.
(b) Nanoscale DBB metastructure based Yagi-Uda antenna. (c) The basic principle of
the Yagi-Uda antenna architecture.
To enable physical understanding, in this chapter we keep to the simplest Yagi-Uda
nanoantenna architecture- comprising only one reflector and one director.
70
Additionally, we assume that the DBBs are surrounded by uniform refractive index of
1.5 mimicking the dielectric membrane underneath and a protective a polymer on top.
§2.4.1. Principle of Yagi Uda Nanoantenna
Figure 2.22 demonstrates the working principle of the Yagi-Uda Nanoantenna
structure. The primitive nanoantenna (dipole embedded in a single DBB), investigated
in the previous section, now serves as the feed element. For reference, in Figure
2.22(a) we show the radiation pattern of the dipole emitting in the feed DBB with the
reflector and director absent- showing a directivity of ~2 to the +X direction. We
emphasize that the interference of the MD and ED mode of the DBB thus allows us to
choose a feed-element that has an intrinsic directionality- a feature that is typically not
present in common Yagi-Uda architectures in both microwave and optical wavelength
range [2.56] that rely mostly on the electric dipole modes of the elements. Figure 2.22
(b) indicates the structure of the Yagi-Uda antenna with a reflector of size
220nmx250nmx220nm. The center to center separation between the reflector and feed,
and between feed and director chosen to be 275nm- allowing a constructive
interference to the +X direction enabling a directivity ~10 at 980nm to the +X
direction- five-fold higher than just the feed element (panel (a)).
To illustrate the origin of the nanoantenna effect, in Fig. 2.22(c), we show the
E-field distribution along the X axis- both to the +X and -X direction. Furthermore,
71
the total E-field is broken down into components resulted by the reflector (blue curve),
feed (red curve) and director (green curve). Note that to the +X direction, the electric
field produced dominantly by the feed and director constructively interfere enhancing
the energy flow to +X direction. On the other hand, to the -X direction, the electric
field produced dominantly by the reflector and feed destructively interfere- reducing
the back-propagation. In addition to the effect of constructive interference shown in
Fig. 2.22(c) that defines the key physics of directionality, we show in Fig. 2.20(d) the
photon local density of states distribution on the XY plane passing through the center
of the three DBBs. Same as Fig. 2.16, we observe that the photon LDOS is dominantly
determined by the symmetry of the magnetic dipole mode. Finally, the operational
effect of the key physics shown in Fig. 2.22(c) and (d) are now shown in panel (e) and
(f). On panel Fig. 2.22(e) we show the spectrum of the directivity along +X direction
for the nanoantenna structure (blue line) in comparison with the only feed element
(black dashed)- demonstrating a significant enhancement of directivity over a large
spectral range of ~200nm owing to the interference of the E-field from the reflector,
feed, and director DBBs as shown in Fig. 2.22(c). We note, however, that the
enhanced directionality invariably results into a reduced E-field confinement and thus
should result in a drop in the Purcell enhancement. This is confirmed in Figure 2.22(f)
that shows the Purcell enhancement spectrum for the nanoantenna structure (blue
curve) in relationship with same for just the feed element (black dashed curve).
72
Figure 2.22. (a) Radiation pattern of a dipole embedded in a single DBB , compared to
dipole embedded in (b) yagi-Uda antenna. (c) Electric field distribution to the +X and
-X direction, shown as a superposition of the E-fields generated by the reflector, feed
and director- indicating that the directionality is resulted by constructive interference
in +X and destructive interference in -X direction. (d) Photon local density of state
distribution at 980nm on the XY plane passing through the center of the DBBs. (e)
Directivity spectrum of the Yagi-Uda antenna (blue) compared with the primitive
single DBB antenna (dashed black). (f) Purcell enhancement spectrum of the Yagi-
Uda and primitive antenna.
73
§2.4.2. Design of the Nanoantenna Structure:
Here we present outcome of systematic study of the optimization of the
nanoantenna waveguide structure. The parameter-set relevant to optimize the response
of the Yagi-Uda nanoantenna structure can be readily narrowed down to a few [2.56].
(1) The size of the reflector element - R
(2) The separation of the reflector from the feed, and
(3) The separation of the director from the feed
Figure 2.23 shows the response of the nanoantenna structure for different sizes of
the size of the reflector element. This is shown in Fig. 2.23(a) and 2.23(b). In these
plots the size of the reflector in the X and Z direction is fixed at 220nm. The feed and
director are fixed at cubic shape of size 220nm. P1 and P2 are fixed at 275nm. Note
that the directivity of the nanoantenna structure is optimal when the reflector size is
~220nm-250nm- elongated in the Y direction. The local density of states- as indicated
in Figure 2.23(b) remain essentially unaffected. These results indicate that the
directionality by the reflector is dominantly contributed by its position with respect to
the feed (P1), and not dominantly by its size.
Thus, we present study of the antenna response as a function of the relative
position of the reflector, feed, and director as captured by the two parameters P1 and P2
shown in Figure. 2.22(b). We vary both P1 and P2 in the range from 275nm to
425nm—an extent by about a wavelength. Further, the lower range is chosen to allow
a surface to surface separation between the DBBs of ~ 50nm that is fabricable using
74
dry etching. The spectrum of the Directivity and Density of States for different
configurations is shown in Figure 2.24. From the results plotted in Figure 2.25, we
draw a response surface of both the directionality and Purcell enhancement, shown in
Fig. 2.23 (a) and (b) respectively. The optimal directionality is achieved when both P1
and P2 are 275nm. We also observe the tradeoff between the Directivity and Purcell
enhancement as the negative correlation between the two response surfaces of Fig.
2.25 (a) and (b).
Figure 2.23. Effect of the size of reflector in the Y direction on (a) Directivity and (b)
Photon LDOS.
75
Figure 2.24. Effect of the pitch of the reflector (P1) and director (P2) on directivity and Green
function (photon LDOS)
Figure 2.25. Response surface of the Directivity and Purcell Factor as a function of P1
and P2, the pitch of the reflector-feed and director-feed.
76
§2.4.3. Effect of Membrane:
Similar to the single DBB primitive nanoantenna discussed in the previous section 2.3,
the realization of the Yagi-Uda antenna structure will also require embedding a low
refractive index oxide membrane underneath the DBBs to prevent photon loss into the
substrate. On Section 2.3 we had established that ~1𝜇𝑚 thickness of the oxide
membrane is sufficient for the Mie mode of the single DBB to be sufficiently isolated
from the underlying high-refractive index GaAs substrate. Since in a DBB
metastructures comprising higher number of DBBs, the collective Mie mode is simply
a linear combination of the Mie modes of the individual DBBs, we expect such ~1𝜇𝑚
thick membrane should also be sufficient for an arbitrary DBB metastructures that
utilize the same electric and magnetic dipole modes. This is confirmed for the Yagi-
Uda nanoantenna structure in the following.
Fig. 2.26, 2.27, and 2.28 show the distribution of the EY Field, HZ Field and SX
(Poynting vector along X) on YZ cross-sectional planes on the DBB’s front and back
surface and on the front and back surface of the director and the reflector DBB
respectively. Particularly, panel (a), (b), (c), and (d) respectively represents the
nanoantenna on bulk GaAs (shown in light blue), on 500nm thick membrane (shown
in light green), 1000nm thick membrane and finally on semi-infinite membrane. Note
that, these three figures are directly comparable to Fig. 2.18, 2.19, and 2.20 in section
2.3- demonstrating that the Yagi-Uda nanoantenna structure provides much higher
directionality compared to the single DBB. Also, we observe, specifically from Fig.
77
2.28, that 500nm-1𝜇𝑚 thickness of the membrane is sufficient for the photon flux to
not be affected by the underlying GaAs substrate.
Figure 2.26. Yagi-Uda nanoantenna on membrane- Plot of |EY| distribution on YZ
cross-section plane on the front and back face of the DBB as well as on the surface of
the reflector and director DBBs for (a) underlying bulk GaAs (light blue) (b) on a
500nm thick membrane of refractive index 1.5 (light green) (c) 1𝜇 𝑚 thick membrane
and (d) the extreme case of semi-infinite membrane. The top of the DBB is assumed to
be of refractive index 1.5 representing a protective polymer.
78
Figure 2.27. Yagi-Uda nanoantenna on membrane- Plot of |HZ| distribution on YZ
cross-section plane on the front and back face of the DBB as well as on the surface of
the reflector and director DBBs for (a) underlying bulk GaAs (light blue) (b) on a
500nm thick membrane of refractive index 1.5 (light green) (c) 1𝜇𝑚 thick membrane
and (d) the extreme case of semi-infinite membrane. The top of the DBB is assumed to
be of refractive index 1.5 representing a protective polymer.
79
Figure 2.28. Yagi-Uda nanoantenna on membrane- Plot of distribution of the x
directional Poynting vector ( SX ) on YZ cross-section plane on the front and back face
of the DBB as well as on the surface of the reflector and director DBBs for (a)
underlying bulk GaAs (light blue) (b) on a 500nm thick membrane of refractive index
1.5 (light green) (c) 1𝜇𝑚 thick membrane and (d) the extreme case of semi-infinite
membrane. The top of the DBB is assumed to be of refractive index 1.5 representing a
protective polymer.
80
§2.4.4. Why use a Nanoantenna?
We established in the preceding studies that with the help the Mie resonant
metastructure based Yagi-Uda nanoantenna it is possible to achieve simultaneous
Purcell enhancement ~5 and a directionality of emitted photons towards horizontal
direction with a directivity of ~10—over a broad wavelength range of ~100-200nm.
Figure 2.29. Schematic drawing indicating photon collection from the nanoantenna in
the horizontal architecture.
The idea is schematically represented in Figure 2.29 above. While the Purcell
enhancement increases the total number of photons emitted from the QD SPS, the
directivity enhances the fraction of the emitted photons that are actually collected over
an effective cross section that represents a guided mode – either on-chip such as a
waveguide, or off-chip such as an optical fiber. Thus, the total number of harvested
photons is proportional to the product of the Purcell enhancement and directivity.
Notably, both the Purcell enhancement and directivity spectrum are broad – allowing
enhanced photon harvesting over a large wavelength range and thus removes any
concern of spectral matching with the emitter.
81
Figure 2.30. Comparison of the spectral width of the response of (a) a typical photonic
crystal L3 cavity [2.6] and (b) DBB metastructure based Yagi-Uda nanoantenna.
This is not true for the conventional approach of photonic crystal. This is
illustrated in Figure 2.30(a) that shows the spectrum of emitted photons from an
ensemble of InAs island QDs in a photonic crystal cavity structure [2.6]. The modes of
the photonic crystal exist only at discrete wavelengths with narrow (typically few
hundred 𝜇 eV) widths. This makes impossible efficiently extract photons from a
specific single QD emitter, and forces one to depends on a huge ensemble (few
thousand/𝜇 𝑚 2
- spread over ~50nm spectral range) of spectrally random quantum dots
to make sure that photons from at least a few of them couples to the narrow modes
and thus are efficiently extracted. This major disadvantage is removed in our
approach, as we show the product of directivity and Purcell enhancement for our
structure in Figure 2.30(b) with a broad response ensuring efficient coupling with a
82
single MTSQD SPS. The broad response of the nanoantenna also facilitates coupling
of the extracted photons to an on-chip waveguide structure as a continuation of the
optical circuit. This is discussed next.
§2.5. Summary and Conclusion
In this chapter we explore the new paradigm of Mie resonance based optical
light manipulating elements to investigate the nanoantenna component – the subpart of
the optical circuit that provides the functions of enhancement of the emission rate of
the SPS and the enhancement of the directionality of the photon emission.
We present Mie theory based analytical and finite element method based
numerical simulation of DBB based nanoantenna structure to identify the key physical
processes that result into the nanoantenna effect. To summarize in a very succinct
way, these are
(a) Directional interference of the E-fields of electric dipole and magnetic
dipole type Mie resonance of the same DBB result in directionality and
Purcell enhancement – even for structures comprising a single DBB- such
as the primitive nanoantenna presented here- demonstrating a Purcell
enhancement ~ 5 and directivity ~2.
(b) Interference of the E-fields of the Mie modes of different DBBs of more
composite structures such as the Yagi-Uda nanoantenna can improve the
83
directionality significantly. This is demonstrated via a simple Yagi-Uda
nanoantenna structure that comprises of one reflector, one feed and only
one director element- resulting in a Purcell enhancement ~4.5 and
directivity~10. We present systematic study of assessing the effect of the
design parameters on the directivity and Purcell enhancement of such
structures.
Notably, unlike the photonic crystal-based approach, the Mie resonance based
approach that typically has a narrow (a few 𝜇 eV) linewidth, the Mie resonance based
nanoantenna structures can provide Purcell enhancement and directivity over a broad
wavelength range of ~10nm. Such broadband response opens up further possibilities
towards integration with optical circuits that provide manipulation of photons towards
on-chip guiding, splitting and recombining. This is discussed next.
To Waveguiding and Beyond: Two paths:
The Yagi-Uda nanoantenna opens up two dominant paths towards
incorporation of further needed functions to realize optical circuits as indicated below
in Figure 2.31. Owing to the broadband nature of the spectral response, the nanantenna
can be efficiently coupled to a ridge waveguide structure, as indicated in the left panel.
Alternatively, following the director element of the nanoantenna, the DBB array can
be continued resulting in a regular array of DBBs that provide the needed waveguiding
function using the same Mie mode but of this extended metastructure (i.e.
84
nanoantenna plus waveguide). Indeed, further manipulation of the photons such as
beam-splitting and beam-combining for photon interference can be incorporated with
continued expansion of the DBB metastructure and usage of a single Mie mode to
provide all five functionalities in systems like the one shown in Fig. 2.1. These aspects
are discussed next in chapter 3.
Figure 2.31. The nanoantenna structure enables realization of optical circuits by
coupling it to continuous waveguides or DBB array waveguides – discussed in the
next chapter.
85
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Chapter 3: From Nanoantenna towards Optical Circuits--
Nanoantenna-Waveguide
§3.1. Motivation and Background
In chapter 2 we showed that a collective Mie resonance of a DBB
metastructure in the form of a minimal nanoantenna can provide enhancement of the
emission rate of an embedded SPS by a factor of ~6 and emission directionality such
that ~50% photons are emitted into an effective cross-section of on-chip waveguide.
Figure 3.1. A schematic of the paradigm of on-chip optical circuits based on DBB
metastructures showing the different needed components.
Unlike the Bragg scattering of photonic crystal cavities, the nanoantenna is
realized exploiting the collective Mie resonance [3.1-3.2] that provides a broadband
response of a bandwidth of ~10-100nm. This paves the path towards design of a
structure that now extends the nanoantenna to an optical circuit- as the broadband
nature of the nanoantenna structure allows efficient coupling with on-chip optical
circuits as shown in Fig. 3.1.
91
The focus of this chapter is to present simulation and experimental work
towards understanding the nature of the collective Mie resonance in such optical
circuits allowing extending the nanoantenna structure to also enable the functions of
waveguiding, beamsplitting and beamcombining [3.1-3.3] —with the same collective
Mie mode of the whole system- thus eliminating mode-mismatch over the whole
optical circuit.
§3.1.1. Key Figures of Merit of an Optical Circuit:
It is important to note that the overall effectiveness of on-chip optical circuits for
quantum information processing applications [3.4-3.5], can be quantified using a few
key commonly accepted figures of merit [3.6]. These relate to how the optical circuit
is able to improve the indistinguishability of photons emitted by an on-chip integrated
photon source, how efficiently the photon is emitted into the desired optical mode of
the horizontal architecture to be propagated in a state-preserving fashion to a region
where its pathway is “split” (traditional notion of beam-splitting) and subsequent
controlled combining with another photon originating from another known source to
interfere or interact with another on-chip emitter. [3.6]. These figures of merits are
listed and discussed below:
(1) Purcell enhancement (FP): Purcell enhancement was discussed in the previous
chapter in the context of the nanoantenna structure- represents the enhancement of the
emission rate of the single photon emitter compared to the situation where it is in a
92
uniform medium. Purcell enhancement is important as typically for solid state single
photon emitters [3.7, 3.8], the dephasing timescale is found to be smaller than the
radiative lifetime which results into only a small part of the wave packet of the emitted
photon to be able to interfere with itself or with other photons. Purcell enhancement
enables shortening the radiative lifetime (T1) of the SPS, so that the temporal extent of
the photon wave-packet becomes comparable to the average time in which the
coherence of the photon is maintained—i.e. the coherence time T2 [3.9]. This is a key
essential needed function as typically the visibility, or intensity contrast in HOM type
interference, whose lower bound can be expressed as [3.10]
𝐼 𝑚𝑖𝑛 𝐼 𝑚𝑎𝑥
~ (1 −
𝑇 2
2𝑇 1
) (3.1)
(2) Coupling efficiency, 𝜷 : The quantity 𝛽 typically refers to fraction of the total
photons emitted by the source that are coupled to a particular photon mode of interest
[3.11]. We here define the quantity coupling efficiency, 𝛽 as the fraction of the total
emitted photons that are coupled to the modes that are involved in on-chip photon
propagation- in our case, the collective Mie resonance of the DBB array.
(3) Bandwidth: The bandwidth of an optical circuit is another important
characteristics as it determines the required tolerance for spectral matching between
the emitted photons within the SPS emission wavelength inhomogeneity and the
optical circuit providing an overlapping spectral range of photon state density
(consequently controlling efficiency of “coupling” between different functions
implemented by the optical circuits).
93
(4) Propagation Loss, 𝜶 : This quantity represents loss of the photons per unit length
of on-chip propagation distance. For semiconductor and dielectric based photonic
circuits such as in Silicon and GaAs, in the optical-NIR wavelength range,
propagation loss of standard waveguide structures and photonic crystal waveguide
structures are usually insignificant over the relevant on-chip distances. Thus, this
factor plays a less significant role in the design but needs to be kept in mind in terms
of the processing protocols and the roughness and contamination introduced during
fabrication processes.
The current status on these figures of merits are summarized in Table 3.1 and Table
3.2 that follow later in this chapter.
Ideal requirement: Typically, the GaAs-InGaAs-GaAs QDs have a dephasing time of
T2~100ps to 200ps, i.e. about 10-fold shorter compared to the radiative lifetime T1 of
~1 ns [3.12, 3.13]. Therefore, one requires a Purcell enhancement of ~10 to ensure
photon indistinguishability. Additionally, the coupling 𝛽 needs to be as close to 100%
as possible. A large bandwidth of 20-30nm of the optical circuit ensures easy spectral
matching between the SPS and the optical circuit as well as a certain amount of
robustness against fabrication errors. Importantly, all three requirements need to be
fulfilled simultaneously-- demonstrating individual near-perfect values in different
implementations sets proof-of-existence but not the viability of the whole system—the
optical circuit.
94
§3.1.2. Current Approaches in the Literature:
Current approaches towards creating optical circuits and the corresponding best
reported figures of merits are summarized in Table 3.1
Table 3.1. Current approaches to efficient extraction/ harvesting of photons to an on-
chip optical circuit.
95
(a) QD Coupled to Standard Ridge Waveguide:
Ridge waveguide based on Silicon, Silicon-Nitride, and III-V -on-insulator platforms
have been demonstrated with extremely low propagation loss down to 0.2dB/cm [3.19,
3.20]. Recent improvements have been caused by two fronts—(1) using reflow-
techniques to improve the surface quality reducing the interfacial light scattering and
(2) lift-off and flip-chip bonding techniques to allow incorporation of underlying SiO2
membrane for a vast category of material platforms such as Si-nitride, Lithium
Niobate, and particularly important- III-V [3.19]. Creating such ridge waveguide
structure around self-assembled island QDs for demonstrating propagation of the
emitted photons on-chip has also been attempted in recent studies. Waveguide modes
are broadband with ~100 nm or more bandwidth- thus allowing coupling with QDs in
a random ensemble such as the island QDs [3.17, 3.21, 3.22, 3.23]. However, lack of
confinement of the electromagnetic modes result in very low/ non-existence Purcell
enhancement produced by such structures.
(b) QD coupled to PhC Cavity-Waveguide
On the other end of the spectrum, Photonic Crystal structures have been heavily
explored to create strong confinement of light by preventing the photons to propagate
into certain parts by Bragg scattering. The Bragg scattering have been proven to
produce strong confinement resulting in ~10
7
Q-factors for cavity structures [3.24].
However, the same property of high Q ensures extremely narrow bandwidth- typically
in the sub-nanometer level. This has caused hindrance in Spectral matching between
the SPSs and the cavity structures as well as matching the cavity mode with an on-chip
96
waveguide to form an optical circuit. From the point of view of holistic design
ensuring all the three criteria of Purcell enhancement, 𝛽 , and bandwidth- the photonic
crystal approach poses challenges of spectral and spatial matching of the mode of the
SPS, cavity and waveguide structure [3.16, 3.25, 3.26].
§3.1.3. Our approach:
In comparison to the two main approaches discussed above, our approach of
exploiting the Mie resonant modes of dielectric metastructures has been geared
towards providing efficiently all five of the needed functions at the same time for the
highest cumulative outcome, even though each function may not be as efficient as the
best reported proof-of-existence implementation in the literature. In the previous
chapter we discussed the nanoantenna structure that provides the Purcell enhancement
and directionality of the emitted photons simultaneously. In this chapter we extend the
nanoantenna by coupling its output to on-chip waveguide, beam-splitter and beam-
combiner – all using the same collective Mie resonance of the whole unit (Fig. 3.2).
This provides an approach to implementing all the needed five functions
simultaneously without any impedance mismatch between them- and at the same time
providing ready spectral matching with the SPS by exploiting the spectral broadness
of the Mie resonance.
97
Figure 3.2. Our approach to on-chip optical circuit providing all the needed functions
via a single collective Mie resonance.
Chapter 3 is organized in the following way:
In Section 3.2, we present the theoretical analysis of the nature of the propagating
mode of the DBB array waveguide structures. Section 3.3 presents experimental
demonstration of such propagating modes using external light as photon source. In
Section 3.4. we extend the simulation studies to such DBB array waveguide being
integrated with the Yagi-Uda nanoantenna discussed in Chapter 2—a first step
towards forming on-chip optical circuits based on the Mie resonant metastructures.
Section 3.5 provides simulation studies of extending the nanoantenna-waveguide
structure to beamsplitting and beam-combining towards essentially a minimum
structural complete unit underpinning optical circuits such as the one shown in Fig.
3.2. It provides all the five needed functions simultaneously towards on-chip photon
98
interference and path entanglement. In Section 3.6, we summarize the findings and
provide concluding remarks.
§3.2. The Nature of the Propagating Mode of a DBB Array:
In Chapter 2 we discussed the nature of the nanoantenna function based on the
interference of the E-fields of the ED and the MD mode of the collective Mie
resonance of the DBB array. As shown in Fig. 3.1, the nanoantenna structure is now
seamlessly extended to the waveguiding structure based on DBB array. It is thus
important to understand the nature of the collective Mie mode in the waveguiding
spatial region of the complete structure. This is fundamentally different from other
approaches as discussed below.
§3.2.1. Existing Approaches to On-Chip Waveguiding
The earliest implementation of waveguiding historically has been in the form of a slab
of higher index material confined by lower-index material forming a ridge structure.
These ridge waveguide structures thus confine and propagate light via repeated total
internal reflection of the propagating photon- and over several decades have been
developed to yield very low-scattering loss structure in both SOI and GaAs waveguide
platforms- and also has been explored to confining light via a periodic medium of
dielectric structures (such as holes) has been more recent- and mostly focused on
photonic crystal structure that rely on operation in the photonic band-gap region also
resulting in reflection of photons by the waveguide boundary. [3.27]
99
The physics of these different approaches was captured in Chapter 1. In Table 3.2
here, the current status of these approaches to the waveguiding function has been
listed.
Table 3.2: Existing approaches to light guiding
Uniqueness of the Mie resonance waveguide
It is important to note that the guiding of light via Mie resonant metastructure is
different from the approach of ridge waveguide, photonic crystal, or even sub-
wavelength grating waveguide as it relies on the principle of coupled resonator optical
100
waveguide (CROW). Such waveguides, as a general concept, was introduced by Yariv
[3.33]. Each individual DBB, in such array architectures, contribute its Mie mode
(dominant magnetic and electric dipole and possibly multipoles) that couple and result
in the collective Mie mode of the structure- just as in a solid electronic states of
individual atoms combine to produce the band-structure. In fact, energy band of such
regular array of Mie resonators has been studied by Ohtaka et. al. [3.34, 3.35](FCC
lattice of Mie resonances). Recently, such waveguide structures have been tested
also—by Bakker nano let. 2017 leading to ~55dB/cm loss on-chip [3.32].
We note that, our approach of Mie resonant functional metastructure looks
beyond just waveguiding and attempts to realize all the needed light manipulating
function exploiting a single Mie mode, as noted in the last section and indicated in Fig.
3.2. However, in the spatial region of the DBB array functioning as a waveguide, the
nature of the collective Mie mode can be understood by studying the photonic band-
structure of an otherwise infinite array of DBBs. This is discussed next.
§3.2.2. Dispersion Characteristics of DBB Arrays
Spherical DBBs: Analytical Solution
In this chapter we discuss the dispersion characteristics of the propagating
wave via a DBB array—that constitutes the waveguide component of the
multifunctional optical metastructure presented in this dissertation.
101
Dispersion characteristics of propagation of a wave via a periodic array of
resonators/ modes is a very well understood problem over several decades [3.33, 3.34,
3.35]. In most situations where the evanescent wave amplitude dies exponentially
outside each vertex- such as in the case of electrons in lattice, or photons in an array of
high Q resonators [3.36], a tight binding approach accounting for one or two nearest
neighbors usually suffices. However, the problem at our hand is very different in that
regard. The mode-mode coupling between the Mie modes of two DBBs occur via real
propagating photons and thus the leading term in such coupling dies ~1/𝑟 , 𝑟 being the
separation between the two DBBs. This results in a scenario, where one needs to
account for many neighbors to find solutions to lossless bands. This will be addressed
in the results presented in the following.
Figure 3.3. Schematic showing an infinite array of spherical DBBs for which we solve
the photonic band-structure in the following to gain understanding on the dispersion
characteristics of the propagating collective Mie mode.
Here, to enable analytical approach to band-structure, once again we exploit
the analytical Mie theory for spherical DBBs that gives us insight on the nature of the
propagating mode of the DBB array. Here, the collective Mode is expressed as a linear
super-position of the Mie modes of the single constituent DBBs. The basic
102
formulation of Mie resonance of spherical DBB was discussed in Chapter 3 section
3.1 and is used here. The array of spherical DBBs is shown in Fig.3.3.
We use the spherical vector harmonics of each DBB as a basis of the net
Electric field of the collective Mie mode. This is expressed as
𝐸 ̅
(𝑟 ̅ ) = ∑∑𝑎 𝑛 ,𝑚 (𝑖 )
𝐸 ̅
𝑇𝐸𝑛 ,𝑚 (𝑟 ̅ − 𝑟 ̅
𝑖 )+ 𝑏 𝑛 ,𝑚 (𝑖 )
𝐸 ̅
𝑇𝑀𝑛 ,𝑚 (𝑟 ̅− 𝑟 ̅
𝑖 )
𝑛 ,𝑚
𝑖 (3.1)
A propagating collective Mie mode of the DBB array can be represented by satisfying
the Bloch periodicity of these multipole mode coefficients defined in eq (3.1):
𝑎 𝑛 ,𝑚 (𝑖 ±1)
= 𝑎 𝑛 ,𝑚 (𝑖 )
𝑒 ±𝑖 𝑘 𝐵 𝑃 (3.2)
and
𝑏 𝑛 ,𝑚 (𝑖 ±1)
= 𝑏 𝑛 ,𝑚 (𝑖 )
𝑒 ±𝑖 𝑘 𝐵 𝑃 (3.3)
Here 𝑘 𝐵 denotes the Bloch wave-vector along the DBB array. The dependence of 𝑘 𝐵
on the energy of the photon constitutes the dispersion characteristics of The Bloch
condition of the coefficients represented by equation (3.2) and (3.3) are combined with
the Mie theory as discussed in detail in the appendix to arrive at a self-consistent
equation from which the dispersion characteristics is computed.
Also, as shown in Chapter 2, in the wavelength of interest, the dominant modes of the
DBBs constituting the metastructures presented in this dissertation are the electric and
magnetic dipole modes. Thus, we limit the calculation of the dispersion characteristics
presented in this section to the ED and the MD modes.
103
To build up a self-consistent equation, we consider the multipole expansion of
the EM wave incident on a particular ith DBB as
𝐸 ̅
𝑖𝑛𝑐 (𝑖 )
(𝑟 ̅ ) = ∑𝑐 𝑛 ,𝑚 (𝑖 )
𝐸 ̅
(𝐽 )
𝑇𝐸𝑛 ,𝑚 (𝑟 ̅ − 𝑟 ̅
𝑖 )+ 𝑑 𝑛 ,𝑚 (𝑖 )
𝐸 ̅
(𝐽 )
𝑇𝑀𝑛 ,𝑚 (𝑟 ̅ − 𝑟 ̅
𝑖 )
𝑛 ,𝑚 (3.4)
Here the superscript (J) on the spherical vector harmonics represent that the radial
component is the spherical J-Bessel function instead of the hankel functions. This are
the radial form of the spherical vector harmonics that is suited for expansion of the
EM wave, when the source sits outside the boundary of the DBB [3.37, 3.38]. The
self-consistent equation is built by considering that the super-position of the scattered
waves of the neighboring DBBs result into the incident wave on the ith DBB. This can
be represented as
[
𝑐 1,1
(𝑖 )
𝑑 1,1
(𝑖 )
] = ∑[
𝑉 1,1
1
(𝑟 𝑖𝑗
) 𝑊 1,1
1
(𝑟 𝑖𝑗
)
𝑊 1,1
1
(𝑟 𝑖𝑗
) 𝑉 1,1
1
(𝑟 𝑖𝑗
)
][
𝑎 1,1
(𝑗 )
𝑏 1,1
(𝑗 )
]
𝑗 ≠𝑖 (3.5)
where 𝑟 𝑖𝑗
= |𝑟 ̅
𝑖 − 𝑟 ̅
𝑗 |. Here 𝑉 1,1
1
(𝑎 ) =
3
2𝑖 𝑒 𝑖𝛽𝑎 𝛽𝑎
(1 +
𝑖 𝛽𝑎
−
1
𝛽 2
𝑎 2
), and 𝑊 1,1
1
(𝑎 ) =
3
2𝑖 𝑒 𝑖𝛽𝑎 𝛽𝑎
(1 +
𝑖 𝛽𝑎
) .
The incident wave for the ith DBB can now be translated to the scattered wave of ith
DBB by using the Mie scattering susceptibility tensor (defined in Appendix A) as
[
𝑎 1,1
(𝑖 )
𝑏 1,1
(𝑖 )
] = [χ][
𝑐 1,1
(𝑖 )
𝑑 1,1
(𝑖 )
] (3.6)
104
Where
[χ] = [
Γ
𝑇𝐸
0
0 Δ
𝑇𝑀
] (3.7)
The Γ
𝑇𝐸
and Δ
𝑇𝑀
are susceptibilities of the TE and TM modes defined as
Γ
𝑇𝐸
=
(−𝑛 𝑗 ̂ 1
(𝛽𝑅 )𝑗 ̂ 1
′
(𝑛𝛽𝑅 )+ 𝑗 ̂ 1
(𝑛𝛽𝑅 )𝑗 ̂ 1
′
(𝛽𝑅 ) )
(𝑛 ℎ
1
(1)
(𝛽𝑅 )𝑗 ̂ 1
′
(𝑛𝛽𝑅 )− ℎ
1
(1)
′
(𝛽𝑅 )𝑗 ̂ 1
(𝑛𝛽𝑅 ))
(3.8)
and
Δ
𝑇𝐸
=
(−𝑛 𝑗 ̂ 1
(𝑛𝛽𝑅 )𝑗 ̂ 1
′
(𝛽𝑅 )+ 𝑗 ̂ 1
(𝛽𝑅 )𝑗 ̂ 1
′
(𝑛𝛽𝑅 ) )
(𝑛 ℎ
1
(1)
′
(𝛽𝑅 )𝑗 ̂ 1
(𝑛𝛽𝑅 )− ℎ
1
(1)
(𝛽𝑅 )𝑗 ̂ 1
′
(𝑛𝛽𝑅 ))
(3.9)
Here 𝛽 represents the vacuum wave-vector of the photon. n represents the refractive
index of the DBB, and R the radius. Also, here 𝑗 ̂ 1
and ℎ
1
(1)
represents the spherical
Bessel function and spherical Hankel function of the first kind, and 𝑗 ̂ 1
′
and ℎ
1
(1)
′
represent their derivatives with respect to their arguments. The origin of these equation
(3.8) and (3.9) are from Mie theory that is discussed in the Appendix A.
Now, equation (3.5) and (3.6) are combined with equation (3.2) and (3.3) to arrive at
the self-consistent equation for the Mie resonance multipole coefficients of the
propagating mode as
[
𝑎 1,1
(𝑖 )
𝑏 1,1
(𝑖 )
] = [χ]∑
[
2𝑉 1,1
1
(𝑙𝑝 )cos(𝑘 𝐵 𝑙𝑝 ) 2𝑖𝑊
1,1
1
(𝑙𝑝 )sin(𝑘 𝐵 𝑙𝑝 )
2𝑖𝑊
1,1
1
(𝑙𝑝 )sin(𝑘 𝐵 𝑙𝑝 ) 2𝑉 1,1
1
(𝑙𝑝 )cos(𝑘 𝐵 𝑙𝑝 )
][
𝑎 1,1
(𝑖 )
𝑏 1,1
(𝑖 )
] (3.10)
𝑁 𝑁 𝑙 =1
Or,
105
[
1 − 2Γ
𝑇𝐸
∑ 2𝑉 1,1
1
(𝑙𝑝 )cos(𝑘 𝐵 𝑙𝑝 )
𝑁 𝑁 𝑖 =1
−2𝑖 Γ
𝑇𝐸
∑2𝑊 1,1
1
(𝑙𝑝 )sin(𝑘 𝐵 𝑙𝑝 )
𝑁 𝑁 𝑖 =1
−2𝑖 Δ
𝑇𝑀
∑2𝑊 1,1
1
(𝑙𝑝 )sin(𝑘 𝐵 𝑙𝑝 )
𝑁 𝑁 𝑖 =1
1 − 2Δ
𝑇𝑀
∑2𝑉 1,1
1
(𝑙𝑝 )cos(𝑘 𝐵 𝑙𝑝 )
𝑁 𝑁 𝑖 =1
]
[
𝑎 1,1
(𝑖 )
𝑏 1,1
(𝑖 )
]
= 0 (3.11)
The propagating collective modes are signified by the nontrivial solution to equation
(3.11). Thus, the dispersion characteristics is calculated by
𝑑𝑒𝑡 [
1 − Γ
𝑇𝐸
∑2𝑉 1,1
1
(𝑙𝑝 )cos(𝑘 𝐵 𝑙𝑝 )
𝑁 𝑁 𝑖 =1
−2𝑖 Γ
𝑇𝐸
∑2𝑊 1,1
1
(𝑙𝑝 )sin(𝑘 𝐵 𝑙𝑝 )
𝑁 𝑁 𝑖 =1
−2𝑖 Δ
𝑇𝑀
∑2𝑊 1,1
1
(𝑙𝑝 )sin(𝑘 𝐵 𝑙𝑝 )
𝑁 𝑁 𝑖 =1
1 − 2Δ
𝑇𝑀
∑2𝑉 1,1
1
(𝑙𝑝 )cos(𝑘 𝐵 𝑙𝑝 )
𝑁 𝑁 𝑖 =1
]
= 0 (3.12)
For simplicity, we define
𝑉 = ∑2𝑉 1,1
1
(𝑙𝑝 )cos(𝑘 𝐵 𝑙𝑝 )
𝑁 𝑁 𝑖 =1
(3.13)
and
𝑊 = −2𝑖 ∑2𝑊 1,1
1
(𝑙𝑝 )sin(𝑘 𝐵 𝑙𝑝 )
𝑁 𝑁 𝑖 =1
(3.14)
Thus, equation (3.12) can be expressed as
(1 − Γ
𝑇𝐸
𝑉 )(1 − Δ
𝑇𝑀
𝑊 )− Γ
𝑇𝐸
Δ
𝑇𝑀
𝑊 2
= 0 (3.15)
106
The dispersion characteristics of the band is thus calculated by finding the zeros of the
function
𝐹 (𝛽 , 𝑘 𝐵 ) =
(1 − Γ
𝑇𝐸
𝑉 )(1 − Δ
𝑇𝑀
𝑊 )− Γ
𝑇𝐸
Δ
𝑇𝑀
𝑊 2
(1 + Γ
𝑇𝐸
)(1 + Δ
𝑇𝑀
)
(3.16)
Here the additional factor (1 + Γ
𝑇𝐸
)(1 + Δ
𝑇𝑀
) in the denominator is introduced to
make sure that the function F is a real quantity for ease of finding its zero [3.39].
Here 𝑁 𝑁 denotes the number of nearest neighbors considered in the calculation
of the bandstructure. Note, unlike the situation where the coupling terms decay
exponentially such as in an array of high-Q cavities [3.36], the mode to mode coupling
term for the Mie modes decays dominantly as~1/r [3.1]. Thus, accounting for only
nearest neighbor does not provide a lossless solution of kB from equation (3.10). We
also note that the solution of the Bloch wave-vector 𝑘 𝐵 as a function of the wavelength
of the photon (represented by 𝛽 =
2𝜋 𝜆 ) from equation (3.12) will, in general, result in
complex roots, since the coefficients of the matrix are in-general complex. The
complex part of the solutions to 𝑘 𝐵 simply represents the overall scattering loss of the
propagating mode of the DBB array. However, as the number of nearest neighbors is
increased, the scattering loss also goes to zero and lossless bands emerge [3.38, 3.39].
As captured in the analysis above, the dominant contribution to the nature of
the collective Mie resonance here is (1) from the Mie resonance of the constituent
DBBs and (2) the pitch of the DBB array (p) determining the mode-mode coupling. In
chapter 2, we had studied the dependence of the MD and ED mode spectrum on the
107
size and shape of the DBBs. Here, we present studies on how the separation between
the DBBs (captured by the parameter p- the center-to-center distance between the
adjacent DBBs for DBBs of a fixed size) affect the nature of the collective Mie
resonance. Continuing the analytical approach for the array of spherical DBBs above,
we thus present here the solution for the lossless bands as a function of the pitch p. We
note here that, like all waveguide structures, for lossless propagating modes to exist,
the propagation vector along the direction of propagation must be larger than the
propagation constant in uniform medium i.e. 𝛽 . Thus, the lossless propagation modes
only exist for 𝑘 𝐵 > 𝛽 . The 𝑘 𝐵 = 𝛽 line in the band-structure as shown in Fig.3.4. is
also referred to the light-line which represents the limiting situation where the lateral
confinement of the waveguide does not exist. No lossless propagating modes can exist
above this light-line in the dispersion relation. Since, for the pitch of the array being P,
the maximum value of 𝑘 𝐵 is
𝜋 𝑃 within the Brilloiun zone, this also sets the limit of P for
existence of lossless modes. We have
𝜋 𝑃 > 𝑘 𝐵 > 𝛽 =
2𝜋 𝜆 (3.17)
Thus,
𝑃 <
𝜆 2
(3.18)
This equation (3.16) sets the limit of the pitch of the DBB array for existence of
lossless propagating modes.
108
To illustrate the effect of the separation between the DBBs on the band structure, we
show the phase space of the energy bands as a function of the center-center pitch (p)
for spherical DBBs of radius 130nm whose response for a single DBB we reported in
Chapter 2.
Figure 3.4. Band-structure of an array of spherical DBBs of radius 130nm and
refractive index of 3.5 for different values of the center-center spacing. Panel (a), (b),
and (c) shows the band structure for p=275nm, 300nm and 375nm in the 𝛽 − 𝑘 𝐵
plane. The dashed line indicates the light cone-𝛽 = 𝑘 𝐵 line. Panel (d)-(f) shows the
corresponding band structure where the vertical axis displays wavelength.
109
We observe from Fig. 3.4(c), (d) and (e) that- as guessed from the understanding of
the requirement of the lossless band expressed via condition shown in equation (16),
as the value of 𝑝 is increased, the lossless bands are shifter to longer wavelengths. The
complete phase space of the photonic band structure is shown in Fig. 3.5.
Figure. 3.5. Phase-space of the photon band-structure showing the wavelength of the
collective Mie mode as a function of its Bloch wave-vector and the pitch (center-
center spacing of the DBBs). The DBBs are chosen to be 130nm in radius and
refractive index ~3.5.
Figure 3.4 and Figure 3.5 indicate that the dispersion characteristic phase-space
indicates that at a particular wavelength, only one collective Mie resonance exist in the
110
regime of the DBB size where the magnetic dipole and the electric dipole is the
dominant Mie modes of the individual DBBs. This is usually not the case for standard
ridge waveguide structures.
§3.2.3. Light Propagation via Array of DBBs
The dispersion characteristics as shown in the previous section reveals that, for a
specific wavelength of light, the DBB array acts essentially as a single mode
waveguide. However, the calculation of dispersion characteristics using the analytical
formulation as shown in the previous section assumes an infinite chain of DBBs which
is not practical, since such DBB array waveguides in an optical circuits will comprise
a finite number of DBBs, for which 𝑘 𝐵 is not a well defined quantity owing to the loss
of perfect translational symmetry. Thus, towards investigating the usefulness of such
DBB array in on-chip light propagation, we now present simulation results of
propagation of light via a finite section of DBB array.
Fig. 3.6 (a) shows the schematic of a finite section of DBB array waveguide
where one end of the waveguide is excited by an incident plane wave as shown in the
schematic. shows light propagation via a finite section of DBB array. With the
incident E-fields towards the Y direction, and the incident H-field towards the Z
direction, the dominant Mie modes that are excited of the constituent DBBs are the
magnetic dipole towards Z (MDz) and electric dipole towards Y (EDy) modes. Note
that these are the same modes that are exploited in the Yagi-Uda nanoantenna function
111
discussed in chapter 2. This will be important towards integration of the nanoantenna
with the waveguide functions with no mode mismatch. We will discuss this towards
the end of this chapter towards the overall objective of on-chip quantum optical
circuits.
Figure 3.6.(a) Light propagation via an array of 40 DBBs of spherical shape. (b)
response in the phase space of varying wavelength of light and pitch of the array.
Here, for the geometry of an array of 40 spherical DBBs shown in Fig. 3.6(a)
excited by a plane incident wave, we calculate the multipole modes corresponding to
the constituent DBBs. Specifically, the multipole mode coefficient of the DBB at the
other end of the DBB (shown in Fig. 3.6(a) by the coefficient 𝑎 1,1
(𝑁 )
) is the important
number as it represents the transfer of the impinging photon to the end DBB. Figure
3.6(b) shows the response surface of 𝑎 1,1
(𝑁 )
as a function of the wavelength of the
112
impinging wave as well as the pitch of the DBB array for a constant DBB radius of
130nm and refractive index of 3.5. The same response surface is plotted separately in
Fig. 3.7 for better visibility.
Figure 3.7. The 𝜆 − 𝑃 phase space response of array of spherical DBBs (same plot as
Figure 3.6(b))- indicating the Fabry-Perot resonant fringes.
Both the plot Fig. 3.6(b) and Fig. 3.7 show two distinct propagation bands
contributed by the collective mode based on the mixing and interference of the EDy
and the MDz modes. Furthermore, for smaller values of P, the propagating mode is
lossless- thus resulting in the Fabry-Perot interference fringes as indicated in Fig. 3.7.
Note that, this basic nature of the response of the DBB array waveguide is also
experimentally observed – reported in Section 3.3 in the following.
113
Cubic DBB Arrays: Finite Element Analysis
As lithographically fabricated shapes are invariably rectangular, we also report
finite element method-based studies of the 𝜆 − 𝑃 phase space response of array of
rectangular DBBs (similar plot to Fig. 3.6 and 3.7). Here we present propagation of
light through an array of 20 cubic DBBs of size 220nmx220nmx220nm and refractive
index 3.5. This is the same size and shape used for implementation of the nanoantenna
structure reported in Chapter 2. As indicated in Fig. 3.8(a), the DBB array is excited
via an incident plane wave. The propagation of light via the array is estimated by
calculating the amplitude of the magnetic dipole Mie mode of the DBB on the other
end. This is achieved via using the multipole decomposition also used in Chapter 2
and discussed in the Appendix A. Fig. 3.8(b) shows the response surface of the MD
mode coefficient (𝑎 1,1
(𝑁 )
) as a function of wavelength and also the DBB array pitch. We
observe the same feature- that the amplitude 𝑎 1,1
(𝑁 )
diminishes as the pitch is increased.
This is consistent with the understanding from the dispersion characteristics of DBB
array above- further indicating that the magnetic dipole mode constitutes the lossless
propagating mode of the DBB array.
114
Figure 3.8. FEM simulation of propagation of a plane incident wave through an array
of 20 cubic DBBs of size 220nmx220nmx220nm and refractive index 3.5. Panel (b)
shows the 𝜆 − 𝑃 phase space of the MD mode amplitude of the last DBB of the array-
a plot like fig. 3.7 for spherical DBBs.
115
To investigate further on the nature of the collective Mie resonance responsible
for the propagation of light in the waveguide structure shown in Fig. 3.8 comprising
array of cubic DBBs, we show the distribution of the E-field and H-field over the
cross-sectional plane of the array for the wavelength values that corresponds to the
peaks of the Fabry-Perot Resonance identified in the spectrum in Fig. 3.8. The
distribution of the x directional E-field and the distribution of the z-directional H-field
for these specific wavelengths are shown in Fig. 3.9 and 3.10 respectively. We observe
that, unlike a standard ridge waveguide, where the propagating mode is confined to the
center of the waveguide cross-section, for the DBB array-based waveguide, the E-field
spreads out from the waveguide cross section. This results in higher interaction
between two DBB array next to each other that can be useful towards creating
directional coupler structure shown later. This fact is further highlighted in Fig. 3.11
that shows the distribution of the EY and HZ fields along with the Poynting vector on
YZ cross-sectional planes of the DBB array for plane wave incidence at 980nm
wavelength. The cross-sectional mode profile has no nodes and EY and HZ maximum
at the center- further confirming that the dominant modes of the dielectric is
represented by electric dipole along Y and magnetic dipole along the X directions.
116
Figure 3.9. Distribution of the Ey field on a XY plane passing the center of the DBB
array. The array here comprises of 20 DBBs of cubic size of 220nm and pitch of
275nm. The Fabry-Perot oscillation along the length of the array are identified with
red dashed curves. The color scale of all the plots are the same and only shown once.
117
Figure 3.10. Distribution of the HZ field on a XY plane passing the center of the DBB
array. The array here comprises of 20 DBBs of cubic size of 220nm and pitch of
250nm. The Fabry-Perot oscillation along the length of the array are identified using
red dashed lines. The color scale of all the plots are the same and only shown once.
118
Figure 3.11. Cross-sectional distribution of the (a) EY and (b) HZ fields and (c)
Poynting vector along x direction, for the linear DBB array of 20DBB length for a
plane incident wave of wavelength 980nm. The mode profile indicate that the E Y and
HZ field attains maximum at the center of the waveguide cross-section- consistent with
the propagating mode being the EDY and MDZ type.
In the distribution of E- and H-fields shown in Fig. 3.9 and 3.10 the Fabry-
Perot oscillations can be seen very clearly as the periodic undulation in the E-field
intensity. These oscillations arise from the interference of the collective Mie resonance
propagating towards +X with the one propagating towards -X directions. Thus, the E-
field and H-field of such Fabry Perot modes show periodicity that is roughly
119
represented by ~|cos (𝑘 𝐵 𝑋 )|. In Figure 3.9 and 3.10 we have indicated these
oscillatory patterns with the wavy red dashed lines. Observing the periodicity of these
oscillations provides us a way to infer the dispersion characteristic of the cubic DBB
Figure 3.12. Partial dispersion characteristics of the DBB array for three different
value of pitch – where the Bloch wave-vector is inferred from the spatial distribution
of the E- and H-fields as indicated in Fig. 3.9-3.10.
array. The dispersion characteristics inferred from Fig. 3.9 and 3.10 are shown in Fig.
3.12. The Fabry Perot interference can be only sustained if the waveguide is lossless
enough if the reflected photon from the end of the waveguide can superpose with the
original photon with near-equal amplitude [3.40]. Thus, the Fabry Perot modes serves
as a signature of lossless propagation of the collective Mie resonance of DBB arrays
as discussed in the next section.
The analytical and numerical studies presented in this section discussing the
physics of the collective Mie resonance in arrays of DBB for light propagation and
120
manipulation based on mode-mode coupling [3.33] and mode interference provides
the major fundamental basis to exploiting Mie resonance to enable all five functions to
our larger objective of optical circuits as laid out in Fig. 3.1. Towards realization of
such optical circuits, within the scope of this dissertation work, we restricted the aim
of the experimental investigations to the wave-guiding function it could, in principle,
be addressed without the need for an actual single photon emitting source in the
pathway. However, even this limited original objective, regrettably, could not be
pursued owing to the sudden shutdown of the GaAs plasma etcher that precipitated a
nine month fruitless effort for finding a reliable alternative GaAs plasma etcher in
reasonable proximity of USC. The decision was thus made to carry out studies on
silicon DBB arrays as its refractive index is sufficiently close to GaAs so as to
maintain the spirit of the GaAs based platform for optical circuits. What follows below
therefore are the results of efforts on fabricating Si DBB arrays on silicon-on-insulator
(SOI) substrates and optical measurements of light propagation through such “wave-
guiding” DBB metastructures. As if the re-directed effort from GaAs to Si and the
attendant loss of time was not enough of a setback to a time-sensitive pursuit like a
PhD dissertation,, by the time the silicon plasma etching protocols were developed and
a few basic DBB arrays fabricated, COVID-19 pandemic driven shutdown of
laboratories came into effect in early March 2020. Consequently, the experimental
studies discussed below are limited to a bare minimum but nevertheless shed light on
the basic worthiness of pursuing the Mie-resonance approach.
121
§3.3. Experimental Studies: Silicon DBB Linear Array Based Waveguiding
As discussed in Chapters 1 and 2, function of on-chip light manipulation
require presence of an underlying low refractive index (n~1.5) oxide layer to prevent
photon loss into substrate and realization of the DBB based optical circuits in the III-V
platform compatible with the MTSQD SPSs thus demand incorporation of underlying
SiO2 layer using flip-chip bonding [3.19, 3.20], or other dielectrics (such as CaF2 and
BaF2) [3.41, 3.42] using lattice-matched epitaxial growth. The needed availability of
fabrication facilities to develop such hybrid (III-V- oxide) integration and fabrication
techniques were unfortunately not met owing to unexpected lock-down of cleanroom
and other lab-facilities, and thus not done within the scope of this dissertation. Thus,
as a proof of the concept experimental demonstration of the Mie resonance based
waveguiding, in this dissertation we take recourse to the alternate path of well-
established technology of Silicon on insulator platform to create Si DBBs on SiO 2
serving as the underlying membrane. Since in the Optical-NIR wavelength of interest,
the refractive index of Si (~3.5-3.8) is very close to that of GaAs (~3.5), probing the
nature of the Mie resonance of Si DBB serve as a useful hint towards understanding
nature of GaAs DBB arrays- fabrication of which will invariably follow as future work
towards realization of MTSQD-DBB optical circuits as envisioned in Fig. 3.1.
Next we present studies on the fabrication of the Silicon DBB arrays and their
optical characterization.
122
§3.3.1. Fabrication and Structural Characterization
The details of the fabrication steps used are captured in Table 3.3 below. As the
starting substrate, we use SOI wafer from Ultrasil (220nm Si on 3𝜇𝑚 SiO2 on Silicon
substrate) to create the DBB arrays. To create the DBB arrays, the top silicon layer is
first etched down to target height of the DBBs and then E-beam lithography and dry
etching are used to, respectively, pattern and carve the DBB arrays in the Silicon. In
the next, we provide the details of the corresponding processes.
Table 3.3. Fabrication Steps for creating the DBB array
1. Stating Substrate:
a. Degrease starting substrate with
Trichloroethylene (TCE), Acetone, Methanol,
and DI water by sonicating for 4 min in each.
This removes organic and inorganic junk on the
substrate.
2. Planar Etch of Silicon Layer
a. Immerse in 50% HF for 30s to remove native
oxide layer.
b. Rinse with DI water for 60s.
c. Etch for 2 min in 55% KOH at room temperature
(20
0
C) [Etch rate ~10nm/min- thus etching away
~20nm of Silicon to achieve the required ~200nm
height of the DBBs].
d. Rinse with DI water for 60s.
123
3. HSQ Spinning
a. Bake the substrate at 120
0
C for 1 min to remove
moisture.
b. On a ~20mmx20m substrate, drop ~300𝜇𝐿 of 6%
HSQ in MBIK (Dow Corning XR1541-006) and
spin at 6kRPM for 40s.
c. Bake at 120
0
C for 1 min to remove excess
solvent (MBIK) from the spinned resist.
4. E-Beam patterning
a. Expose the DBB pattern on the HSQ film with
20kV beam, ~85pA beam current.
b. Etch unexposed HSQ with NaCl(4wt%)
+NaOH(1wt%) aqueous solution for 32s. Then
rinse with DI water.
c. Bake at 350
0
C for 10min. This hardens the
exposed HSQ into high density SiOx mask for
dry etching.
5. DBB Etching
a. Oxford Plasmalab 100 ICP system was used to
etch the Silicon DBBs using the HSQ mask with
SF 6: C 4F 8 Inductively coupled plasma.
b. Etching condition: Pressure- 20mTorr, 500W ICP
Power, 50W RF Power, SF 6: 57 sccm, C 4F 8
33sccm, DC bias~200V. This leads to ~3nm/s
etch rate of Silicon and ~1nm/s etch rate of the
HSQ Mask. Etching was done for 1 min- leading
to a ~200nm etch depth of the Si DBBs.
a.
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E-beam Lithography
The high contrast electron beam resist hydrogen silsesquioxane (HSQ) disolved in
methyl iso-butyl ketone (MBIK) was used for the E-beam patterning. Over last
decade, HSQ has been established as a very high contrast negative e-beam resist and
has grown enormously in popularity in the research community [3.43]. Specifically, a
6% solution manufactured by Dow Corning (Dow Corning XR-1541-006) is used. The
SOI substrate is first treated with 49% HF solution for 30s to remove native oxide.
The substrate is spin-coated with the resist by dropping ~300𝜇𝐿 of HSQ on a ~1 inch
square substrate and spinning at 6kRPM for 40s- this results in a spinned HSQ
thickness of ~65nm. The spin-coated substrate is then baked for 1min at 120
0
C to
remove excess solvent (MBIK) before proceeding to E-beam lithography.
The E-beam lithography was performed using a Raith e-line 150 system. The
HSQ resist is exposed with e-beam of energy 20kV and aperture of 20𝜇𝑚 leading to a
beam current ~85pA. At best focus, the electron beam can achieve a spot size of ~1 to
5nm. Thus, a step size of 2nm for raster scan of the beam was used to expose the DBB
patterns on the HSQ film. The optimal area dose for the ~200nm square patterns is
~3000𝜇𝐶 /cm
2
. Thus, exposing the HSQ pattern for an array of 40DBBs takes <1s to
expose.
After exposure of the HSQ, the unexposed HSQ is then etched away using the
developer solution NaCl(4%) +NaOH(1%). It has been shown in the literature that this
solution results in very high contrast between the exposed and unexposed parts of the
HSQ [3.45]. The development of the HSQ patterns is a more complicated process
125
compared to the standard developing of photoresists. In addition to etching away the
unexposed HSQ, the NaCl(4%) +NaOH(1%) solution at the same time helps cross-
linking the exposed part of HSQ resulting in a high density SiOx as the final chemical
structure of the resist [3.43]. This is important as these cross-linked HSQ mesas are
resistant to most etching processes- wet or dry, and thus can be directly used as the
etching mask without any need for additional pattern transfer. We do the same in this
dissertation – using the cross-linked HSQ mesa as etching masks for the dry-etching of
Silicon- discussed next.
Dry etching of Si on SOI wafers using SF6/C4F6 ICP RIE
To create the narrow gap of ~50nm between subsequent DBBs, we use highly
anisotropic etching with Oxford Plasmalab 100- Inductively Coupled Plasma Reactive
Ion Etching (ICP-RIE) based on SF6/C4F8 chemistry. The process we use is commonly
known as the “pseudo-Bosch” process [3.45]. Unlike the Bosch process, where the SF6
and the C4F8 gasses are introduced in an alternating way [3.46] – creating cycles of
etching and sidewall passivation to create deep trenches with vertical sidewall, we
introduce the two gasses simultaneously [3.47]. For smaller depths, this improves the
sidewall angle – unlike the Bosch process- where undulations in the sidewall are
observed. Since in our case the etch depth is only 200nm, the pseudo-Bosch process
yields very good smoothness of the surface of the DBBs (below the resolution of SEM
~5nm) with vertical sidewalls and narrow (~50nm) spacing needed between the DBBs.
The Plasma density (controlled by the pressure, gas flow rates and the ICP
Power) and the impinging ion’s kinetic energy and direction (controlled by the bias
126
potential applied between the sample and the plasma) are chosen to achieve smooth
vertical sidewall with negligible undercutting. We use an ICP power of 500W and
chamber pressure of 20mTorr during the etching process. The ion’s bombardment
energy and direction are controlled by the RF power- 20W- resulting in a bias
potential of ~200V that accelerates the ions towards the substrate.
Representative scanning electron microscope (SEM) images of the resultant
DBB arrays are shown in Fig. 3.13, indicating the ~55nm wall-to-wall gap between
the nearest neighbor walls of the DBBs and the target size and shape of the DBB of
~200nm cube shapes. We note that the etching is done in a way that the DBBs sit on
the ~3𝜇𝑚 thick layer of SiO2 (as shown in Table 3.3) that serves as the needed
underlying low-refractive index membrane for on-chip light propagation. In the next
section we present studies to probe such on-chip photon propagation via the fabricated
DBB array structures.
127
Figure 3.13. Fabricated DBB array on SOI wafer. (a) Top-view of the HSQ mesas on
SOI used as etching masks. (b) Tilted SEM image of the fabricated DBB arrays
comprising DBB of size ~200nmx200nmx200nm and separation of ~50nm. Panel (c)
and (d) shows top-view and side-view of the constituent BDBs of the array.
128
§3.3.2. Optical Measurement Setup and Results
The experimental studies of the waveguiding function in DBB arrays were carried out
using an external super-continuum laser source over the wavelength-range of interest
to excite the array from /at one end and detecting the propagated photons at / from the
other end. As shown in Fig. 3.14, two different excitation / detection geometries are
studied:- (a) vertical excitation and detection, and (b) horizontal excitation and
detection geometry.
Figure 3.14. Two experimental measurement geometry studied for characterization of
the DBB array in this dissertation work.
The vertical configuration of Fig. 3.14(a) has the distinct advantage of optical
access anywhere on the chip that is specifically important for large scale optical
circuits [3.48]. However, the incident photons suffer huge insertion loss since coupling
efficiency of a vertical excitation beam to the waveguiding mode is typically less than
1%. To enable more effective vertical coupling, typically the waveguides are flared out
to grating couplers of lateral dimension of ~10𝜇𝑚 that can provide coupling efficiency
of 50% or higher [3.48, 3.49]. However, low (~10-20nm) bandwidth [3.48, 3.49] of
such grating couplers make them unsuited for our purpose of characterization of the
collective Mie resonance of the DBB array over several hundred nm wavelength
129
range. In the studies reported in this dissertation we thus did not implement a grating
coupler in the vertical excitation/detection geometry measurement. However, as
shown in Section 3.3.2.a next, even with the <1% insertion efficiency the resultant
signal to noise for the transmission efficiency measurement was good to spectrally
resolve and identify the collective Mie mode of the DBB array.
In contrast, the horizontal geometry of Fig. 3.14(b) has the distinct advantage
of coupling light in and out of the waveguide mode with much higher efficiency that
has been reported to go as high as ~97% [3.50] for adiabatic coupling with tapered
fibers. For measurement purposes, the horizontal geometry has the added demand of
the DBB array needing to be <10𝜇𝑚 away from the substrate edge to allow access for
the detection fiber. In the scope of this dissertation work we have implemented such
measurement geometry using a combination of horizontal optical microscope and
lensed single mode optical fiber to realize horizontal coupling to the waveguiding
mode with ~10-20% efficiency [3.51, Appendix C]. As discussed in Section 3.3.2.b,
such horizontal geometry is exploited to measure transmission spectrum through DBB
arrays as a function of wavelength for varying lengths of the DBB array to probe the
nature of the Mie resonance and its propagation loss.
§3.3.2. a. Vertical Optical Access via Objective Lens:
To get a quantitative perspective on the expected transmission efficiency in the
vertical geometry of Fig. 3.14(a), we carried out finite element method-based
simulations of light propagation via DBB array mimicking the vertical geometry. As
indicated in Fig. 3.15.(a)- an array of 40DBBs is excited using a plane wave vertically
130
Figure. 3.15. (a) Simulation geometry of array of 40DBBs of size 200nm cube and
refractive index 3.8 mimicking Si- with pitch of the array 250nm and surrounding
medium 1.5 representing the SiO2 membrane and surrounding microscope oil for the
measurement. (b) The MD and ED mode amplitudes along the DBB array at 900nm.
(c) The expected spectrum of collected photons in the vertical-vertical measurement
geometry. This shows good resemblance with the experimentally acquired spectrum.
incident on a ~2𝜇𝑚 spot at one end of the array. The collected photons are mimicked
by integrating the Poynting vector over a ~2𝜇𝑚 diameter surface on the other end of
the DBB array as indicated in Fig. 3.15(a). We carried out the multipole
decomposition of the DBB array and found that indeed, the ED and MD modes are
responsible for the propagation – shown in Fig. 3.15 panel (b). The estimated
131
transmission spectrum of the photon flux is shown in Fig. 3.15(c). Note that, owing to
the <1% coupling efficiency in the vertical measurement geometry the expected
overall transmission is ~10
-3
. However, the shape of the transmission spectrum clearly
reveals the propagating Mie resonance modes. The measured data presented next are
seen to be consistent with this analysis.
The vertical excitation-detection geometry measurements are carried out using
a custom-made far-field scattering spectroscopy set-up spanning a broad wavelength
range of ~500nm-1100nm limited by the detector. The measurement set-up is
schematically shown in Fig. 3.15.(a). One end of the DBB array is excited using a
vertical excitation beam and the propagated photons via the DBB are collected, also
vertically, from the other end of the DBB array.
More detailed description of the instrumentation can be found in Appendix C. Briefly
the geometry involves:
(1) Con-focal excitation geometry that allows exciting a spatial region of diameter
~2𝜇𝑚 around one end of the DBB array.
(2) Con-focal detection geometry that allows collecting photons from a spatial
region of diameter ~2𝜇𝑚 around the other end of the DBB array.
(3) Additionally, an adjustable pinhole in the angular-space of the collecting
Objective lens allows collecting photons that are only coming out in any
particular direction within the numerical aperture of the Objective lens- here
±70
0
with respect to the vertical direction. As we will show later, this angle
132
dependent photon collection allows get a sense of the directionality of the
photon propagation,
Figure 3.16.- (a) Experimental measurement geometry and (b) optical setup.
We present far-field scattering spectroscopy measurements on array of
40DBBs (~10𝜇𝑚 on-chip length) size ~200nm cube and nearest neighbor wall-to-wall
separation ~50nm. The results are divided into two categories (1) the detection is
integrated over the whole 70
0
collection cone of the objective lens and (2) the pinhole,
as shown in Fig. 3.16 is engaged to selectively collect the photons in specific
133
directions to measure the directionality of the photons coming out of the DBB array.
The results are presented next:
The measurement geometry overlayed on the SEM image of the DBB array is
shown in Fig. 3.17(a). Here the excitation laser from the supercontinuum source is
brought in and focused on one end of the DBB array with a ~2𝜇𝑚 diameter spot and
photons from ~2𝜇𝑚 diameter spot on the other end of the DBB are collected via the
confocal geometry and spectrally analyzed using the Acton 300i spectrometer. The
confocal geometry ensures that in order to get collected into the detector, the photon
must travel via the DBB array – as all other background scattering was brought down
to an insignificant level with ~3 orders of magnitude isolation. The intensity of the
collected photons as a function of the wavelength thus gives an indirect probe to the
propagation via the DBB array.
In Figure 3.17 (b), (c) and (d) we show the measured response on three
different arrays of 40DBBs on the same substrate- separated by ~20𝜇𝑚 distance. Both
the wavelength response and the absolute magnitude of these three spectra are found
same within ~10%- thus demonstrating the reproducibility of the DBB array. Further,
on Fig. 3.17 (e) we plot these three spectra overlaid on top of each other- they reveal
Fabry-Perot fringes in the same spectral location for each one of them. The presence
of the Fabry-Perot fringes indicates existence of propagating modes via the DBB
array. Note, the spectral feature of this propagation band is very similar to simulation
results shown in Fig. 3.7 and 3.8- indicating these are the ED and MD collective mode
of the DBB array.
134
Figure 3.17.-(a) Measurement geometry of the angle-integrated photon collection. (b),
(c) and (d) Result from three distinct DBB arrays of experimental measurement of far-
field angle-dependent scattering spectroscopy with vertical incident and vertical
detection geometry (e) The three spectra plotted on the same graph- showing the
existence of the Fabry-Perot fringes in all the three spectra. The existence of these
fringes indicates lossless/ near-lossless on-chip propagation.
(2) Angle Resolved Photon Collection:
We further improved the experimental set-up by adding a pinhole on a translational
stage in the plane that is optically reciprocal to the Field plane of the microscope. This
135
is indicated in both the simplified geometry and the optical set-up shown in Fig. 3.15.
Here the position of the pinhole directly corresponds to a specific angle of the emitted
photons.
Figure 3.18 (a) Angle dependent photon collection- simulation and experimental result
(b) Key physics governing the photon directionality.
We measured the angular distribution of the photon flux coming out of the DBB array
over the whole 70
0
collection cone of the NA1.41 objective lens. This unfortunately
excludes the horizontal direction which is most relevant towards applications of on-
chip optical circuit but gives a sense of the distribution of the photons over the
collection cone of the objective, indirectly giving a sense of directionality. The
measured angular distribution at the 900nm wavelength – is shown in Fig. 3.18(a)-
compared with the simulation result. We see that most of the photons are emitted to
one side of the numerical aperture- which indicates directionality of the emitted
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photons towards the axis of the DBB array owing to the interference between the MDz
and EDy modes as discussed in Chapter 2 and captured here again in Fig. 3.18(b).
These measurements revealed the normalized frequency response of the
collective Mie mode of the DBB array and thus allow to deduce the nature of the Mie
mode involved to be magnetic and electric dipole mode. However, the large loss in
number of photons both for coupling from the excitation beam to the collective Mie
resonance, and collecting the waveguided photons vertically results in an effective
transmission of ~10
−3
to ~10
−4
. This is remedied in the next section where we
present experimental attempts to measure transmission via the DBB array in horizontal
geometry shown in Fig. 3.14(b).
§3.3.2. b. Horizontal Excitation-Detection Measurement using a Lensed Fiber
The geometry of the horizontal excitation-detection measurement, as shown in
Fig. 3.14(b), is enabled by a horizontal microscope objective allowing excitation of
one end of the DBB array and a lensed optical fiber allowing photon collection in the
horizontal geometry. The detailed instrumentation for this measurement is discussed in
Appendix C, but a brief schematic is shown in Figure 3.19. The broadband excitation
from the super-continuum laser (400nm-2000nm) is brought in as a gaussian beam via
a 50X objective resulting in a 5𝜇𝑚 spot size and a ~5
0
beam divergence angle to
selectively excite end of an array. The propagated photons are collected using a lensed
single mode optical fiber with focal length of 10𝜇𝑚 and focal spot size of 2𝜇𝑚 . The
photons collected into the optical fiber are spectrally filtered using a 1200g/mm, 30cm
focal length Spectrometer and finally detected using Si CCD detector allowing
137
~500nm-1100nm detection window. The excitation beam is brought in at an angle of
~30
0
to the DBB array to avoid direct coupling of the excitation beam to the detection
optical fiber.
Figure 3.19. Experimental geometry for measurement of transmission efficiency via
DBB array in horizontal geometry using Objective lens for excitation and lensed
optical fiber for detection.
Figure 3.19 also indicates the three major factors in the overall measured
transmission efficiency- i.e. insertion efficiency, propagation efficiency, and collection
efficiency. However, unlike the measurement with vertical geometry, in this case a
significant fraction of insertion and collection can be achieved. To get a quantitative
understanding of this, we carried out finite element method-based simulations
mimicking the experimental geometry of Fig. 3.20. As expected, the simulations
reveal, at ~930nm, the magnetic dipole mode with the magnetic field distribution
shown in Fig, 3.20(b). The simulation indicates that in the excitation geometry, ~7%
of the excitation photons are coupled to the collective Mie resonance of the DBB
array. At the other end of the array, to estimate the photon collection efficiency, the
138
mode overlaps between the collective Mie resonance of the DBB array and the mode
of the single mode optical [Details in Appendix C]. Theoretically, a ~20-25% photon
collection efficiency from the collective Mie resonance to the horizontal optical fiber
is estimated. Based on the finite element method simulation of electromagnetic field
corresponding to the geometry shown in Fig. 3.20(a), the expected transmission
efficiency spectrum is shown in Fig. 3.20(c). An overall transmission efficiency of
~1% (a factor of 100 improvement compared to the vertical geometry) is expected.
Figure 3.21(a) shows the measured transmission via DBB arrays of targeted
size ~200nm cube and separation ~50nm in the horizontal measurement geometry
same as the simulated array of Fig. 3.20(a). As indicated in Fig. 3.21(a), we observe
distinct magnetic dipole resonance at ~785nm with, a peak transmission efficiency of
~1%. The peak transmission efficiency observed agrees with the expected
transmission efficiency, however the observed magnetic dipole mode peak is ~100nm
blue-shifted compared to the expected peak. Based on the studies of the dependence of
139
Figure 3.20. (a) Simulation geometry mimicking the horizontal excitation/detection
measurement on DBB array waveguides shown in Fig. 3.14(b). (b) Magnetic field
distribution on the XY plane passing through the center of the DBBs indicating the
collective magnetic dipole mode of the DBB array at ~930nm. (c) Estimated overall
transmission spectrum including effect of insertion efficiency and collection efficiency
of the photons into the optical fiber.
the magnetic dipole mode wavelength on the size and shape of the DBBs as reported
in Chapter 2, Section §2.3.1, the wavelength discrepancy is most likely a result of
deviation of the fabricated size of the DBB from the expected and simulated size of
140
200nm by ~10-20nm, i.e about 5% to 10%. Clearly this can be reduced significantly
with optimization of the fabrication protocols.
Figure 3.21. (a) Measured transmission via three DBB arrays of length 24𝜇𝑚 , 41𝜇𝑚
and 60𝜇𝑚 - showing the collective Mie resonance around 800nm and ~1% overall
transmission efficiency. (b)The peak transmission as a function of the length of the
DBB array 12 different DBB arrays of different lengths in the range of 24𝜇𝑚 and
60𝜇𝑚 . The dashed straight line indicates ~70 to 100dB/mm propagation loss.
Furthermore, Figure 3.21(a) presents representative transmission spectrum
collected on DBBs arrays of different lengths ranging from ~24𝜇𝑚 to ~60𝜇𝑚 . The
141
results show a steady drop in the peak transmission efficiency as a function of the
length of the array from which the propagation loss is quantified. Figure 3.21(b) shows
the peak transmission coefficient as a function of DBB array length for the five DBB
arrays whose spectrum are shown in Fig. 3.21(a) and other DBB arrays of same target
DBB size of ~200nm cube and ~50nm separation. The linear fit of the peak
transmission indicates a propagation loss of ~100dB/mm.
There are two dominant sources to such propagation loss—(1) Surface
roughness, particularly owing to the increased surface area because of DBB array
etching and (2) Scattered light within the SiO2 underlying substrate – both resulting in
dissipation of the Mie resonance into the radiation continuum. In literature, the best
reported value of propagation loss at around 900nm for Si DBB array waveguide
structures [3.32] is ~5.5dB/mm. Thus, for our case, further optimization of the
fabrication process is needed to improve the surface quality and thus reduce loss.
However, we should not loose sight that towards the final objective of realization of
the on-chip quantum optical circuits as shown in Fig. 3.1, one does not need on-chip
propagation of ~100𝜇𝑚 scale and only ~10𝜇𝑚 scale at best, since the Mie resonance
allows realization of the light manipulating functions in a much lower footprint as we
will explore later in this chapter.
The experimental work in this section provides demonstration of the nature of
the collective Mie resonance – particularly the magnetic dipole mode resulting in on-
chip light propagation. This, combined with the physics of nanoantenna discussed in
Chapter 2 provides now to seamlessly combine these two functions into the
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nanoantenna waveguide unit. We refer to this as the nanoantenna-waveguide structure-
and address simulation studies on this structure in the next section.
§3.4. Nanoantenna-Waveguide
So far we have discussed three functionalities needed for creating optical
circuits: in Chapter 2 nanoantennas that exploit directional interference of Mie
resonance of the constituent DBBs to provide (i) enhancement of the emission rate of
the SPS and (ii) enhancement of the emission directivity, both over a broad
wavelength range of ~10nm. In the preceding subsection, we have discussed DBB
array waveguide that work via coupled-resonator optical waveguides exploiting the
electric and magnetic dipole modes of the DBBs. In this section, we discuss the
integration of the above two light manipulating elements into a single unit. We refer to
it as the nanoantenna-waveguide. As summarized in Table 3.1, the nanoantenna-
waveguide metastructure is unique in the literature and has the distinct advantage of
providing the enhancement of emission rate, enhancement of directionality and
lossless propagation over a large wavelength range that is at least an order of
magnitude larger than the photonic crystal approach. In addition to the spectral width,
the nanoantenna and DBB array waveguide exploit the same collective electric and
magnetic dipole mode of the DBBs. Thus, in a combined nanoantenna-waveguide
unit, there is no concept of mode-mismatch. This allows extending the nanoantenna
into nanoantenna waveguide without any impedance mismatch.
143
§3.4.1. Transition from Nanoantenna to Nanoantenna-Waveguide
To understand how the nanoantenna structure is readily extended to
nanoantenna-waveguide towards more complex optical circuits, we present the here
finite element method-based simulation of the response of nanoantenna integrated
with a varying number of DBBs in a linear array waveguide structure. The design of
the nanoantenna is kept consistent with the design achieved in Chapter-2. Thus, the
Yagi-Uda nanoantenna comprises of the reflector DBB of size 220nmx250nmx220nm
in the X, Y and Z direction respectively. The Feed and the Director DBBs are both
cubic shape of size 220nm. The SPS transition dipole is placed 50nm from the center
of the feed DBB. The surface to surface separation between the DBBs are taken to be
55nm as determined in the design of the nanoantenna structure in Chapter 2. Now, the
nanoantenna structure is appended by DBB array – also of cubic shaped DBB of size
220nm and separation 55nm- as determined in the study of the DBB array waveguide
in the last two sections of this chapter. This is represented in Fig. 3.22(a). On Fig.
3.22(b-f) we show the E-field distribution on a XY plane passing through the center
plane of the DBB array for the different numbers of DBBs in the waveguide section.
From the E-field distribution we observe that the nature of the collective Mie
resonance remains the same as the number of DBBs on the waveguide part is
increased.
To understand further the nature of the propagating mode of the nanoantenna-
waveguide structure, we plot next in Fig, 3.23-3.25 the cross sectional EY, HZ and
Poynting vector along X along the YZ cross-section of the nanoantenna structure
144
revealing the mode-profile of the collective Mie mode. We find that both the EY, HZ
obtains maximum at the center of the YZ cross-section of the DBBs. This is consistent
with the fact that collective Mie mode of the DBB array comprises of the EDY and
MDZ modes. This is also consistent with the mode profile of the DBB array
waveguide structure shown in section 3.2.3 above.
Figure 3.22.(a) Nanoantenna-Waveguide structure- the Yagi-Uda nanoantenna
structure discussed in Chapter 2 now integrated with a waveguide segment of N
DBBs- (b)-(f) shows the E-field distribution for a 1debye transition dipole emitting at
980nm for different numbers of DBBs in the waveguide segment- 1,2,3,5,and 10
respectively. E-field symmetry indicates that the same collective Mie mode based on
Magnetic dipole and electric dipole mode is now extended for the whole
“nanoantenna-waveguide” unit.
145
Figure 3.23. Distribution of the EY field on YZ cross-sectional planes for the
nanoantenna-waveguide structure with varying number of DBBs in the waveguide
section shown in panel (a): only Yagi-Uda nanoantenna (b) nanoantenna+1DBB (c)
Nanoantenna+2DBBs (d) Nanoantenna+5DBBs and (e) Nanoantenna+10DBBs for a
transition dipole of 1 debye strength radiating at 980nm representing the SPS.
146
Figure 3.24. Distribution of the HZ field on YZ cross-sectional planes for the
nanoantenna-waveguide structure with varying number of DBBs in the waveguide
section shown in panel (a): only Yagi-Uda nanoantenna (b) nanoantenna+1DBB (c)
Nanoantenna+2DBBs (d) Nanoantenna+5DBBs and (e) Nanoantenna+10DBBs for a
transition dipole of 1 debye strength radiating at 980nm representing the SPS.
147
Figure 3.25. Distribution of the x-directional Poynting vector on YZ cross-sectional
planes for the nanoantenna-waveguide structure with varying number of DBBs in the
waveguide section shown in panel (a): only Yagi-Uda nanoantenna (b)
nanoantenna+1DBB (c) Nanoantenna+2DBBs (d) Nanoantenna+5DBBs and (e)
Nanoantenna+10DBBs for a transition dipole of 1 debye strength radiating at 980nm
representing the SPS.
148
Figure 3.26.- Evolution of Purcell enhancement as the nanoantenna structure is
extended gradually to nanoantenna waveguide.
The evolution of the relevant figures of merit (e.g. Purcell enhancement,
coupling efficiency) are shown in Fig. 3.26 and Fig. 3.27. On panel (a) to panel (f) we
show the Purcell enhancement and coupling efficiency as the nanoantenna structure is
gradually transformed into a nanoantenna waveguide. In section 1 of this chapter, we
defined the figures of merit. Here we show the evolution of the Purcell enhancement
(or, enhancement of the photon density of states at the location of the QD) (Figure.
149
3.26) and evolution of the coupling efficiency 𝛽 - representing what fraction of the
total emitted photons by the QD SPS is coupled to the collective Mie mode of the unit
(Fig. 3.27).
Figure 3.27. - Evolution of coupling efficiency (𝛽 ) as the nanoantenna structure is
extended gradually to nanoantenna waveguide.
150
We observe from Fig. 3.26 and Fig. 3.27 the following key facts:
(1) As we append the nanoantenna structure to nanoantenna waveguide, the
Purcell enhancement slightly increases. In fact, in the presence of the DBB
array waveguide, the density of states is enhanced owing to the Fabry Perot
resonance in the finite section of the DBB array waveguide structure. Since
this enhancement in density of states is caused by the super-position between
the forward-propagating photons and the reflected backward-propagating
photon in the waveguide section, the characteristic length scale for such effect
is determined by the Bloch wave-vector of the photon in the waveguide
section. In section 3.2 we have discussed the origin and nature of these Fabry-
Perot fringes for the collective Mie resonance of the DBB array.
(2) The fraction of the emitted photons coupled to the collective Mie resonance
remain mostly unchanged but increase slightly. This is tied to point (1), as the
increased local density of states of photons result in higher degree of coupling
of the emitted photon into the collective Mie resonance of the unit.
This is captured in Figure 3.28 where we show the effect of transition from
nanoantenna to nanoantenna-waveguide in a more compact form- plotted the resultant
peak Purcell enhancement and peak coupling factor as a function of the number of
DBBs added to the nanoantenna structure.
151
Figure. 3.28.- Peak Purcell enhancement and coupling efficiency (𝛽 ) as a function of
number of DBBs in the array extended from the nanoantenna. Both the figures of
merit saturate over ~5 DBBs (~1.5-2 𝜇𝑚 ) distance.
These two observations (1) and (2) above indicate that, in going from the
nanoantenna to the nanoantenna-waveguide, the local density of states of the
collective Mie resonance at the location of the SPS is improved- resulting in both
higher Purcell enhancement and higher photon coupling efficiency to the collective
Mie mode. To address this, next we study the spectral and spatial distribution of the
photon local density of states.
152
§3.4.2. Local Density of States of Photon:
As discussed in chapter 2, the photon local density of states is one of the most
important physical consequence of a light manipulating element that decides how it
couples to / extracts photon from a photon emitter. From the classically computed
electromagnetic Green function defined in Chapter 2, Section 2.2.1, the local density
of states is given by
𝜌 𝐿𝐷𝑂𝑆
(𝑟 ,𝜔 ) =
𝜔 𝜖𝜋
𝐼𝑚 (𝑝 ̂ ⋅ 𝐺 ̿(𝑟 ̅ ,𝑟 ̅
𝑆𝑃𝑆 ,𝜔 )⋅ 𝑝 ̂) (3.21)
In Figure 3.29, we present the distribution of 𝜌 𝐿𝐷𝑂𝑆
at the SPS emission
wavelength of 980nm on XY plane passing through the center of the DBBs for the
nanoantenna-waveguide structure where the waveguide comprises of 10 DBBs. The
position of the SPS transition dipole inside the feed DBB is chosen at the local
maximum of photon density of states- indicating the position of the SPS at 50nm from
the center of the DBB. It is not surprising that this configuration is identical to the
position of the source dipole in the nanoantenna structure as determined in Chapter 2.
153
Figure 3.29. Spatial distribution of the Local density of photon states at fixed
wavelength of 980nm for the (a) nanoantenna-waveguide structure where the Yagi-
Uda nanoantenna is combined with array of 10DBBs as waveguide. The nanoantenna
comprises of reflector of size 220nmx250nmx220nm. The Feed DBB, director, and
waveguide DBBs are of cubic shape and size 220nm. The surface-to-surface
separation between the DBBs are 55nm. Note- in the feed DBB- the DBB bearing the
QD in our design, the density of states obtains a maximum ~50nm from the center of
the DBB- at which point the QD is placed.
To further understand how the nanoantenna waveguide is different from only
the nanoantenna- in Fig. 3.30 we present the photon LDOS at the location -50nm from
the center of the DBB as indicated in the inset. We observe that, in the presence of the
waveguide DBBs, the LDOS is enhanced, which can be attributed to the Fabry-Perot
resonance of the collective Mie mode in the waveguide segment—as apparent from
the fringes appearing in Fig. 3.30.
154
Figure 3.30. Spectrum of local photon density of states at the location of the SPS
transition dipole for different numbers of DBBs in the DBB array integrated with the
nanoantenna. Vertical division for each of the plots is 12× 10
30
states/m
3
/ J.
§3.4.3. Response of the Nanoantenna-Waveguide:
Based on the understanding acquired above, the finite element method based
simulated E-field distribution of a nanoantenna structure comprising array of 20 DBBs
as a part of the waveguide is presented below in Fig. 3.31. The nanoantenna structure
is based on the design presented in Chapter 2. It comprises of a reflector element of
size 220nmx250nmx220nm; and feed and director elements being cubic DBBs of size
220nm. The center to center separation between the reflector and the feed, and also the
155
feed and the director is kept being 275nm based on the design shown in Chapter 2.
However, in this case, the director DBB is appended by an array of 20DBBs – each of
them of cubic shape of size 220nm and center-center separation 275nm. The emitter
transition dipole, approximated as a point electrical dipole towards the Y direction is
placed within the feed DBB of the antenna, 50nm away from the center of the DBB-
as was determined from Chapter 2 for optimal coupling of the emitted photons to the
electric and magnetic dipole mode of the DBBs. The MD and ED excitation of each of
the DBBs 𝜋 phase shift apart- resulting in directionality of the emitted photons
propagating along the DBB array. Once coupled to the DBB array waveguide, the
photon propagates along the array with minimal to no loss based on the principle of
discussed in Section 3.2 above. This results in directionality in propagation. On panel
(b) we have shown the angular distribution of the Ponting vector at a surface ~5×p
distant from the SPS dipole. The directionality of the emitted photon is very evident
from this plot. By integrating the pointing vector over the cross section of the mode
we determine that 𝛽 , the fraction of the emitted photons that are coupled to the
collective Mie resonance is ~40%-60%. The E-field distribution of the collective Mie
mode combined the electric and magnetic dipole modes is shown in Fig. 3.31(c). Note,
that the nature of the E-field distribution is similar to Fig, 3.9- the collective mode of
the waveguide structure- except for the first three blocks that represents the
nanoantenna. The nanoantenna here, as discussed in chapter 2, enhances the E-field at
the location of the SPS emitter- resulting in the Purcell enhancement- which is shown
in panel (d) – showing a 10nm bandwidth at 980nm wavelength.
156
Figure 3.31. (a) Yagi-Uda nanoantenna-waveguide structure. The SPS transition
dipole is embedded in a cubic DBB of size 220nm. The reflector DBB of the
nanoantenna is of size 220nm×250nm×220nm, where as the director DBB and the
waveguide DBBs are of cubic shape of size 220nm. (b) Angular distribution of the
photon flux on a spherical surface (radius ~ 500nm) surrounding the nanoantenna for a
radiating dipole of strength 1-debye at 980nm. The asymmetry in this angular
distribution indicates the nanoantenna effect. (c) Finite element method-based
calculated spatial distribution in the nanoantenna-waveguide of the electric field for a
1-debye point oscillating electric dipole source representing the SPS emitting at
980nm. (d) Purcell enhancement spectrum indicating broadband response with Purcell
enhancement ~7.
157
The nanoantenna-waveguide serves as a building unit that provides efficient
coupling of emitted photons from an on-chip single photon emitter into horizontal
optical circuit architecture. Thus, the nanoantenna-waveguide structure can be
exploited to guide the emitted photons on-chip towards distant location and potentially
be coupled directly to another SPS. We will discuss this approach to create
communication and entanglement in details in chapter 4 and 5. Alternatively, the
emitted photons from distinct emitters can be brought together using splitter/combiner
circuits to create on-chip photon interference. Establishing such photon interference
allows creating entangled photon state/ emitter state exploiting the Bunching property
of identical photons and thus allows realizing on-chip entangled photon state towards
quantum information processing based on Linear Optical Quantum Computation-
LOQC (Knill-Laflamme Milbourne protocol) and also measurement based cluster
state quantum computation. In the next section, we discuss simulation studies of such
beam splitter/combiner circuits.
§3.5. Beam-Splitting and Beam-Combining
In this section we extend the design of DBB metastructures that allow
interference of photons originated from two distinct SPS that allow communication
between those two SPSs towards quantum entanglement. Interference of light- mostly
in the classical domain has been an essential tool for optical signal processing for
classical and quantum information processing. The dominant approach to realizing
such on-chip optical circuit components has been via directional coupler- where two
158
waveguide sections (typically ridge waveguide) are brought close to each other so that
the evanescent component of the propagating mode couple – allowing the two
waveguide sections to exchange energy in a controlled manner. Directional couplers
are extensively studied in the context of conventional photonics based on Ridge
waveguide where two waveguides, propagating parallel to each couple via evanescent
field and thus produces splitting with any desired ratio over ~50-100𝜇𝑚 distances
[3.6, 3.21, 3.23].. Such directional coupler has been shown to mimic the function of a
beam splitter in the context of the optical circuits that follow KLM schemes to
implement quantum information processing [3.4].
§3.5.1. Conventional Approach to On-Chip Beam Splitter: Directional Coupler
We note that the Mie resonant metastructures discussed in this dissertation is
fully compatible to the approach of directional coupler. In fact, since the propagating
mode of the DBB array is dominantly magnetic dipole – and thus has the E-field
distribution close to the surface of the DBBs, it is easier to couple two parallel DBB
array waveguides to form the directional coupler. To investigate, we provide finite
element method based simulated results of a nanoantenna waveguide structure
continued to a directional coupler and study the splitting ratio as a function of the
spacing between the two adjacent DBB arrays for a fixed coupling distance. This
structure is shown in Fig. 3.32(a).
159
Figure 3.32. Conventional directional coupler structure using DBB array. The
nanoantenna and DBB array waveguide has same design as the structure presented in
Fig. 3.26. The coupling region is ~1𝜇𝑚 in length. The surface-surface separation
between the two parallel sections of the DBB arrays in the coupled region (𝑑 𝐺𝑎𝑝
) is
varied and the resultant E-field distribution is shown in the different panels.
160
In Figure 3.32 panel (b)-(e) we show the E-field distribution of a dipole emitting at
980nm from the nanoantenna of one of the branches- as the spacing between the DBB
arrays are varied from 500nm to 200nm. The resultant ratio between the Poynting
vector at each arm at the output is shown in Fig. 3.33 as a function of spacing between
the DBBs.
Figure 3.33. The ratio of photon flux (integrated Poynting vector over the waveguide
cross-section) at the two output branches of the structure shown in Fig. 3.31- when
only one of the SPSs is emitting photon- plotted as a function of the surface-surface
separation of the two parallel DBB array waveguides in the coupling section.
As indicated in section 3.2, owing to the magnetic dipole mode as the
dominant constituent mode, the E-field distribution is closer to the surface of the DBB
compared to standard ridge waveguide structures. This results into higher degree of
coupling between the two parallel branches of DBB array waveguide. Thus, we see, at
a separation of 200nm, 1:1 splitting is achieved over ~1𝜇𝑚 distance. By contrast in
161
standard ridge waveguide, typically a spacing of ~50nm over few hundred-micron
distance is needed [3.23] to achieve similar on-chip beamsplitting functionality. Thus,
the structure shown in Fig. 3.32 is a viable ideal candidate to realize photon
interference over small micron scale on-chip footprint towards Hong-Ou-mandel
interference between photons originating from two distinct SPSs. Further systematic
studies of such HOM on-chip however was not carried out under this dissertation work
and will be part of future work.
§3.5.2. Alternative Architecture of Nanoantenna / Waveguide / Beam-Splitter/
Beam-Combiner System:
• Y-Shaped Balanced Beam Splitter
Alternative to the directional coupler structure as discussed above, we also
explore a new architecture towards implementation quantum optical circuits. This is
done by directly coupling the Mie resonance of one DBB to two neighbors- exploiting
structures mimicking Y-shape junctions to split the propagating photons into two
branches or merging two propagating photons on-chip. The overall structure is
schematically shown in Fig. 3.34(a)- which shows conceptually an optical circuit that
represents all the needed light manipulating functions—i.e. SPS emission rate
enhancement, SPS emission directionality, on-chip propagation, beam splitting and
beam combining based on the same single collective Mie resonance over the whole
unit. Compared to the conventional approach to beam splitting that, as discussed
earlier, is implemented via 2-input 2-output directional coupler structure, the
architecture shown in Fig. 3.34 is unique- as one DBB array waveguide is split into
162
two connecting DBB array waveguides- such that the collective Mie mode of the
whole unit is equally split between the two branches. Such structure, we show, can
produce on-chip single photon interference from distinct SPSs resulting in
entanglement between the two distant on-chip SPSs.
The beam splitter part is highlighted in Fig. 3.34(a). Here, the input DBB array
is coupled to the two out-going DBB array at ~60
0
angle to each other. To enhance the
mode-mode coupling at the junction, we use a cylindrical shaped DBB at the junction
of radius. whose size and shape are so chosen to also have the magnetic dipole and
electric dipole as the dominant Mie modes at the wavelength of interest. Figure
3.34(b) shows the E-field distribution at 980nm for the beamsplitter part of the overall
structure.
On Figure 3.34(c) we show the distribution of the Poynting vector in the input
branch and the two output branches of the beam splitter at 980nm. By integrating the
Poynting vector over the cross-section of the DBB array waveguide, we determine a
scattering loss of ~50% of the incident photons at the junction. The rest of the photons
that are not lost are equally split between the two branches. We note that, there is a
~50% scattering loss at the junction- thus this beamsplitter is in no way as efficient
compared to a directional coupler. However, compared to the footprint of the
directional coupler which is typically ~50-100𝜇𝑚 [3.21], here the footprint is
significantly reduced to few hundred nanometers.
163
Figure 3.34 (a) Schematic showing the nanoantenna-waveguide-beamsplitter-
combiner circuit. The junction comprises of a DBB of cylindrical shape of diameter
230nm. The rectangular DBB size and pitch are identical to the nanoantenna-
waveguide structure discussed before. (b) E-field distribution of the collective Mie
mode of the whole circuit when only SPS1 transition dipole is emitting. (b) E-field
distribution of the collective Mie mode in the spatial region of the beam-splitter that
participates in the equal splitting of the photons into the two branches. (c) Cross
sectional Poynting vector distribution of the input waveguide section and the two
output branches of the beamsplitter. (e) E-field distribution of the collective Mie mode
in the spatial region of the beam-combiner. (c) Cross sectional Poynting vector
distribution of the two input waveguide sections and the output branch for the
beamcombiner part.
164
As shown in the schematic of Fig. 3.34(a), the beamsplitter is seamlessly
continued to the beamcombiner-where the same collective Mie resonance is now
exploited to merge photons from two distinct SPSs into a single branch. This is shown
in the E-field distribution Fig. 3.34(e) via plotting the E-field distribution of the where
only light emitted by SPS1 is merged into the common branch. On panel (f) we show
the cross-sectional Poynting vector distribution of the input and the merged branches
of the beam-combiner part. By symmetry the photons emitted from SPS2 are merged
into the common branch in similar way- resulting in photon interference in the
common branch from two distinct on-chip SPSs. This is discussed next.
§3.5.3. Beam-Splitter-Combiner and Photon Interference:
The nanoantenna-waveguide-beamsplitter-combiner unit thus stands as a unit that
represents all the needed light manipulating functions in a quantum optical network –
i.e. enhancement of the emission rate of the SPS, enhancement of the emission
directionality, on-chip propagation, beamsplitting and recombining towards on-chip
photon interference. Overall response of such a structure is noted in Fig. 3.35. Here
panel (b) shows the E-field distribution in the nanoantenna part- resulting in an
enhanced local density of states leading to a Purcell enhancement ~5 with bandwidth
~10nm- as shown in panel (c). Such Purcell enhancement will play a vital role in
improving the indistinguishability between the emitted photons to facilitate photon
interference. The on-chip photon interference is enabled by the beamsplitter- and
beam-combiner parts. To this end, on panel (d) and panel (e) respectively we show the
165
E-field distribution when only SPS 1 is emitting and when SPS1 and SPS2 are
emitting in phase. The cross-sectional Poynting vector distribution in the common
branch of the beam combiner part corresponding to panel (d) and panel (e) are shown
as insets. Note that the calculated Poynting vector in the combined branch when both
SPSs are emitting is a factor of 4 higher compared to when either of the single SPS
were emitting. This indicates interference at the common branch. Such interference
paves the way for the realization of path-entanglement between the two on-chip SPSs
[3.4].
However, from panel (d) of Fig. 3.35, and also from Fig. 3.34(f) we observe
that the beam-combing works in a way that the photon from SPS1 does not get
reflected and reach back to SPS2- and vice versa. Thus, we see that that there is no
direct communication between the SPSs- each SPS still decay incoherently with
respect to each other. For on-chip photon interference in a deterministic manner, it is
required for the SPSs to emit photon that are coherent with respect to each other.
We attempt to tackle this issue in Chapter 4 and 5 by exploiting the collective
Mie resonance of the DBB metastructures to establish direct communication between
the on-chip SPSs. Moreover, understanding this process of coherence demand more
than the classical Maxwell equation-based approach that is shown in this chapter.
These lines of investigations are presented next in chapter 4 and chapter 5.
166
Figure 3.35. (a) Nanoantenna-waveguide-beamsplitting-combining circuit based on
rectangular DBBs. (b) Distribution of Ey around the SPS1 when only SPS1-transition
dipole is emitting. (c) Purcell enhancement spectrum when only SPS1 is emitting. (d)
The Ey distribution of the collective Mie resonance of the whole unit when the only
SPS1 is emitting: showing the splitting of the photon at the Y-junction. (e) The Ey
distribution of the collective Mie resonance of the unit when the both the SPSs are
emitting—showing recombining of the photons in the common branch.
167
§3.6. Conclusions and Next Steps.
To sum, in this chapter we presented studies on on-chip optical circuits based on the
paradigm of exploiting Mie resonances of dielectric building block based
metastructures as co-designed light manipulating units (LMUs) whose collective Mie
resonance provides all five required on-chip emitted photon manipulation functions
needed in their inter-connected network called an optical circuit. Such control and
manipulation can be carried down to the single photon level. We have extended the
concepts and designs of the nanoantenna unit presented in Chapter 2 to include wave-
guiding, beam splitting and beam combining functions needed for a holistic design of
all dielectric metastructures that act as optical circuits. It is particularly important to
emphasize and note that Mie resonance based optical circuits do not involve discrete
components to provide all the needed light manipulating functions- i.e. enhancement
of emission rate of on-chip SPS, enhancement of emission directionality, on-chip
propagation, splitting and recombining towards the objective of on-chip photon
interference for quantum information processing. The Mie resonance based optical
circuits thus do not face the issue of “impedance-matching” between the different
“components” of the circuit. Although with the advances in machine learning and
artificial intelligence based approaches to inverse design the impedance mismatch
problem can likely be addressed sufficiently in approaches such as the photonic crystal
platform, the Mie resonance platform is a “direct problem” problem approach not
requiring the same notion and thus criteria of finding optimal design.
168
The approach that we introduced and pursued in this dissertation, particularly
detailed in this chapter, is unique as not only are all the needed light manipulation
functions noted above provided by the same collective Mie mode but these
functionalities can be provided over a broad band of 10nm to 100nm bandwidth, thus
accommodating the inherent inhomogeneity of the single photon emission wavelength,
narrow as it may be. These two remarkable features help eliminate spectral mismatch
between the Mie resonant optical circuit and the on-chip SPS, and also between the
different components of the optical circuit themselves.
Specifically, supplementing the studies on nanoantenna in chapter 2, we have
presented detailed study on the nature of the propagating modes of a finite DBB array
based on analytical approach with Mie theory and also numerical approach of finite
element method-based analysis. We have established the basic physics that, the
propagating collective Mie mode of a DBB array is also a superposition of Magnetic
dipole and electric dipole mode- similar to the nanoantenna structure. This builds the
foundation towards seamless integration of the nanoantenna-waveguide structure.
We have presented Fabrication of such DBB arrays in the platform of Silicon-
on-insulator and presented optical characterization of photon propagation/transfer
from one end of such DBB array to the other via the collective Mie mode exploiting
home-build far-field angle-resolved scattering spectroscopy system. The studies
confirm the understanding from simulations and numerical calculations.
Based on the foundation set above, we have presented the nanoantenna-
waveguide unit that exploits the collective Mie resonance and provides the
169
enhancement of the emission rate and enhancement of emission directionality along
with lossless propagation – thus forming the key building unit of an optical circuit
towards achieving photon interference from distinct on-chip SPSs. To this end we
have presented simulation studies on beam splitting and combining based on the
conventional directional coupler as well as an alternate approach of exploiting Y-
shaped junctions that allows splitting and recombining in few-hundred nanometer
footprint.
We note that, towards achieving on-chip photon interference towards quantum
entanglement, it is necessary that the two SPSs emit single photons coherently with
respect to each other. In the beamsplitter-combiner architecture this is not realized as
the photon from one SPS never reach directly the other SPS. In the next chapter we
present a possible approach that allows the same collective Mie resonance of the DBB
metastructure to establish direct communication between the two SPSs and thus force
them to emit in coherence. Systematic study of such SPS-SPS coupled design and
quantum mechanical approach to probe emergence of SPS-SPS coherence in such
systems will be discussed.
170
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175
Chapter 4: SPS-SPS Coupling and Super Radiance in Mie-
Resonance Based Optical Circuits: Classical Approach
§4. 1. Introduction and Motivation
In chapter 3, we presented simulations of the spatial distribution of the electric
field associated with the excitation of a dominantly magnetic dipole of the collective
Mie resonance of an all dielectric metastructure of the type recaptured below as
Fig.4.1. The relatively broad-band ( a few nm) collective Mie resonance, simulated to
be excited by the electric dipole mimicking a two-level electronic system emitting
photons within the bandwidth (such as from mesa-top single quantum dot (MTSQDs)
noted in chapter 1), provides the necessary functions (1) enhancement of the emission
rate of a MTSQD SPS (2) enhancement of the emission directionality, (3) on-chip
waveguiding, (4) splitting and (5) recombining the emitted photons from distinct SPSs
towards on-chip photon interference [4.1, 4.2].
Figure 4.1. The nanoantenna-waveguide-splitter-combiner metastructure representing
DBB based quantum optical circuits.
176
This last action- - recombining- - embodies the essence of a quantum optical circuit as
controlled phase relationship of the two combining photons arriving from two distinct
sources in the “circuit” containing multiple sources underpins interference and
entanglement. Thus, in this chapter we address- still using classical electromagnetism-
the formulation of a description involving two or more photons and the impact of
coherence between the single photon emitters. This controls preparation of entangled
states and manipulation towards quantum information processing [4.3] and metrology
[4.4].
The starting point of examining such quantum optical circuits based on the Mie
resonant metastructures has been established in the preceding two chapters where
emission and manipulation of single photons is understood using the concepts of E-
and H-field as well as the local density of states of single photon states calculated via
the electromagnetic Green function [4.6, 4.7] of the collective Mie resonance of the
DBB array [4.1]. However, the objectives of understanding on-chip photon
interference and entanglement, by definition, require multiple polariton states- thus
requiring description of multiple SPSs coupled to the Mie mode. The efforts towards
understanding the response of such multiple SPS coupled to the DBB metastucture
based optical circuit require, in the final analysis, accounting for the quantum nature of
both the emitters and the emitted photons.
Descriptions of controlled interaction between multiple emitters and multiple
well-defined photon states has been pursued for several decades [4.8, 4.9] with the aim
of understanding many fundamental quantum phenomena emergent from the interplay
177
of the fermionic nature of the emitters and bosonic nature of the optical fields [4.10],
the most focused upon likely being the emergence of entangled coherent states with
non-classical statistics. For our purposes of exploring such effect for SPS-DBB
metastructure based circuits, we divide our efforts into two steps. First, for a system
involving multiple emitters, the emission and manipulation of a single photon can still
be considered classical [4.11] and thus response understood via classical Green
functions. From such coherent SPS-SPS coupled states, the rate and dynamics
corresponding to emission of a single photon is significantly modified. This is
important in the context of optical quantum information processing as such coherence
between SPSs forms the basis of entanglement and enables photon interference. The
single decay rate of such coherent SPS-SPS coupled state is discussed in this chapter
with the help of classical Green function. The investigation of phenomena involving
many photons states continues in Chapter 5 where we will quantize the photon fields
corresponding to Mie resonance to finally address many-photon processes such as the
spontaneous emergence of such SPS-SPS coherent states.
§4.1.1. Importance of Coherence of the On-Chip SPSs
Establishing coherence between on-chip SPSs is the key towards on-chip SPS-
LMU coupled quantum information processing systems. Fig. 4.2 pictorially shows the
fundamental obstacles that stand in front of achieving on-chip photon interference.
Here a SPS, represented by a QD is optically excited and caused to emit a single
178
photon. However, there lies several uncertainties in the nature of the wave packet
emitted that directly affects the result of photon interference.
Figure 4.2. A schematic drawing of the typical excitation and photon emission in a QD
SPS- indicating the key processes that result in dephasing of the emitted photon.
◼ The fastest timescale is given by the exciton energy- which results in
oscillation of the E-field – typically at ~1eV, or ~0.3fs timescale.
◼ Intensity of the emitted photon decays with a timescale of T1- the radiative
lifetime- typically ~1ns for the semiconductor QDs [4.12, 4.13].
◼ Owing to effect of random phonon scattering and random stark shift due to
accumulated charge [4.14, 4.15], the phase of the emitted photon is gradually
randomized over time- becoming completely uncertain after a time T2
*
- also
known as the pure dephasing time. The overall effect of the dephasing and
amplitude decay results in the concept of coherence time for a photon, denoted
179
as T2. For semiconductor QDs, the coherence time is found to be typically
~100-200ps [4.16].
For photon interference, we need the coherence time to be comparable to the radiative
lifetime [4.17, 4.18]. This was the motivation behind Purcell enhancement that is
demonstrated in Chapter 2 and 3.
In addition to the above, there are also two uncertainties that are captured in Fig. 4.3.
◼ After optical excitation, there is a delay during the relaxation process of the
exciton to typically the ground state exciton in the QD SPS- resenting in a time
jitter in photon emission indicated by 𝜏 𝑅 .
◼ In addition, the process of spontaneous emission creates a random global phase
of the wave packet- shown using the 𝑒 𝑖𝜙
factor in the front- represents the
phase synchronization of the emitted photon.
The time jitter caused by the exciton relaxation to the ground level in the SPS and
can be eliminated by using resonant excitation of the QD SPSs. However,
uncertainty in global phase of the photon wave packet introduced by the
spontaneous decay process itself is an intrinsically random quantity and thus
prevents synchronization of the E-field of the emitted photons by two or more
emitters.
180
§4.1.2. Synchronization in Collective Emission- Background
The concept of synchronization of two coupled oscillators dates back as early as
1665 when Huygens observed that the phase of the oscillation of two coupled clock-
pendulums become locked over time [4.19, 4.20]. The task of synchronization
between two or more single photon emitters is conceptually no different. Historically
this has been observed/demonstrated exploiting the overlap of the near-field of the
emitted photon by one emitter with the electronic state of the other emitters [4.21,
4.22]. The first physical observation synchronization of such collective emission was
possibly by E. Jelly in 1937 [4.23], in the context of reduction of linewidth of the
fluorescence from J-aggregates formed by particular dye molecules. Later-on, in
1950’s similar phenomenon has been exploited in nuclear magnetic resonance studies
[4.21, 4.24], and also towards study of super-luminescence from low-pressure gas
molecules [4.21]. Under coupling by the near field, the coupling strength between two
emitters falls ~
1
𝑅 6
– [4.22]- requiring very closely placed emitters to achieve the
coupling strength comparable to the dephasing rate.
However, with the advent of photonics it became possible to confine a photon in a
waveguide like structure and propagate it over long distances without significant loss
in the field strength. While the generic theory of coupling between emitters mediated
by a photon mode has been explored quite well, the specific architectures that has been
studied, both theoretically and experimentally, are summarized in Table 4.1.
181
Table. 4.1. Current status in the literature of super-radiance/ emitter-emitter coupling in on-
chip optical circuits.
As indicated, the existing work in the literature can be divided in 4 different
categories:
182
(1) Multiple emitters coupled to the same photonic crystal cavity: Refs. [4.25,
4.26, 4.27, 4.28].
(2) Each emitter coupled to its own PhC cavity, and the cavities coupled between
themselves via evanescent wave: Ref. [4.29]
(3) Multiple emitters coupled to the same waveguide. Ref. [4.30, 4.31].
(4) Each emitter coupled to its own PhC cavity, and the cavities coupled between
themselves via a waveguide segment: Ref. [4.32].
As indicated in Table 4.1, most of the attempts to realize coupling between the
on-chip single photon emitters has been focused on coupling multiple emitters to the
same cavity or the same waveguide. This has been caused by the hard task of
achieving mode-matching between the different emitters and the photonic
components, and in between the cavities and waveguides themselves—particularly
owing to the narrow linewidth of the prevalent photonic crystal-based structures.
§4.1.3. Our Approach
In contrast, in the system proposed by us, the cavity and the waveguide
functions are realized using a single broadband collective Mie resonance. This
eliminates the mode matching problem and enables realization of larger systems
comprising of multiple single photon emitters coupled to the photonic circuits. A
simple realization of achieving two QDs coupled via the collective Mie resonance is
183
shown in Fig. 4.4- that comprises of two nanoantenna-waveguide structure facing and
coupled to each other, resulting in directional emission from one QD towards the other
QD and the emitted photon is collected by the lossless propagating waveguide-section
in-between allowing large separation between the SPSs.
Figure 4.3. Our Approach of MTSQD-MTSQD Coupled structure via a collective Mie
mode of the DBB array.
The work presented in this chapter hinges on, but is not confined to, the structure
schematically shown in Fig. 4.3.
In literature, the process of collective spontaneous emission by coupled emitter
has been analyzed for narrow-linewidth cavity modes formulated as a near-Unitary
Hamiltonian. [4.25]. This approach however does not work for spectrally broad modes
such as in the case of the Mie resonance. To study the collective spontaneous emission
of emitters under the influence of spectrally broad photon mode we here invoke Green
function of the broad photon mode. Notably, Green function can be attributed as the
most general property of any arbitrary spectrally broad photon mode and is intimately
tied with the concept of density of states of photons as we have explored in Chapter 2.
Classically computed Green function exactly represents the photon density of states
[4.6, 4.7] as well as exactly represents propagation of a single photon. The propagation
of a photon under the presence of multiple emitters is historically dealt with by
184
invoking Dyson expansion – such as approaches by Goldhaber [4.33, 4.34] and Low
[4.33, 4.35]. Recently such multiple scattering approach has been adopted in the
context of nanophotonic systems [4.26, 4.31, 4.32, 4.36]- to study emission of a single
photon by a collection of SPSs. However, typically they do not address decay rates of
the processes where more-than-one excited states are involved. We will address
dynamics involving multiple excitation later in Chapter 5.
§4.2. Classical Formulation of Super Radiance using EM Green Function
Similar to Chapter 2 and 3, in the classical picture the SPS is approximated as
a point electrical dipole- representing its transition dipole moment 𝑝 ̅ =< 𝑔 |−𝑒 𝑟 ̅ |𝑒 >
where |𝑒 > and |𝑔 > represent the excited state and ground state of the SPS emitter.
The emission of a single photon from this transition dipole representing the SPS is
manifested by the projected photon local density of states [4.6, 4.7]. A very useful
quality about photon density of states is that it is a quantity that is obtained completely
classically exploiting the projected Green function along the direction of the
oscillation of the dipole. In Chapter 2, we provided the formulation of radiative decay
rate of a single photon emission exploiting classical Green function. In this approach,
the Green function from Maxwell equation is modified to include multiple scattering
events from the SPS acting as a point scatterer and thus exactly represents the
propagator of a single photon emitted by the SPS. Importantly the poles of the Green
function in the frequency domain reveal the characteristic oscillatory and decaying
behavior of the eigenmodes that describe such decay [4.6, 4.7, 4.31]. This was utilized
185
in Chapter 2 and 3 to understand emission rate enhancement of a single emitter in the
presence of dielectric Mie metastructure environment. In this section we extend this
approach to account for multiple emitters – thus paving the way to classically
understand enhanced emission rate of a collection of emitters coupled to the Mie
resonant metastructures. The formulation utilizes the concept of classical Green
function derived from Maxwell equation to find the dynamics of emission and
scattering of classical E&M wave from a point-like entity- here the SPS. The
flexibility in the approach allows include multiple point -like scatterers and thus the
behavior of classical Green function in the single photon limit in the presence of
multiple SPSs.
§4.2.1. Green function for Multiple SPS Emitters
In Chapter 2 we discussed the modified electromagnetic Green function to capture the
effect of a single emitter involved in the scattering process of the emitted single
photon to derive the Purcell enhancement of the single emitter. The formulation is
readily extended to capture the effect of single photon scattering by two emitters such
as in the back-to-back nanoantenna waveguide shown in Fig. 4.4. The detailed
derivation is shown in Appendix B.3- here the key steps are noted. The modified
Green function that includes the effect of both the emitters is denoted by 𝐺 ̿ (2)
and
defined by
𝐸 ̅
(𝑟 ̅ , 𝜔 ) = 𝐺 ̿ (2)
(𝑟 ̅ , 𝑟 ̅
1
, 𝜔 ) ⋅ 𝑝 ̅
1
(𝜔 ) + 𝐺 ̿ (2)
(𝑟 ̅ , 𝑟 ̅
2
, 𝜔 ) ⋅ 𝑝 ̅
2
(𝜔 ) (4.1)
186
Where 𝑝 1
(𝜔 ) and 𝑝 2
(𝜔 ) at position 𝑟 ̅= 𝑟 ̅
1
and 𝑟 ̅ = 𝑟 ̅
2
, respectively represents the
transition dipole moments corresponding to the exciton transition of SPS1 and SPS2.
The multiple scattering process is schematically shown in Fig. 4.4 where the wiggle
arrow depicts the photon propagator 𝐺 ̿ defined using Maxwell equation (Appendix
B.1).
By including the multiple scattering using a Dyson sequence, the second order Green
dyadic is derived as [Details in Appendix B],
𝐺 ̿ (2)
(𝑟 ̅ , 𝑟 ̅
1
, 𝜔 )
=
(𝜔 2
− 𝜔 1
2
)[𝐺 ̿ (𝑟 ̅ , 𝑟 ̅
1
, 𝜔 )(𝜔 2
− 𝜔 2
2
− 2𝜔 2
𝑋 2
(𝜔 )) + 𝐺 ̿ (𝑟 ̅ , 𝑟 ̅
2
, 𝜔 ) (𝑝 ̂ 2
𝑝 ̂ 1
)
𝑝 2
𝑝 1
2𝜔 2
𝐽 2,1
(𝜔 )]
(𝜔 2
− 𝜔 2
2
− 2𝜔 2
𝑋 2
(𝜔 ))(𝜔 2
− 𝜔 1
2
− 2𝜔 1
𝑋 1
(𝜔 )) − 4𝜔 1
𝜔 2
𝐽 2,1
(𝜔 )𝐽 1,2
(𝜔 )
(4.2)
where
𝑋 𝑖 (𝜔 ) =
𝑝 𝑖 2
ℏ
𝐺 𝑖 ,𝑖 (𝜔 ) =
𝑝 𝑖 2
ℏ
𝑝 ̂ 𝑖 ⋅ 𝐺 ̿ (𝑟 ̅
𝑖 , 𝑟 ̅
𝑖 , 𝜔 ) ⋅ 𝑝 ̂ 𝑖 (4.3)
𝐽 𝑖 ,𝑗
(𝜔 ) =
𝑝 𝑖 𝑝 𝑗 ℏ
𝐺 𝑖 ,𝑗 (𝜔 ) =
𝑝 𝑖 𝑝 𝑗 ℏ
𝑝 ̂ 𝑖 ⋅ 𝐺 ̿ (𝑟 ̅
𝑖 , 𝑟 ̅
𝑗 , 𝜔 ) ⋅ 𝑝 ̂ 𝑗 (4.4)
Here 𝑋 𝑖 (𝜔 ) physically represents the interaction energy of an emitter onto itself owing
to the environment and is responsible for decay of the single emitter. On the other
hand, the term 𝐽 𝑖 ,𝑗
(𝜔 ) represents the interaction energy and is responsible for the
exchange of exciton between the two excitons. Note, that by Reciprocity theorem of
187
Figure 4.4. Representing the second-order Green function when the both the emitters
interact with the emitted photon.
electromagnetism [4.38], we have 𝐽 1,2
(𝜔 ) = 𝐽 2,1
(𝜔 ) = 𝐽 (𝜔 ), say. The super-radiance
and sub-radiance are a result of combination of these two effects- as discussed next.
§4.2.2. Super-radiant and Sub-radiant States
The formulation of the Green function including all possible scattering events by two
distinct SPSs as derived in eqn. (4.11) now allows to calculate the characteristic
oscillatory and decay behavior of the E-field as a function of time by studying the
poles of the Green function [4.7, 4.31, 4.36]. These complex poles of 𝐺 ̿ (2)
are
therefore given by the equation
(𝜔 2
− 𝜔 2
2
− 2𝜔 2
𝑋 2
(𝜔 ))(𝜔 2
− 𝜔 1
2
− 2𝜔 1
𝑋 1
(𝜔 )) − 4𝜔 1
𝜔 2
𝐽 2,1
(𝜔 )𝐽 1,2
(𝜔 )
= 0 (4.5)
We further simplify here assuming that
(a) The emitters are identical, i.e. 𝜔 1
= 𝜔 2
= Ω, and 𝑝 1
(𝜔 ) = 𝑝 2
(𝜔 )𝑒 𝑖𝜃
.
188
(b) The emitters are symmetrically placed with respect to the light manipulating
metastructure, i.e. 𝑋 1
(ω) = 𝑋 2
(ω) = 𝑋 (𝜔 ) ,say.
Here although the emitters are identical, a generic phase shift 𝜃 is assumed between
them. With this, the complex poles of 𝐺 ̿ (2)
, under the weak coupling limit are now
given by
(𝜔 2
− Ω
2
− 2Ω𝑋 (Ω))
2
= 4Ω
2
(𝐽 (Ω))
2
(4.6)
⇒ 𝜔 = ±(Ω + 𝑋 (Ω) ± 𝐽 (Ω)) (4.7)
Thus, the complex poles are
𝑠 𝑝𝑜𝑙𝑒𝑠 (+)
= ±(−Γ
(+)
+ 𝑖 (Ω + Δ
(+)
) (4.8)
𝑠 𝑝𝑜𝑙𝑒𝑠 (−)
= ±(−Γ
(−)
+ 𝑖 (Ω + Δ
(−)
) (4.9)
where Γ
(+/−)
= 𝐼𝑚 (𝑋 (𝜔 ) ± 𝐽 (Ω)), and Δ
(+/−)
= 𝑅𝑒 (𝑋 (Ω) ± 𝐽 (Ω)).
Equation (4.8) and (4.9) points to a pair of complex poles whose separation in the
complex plane is denoted by the complex number 2𝐽 (Ω). Importantly these two
solutions of the poles depend on 𝜃 , the phase shift between the two emitters which also
results in the complex phase of 𝐽 (Ω). The phase 𝜃 can be so chosen that the two
solutions 𝑠 𝑝𝑜𝑙𝑒𝑠 (+)
and 𝑠 𝑝𝑜𝑙𝑒𝑠 (−)
represents the fastest and the slowest decay processes
respectively. In that condition, we have Γ
(+/−)
= 𝐼𝑚 (𝑋 (𝜔 ) ± |𝐽 (Ω)|)- the super-
radiant decay rate and the sub radiant decay rate respectively. The lamb shift
corresponding to these two decay processes, under this condition, becomes equal-
189
Δ
(+/−)
= 𝑅𝑒 (𝑋 (Ω)). These complex poles corresponding to the super-radiant and the
sub-radiant states are shown in Fig. 4.5.
Figure 4.5. complex poles corresponding to the super-radiant and sub-radiant emission
of the two coupled QDs.
The Purcell enhancement for the super-radiant state can now be expressed as
𝐹 𝑝 𝑆𝑢𝑝 =
Γ
(+)
Γ
𝐻𝑜𝑚 =
6𝜋 𝜖 0
𝑐 3
𝑛 𝑖 Ω
2
[ 𝐼𝑚 (𝑝 ̂ 1
⋅ 𝐺 ̿ (𝑟 ̅
1
, 𝑟 ̅
1
, Ω) ⋅ 𝑝 ̂ 1
) + |(𝑝 ̂ 1
⋅ 𝐺 ̿ (𝑟 ̅
1
, 𝑟 ̅
2
, Ω) ⋅ 𝑝 ̂ 2
)
𝑝 2
𝑝 1
|]
(4.10)
and corresponding to the sub-radiant state-
𝐹 𝑝 𝑆𝑢𝑏 =
Γ
(−)
Γ
𝐻𝑜𝑚 =
6𝜋 𝜖 0
𝑐 3
𝑛 𝑖 Ω
2
[ 𝐼𝑚 (𝑝 ̂ 1
⋅ 𝐺 ̿ (𝑟 ̅
1
, 𝑟 ̅
1
, Ω) ⋅ 𝑝 ̂ 1
) − |(𝑝 ̂ 1
⋅ 𝐺 ̿ (𝑟 ̅
1
, 𝑟 ̅
2
, Ω) ⋅ 𝑝 ̂ 2
)
𝑝 2
𝑝 1
|]
(4.11)
190
We note that the super-radiant state and the sub-radiant state as shown above
are the normal modes corresponding to emission of a single photon from a collective
excited state of the two SPSs under the assumption that only one emitter can be in the
excited state at a time. Under this condition, the phenomenon is still classical- which is
why the classical Green function gives a satisfactory picture of the decay rate. Thus,
this classical formulation can be now exploited to design and study the collective
spontaneous single-photon emission rates of coupled SPSs in Mie resonant
metastructures. This we present next. We note, however, that as we consider the
possibility of the two emitters both being in the excited state, the classical picture fails
and a quantum formulation is needed – which we will address later in this chapter.
§4.3. Design of SPS-SPS Coupling Metastructure
In this section, we employ the formulation developed in the last section on classical
solution of the super-radiant and sub-radiant decay of coupled emitters to explore the
response of coupled SPSs in Mie resonant Metastructures. As evident from the
analysis in the last section, the quantity of interest here is the Green function of the
electromagnetic wave from emitter 1 to emitter 2, i.e. the 𝐺 ̿ (𝑟 ̅
1
, 𝑟 ̅
1
, Ω) term.
On Chapter 3 we had shown that in the nanoantenna-waveguide-beamsplitter-
combiner structure, the photon emitted from SPS1 does not directly reach SPS2 and
vice versa. Direct coupling can, however, be readily achieved by exploiting a pair of
nanoantenna-waveguide units connected directly via a common waveguide section.
191
This is the simplest system in which the two SPSs can directly couple mediated by a
collective Mie mode of the DBB array.
Here we present numerical solution of the Green function defined in equation
(3) for the system shown in Fig. 4.3 and repeated in Fig.4.6(a) employing finite
element method. The resulting E-field distribution for a single QD emitting in such
structure based on the zeroth order Green function, is shown in Fig.4.6(b) and 4.6(c).
Here panel (b) shows the XY plane cross-section of the distribution of 𝐸 ̅
𝑆𝑃𝑆 1
(𝑟 ̅ ) =
𝐺 ̿ (𝑟 ̅ , 𝑟 ̅
1
, 𝜔 ) ⋅ 𝑝 ̅
1
(𝜔 ). Similarly panel (c) shows the E-field distribution produced by the
second QD only, i.e., 𝐸 ̅
𝑆𝑃𝑆 2
(𝑟 ̅ ) = 𝐺 ̿ (𝑟 ̅ , 𝑟 ̅
2
, 𝜔 ) ⋅ 𝑝 ̅
2
(𝜔 ).
From equation (4.10) and (4.11) we know that the Green propagator values
that affect the radiative decay process of the coupled SPSs are 𝐺 ̿ (𝑟 ̅
1
, 𝑟 ̅
1
, 𝜔 ), and
𝐺 ̿ (𝑟 ̅
1
, 𝑟 ̅
2
, 𝜔 ). In panel (d) and (e) we thus show these two green functions as a function
of wavelength. The Green function at the location of the SPS itself- plotted in Panel
(d) determine the local photon density of state at each of the emitters if the other
emitter was not present. The effect of the other emitter is captured in panel (e),
showing the Green propagator from one SPS to the other.
192
Figure 4.6. (a) schematic of the back-to-back nanoantenna waveguide structure. E-
field distribution of when only (b) SPS1 and (c) SPS2 is emitting a photon. (b) and (c)
thus represents the classically computed zeroth order Green function as discussed in
the previous section. (d) Spectrum of the Green function component 𝐺 11
(𝜔 ) = 𝑝 ̂ 1
⋅
𝐺 ̿ (𝑟 ̅
1
, 𝑟 ̅
1
, 𝜔 ) ⋅ 𝑝 ̂ 1
- at the location of the emitter QD (e) Spectrum of the Green function
component 𝐺 12
(𝜔 ) = 𝑝 ̂ 1
⋅ 𝐺 ̿ (𝑟 ̅
1
, 𝑟 ̅
2
, 𝜔 ) ⋅ 𝑝 ̂ 2
– representing the propagator from one QD
to the other.
Super-radiant and Sub-radiant Emission Rate:
Based on the Green function shown in Fig. 4.6(d) and (e), we employ equation (4.10)
and (4.11) to calculate the decay rate of the super-radiant and the sub-radiant states.
They are shown in Fig. 4.7. The super-radiant and sub radiant decay rates are shown
193
with the reference of the Purcell factor of a single SPS (shown with the black dashed
line) for the same structure show in Fig, 4.6(a). We find that the super-radiant
emission rate is factor of ~1.7 higher compared to a single emitter case. Consequently,
the sub-radiant decay rate is ~0.3 times of the single emitter case.
Figure 4.7. The emission rate of the sub-radiant and super-radiant states for the system
shown in Fig. 4.6(a) exploiting equation (4.10) and (4.11).
Study on Dependence on SPS-SPS Distance:
The coupling between the SPSs revealed via the Super-radiant emission rate
enhancement shown in Fig. 4.7 occurs via the collective Mie mode of the back-to-back
nanoantenna waveguide structure. Since in the waveguide section, the collective Mie
mode is lossless, we expect such super-radiance to be realizable over arbitrary on-chip
distances. To investigate, in Fig. 4.8(a) and (b), we show the dependence of the Green
194
function propagator as the distance between the two SPSs are varied for the back-to-
back nanoantenna waveguide structure. The real and the imaginary part of the Green
function both show oscillatory behavior. The absolute value of the Green function,
however, does not fall off with distance. This indicate that the coupling term J in
equation (4.4) allows realization of long-range super-radiance.
For comparison, Fig. 4.8(c) indicate equivalent scenario for two emitters coupled to a
simple ridge waveguide, and Fig. 4.8(d) shows the SPS-SPS Green function for this
situation. Note that the presence of the guided mode in both the DBB back-to-back
nanoantenna waveguide and the ridge waveguide results significantly higher and
distance independent SPS-SPS interaction compared to two emitters in uniform
medium shown in Fig. 4.8(e) and (f). However, the nanoantenna structure results in
higher directionality and enhancement of local density of states compared to the ridge
waveguide. This results in higher value of the Green function in Fig. 4.8(b) compared
to Fig. 4.8(d). From this observation, the effect of the nanoantenna structure in
enhancing the photon directionality- ultimately enhancing the SPS-SPS coupling is
evident.
195
Figure 4.8. Dependence of the Green function on the distance between the two
emitters. (a) and (b) Back-to-back nanoantenna waveguide structure where the number
of DBBs of the waveguide section is varied to change the emitter-emitter distance. For
reference, (c) and (d) two emitters coupled via a continuous waveguide; and (e) and (f)
via free-space propagation in a uniform medium.
196
§4.4. Summary and Conclusions
In this chapter we have explored single photon decay of coherent collective excited
state of SPSs coupled via collective Mie resonance of DBB metastructures. We have
built upon the foundation of the modelling and simulation based on Maxwell
equations of the DBB array metastructures in Chapter 2 and Chapter 3 that help
understand the manipulation of a single photon dominantly based on the
electromagnetic Green function and the traditional one-photon density of states.
Extending from it, the electromagnetic Green propagator was modified to include
photon scattering from both the emitters coupled via the back-to-back nanoantenna
waveguide structure. We demonstrated that the coherent super-position of the two
transition dipoles of the two SPSs can result in either super-radiant decay rate (1.7x
single emitter decay rate) or subradiant decay rate (0.3x single emitter decay rate)
depending on the phase between the two transition dipoles. Further, we demonstrated
that the SPS-SPS coupling effect for the DBB metastructure is separation distance
independent and also results in significantly higher coupling compared to SPSs
coupled via a standard ridge waveguide structure.
The premise of this chapter has been approximating the emitters as classical
point dipoles and the emitted photon as classical electromagnetic field. This is a valid
assumption when examining single photon emission from an assumed existent
collective excited state comprising many emitters [4.36]. However, in the classical
approach, we simply did not address the origin of such a collective excited state
starting from a state where both SPSs are initially excited. The classical picture breaks
197
down as soon as more than one photon is present in the system [4.8]. This is because
of the added requirement of the symmetrization of the many-photon state, a purely
quantum mechanical property that does not have any classical analog. To analyze the
response of the quantum optical circuits in the many-photon regime, we need to apply
field quantization on the emitted many-photon states. This is discussed in Chapter 5
next.
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201
Chapter 5: Emergence of SPS-SPS Coherence- Quantum
Approach
§5.1. Introduction and Motivation
In the preceding chapters we showed that the collective Mie resonance of
suitably designed DBB metastructures with appropriately embedded single photon
sources (Fig. 5.1(a)) can provide the needed five basic on-chip light manipulating
functions of the SPS emitting preferentially into a Mie mode that defines desired
horizontal propagating direction while enhancing the emission rate (over that into
vacuum), on-chip propagation to the desired distance, splitting, and subsequent
Figure 5.1. (a) A generic on-chip SPS-DBB integrated quantum optical circuit (b) The
specific simple two-SPS coupled back-to-back nanoantenna waveguide structure
investigated in this chapter.
202
recombining at the desired distance. This is the repertoire needed for implementing
on-chip controlled interference and entanglement between photons, as well as photon
mediated entanglement between distant emitters [5.1].
Specifically, in Chapter 4, the collective Mie mode was demonstrated to be
viable means to establish long-distance on-chip coupling between distinct SPSs. We
analyzed the response of a simple representative back-to-back nanoantenna-waveguide
circuit as shown in Fig. 5.1(b) and theoretically demonstrated super-radiant single-
photon emission from a collectively excited state of the two SPSs. However, the
classical approach in Chapter 4 does not address how such coherent collective SPS-
SPS coupled states are created. Creating such coherent coupled states of multiple SPSs
starting from an otherwise product state where all the SPSs are in excited state is a key
milestone as it allows both preparation of entangled states as well as enables on-chip
photon interference by synchronizing distinct photons emitting from distinct SPSs
[5.2, 5.3].
To this end we present a formulation to address a classical and quantum
description of such multiple-exciton states of the SPS-DBB optical circuits. We extend
the concept of classical Green function [5.4] to a semiclassical quantum mechanical
model that accounts for the exchange of identical Fermions/Bosons (a purely quantum
mechanical phenomenon) and allows us to analyze the emergence of entanglement in
the SPS-DBB Metastructure based systems. We demonstrate that such coherent
emission of multiple SPSs can be achieved exploiting the same collective electric and
magnetic dipole Mie resonance of the DBB based quantum optical circuits.
203
The overall scheme of this chapter in the context of our larger objective is presented in
the flowchart in Fig. 5.2.
Figure 5.2. Flowchart capturing the content and interconnection of the different
chapters to the overall theme of this dissertation: on-chip quantum optical circuits.
Uniqueness of the Approach in this Chapter:
Typically, the collective spontaneous emission by coupled emitters has been studied
for spectrally narrow cavity modes [5.5-5.7]- thus representing the photon state as a
204
single delta-function like mode. In contrast, Mie resonance is spectrally broad and thus
emission into a continuous spectrum of modes needs to be considered. A viable way
to account for the effect of a spectrally broad electromagnetic environment is to
exploit the notion of Green function to formulate the Hamiltonian [5.7-5.11]. Green
function for the propagation of a photon under the presence of multiple emitters is
modified by invoking Dyson expansion – such as approaches by Goldhaber [5.12,
5.13] and Low [5.12, 5.14]. Recently such multiple scattering approach has been
adopted in the context of nanophotonic systems [5.8, 5.10] to study emission of a
single photon by a collection of SPSs. However, typically the authors do not address
decay rates of the processes where more-than-one excited state is involved. In
contrast, here we extend multiple scattering approach of Green function to formulate a
semi-classical Schrodinger equation that is applicable to Hilbert spaces of arbitrary
number of excitations. Further, to address emergence and evolution of entanglement
we formulate the decay dynamics based on the Von-Neumann Lindblad Master
equation approach [5.9, 5.12, 5.15, 5.16] that has been well established in the literature
for solution of the density matrices including both coherent and incoherent decay
processes [5.16]. Further we exploit the concept of Von-Neumann entropy [5.17] to
understand entanglement in such systems. We find that the entropy of the two-emitter
system is lower compared to the entropy of each of the single emitters- thus
demonstrating entanglement of the coupled emitter system. The approach presented
here provides a tool to understand the nature of the MTSQD-Mie resonant
metastructure systems in the context of realizing quantum optical circuits and the
results presented attest to its viability.
205
§5.2. Formulation of Coherent Emission: Quantum Approach
Single SPS Emitter
To get a quantitative understanding on the behavior discussed above of the two
coupled SPSs, let us first formulate the Jaynes Cumming [5.18-5.19] type Hamiltonian
for the nanoantenna-waveguide system that we classically studied in Chapter 2- a
system comprising only one SPS coupled to a collective Mie resonance as shown in
Fig. 5.3.
Figure 5.3. A single SPS with the nanoantenna-waveguide from the quantum
mechanical point of view.
The overall Hamiltonian is expressed in terms of the Hamiltonian of the SPS (𝐻 𝑆𝑃𝑆 ),
the Hamiltonian corresponding to the Mie modes of the DBB array (𝐻 𝑀𝑖𝑒 ) and also
additional radiation modes of the photon that are not part of the Mie resonance and
decays into the continuum (𝐻 𝑅 ).
𝐻 = 𝐻 𝑆𝑃𝑆 + 𝐻 𝑀𝑖𝑒 + 𝐻 𝑅 + 𝐻 𝑆𝑃𝑆 −𝑀𝑖𝑒 + 𝐻 𝑆𝑃𝑆 −𝑅 (5.1)
206
The SPS here is taken as a two-level system defined by the exciton creation operator
𝜎 †
. Thus, the Hamiltonian of the SPS is
𝐻 𝑆𝑃𝑆 = ℏΩ(t)𝜎 †
𝜎 (5.2)
The exciton energy, here denoted by Ω(t), represents the emission energy of the
photon at any point of time. The generic time-dependence of Ω(t) is useful to mimick
the effect the spectral wandering of the excitation peak owing to random stark-shift by
accumulated charges in the material surrounding the QD SPS- that has been shown to
play an important role in controlling the indistinguishability of the emitted photons
with respect to each other [5.20, 5.21]. In the current scope of the analysis in this
chapter we have ignored the effect of spectral wandering and assumed Ω to be
constant over time and accounting for the spectral wandering will be part of future
work.
To express 𝐻 𝑀𝑖𝑒 we follow the canonical quantization of electromagnetic field
based on the definition of the Hamiltonian based on electromagnetic energy density
𝐻 𝐸𝑀
=
1
2
∫(𝜖 (𝑟 ) |𝐸 (𝑟 ̅ )|
2
+ 𝜇 (𝑟 ) |𝐻 (𝑟 ̅ )|
2
)
𝑉 . Following the quantization of the
magnetic vector potential [5.22, 5.23], we can represent the Hamiltonian
corresponding to the Mie mode in terms of the ladder operators of the collective Mie
mode denoted as,
𝐻 𝑀𝑖𝑒 = ∑ ℏ𝜔 𝑎 𝜔 †
𝑎 𝜔 𝜔 (5.3)
207
Note that in equation (5.3) we exploit the fact that at any particular frequency 𝜔 there
exists only a single collective Mie mode. In a similar way, the Hamiltonian
corresponding to the leaky modes are expressed as:
𝐻 𝑅 = ∑ ℏ𝜔 𝑏 𝜔 ,𝑘 ̂
,𝜂 †
𝑏 𝜔 ,𝑘 ̂
,𝜂 𝜔 ,𝑘 ̂
,𝜂 (5.4)
where 𝑎 𝜔 †
represents the creation operator of a photon in the collective Mie mode at
the frequency 𝜔 and 𝑏 𝜔 ,𝑘 ̂
,𝜂 †
represents creation of a photon in the radiation modes with
the indices 𝜔 , 𝑘 ̂
, 𝜂 representing the sum over all possible degree of freedom such as
frequency, direction of propagation and polarization respectively. Unlike very narrow
bandwidth high Q photonic crystal cavities [5.5], the Mie mode exist not as a sharp
energy level, but as a continuous band. Also, as established in Chapter 3, the Mie
resonance is a single mode at any particular frequency. Thus, the summation indices in
the Hamiltonian of the Mie mode do not contain any other term than the energy.
The interaction of the SPS to Mie mode and the radiation modes can be generically
expressed by the interaction term in a typical Jaynes Cumming model as
𝐻 𝑆𝑃𝑆 −𝑀𝑖𝑒 = ∑ ℏ𝑔 𝜔 (𝑎 𝜔 †
𝜎 + 𝑎 𝜔 𝜎 †
)
𝜔 (5.5)
and
𝐻 𝑆𝑃𝑆 −𝑅 = ∑ ℏ𝜅 𝜔 (𝑏 𝜔 ,𝑘 ̂
,𝜂 †
𝜎 + 𝑏 𝜔 ,𝑘 ̂
,𝜂 𝜎 †
)
𝜔 ,𝑘 ̂
,𝜂 (5.6)
With this, the overall Hamiltonian thus takes the form
208
𝐻 = Ω𝜎 †
𝜎 + ∑ ℏ𝜔 𝑎 𝜔 †
𝑎 𝜔 𝜔 + ∑ ℏ𝜔 𝑏 𝜔 ,𝑘 ̂
,𝜂 †
𝑏 𝜔 ,𝑘 ̂
,𝜂 𝜔 ,𝑘 ̂
,𝜂 + ∑ ℏ𝑔 𝜔 (𝑎 𝜔 †
𝜎 + 𝑎 𝜔 𝜎 †
)
𝜔 + ∑ ℏ𝜅 𝜔 ,𝑘 ̂
,𝜂 (𝑏 𝜔 ,𝑘 ̂
,𝜂 †
𝜎 + 𝑏 𝜔 ,𝑘 ̂
,𝜂 𝜎 †
)
𝜔 ,𝑘 ̂
,𝜂 (5.7)
Let us now investigate what the interaction energy 𝑔 𝜔 means. The energy term arises
from the interaction of the E-field of the collective Mie mode with the SPS transition
dipole and can be expressed as
𝑔 𝜔 = 𝑝 ̅ ⋅ 𝐸 ̅
𝑀𝑖𝑒 (𝑟 ̅ , 𝜔 ) (5.8)
Here 𝐸 ̅
𝑀𝑖𝑒 (𝑟 ̅ , 𝜔 ) is the Mie resonance electric field eigenfunction corresponding to the
frequency 𝜔 , and defines the field operator of the collective Mie mode as,
𝐸 ̂
𝑀𝑖𝑒 (𝑟 ̅ , 𝜔 ) = 𝐸 ̅
𝑀𝑖𝑒 (𝑟 ̅ , 𝜔 )𝑎 𝜔 †
(5.9)
Furthermore, the Green function directly correspond to the eigenmodes 𝐸 ̅
𝑀𝑖𝑒 (𝑟 ̅ , 𝜔 ) as,
𝐺 ̿ 𝑀𝑖𝑒 (𝑟 ̅
1
, 𝑟 ̅
2
, ω
1
) = ∑
𝐸 ̅
𝑀𝑖𝑒 (𝑟 ̅
1
, 𝜔 )⨂𝐸 ̅
𝑀𝑖𝑒 (𝑟 ̅
2
, 𝜔 )
𝜔 2
− ω
1
2
𝜔 (5.10)
The superscript “Mie” here indicates the Green propagator only via the collective Mie
modes of the DBB array. An important property of Green function is that it is also
completely defined from classical Maxwell equation, and thus serves as the link
between the classical approach in chapter 2 and 3, and the quantum mechanical
approach in this section. We have studied the Green function extensively in Chapter 2
as well as in Chapter 4. For a SPS transition dipole with a specific direction given by
209
the vector 𝑝 ̅ , the projected Green function that contributes to the radiative transition
[5.24] is given by
1
ℏ
2
𝑝 ̅ ⋅ 𝐺 ̿ 𝑀𝑖𝑒 (𝑟 ̅
𝑆𝑝𝑆 , 𝑟 ̅
𝑆𝑃𝑆 , Ω) ⋅ 𝑝 ̅ =
1
ℏ
2
∑
𝑝 ̅ ⋅ 𝐸 ̅
𝑀𝑖𝑒 (𝑟 ̅
𝑄𝐷
, 𝜔 )⨂𝐸 ̅
𝑀𝑖𝑒 (𝑟 ̅
𝑄𝐷
, 𝜔 ) ⋅ 𝑝 ̅
𝜔 2
− Ω
2
𝜔 =
1
ℏ
2
∑
|𝑔 𝜔 |
2
𝜔 2
− Ω
2
𝜔 (5.11)
Figure 5.4. Contour for integrating the expression for Green function in the 𝜔 -sum in
equation (5.11).
Contrary to modes of a high Q cavity such as in photonic crystals the Mie resonance
offer a spectrally broad mode of width few hundred meV- much higher compared to
the timescale of the photon emission - ~10𝜇𝑒𝑉 . Thus, for all practical purpose the
summation over 𝜔 can be replaced by an infinite integral. By doing this, we can derive
via performing a contour integration as shown in Fig. 5.4.
𝑝 ̅ ⋅ 𝐺 ̿ (𝑀𝑖𝑒 )
(𝑟 ̅
𝑆𝑝𝑆 , 𝑟 ̅
𝑆𝑃𝑆 , Ω) ⋅ 𝑝 ̅ = ∫
|𝑔 𝜔 |
2
𝑑𝜔 𝜔 2
− Ω
2
= 2𝜋𝑖 |𝑔 Ω
|
2
(5.12)
210
This relates the concept of the photon local density of states to the coupling energy
𝑔 Ω
. In chapter 2 we had established that the density of states of photons projected
along the dipole orientation 𝑝 ̂ is given by
𝜌 𝐿𝐷𝑂𝑆
(𝑀𝑖𝑒 )
(𝑟 , Ω) =
1
𝜋 𝐼𝑚 (𝑝 ̂ ⋅ 𝐺 ̿ (𝑀𝑖𝑒 )
(𝑟 ̅
𝑆𝑝𝑆 , 𝑟 ̅
𝑆𝑃𝑆 , Ω) ⋅ 𝑝 ̂ ) = 2
|𝑔 Ω
|
2
𝑝 2
(5.13)
Note that the continuum radiation modes also contribute to the local density of states
of the photon at the location of the SPS and thus result in emission of photons that are
not coupled to the collective Mie mode of the nanoantenna waveguide. Thus, the
photon density of states corresponding to the radiation modes at the location of the
SPS can be expressed in a similar way as
𝜌 𝐿𝐷𝑂𝑆
(𝑅𝑎𝑑 )
(𝑟 , Ω) =
2
𝑝 2
∑|𝜅 Ω,𝑘 ̂
,𝜂 |
2
𝑘 ̂
,𝜂 (5.14)
The overall density of states is a sum of the density of states provided by the Mie
resonance and the radiation modes, and determines the overall radiative decay rate of
the SPS. Thus, the overall density of states is given by,
𝜌 𝐿𝐷𝑂𝑆
(𝑟 , Ω) = 𝜌 𝐿𝐷𝑂𝑆
(𝑀𝑖𝑒 )
(𝑟 , Ω) + 𝜌 𝐿𝐷𝑂𝑆
(𝑅𝑎𝑑 )
(𝑟 , Ω) ∝ 2
|𝑔 Ω
|
2
𝑝 2
+
2
𝑝 2
∑|𝜅 Ω,𝑘 ̂
,𝜂 |
2
𝑘 ̂
,𝜂 (5.15)
Under spectrally broad density of states such as in the case of the Mie mode, the
spectral width of the density of states are much larger compared to the decay rate
limited linewidth of the SPS. The reservoir correlation function, also found by the
Fourier transform of 𝑔 Ω
and 𝜅 Ω,𝑘 ̂
,𝜂 are thus sharp delta function and the Markov
211
condition apply [5.16] for the decay of the SPS. Thus, the dynamics of the SPS
decaying can be simplified by averaging out over the continuum of the Mie mode and
the radiation modes and a Master equation can be invoked as
𝑑 𝑑𝑡 𝜌 = −
𝑖 ℏ
[𝐻 𝑆𝑃𝑆 , 𝜌 ] + Γ ℒ
𝜎 [𝜌 ] (5.16)
where ℒ
𝜎 [𝜌 ] = (2 𝜎𝜌 𝜎 †
− 𝜎 †
𝜎𝜌 − 𝜎 †
𝜎𝜌 ) is the Lindblad super-operator applied on
the density matrix of the SPS state only. The rate Γ can be expressed in terms of the
density of states of the photon states at the location of the SPS defined using the Fermi
golden rule, i.e. Γ =
2𝜋 ℏ
𝑝 2
𝜌 𝐿𝐷𝑂𝑆
(𝑟 𝑆𝑃𝑆 , Ω). Since in this particular situation of a single
SPS, the Hamiltonian 𝐻 𝑆𝑃𝑆 imposes only a global phase on 𝜌 , the solution of the
master equation in equation (5.16) is actually trivial and results in a exponentially
decaying density matrix elements. Thus, the general solution to 𝜌 for the decay of the
SPS starting at the excited state |𝑒 ⟩ can be readily expressed as
𝜌 (𝑡 ) = |𝑒 ⟩⟨𝑒 |𝑒 −2Γ𝑡 + |𝑔 ⟩⟨𝑔 |(1 − 𝑒 −2Γ𝑡 ) (5.17)
In this analysis we have exploited the spectral broadness of the Mie resonance
to course-grain to arrive at the Master equation. In some situations, however, to get a
better physical insight, it may be useful to not approximate 𝑔 Ω
as a uniform frequency
distribution (thus a reservoir correlation function that is not a delta function in time) it
may be useful to solve the Hamiltonian in equation (5.4) exactly by using the mode
expansion of the Mie modes as shown in eqn.(5.9). Such analysis however is
addressed in this dissertation work and will be part of future work.
212
Effect of the Mie Resonance
It should be noted that in equation (5.15) the term 𝑔 Ω
physically corresponds to decay
into the guided mode and the terms 𝜅 Ω
corresponds to the decay into the lossy
radiative modes. Compared to a conventional structure such as a ridge waveguide, the
presence of Mie resonance affects these two terms in the following way-
First: The interference of the electric and magnetic dipole modes of the DBBs results
in a forward directionality, as shown in Chapter 2, resulting in preferential emission of
the emitted wave into the collective Mie resonance of the nanoantenna waveguide.
Second, the local density of state enhancement also by the magnetic Mie resonance
enhances the interaction energy with the SPS emitter- thus increasing 𝑔 Ω
. The
combined effect of both the enhancement of emission rate and enhancement of
directionality results in the enhancement of the SPS-SPS Green function shown in
Chapter 4 as a contrast between Fig. 4.9(b) and (d). We further highlight the effect of
the Mie resonance here by showing in Fig. 5.5 comparison between a nanoantenna-
waveguide and a conventional ridge waveguide shown in Fig. 5.5(a) and (b). In Figure
5.5(c) and (d), the cross-sectional Poynting vector distribution for a 1 debye transition
dipole emitting at 980nm is plotted in the same color-scale- clearly demonstrating that
the photon directionality is much higher in the nanoantenna-waveguide compared to
the ridge waveguide. For the case of the nanoantenna waveguide, ~50% of the emitted
photons are coupled to the collective Mie mode of the DBB array, compared to ~25%
in the case of the ridge waveguide.
213
Figure 5.5. (a) and (b) represents the DBB array nanoantenna waveguide structure
(same as in Chapter 3) and the conventional ridge waveguide structure of
220nmx220nm cross section coupled to a single SPS. (c) and (d) shows the cross-
sectional Poynting vector distribution for the Nanoantenna waveguide and the
conventional waveguide for the forward direction and reverse direction- demonstrating
the nanoantenna effect. (e) and (f) shows the E-field distribution on the XY plane
passing though the center of the DBBs- indicating a higher E-field enhancement and
higher coupling to the guided mode in the case of the nanoantenna waveguide
compared to the standard ridge waveguide. Panel (e) and (f) share the same color-
scale to the right.
Additionally, Fig. 5.5(c) and (d) shows the plot of the Green function on the
XY plane passing through the SPS for these two cases. Evidently, the Mie resonance
results in a larger E-field at the location of the SPS- thus results in higher local photon
density of states, enhancing the interaction of the SPS with the collective Mie
resonance represented by the interaction 𝑔 Ω
in this analysis.
214
Two Coupled SPSs
When two emitters are coupled to the Mie mode, such as in a back to back
nanoantenna waveguide structure in Fig. 5.1, however, the situation is no longer
trivial. The overall Hamiltonian of the system, shown in Fig. 5.6, now can be
expressed as
𝐻 = 𝐻 𝑆𝑃𝑆 1
+ 𝐻 𝑆𝑃𝑆 2
+ 𝐻 𝑀𝑖𝑒 + 𝐻 𝑅
+ 𝐻 𝑆𝑃𝑆 1−𝑀𝑖𝑒 + 𝐻 𝑆𝑃𝑆 1−𝑅 + 𝐻 𝑆𝑃𝑆 2−𝑀𝑖𝑒 + 𝐻 𝑆𝑃𝑆 2−𝑅 (5.18)
Here 𝐻 𝑆𝑃𝑆𝑖 = Ω𝜎 𝑖 †
𝜎 𝑖 , for i=1,2. Invoking the same broadband nature of the Mie mode,
we can invoke the Markov approximation, enabling Master equation by taking in all
the photon state added up in the form of the Lindblad operators. The overall Master
equation takes the generic form
𝑑 𝑑𝑡 𝜌 = −
𝑖 ℏ
[𝐻 𝑆𝑃𝑆 1
+ 𝐻 𝑆𝑃𝑆 1
, 𝜌 ]
+ ∑ Γ
𝑖 ,𝑗 (2 𝜎 𝑖 𝜌 𝜎 𝑗 †
− 𝜎 𝑖 †
𝜎 𝑗 𝜌 − 𝜎 𝑖 †
𝜎 𝑗 𝜌 )
2
𝑖 ,𝑗 =1
(5.19)
We have ignored here any direct coupling between the two SPSs and assumed that the
coupling is always mediated by a real photon, either in the Mie resonance of the DBB
array, or in the radiation modes of the empty space around. However, if the two SPSs
are close within ~100nm, the direct coupling via dipole-dipole interaction cannot be
ignored anymore. The direct dipole-dipole coupling energy results in interaction
energy that falls with ~1/R
6
[5.25] and thus does not belong to the Hilbert
215
Figure 5.6. System of two SPSs coupled via the collective Mie mode, but also
independently decaying into radiation continuum.
space of the radiation fields. It is the process responsible for non-radiative energy
transfer that can be extremely relevant for certain systems involving colloidal QDs
[5.26] and stacked pair of semiconductor QD SPSs [5.27]. In our analysis, however,
since the objective is to establish coupling between two SPSs that are siting far from
each other, we ignore any direct near-field coupling.
The cross terms in the Lindblad operator, i.e. the term Γ
1,2
and Γ
2,1
represent
the rate of the processes of coupling between the two SPS mediated by the continuum
of the Mie mode. Physically this is represented by a photon emitted by one of the SPS
to the Mie mode and reabsorbed by the other SPS. In general, since along this process,
the photon can actually bounce back and forth between the two SPSs and thus the
SPSs can periodically exchange energy mediated by the Mie mode, Γ
1,2
and Γ
2,1
will
be in general complex numbers, containing both decaying (real part) and oscillating
(imaginary component).
216
However, the Lindblad decay terms are enormously simplified if we do a
simple basis transformation of the coupled SPSs into the basis of the super-radiant and
sub radiant collective states. These states are defined as a coherent super-position of
the excited state of each emitter, defined as,
𝜎 (𝑆𝑢𝑝 )
†
=
1
√2
(𝜎 1
†
+ 𝑒 𝑖𝜃
𝜎 1
†
), and, 𝜎 (𝑆𝑢𝑏 )
†
=
1
√2
(𝜎 1
†
− 𝑒 −𝑖𝜃
𝜎 1
†
) (5.20)
The relative phase 𝜃 between the creation operators of the two SPSs is required such
that the coupling energy term 𝐽 1,2
(𝜔 )=
𝑝 2
𝑒 𝑖𝜃
ℏ
𝑝 ̂
1
⋅ 𝐺 ̿ (𝑟 ̅
1
, 𝑟 ̅
2
, 𝜔 ) ⋅ 𝑝 ̂
𝑗 2
is purely real,
leading to the two normal modes- the maximum and minimum decay rate
combinations. Table 5.1 summarizes the change in the basis and its effect on the
Unitary part of the evolution in the Master equation.
217
Table 5.1. Basis transformation, from single to collective, and its impact on the
Hamiltonian
Basis of individual SPSs Basis of Super-radiant and Sub-radiant
SPS Basis States:
{|𝑒 1
𝑒 2
⟩, |𝑒 1
𝑔 2
⟩, |𝑔 1
𝑒 2
⟩, |𝑔 1
𝑔 2
⟩ }
Unitary part of the Hamiltonian of the two
SPSs
𝐻 𝑆𝑃𝑆 1,𝑆𝑃𝑆 2
= 𝐻 𝑆𝑃𝑆 1
+ 𝐻 𝑆𝑃𝑆 2
= Ω( 𝜎 1
†
𝜎 1
+ 𝜎 2
†
𝜎 2
)
Unitary part of the Hamiltonian in the
Matrix Form
[𝐻 𝑆𝑃𝑆 1,𝑆𝑃𝑆 2
] = [
2𝛺 0 0 0
0 𝛺 0 0
0 0 𝛺 0
0 0 0 0
]
SPS Collective Basis States:
{|𝑒 1
𝑒 2
⟩, |𝜙 𝑆𝑢𝑝 ⟩, |𝜙 𝑆𝑢𝑏 ⟩, |𝑔 1
𝑔 2
⟩ }
Where |𝜙 𝑆𝑢𝑝 ⟩ =
1
√2
(𝜎 1
†
+ 𝑒 𝑖𝜃
𝜎 1
†
) |𝑔 1
𝑔 2
⟩
And |𝜙 𝑆𝑢𝑏 ⟩ =
1
√2
(𝜎 1
†
− 𝑒 −𝑖𝜃
𝜎 1
†
) |𝑔 1
𝑔 2
⟩
Unitary part of the Hamiltonian of the two
SPSs
𝐻 𝑆𝑃𝑆 1,𝑆𝑃𝑆 2
= Ω( 𝜎 (𝑆𝑢𝑝 )
†
𝜎 (𝑆𝑢𝑝 )
+ 𝜎 (𝑆𝑢𝑏 )
†
𝜎 (𝑆𝑢𝑏 )
)
Unitary part of the Hamiltonian in the
Matrix Form
[𝐻 𝑆𝑃𝑆 1,𝑆𝑃𝑆 2
] = [
2𝛺 0 0 0
0 𝛺 0 0
0 0 𝛺 0
0 0 0 0
]
218
We observe that the unitary part of the Hamiltonian remains unchanged under
this transition. However, the basis transition has significant effect on the incoherent
decay rates. Since the super-radiant and sub-radiant states are orthogonal normal
modes of the coupled SPS system, they act like independent decay channels. This is
schematically represented in the following.
Figure 5.7. Decay pathways for the two-SPS coupled system.
Each of the red arrow in the figure above represents a decay process of emission of a
single photon. The rate of each of these processes can thus be achieved from purely
classical Green function-based approach presented in Section 4.3.
219
The overall Hilbert space, in Fig. 5.7 is divided into three levels, ℋ
(2)
, ℋ
(1)
, and ℋ
(0)
based on the number of excited SPSs. The super-script indicates the number of
emitters in the excited state. Thus, we may write
ℋ
(2)
= 𝑠𝑝𝑎𝑛 {|𝑒 1
𝑒 2
⟩} (5.21)
ℋ
(1)
= 𝑠𝑝𝑎𝑛 { |𝑒 1
𝑔 2
⟩, |𝑔 1
𝑒 2
⟩ } = 𝑠𝑝𝑎𝑛 {{ |𝜙 𝑆𝑢𝑝 ⟩, |𝜙 𝑆𝑢𝑏 ⟩} (5.22)
and,
ℋ
(0)
= 𝑠𝑝𝑎𝑛 {|𝑔 1
𝑔 2
⟩} (5.23)
The scattering of the single photon corresponding to each of the decay
channels shown in Fig. 5.7 is shown in Fig. 5.8. When the system decays from the
ℋ
(2)
to ℋ
(1)
, initially both of the emitters are in the excited state. Once an emitter
emits a single photon, the photon can not get scattered by the other emitter- since the
other emitter is still in the excited state. Thus, the E-field created by this decay process
is classically described by a Green function that involve only scattering by one emitter
and thus has poles 𝜔 = ±(Ω + 𝑋 (Ω)) [Appendix B]. On the other hand, when the
system decays from a state in ℋ
(1)
to the ground state, such as decaying from |𝜙 𝑆𝑢𝑝 ⟩,
or |𝜙 𝑆𝑢𝑏 ⟩ to the ground state |𝑔 1
𝑔 2
⟩, both emitters may scatter the emitted photon.
This causes the classical E-field produced by the emitted photon to follow a response
that is described by a Green function that involve scattering by both emitters and thus
has two pairs of poles, 𝜔 = ±(Ω + 𝑋 (Ω) ± |𝐽 (Ω)|) [Appendix B]. These two pairs of
poles represent, respectively the complex frequency of the super-radiant and the sub-
radiant decay channels.
220
Figure 5.8. Decay pathways of the Hamiltonian and their connection to the multiple
scattering Green function as discussed in the last section.
Thus, under the transformed basis of the super-radiant and sub-radiant states, the
master equation from eqn. (5.19) can be simplified into,
𝑑 𝑑𝑡 𝜌 = −
𝑖 ℏ
[𝐻 𝑆𝑃𝑆 1,𝑆𝑃𝑆 2
, 𝜌 ]
+ Γ(2 |𝜙 𝑆𝑢𝑝 ⟩⟨𝑒 1
𝑒 2
|𝜌 |𝑒 1
𝑒 2
⟩⟨𝜙 𝑆𝑢𝑝 | − |𝑒 1
𝑒 2
⟩⟨𝑒 1
𝑒 2
|𝜌
− 𝜌 |𝑒 1
𝑒 2
⟩⟨𝑒 1
𝑒 2
|)
+ Γ(2 |𝜙 𝑆𝑢𝑏 ⟩⟨𝑒 1
𝑒 2
|𝜌 |𝑒 1
𝑒 2
⟩⟨𝜙 𝑆𝑢𝑏 | − |𝑒 1
𝑒 2
⟩⟨𝑒 1
𝑒 2
|𝜌
− 𝜌 |𝑒 1
𝑒 2
⟩⟨𝑒 1
𝑒 2
|)
221
+Γ
+
(2 |𝑔 1
𝑔 2
⟩⟨𝜙 𝑆𝑢𝑝 |𝜌 |𝜙 𝑆𝑢𝑝 ⟩⟨𝑔 1
𝑔 2
| − |𝜙 𝑆𝑢𝑝 ⟩⟨𝜙 𝑆𝑢𝑝 |𝜌
− 𝜌 |𝜙 𝑆𝑢𝑝 ⟩⟨𝜙 𝑆𝑢𝑝 |)
+Γ
−
(2 |𝑔 1
𝑔 2
⟩⟨𝜙 𝑆𝑢𝑏 |𝜌 |𝜙 𝑆𝑢𝑏 ⟩⟨𝑔 1
𝑔 2
| − |𝜙 𝑆𝑢𝑏 ⟩⟨𝜙 𝑆𝑢𝑏 |𝜌
− 𝜌 |𝜙 𝑆𝑢𝑏 ⟩⟨𝜙 𝑆𝑢𝑏 |) (5.24)
Note, there are no longer any cross-terms between the different basis states. That is to
say, the sub-radiant, and super-radiant states, once created, radiate without any energy
exchange between them. The decay rates in equation (5.24) can be summarized as
below,
i- Index
representing the
decay channels
Decay Operator Decay Rate:
1 |𝑒 1
𝑒 2
⟩⟨𝜙 𝑆𝑢𝑝 |
Γ = 𝐼𝑚𝑋 (Ω)
2 |𝑒 1
𝑒 2
⟩⟨𝜙 𝑆𝑢𝑏 | Γ = 𝐼𝑚𝑋 (Ω)
3
|𝜙 𝑆𝑢𝑝 ⟩⟨𝑔 1
𝑔 2
|
Γ
+
= 𝐼𝑚𝑋 (Ω) + |𝐽 (Ω)|
4
|𝜙 𝑆𝑢𝑝 ⟩⟨𝑔 1
𝑔 2
|
Γ
−
= 𝐼𝑚𝑋 (Ω) − |𝐽 (Ω)|
The Master equation for a general 𝜌 - the density matrix is readily solved from
equation (5.24). The diagonal terms are given as-
𝜌 1,1
(𝑡 ) = 𝜌 1,1
(0)𝑒 −4Γ𝑡 (5.25)
222
𝜌 2,2
(𝑡 ) = 𝜌 2,2
(0)𝑒 −2Γ
+
𝑡 +
2Γ𝜌 1,1
(0)
(2Γ
+
− 4Γ)
(𝑒 4Γ𝑡 − 𝑒 −2Γ
+
𝑡 ) (5.26)
𝜌 3,3
(𝑡 ) = 𝜌 3,3
(0)𝑒 −2Γ
−
𝑡 +
2Γ𝜌 1,1
(0)
(2Γ
−
− 4Γ)
(𝑒 4Γ𝑡 − 𝑒 −2Γ
−
𝑡 ) (5.27)
𝜌 4,4
(𝑡 ) = 1 − 𝜌 1,1
(𝑡 ) − 𝜌 2,2
(𝑡 ) − 𝜌 3,3
(𝑡 ) (5.28)
As expected, the diagonal terms of the density matrix represent probability values for
the system to residing on the basis states and thus show simple exponential
dependence with the characteristic decay rates of the system. Whereas the off-diagonal
terms of the density matrix are given as
𝜌 1,2
(𝑡 ) = 𝜌 1,2
(0)𝑒 −(2Γ+Γ
+
)𝑡 𝑒 −𝑖 (Ω−Δ)𝑡 (5.29)
𝜌 2,1
(𝑡 ) = 𝜌 2,1
(0)𝑒 −(2Γ+Γ
+
)𝑡 𝑒 𝑖 (Ω−Δ)𝑡 (5.30)
𝜌 1,3
(𝑡 ) = 𝜌 1,3
(0)𝑒 −(2Γ+Γ
−
)𝑡 𝑒 −𝑖 (Ω+Δ)𝑡 (5.31)
𝜌 3,1
(𝑡 ) = 𝜌 3,1
(0)𝑒 −(2Γ+Γ
−
)𝑡 𝑒 𝑖 (Ω+Δ)𝑡 (5.32)
223
𝜌 1,4
(𝑡 ) = 𝜌 1,4
(0)𝑒 −2Γ𝑡 𝑒 −2𝑖 Ω 𝑡 (5.33)
𝜌 4,1
(𝑡 ) = 𝜌 4,1
(0)𝑒 −2Γ𝑡 𝑒 2𝑖 Ω 𝑡 (5.34)
𝜌 2,3
(𝑡 ) = 𝜌 2,3
(0)𝑒 −(Γ
+
+Γ
−
)𝑡 𝑒 −2𝑖 Δ 𝑡 (5.35)
𝜌 3,2
(𝑡 ) = 𝜌 3,2
(0)𝑒 −(Γ
+
+Γ
−
)𝑡 𝑒 2𝑖 Δ 𝑡 (5.36)
𝜌 2,4
(𝑡 ) = 𝜌 2,4
(0)𝑒 −Γ
+
𝑡 𝑒 −𝑖 (Ω+Δ) 𝑡 (5.37)
𝜌 4,2
(𝑡 ) = 𝜌 4,2
(0)𝑒 −Γ
+
𝑡 𝑒 𝑖 (Ω+Δ) 𝑡 (5.38)
𝜌 3,4
(𝑡 ) = 𝜌 3,4
(0)𝑒 −Γ
−
𝑡 𝑒 −𝑖 (Ω−Δ) 𝑡 (5.39)
𝜌 4,3
(𝑡 ) = 𝜌 4,3
(0)𝑒 −Γ
−
𝑡 𝑒 𝑖 (Ω−Δ) 𝑡 (5.40)
The above solution satisfies the following boundary conditions:
1. At 𝑡 = 0 , the density matrix is same as the specified initial state.
2. At any time, 𝑇𝑟𝑎𝑐𝑒 (𝜌 ) = 1.
3. At 𝑡 = ∞, 𝜌 (𝑖 ,𝑗 )
≠ 0 only if the element i,j represents a state where both the
emitters are in ground states. (in other words, the emitters eventually decay).
224
Next, we employ this master equation approach and explore the evolution of the SPS-
SPS coupled states.
§5.2.1. SPS-SPS Energy Transfer
With the help of the theoretical framework established above, we now come
back to the system of coupled SPSs in the back-to-back nanoantenna waveguide
structure. As it was shown using the classical Maxwell equation-based analysis in
Chapter 4, Fig. 4.8, we have the super-radiant emission rate ≈ 1.7 times the radiative
decay rate of a single SPS, and the sub radiant emission rate is 0.3 times a single
emitter. i.e. Γ
+
~1.7 Γ, Γ
−
~0.3 Γ. We further note that the Purcell enhancement of a
single emitter (even in the absence of the other emitter), we found, is ~5. The decay
rate of a single emitter in bulk medium is known to be ~1/ns for the case of the mesa-
top single quantum dots. Thus, in the analysis ahead, we assume Γ ≈ 5 ×
1
𝑛𝑠
= 5/𝑛𝑠 .
We first present a simple situation where only one SPS is excited at any point
of time- thus utilizing only the single-excitation Hilbert space to represent the system.
The physical situation is represented in Fig. 5.9, where we assume that at 𝑡 = 0 only
SPS1 is in the excited state whereas SPS2 is in the ground state. As SPS1 decays
under the nanoantenna-waveguide it is embedded in, the emitted photon couples to the
collective Mie mode, and thus can propagate to SPS2 and excite it. This results in a
back-and-forth oscillatory behavior of the occupation probability of the exciton
between SPS1 and SPS2. Schematically this process is represented in Fig. 5.9(b).
225
We employ the general solution of the density matrix of the two-SPS system
under this initial condition to derive the probability of occupation of the state | 𝑒 1
𝑔 2
⟩
and | 𝑒 2
𝑔 1
⟩ shown in Fig. 5.9(c) with the blue and the red dashed lines respectively.
The oscillatory behavior can be seen with a period of ~300ps- representing a ~5𝜇𝑒𝑉
coupling energy between the SPSs. Fig. 5.9(d) shows the dynamics of the same
process in the basis of the super-radiant and sub-radiant states, i.e. | 𝜙 𝑆𝑢𝑝 ⟩ and | 𝜙 𝑆 𝑢𝑏
⟩
respectively. In this basis, we note that the oscillatory behavior is not present and the
coupled SPS-SPS states are seen to be decaying through two effectively independent
channels. This is consistent with the decay channels that we discussed earlier in this
chapter shown in Fig. 5.7.
Establishing such coherent energy transfer between two or more on-chip SPSs
have significant implications in the context of quantum optical circuits in preparation
and manipulation of entangled states based on multiple SPSs and photons. In the
literature, such SPS-SPS energy transfer in the single excitation domain is mostly
explored in the context of photonic crystal structures- where one can achieve higher
~100𝜇𝑒𝑉 [5.5] coupling strength at the expense of very narrow-band response of
linewidth ~0.2nm in wavelength. The analysis approach that is developed in this
chapter however is not specific to any platform.
226
Figure 5.9. Exchange of photon between two SPSs mediated by the Mie resonance—
simulation using the master equation approach when at t=0, only SPS1 is in the
excited state. (b) shows the overall decay process- involving energy exchange between
the two SPSs. This is seen in the occupation probability calculated from the Master
equation shown in (c) and (d)- in the single SPS basis and the super-radiant-and sub-
radiant basis respectively.
As we mentioned earlier in this chapter, the physically more relevant question is not
the decay of the super- and sub-radiant states but their emergence. The Master
227
equation-based approach allows us to probe the decay dynamics when initially both
the SPSs are in the excited state. This is shown next.
§5.2.2. SPS-SPS Collective Emission
In this section we explore the dynamics of the collective photon emission by the two
SPSs coupled via the back-to-back nanoantenna waveguide structure shown in Fig.
5.10(a). Contrary to section 5.2.2. above, we now explore the scenario when both the
SPSs are in the excited state, i.e. the system starts at the state | 𝑒 1
𝑒 2
⟩. The decay
process is schematically represented in Fig. 5.10 (b) and (c). We note that decay of the
initial state | 𝑒 1
𝑒 2
⟩ results in a super-position of the sub-radiant and the super-radiant
state. The calculated occupation probability values for the constituent states are shown
in Fig. 5.10 (d) and (e). We note that the decay of the initial state results in both the
super-radiant and the sub-radiant state with comparable probability. However, the
super-radiant state decays fast- thus leaving the system majorly in a sub-radiant state
after a period of time. This asymmetry between the super-radiant and the sub-radiant
state is the basis for emergence of entanglement in such coupled SPS systems.
To understand the origin and effect of such entanglement in a physical system, it is
now important that we include, in our analysis, the notion of a measurement outcome.
This is shown in the next section.
228
Figure 5.10. (a) SPS-SPS coupled system in the back-to-back nanoantenna waveguide
structure with (b) and (c) indicating the decay process when the initial state is | 𝑒 1
𝑒 2
⟩.
(d) and (e) show the occupation probability of the basis states in the single-SPS basis
and the super-and sub-radiant basis respectively.
229
§5.3. Effect of Measurements —Inclusion of the Detector States
The semiclassical Hamiltonian as derived in section 5.2 alongside with the Master
equation approach to address both coherent and incoherent transition between states in
the presence of the photon vacuum fluctuation acting as a bath allows calculating the
state of the system as a function of time starting from an arbitrary initial state—and
thus allows us to estimate the information content in the density matrix of the system.
In real applications, however, the such information is inherently dependent on
performing certain measurements on the emitted photons.
Typically, the state of photon can be accounted for in the Hilbert space by accounting
for the all the combinations of polatization and phase of the photon as a wavepacket.
In our model, we have so far integrated over the degree of freedom of the Mie
resonance to form the Master equation. Thus, to understand effect of measurements on
the emitted photons by the coupled QD system, we must account for the degree of
freedom associated with the detectors.
Thus, the system view shown in Figure 5.10 (a) now needs to be extended to one
shown in Fig. 5.11. As the detector states are included in the Hilbert space, this
requires extending the density matrix to include the detector states. We illustrate this
next by starting with the system with only one detector, shown in Fig. 5.11 (a).
§5.3.1. One Detector: For the system shown in Fig. 5.11 (a)- including two SPSs and
one detector, each approximated as a two-level system, the overall Hilbert space is
now given by
230
ℋ = ℋ
𝑆 ⨂ℋ
𝐷 = (ℋ
(2)
⊕ ℋ
(1)
⊕ ℋ
(0)
)⨂ℋ
𝐷
Where ℋ
(2)
, ℋ
(1)
, and ℋ
(0)
are same as defined in Fig. 5.7 and ℋ
𝐷 =
𝑠𝑝𝑎𝑛 {|𝑒 𝑑 ⟩, |𝑔 𝑑 ⟩} is the Hilbert space for the detector state. Here |𝑔 𝑑 ⟩ represents the
Figure 5.11. A schematic indicating the detector state included into the Hilbert space
of the system. (a) including the state of a single detector represented by a 2 level
system and (b) two 2-level systems representing two detector arranged in HBT set-up.
detector being in the ground state – i.e. no photon detected and |𝑒 𝑑 ⟩ is assumed to be a
state when the detector has been triggered by incident photon. Under this Hilbert
space, the density matrix is now represented in Fig. 5.12.
231
Figure 5.12. Structure of the density matrix when the two emitters are joined by a
single detector, also approximated as a two level system.
As shown in Fig. 5.12, the off-diagonal Block matrices are assumed to be identically
zero. This is appropriate for typical detectors such as Avalanche photo diodes or
super-conducting nanowire detector where the process of detection of a photon
involves a large number of electrons and phonons- and therefore are by definition
classical in nature. However, as we account for the detector states as part of a general
Hilbert space, the formulation equally applies for photon detection where the detector
may not decohere- such as in the case of weak measurements. The density matrix
solution from master equation for this situation with the detector is derived readily
from our original Master equation solution of the density of states shown in equation
232
(5.22)-(5.37). To do this we define the effective detection efficiency 𝜂 − as the
probability that a photon emitted by the optical circuit triggers a photon-detection
event at the detector. Such photon detection event is schematically shown by the red
and magenta arrow in Fig. 5.12- the red representing the process of the two SPSs, both
staring at the excited state emits a single photon which is detected, and the magenta
representing the process of the two SPSs starting at a single excited state in ℋ
(1)
emits
a photon which is detected.
Under our objective of emergence of coherence and entanglement, the relevant
process is to start with two SPSs both at excited state- thus the process shown in the
red arrow in Fig. 5.12. After a click (event of detection) is observed at the detector, the
system resides in the Block shown in red in fig. 5.12, represented by 𝜌 𝑅
𝜌 𝑅 =
1
𝜌 6,6
+ 𝜌 7,7
[
𝜌 6,6
𝜌 6,7
𝜌 7,6
𝜌 7,7
] (5.41)
Here the additional factor in the start ensures that 𝑇𝑟 (𝜌 𝑅 ) = 1- thus ignoring all events
where the detector does not click.
Extending the density matrix solution shown earlier in eqn (5.25)-(5.40), we can
derive
𝜌 6,6
(𝑡 ) = 𝜌 6,6
(0)𝑒 −2Γ
+
𝑡 +
2Γ(𝜂 𝜌 1,1
(0) + 𝜌 5,5
(0))
(2Γ
+
− 4Γ)
(𝑒 4Γ𝑡 − 𝑒 −2Γ
+
𝑡 ) (5.42)
233
𝜌 7,7
(𝑡 ) = 𝜌 7,7
(0)𝑒 −2Γ
−
𝑡 +
2Γ(𝜂 𝜌 1,1
(0) + 𝜌 5,5
(0))
(2Γ
−
− 4Γ)
(𝑒 4Γ𝑡 − 𝑒 −2Γ
−
𝑡 ) (5.43)
𝜌 6,7
(𝑡 ) = 𝜌 6,7
(0)𝑒 −(Γ
+
+Γ
−
)𝑡 𝑒 −2𝑖 Δ 𝑡 (5.44)
𝜌 7,6
(𝑡 ) = 𝜌 7,6
(0)𝑒 −(Γ
+
+Γ
−
)𝑡 𝑒 2𝑖 Δ 𝑡 (5.45)
Equation (5.42)-(5.45) shows the general analytical functions defining the reduced
density matrix 𝜌 𝑅 - the density matrix of the coupled SPSs after a single photon has
been detected from the starting state of |𝑒 1
𝑒 2
⟩. We show in Fig. 5.13 the key results.
The schematic of the system is repeated in Fig. 5.13(a) with Fig. 5.13 (b) showing the
decay pathways. We show in Fig. 5.13 (d) the probability of the two SPSs being in the
super-radiant and the sub radiant state as a function of time. Notably, the super-radiant
part emits a second photon quickly and decays to the ground state, whereas the sub-
radiant part decays slowly. This asymmetry induces a correlation between the two SPS
states.
To quantify this correlation, we study the information content of the resultant
state after the detection of a single photon under the process shown in Fig. 5.13(b). A
measure of such information content is the entropy of the system, denoted by [5.17]
𝑆 (𝜌 ) = −𝑡𝑟 (𝜌 log
2
𝜌 ) (5.46)
234
Figure 5.13. (a) Two SPSs coupled via the back-to-back nanoantenna waveguide
structure in the presence of a single detector (b) decay pathways when the system
starts with both SPSs excited. (c) The probability of the super-radiant and the sub-
radiant state once a single photon has been detected. (d) The Von-Neumann entropy of
the two SPSs, and a single SPS, and their difference indicating “shared information”
between the two SPSs.
235
Thus, the entropy content of the two SPS system after detection of the first photon is
𝑆 (𝜌 𝑅 )- shown with the red curve in Fig. 5.13 (d). However, if we trace 𝜌 𝑅 over SPS2,
we get the information content in only SPS1 – i.e. 𝑆 (𝑇 𝑟 2
(𝜌 𝑅 ))- shown using the blue
dashed line in Fig. 5.13 (d). The difference between those two denote the shared
information between the two SPSs which is the notion of entanglement. This is shown
in Fig. 5.13 (d) using the black line- indicating that once a photon is detected by the
detector, over time entanglement emerges between the two SPSs.
An established method to measure the existence of such entanglement is to
exploit two-photon correlation measurements. The underlying concept behind such
measurement is- once a single photon has been detected by the detector and the SPS-
SPS coupled system is in the entangled state, the time it takes for the second photon to
come out is faster compared to the standard radiative decay rate of the SPS owing to
the existence of the super-radiant state. This faster timescale emission of the 2
nd
photon is revealed by measuring the correlation between the detection time of the 1
st
photon and the 2
nd
photon. This is illustrated next.
§5.3.2. HBT Measurement Outcome- Two Detectors:
We now consider the system shown in Fig. 5.14(a)- i.e. the SPS-SPS coupled back-to-
back nanoantenna waveguide in the presence of two detectors in HBT arrangement.
The structure of the 16x16 dimensional density matrix is shown in Fig. 5.14 (b). The
general solution of the density matrix in eqn(4.43)-(4.54) can be readily extended to
236
the case with two detectors, however, the basic physical process can be much easily
understood. The red arrows in Fig. 4.21(b) show the processes which are responsible
for registering a click in the start detector, and registering the click in the stop detector.
Using this, we can express the expected outcome of a photon correlation measurement
as a probability distribution of 𝜏 - the interval between the detection of the two photons
as
𝑔 (2)
(𝜏 ) = 𝜂 2
𝑒 −2Γ
+
|𝜏 |
∫ 𝑑𝑡 2𝜂 1
Γ 𝜌 (1,1)
(𝑡 )
∞
0
+ 𝜂 2
𝑒 −2Γ
−
|𝜏 |
∫ 𝑑𝑡 2𝜂 1
Γ 𝜌 (1,1)
(𝑡 )
∞
0
(5.47)
However, from the solution of density matrix in Eqn (5.25), we know
𝜌 (1,1)
(𝑡 ) = 𝑒 −4Γ𝑡 (5.48)
Thus, from eqn. (5.44) and (5.45) we get:
𝑔 (2)
(𝜏 ) =
𝜂 1
𝜂 2
2
(𝑒 −2Γ
+
|𝜏 |
+ 𝑒 −2Γ
−
|𝜏 |
) (5.49)
Here 𝜂 1
, 𝑎𝑛𝑑 𝜂 2
are the overall detection efficiencies of the two detectors.
Qualitatively this indicates that the peak at 𝜏 = 0 has two exponentials hidden in it-
one governed by the super-radiant decay channel and other governed by the subradiant
decay channel. Clearly, the peak-width is governed by the super-radiant decay which
is ~1.7 times the decay rate of a single emitters. Thus, a factor of 2 narrower
𝑔 (2)
(0)peak compared to the nonzero g2 peaks will demonstrate the existence of
super-radiance.
237
Figure 5.14. (a) SPS-SPS coupled system and two detectors in HBT set-up. (b)The
transitions in the 16x16 dimensional density matrix that correspond to the triggering of
the start and the stop detectors in an HBT measurement.
Figure 5.15 shows the expected 𝑔 (2)
(0)peak for a standard situation where the decay
rate of the emitters in uniform medium is taken to be 1ns
-1
, and the Purcell
238
Figure 5.15. Expected coincident count rate (normalized per excitation pulse) for 2Γ =
5 /ns , Γ
+
~1.7 Γ, Γ
+
~0.3 Γ. Shown in (a) linear scale and (b) log-scale- clearly
showing the super-radiant and sub-radiant decay channels.
enhancement of a single emitter is taken to be 5. Thus 2Γ = 5 /ns and Γ
+
~1.7 Γ,
Γ
+
~0.3 Γ are the super-radiant and sub-radiant decay rates that are indicated by the
Green function solutions of the back-to-back nanoantenna waveguide structure. We
further show the calculated coincidence count histogram in log-scale in Fig. 5.15 (b)
that reveals the two different exponential trends- i.e. the super-radiant and sub radiant
decay channels behind the resultant two-photon correlation. The origin of such
239
signature is specific to emergence of entangled SPS-SPS coupled states and thus can
be used in an actual HBT type measurement on two on-chip SPSs coupled via on-chip
optical circuit to identify collective emission from the coupled SPSs and emergence of
SPS-SPS entangled states.
§5.4. Summary and Conclusions
To summarize, in this chapter we have explored collective emission of photons
for an analytically tractable system of two single photon sources coupled via the
collective Mie resonance of DBB metastructures. Such Mie resonance mediated
collective emission processes dictate how two or higher number of photons can be
created in an on-chip optical circuit with the degree of coherence needed for optical
quantum information processing such as double the resolution for two photon states
compared to the normal single photon imaging.. We have built upon the foundation of
the classical modelling and simulation based on Maxwell equation of the DBB array
metastructures presented in chapter 2 to 4 that help understand the manipulation of a
single photon dominantly based on the electromagnetic Green function and photon
density of states. In this chapter we have extended the methodology used in Chapter 4
to include processes involving more than one photon states that are non-classical in
nature. For the model system of two coupled SPSs, using this approach, we
demonstrate that the radiative decay rate of the state where both the emitters are in the
excited state to a coherent superposition of single excited state is same as decay rate of
isolated emitters—i.e. the sub-radiant and super-radiant decay rates only emerge at a
later time during the decay process. This is consistent with understanding and
240
experimental evidence of super-luminescence from closely packed emitters as in low
pressure gas, abundantly present in numerous studies in the last century [5.28]. Our
semiclassical model allows to explicitly derive the density matrix equations in the
general many excitation Hilbert space for emitters at a distance coupled by a photon
mode to understand such emergence of coherence and entanglement that has
relevance towards quantum information processing.
Specifically, the following was accomplished:
1. The semiclassical model based on Von-Neumann Master equation has been
formulated to capture the time-evolution of the two-SPS coupled system in the
many-particle Hilbert space. Electromagnetic Green function with a Dyson
series based multiple scattering approach has been exploited to capture the
effect of enhancement of the photon DOS and thus enhancement of radiative
decay rate of the emitters in the presence of itself and others in a quantum
mechanical way that accounts for the fermionic nature of the electronic states
of the emitters.
2. Based on the model discussed above, we demonstrated, for the SPS-DBB
back-to-back nanoantenna waveguide system, emergence of the coherent
super-position states (sub-radiant and super-radiant) starting from a product
state where both the SPSs are initially in excited state and decay collectively
under the influence of the collective Mie mode (electric and magnetic dipole)
of the DBB array. We show that the emission rate of the super-radiant state is
enhanced compared to the emission rate of a single SPS by a factor of 1.7,
241
while the emission rate of the sub-radiant state is reduced to a factor of 0.3. We
show the dependence of the super-radiance emission rate on distance between
the two SPSs. The study reveals that owing to the lossless nature of the
collective Mie mode, super-radiant decay rate is independent of the emitter-
emitter separation. This is very important feature towards realization of more
complex circuits.
3. To understand the outcome of measurement on such coupled SPS state we
have extended the Master equation based approach to include the detector
states as a part of the Hilbert space and demonstrated the emergence of SPS-
SPS entangled state after detection of a single photon from a SPS-SPS coupled
state that is initially prepared at both SPS excited. The calculation of Von-
Neumann entropy indicates emergence of correlation and entanglement.
Furthermore, by including two detectors mimicking the Hunbury-Brown and
Twiss system, we have shown that the such entanglement can be detected by
photon correlation measurement that reveal a sharp peak at 𝑔 (2)
(0) whose
width is determined by the super-radiant decay rate.
The studies presented in this chapter attempt to address the key fundamental
question of the emergence of coherence of on-chip SPSs coupled via a collective Mie
mode. The formulation developed and demonstrated however is a general approach
that is readily extended to system comprising more than two SPS emitters and any
arbitrary electromagnetic environment for which the classical electromagnetic Green
242
function is known. We demonstrated emergence of entanglement in the SPS-DBB
metastructures, but its implication towards practical quantum information processing
has not been a part of this chapter and will be a part of future work.
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244
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245
Chapter 6. Conclusions and Outlook
In this dissertation we introduced and presented studies on a new class of on-chip
light manipulating platform that exploits collective Mie resonances of metastructures
made of subwavelength scale dielectric building blocks (DBBs) to provide functions
essential to the realization of quantum optical circuits built on-chip around co-
designed single photon source (SPS) arrays to manipulate emitted single and multiple
photons for quantum information processing. The Mie resonance approach is
conceptually unique as it introduces a platform comprising a single metastructure
whose collective mode provides all needed functions of an optical circuit but without
the concept and use of individual components whose existence and inter-
connectedness has traditionally defined the very perception of a “circuit”, viz the
ubiquitous electrical circuit built out of transistors, resistors, capacitors, and
inductors;. Its counterpart, the sought after optical circuit being pursued with the same
approach of interconnecting components: photon source, cavity, antenna (to give
directivity to emission), waveguide, beam-splitter, and beam combiner as the
dominant components with a few specialized components thrown in. The Mie
resonance approach provides the above noted five functions without the concept of
“components” being “matched” to be interconnected into an efficiently functioning
circuit. Rather one and the same “component”—the metastructure-- through its single
Mie mode providing all needed functions depending upon the physical location of the
traveling photon wave packet in the metastructure. Simply stated, the traditional
246
physical entity dubbed “component” is replaced by function. Thus, component-less
optical circuits.
The incentive to pursue this function-based metastructure approach to
manipulating light emitters at the level of single photons came from the development
of on-chip single photon sources in spatially regular arrays in the form of the mesa-
top single quantum dots [6.1-6.5] with remarkable spectral uniformity, thus injecting
some hope of potentially creating scalable on-chip optical circuits [6.6], as shown
schematically in Fig 6.1. Depicted is the uniqueness of the metastructure comprising
the blue blocks whose size, shape, and placement geometry decides the relative roles
and trade-offs between the various functions to define an overall optimized circuit
performance. Optimizing the key functionalities allows realization of on-chip photon
interference and entanglement, as well as photon mediated SPS-SPS coupling- as we
have shown theoretically.
Figure. 6.1. A representative schematic of the MTSQD-DBB on-chip quantum optical
circuit that has been investigated in this thesis.
247
The specific achievements of the studies reported in this dissertation can be succinctly
summarized as follows:
(1) Starting with studies of the response of a single DBB, we established the properties
of Mie resonance and its behavior for an array of DBBs for single photon
manipulation and identified the key physics responsible for enhancement of the
emission rate of the SPS and directional manipulation of the emitted photon (Chapter
2). It is to this end, the engineering approach to the design of such DBB array
metastructure is presented, which culminates in the nanoantenna-waveguide-splitter-
combiner unit (Chapter 3)- the simplest DBB array optical circuit that provides all the
needed functions of enhancement of emission rate of SPS, enhancement of emission
directionality, propagation, splitting and combining in a mode-matched way towards
on-chip photon interference and entanglement.
(2) Towards the ultimate goal of quantum optical circuits using the Mie resonance
approach, we have explored the dynamical response of multiple SPSs coupled to the
same DBB array metastructures in a classical (Chapter 4), and quantum (Chapter 5)
regime. To such end, an analysis is developed that exploits the classical
electromagnetic Green function to achieve a quantum model of many-SPS system that
demonstrates the emergence of entanglement (Chapter 5) in the SPS-DBB quantum
optical circuits and also provides a viable general approach applicable to do so to other
classes of optical circuits as well.
248
In the following we capture the major conclusions drawn from the studies undertaken
in this dissertation:
§6.1. Mie Resonant Metastructures Established as Viable Approach to On-Chip
Light Manipulating Elements and Units
Mie resonance was identified as a viable platform for integration with on-chip
single photon source, such as the mesa-top single quantum dots (MTSQDs) [6.1-6.5]
as depicted in Fig. 6.1 resulting in on-chip quantum optical circuits for quantum
information processing. Towards establishing this approach, we started by exploring
the physical nature of the Mie resonance of DBBs. DBBs of size comparable to the
wavelength (~𝜆 /𝑛 ) are shown to exhibit strong magnetic dipole alongside with electric
dipole with comparable scattering cross section and bandwidth ~ 𝜆 /10. The spectral
overlap of these two dipoles has allowed designed interference between them,
producing directionality. This is demonstrated by response of the primitive
nanoantenna in Chapter 2 through engineering interference of these two modes for a
dipole representing the SPS transition embedded in a cubic DBB of size 220nm and
refractive index 3.5 representing GaAs-- resulting in a directionality enhancement of
factor of 2. We investigated the coupling of the SPS to the Mie resonance through the
notion of the local density of photon states (LDOS) projected along the orientation of
the transition dipole (also known as partial LDOS) using the notion of electromagnetic
Green function and demonstrated that the LDOS enhancement owing to the dominant
magnetic dipole mode results in a Purcell enhancement ~ 5. Such Purcell
249
enhancement, we note, enables shortening the radiative lifetime, typically~1ns for III-
V semiconductor-based quantum dot SPSs such as the MTSQDs, to ~200ps,
comparable to the typical dephasing timescale- thus enabling indistinguishable photon
emission needed for creating interference [6.7, 6.8].
We established that the nature of the Mie resonance is different from the
conventional structure of photonic crystal [6.9] in both, the fundamental physics and
its practical applicability points of view. Compared to photonic crystal cavity
structures that can achieve a quality factor ~10
5
with a typical bandwidth ~0.1nm [6.9,
6.10] the DBB metastructures are limited by the low Q ~100 of the Mie resonances
and thus do not achieve strong field localization. This results in weak coupling with
the SPS. However, the same physics of the Mie resonance is also responsible for
achieving large bandwidth ~10-100nm and thus fast response of ~ps scale [6.11].
More importantly the spectrally overlapping electric dipole and magnetic dipole mode
allow manipulation of the photon propagation even without strong field localization.
These features make it a viable candidate for certain applications of on-chip quantum
optical circuits that rely specifically on weak coupling (such as optical quantum
computation, quantum key distribution) with the added advantage of ease of spectral
matching with the on-chip SPS and also when integrated with other elements of
optical circuits.
Towards realization of the optical circuits targeted in the III-V platform of
integrable on-chip MTSQD SPSs as on-chip source [6.1-6.6], we reported in Chapter 2
studies testing the effect of substrate on the function of the MTSQD-DBB
250
metastructures. We found out that realization of the optical circuit requires ~1𝜇𝑚 thick
low-index (n~1.5) membrane structure underneath the DBBs to create enough
refractive index contrast with GaAs (n~3.5) and prevent photon loss. We therefore
conclude that realization of the MTSQD-DBB metastructure based optical circuits will
demand relying on the established platform of III-V on insulator that exploit flip-chip
bonding process to have the active III-V layer sitting on a high quality SiO2 membrane
[6.12-6.13]. Alternatively, dielectrics such as CaF2 and BaF2 that can be epitaxially
grown on GaAs [6.14-6.16] also can be exploited to incorporate the underlying
dielectric membrane structure.
Towards realization of the optical circuits as shown in Fig. 6.1, we presented
simulated response of the simplest unit comprising the nanoantenna-waveguide
structure- that combines the function of enhancement of emission rate and
enhancement of emission directionality as well as basic waveguiding of the emitted
photon from the SPS. To this end, a Purcell enhancement of ~5 along with photon
coupling efficiency to the Mie mode of ~50% is found. More importantly, both
simulation and, within the scope of this dissertation, the limited experimental
measurements of light transmission via DBB array indicate that both, the nanoantenna
and waveguide functions exploit the same collective Mie mode of the whole unit-
which simply attests to what is unique to the Mie resonance approach; the
metastructure provides light manipulating functions as a function of spatial location of
the photon wavepacket and the concept of individual components providing specific
functions simply has been done away with.
251
The nanoantenna and the DBB array waveguide- since demonstrated to exploit
the same collective Mie mode-- are easily integrated to form an unique single element
– the nanoantenna-waveguide structure that, we theoretically demonstrate, provides
enhancement of the SPS decay rate by ~5, enhancement of the emission directionality
and subsequent propagation, splitting and combining of the emitted photons via the
same collective Mie mode to enable the needed architecture and functionalities for
interference between photons from distinct on-chip SPSs. Importantly all functions
implemented via a single collective Mie mode, spread over a broad wavelength range
of ~10nm ensures mode matching of the SPS emission (within its much smaller
inhomogeneity) to the DBB metastructure (the optical circuit) and amongst different
parts of the optical circuit themselves. Furthermore, the nanoantenna-waveguide is
shown to be seamlessly extended to (a) beamsplitter-beamcombiner and (b) directional
coupler structure exploiting the same collective electric and magnetic dipole modes.
This provides a viable platform for achieving on-chip photon interference towards
optical quantum information processing.
§6.2. Mie Resonance Established as a Viable Approach for Collective Emission
and Entanglement of Multiple SPSs
The foundation of physics of Mie resonance and the engineering design
approach for optical circuit such as in Fig. 6.1 are in the next part of this dissertation
work extended towards objective of optical quantum information processing that
requires manipulation and interference of photon emitted from distinct on-chip single
photon emitters interconnected via the DBB array metastructures. To this end we
252
presented systematic study of a simple structure of two coupled SPSs via back-to-back
nanoantenna waveguide structure (Chapter 4 and Chapter 5) as shown in Fig. 6.2.
Figure 6.2. The Back-to-back nanoantenna-waveguide structure studied in this
dissertation towards SPS-SPS coupling and entanglement via Mie mode.
The approach developed to study this, the simplest, two SPS optical circuit is,
however, completely general and can be broadly divided in two categories –(i)
classical approach describing emission and propagation of a single photon from
collective excited state of multiple SPSs and (ii) quantum mechanical approach that
describes emission and propagation of multiple photons.
First, we present simulation studies on the back-to-back nanoantenna waveguide
unit where the collective electric and magnetic dipole mode of the DBB array is
exploited to now achieve direct coupling between the two SPSs. The multiple
scattering-based analysis of the electromagnetic Green function reveal that the two
SPSs can be a coherent super-position state leading to ~1.7 times the decay rate of a
single SPS (Super-radiance)- indicating a coupling strength of ~7𝜇𝑒𝑉 . As a
comparison, this coupling is a factor of 3-4 higher compared to an equivalent
continuous ridge waveguide structure. Furthermore, the super-radiant enhancement
was found to be independent of the distance between the two SPSs- thus indicating the
possibility of achieving SPS-SPS coupling and coherence over large on-chip distance.
253
As a final step of the work presented in this dissertation, we addressed the origin
of the collective excited states such as the super-radiant and sub-radiant states starting
from a trivial product state of the two SPSs. This by definition needed understanding
many-exciton/photon evolution. The dynamics of evolution of a two-particle (Boson/
Fermion) system is nonclassical in nature as the effect of symmetrization/
antisymmetrization has no effective analog in classical physics. Thus, we invoked a
fully quantum mechanical approach. We demonstrated that the classical Green
function and local density of photon states can still be used to course-grain the effect
of the spectrally broad single Mie mode- resulting in a Von-Neuman Lindblad master
equation [6.27, 6.28] to exactly represent the evolution of the density matrix of the two
SPS system in the generalized two-body Hilbert space.
This has allowed exploring the decay process of the two SPSs stating from
situations where both SPSs are in excited state, or only one in excited state. If only one
SPS is in excited state, a SPS-SPS energy transfer with a timescale ~300ps is
demonstrated. When both SPSs are initially in the excited state, the semiclassical
model reveals emergence of coherence between the two SPSs over time. Such SPS-
SPS coherence, we theoretically demonstrate, can be revealed by measuring the two
photon correlation, g
(2)
from the coupled SPSs where a sharp peak at g
(2)
(0) indicates
SPS-SPS coherent emission.
The studies presented in this chapter not only demonstrates the viability of the
MTSQD SPS-DBB metastructures based optical circuits in the many-coupled SPS
domain to create entanglement, but also provides a general approach to study any
254
other platform of on-chip optical circuit, so long as a classical EM Green function can
be defined.
§6.3. Outlook
The Mie resonance-based approach to the quantum optical circuits is not only
unique in the existing literature but is demonstrated to be capable of achieving all the
needed light manipulating functions such as enhancement of emission rate of SPS,
enhanced directionality, propagation, splitting and combining exploiting the same
mode in a perfectly mode-matched way—a specific advantage over the conventional
photonic crystal based, or ridge waveguide-based architectures. The richness of the
underlying physics as well as the engineering advantage of realizing all the needed
light manipulating function by exploiting the same broadband collective dipole
(electric + magnetic) encourages further investigations and studies of this platform.
§6.3.1. Extending the Quantum Approach to More Than 2 SPSs
Towards the long-term objective of on-chip optical quantum information
processing systems, a critical step to be taken is to understand the response of the
optical circuit in the many-excitation domain. We have started this process for the
SPS-Mie resonant metastructure based approach in Chapter 5 by formulating exactly
the dynamics of an arbitrary SPS-SPS coupled state in the two-body Hilbert space.
Typically, such approach in the literature [6.18] has only been in the basis of
phenomenologically representing the decay rate of the different collective states-
plugged into the Master equation. However, the real obstacle lies in connecting those
255
decay rates to the dielectric environment. We have attempted to do so in this
dissertation for the two-emitter situation by exploiting the pole analysis of the multiple
scattering Green function that can be calculated with high accuracy for any arbitrary
optical circuit. Thus, extending the Dyson expansion to higher number of emitters is
clearly a viable approach.
§6.3.2. General Quantum Approach to Mie Scattering
Another major next step can be readily identified on the theoretical front. In
the master equation formulation in Section 5.2, we course-grained over the broad
spectrum of the Mie resonance and assumed a perfect Markov approximation. This
allowed us to sum over the frequency spectrum of the Mie resonances and arrive at the
average effect expressed via the density of states. However, to understand the Mie
resonant metastructures aimed at the goal of quantum optical circuits, a quantum
mechanical approach to Mie scattering must be developed, allowing to express the
propagating photons via the collective Mie resonance for arbitrary non-classical states.
Such a formulation would be critical towards further development of the DBB array
metastructures.
6.3.3. Topological Photonics and Applicability of Mie Resonance
Over the past few decades, topological properties of photonic metasurfaces has
emerged as an indispensable degree of freedom. Topological properties are a global
property of the mode of the entire photonic structures- thus resistant to local
disturbances such as fabrication faults. In photonic crystal architecture, by varying the
256
lattice constant of the photonic crystal in different regimes, emergent topological edge
state [6.19] and corner state [6.20] have been demonstrated as a viable platform for
waveguide/ cavity functions. Compared to photonic crystal, the DBB metastructures
allow fine-tailoring the interaction between the mode-mode coupling between the
neighboring DBBs- thus provide more direct control on the dispersion characteristics
and potentially allows more rich topological states. Recently Mie resonance based
topological metasurface was demonstrated for ultrafast polarization conversion [6.21].
Exploring Mie resonant topological states for manipulation of single photons on-chip
towards quantum information processing is a rich and unexplored area that will be
explored in future.
§6.3.4. Integration of Nanoantenna with MTSQD SPSs
A key next step towards realization of the SPS-DBB metastructures
theoretically studied in this dissertation work is its integration with on-chip SPSs. To
this end, the mesa-top single quantum dots (MTSQDs) have been recently
demonstrated to be spectrally uniform and spatially regular single photon sources [6.1-
6.4]. Recently, planarizing overgrowth on these MTSQDs was also demonstrated [6.5]
that resulted in a planar substrate with buried spectrally uniform MTSQD SPSs with
positions known with ~nm scale precision. Such a platform, as indicated in Fig. 6.3,
provides a perfect starting point towards integration with the light manipulating DBB
metastructures to realize the long-sought on-chip quantum optical circuits.
257
Figure 6.3. Planarized MTSQD SPSs towards integration with DBBs.
On a concluding note, in this dissertation we have stepped away from the
conventional structure and explored a new approach to on-chip light manipulation
based on Mie resonance instead of Bragg scattering. Classical electromagnetic and
quantum mechanical analysis and experimental work partaken to understand the
physics and design principles of this new approach to on-chip light manipulation has
revealed both its limitations and its advantages over conventional approach of
photonic crystals. The philosophy behind the exploration in this dissertation has been
not to focus on a single functionality- such as cavity or waveguide and achieve the
best device design for it, but rather to have eye on the holistic need of light
manipulation for realization of a complete on-chip quantum optical circuit for
quantum information processing. We hope the new grounds uncovered in these studies
will prepare further explorations onto uncharted territory in the field of on-chip optical
quantum information processing.
258
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270
Appendix A. Mie Theory of Spherical DBBs
Historically Mie theory started with the need understand light scattering by small
particles [A.1]. It is to this, the analytical solutions to light scattering by particles with
spherical symmetry was given by Mie in 1908 [A.2]. In this appendix, we discuss the
analytical calculation method adopted to understand the Mie resonance for an isolated
spherical scatterer, as well as the methodology to find the collective response of an array
of them. This approach gives a sense of the nature of the Mie resonance that is applied
in Chapter 2 and Chapter 3 for spherical DBBs. Also, the understanding on the nature
of the Mie resonances for spherical particles lays the foundation to understand such
modes for particles in non-spherical shape such as the cubic/rectangular DBBs
dominantly studied in this dissertation.
We start with laying out the general system under investigation and the
fundamental assumptions in our model. The system under investigation comprises an
assembly of non-overlapping spherically symmetric scatterers, as shown in Figure 1,
with their center positions denoted by {r1, r2,…etc.}. These spheres scatters light from
an external source emitting at a particular wavelength 𝜆 . Under the linear regime of the
dielectric, it is assumed that that the scattered wave from the spheres does not affect the
source itself. Thus, one can write,
𝐸 ̅
𝑇𝑜𝑡𝑎𝑙 = 𝐸 ̅
𝑆𝑜𝑢𝑟𝑐𝑒 + 𝐸 ̅
𝑆𝑐𝑎𝑡𝑡𝑒𝑟𝑒𝑑 ( A.1)
And same relation holds for the magnetic field as well.
271
Figure A.1. Schematic showing an array of non-overlapping spherical scatterers excited
by a source (red wave-front).
In Mie theory, the scattered electromagnetic field is expanded in the basis of the
vector spherical harmonics. These vector spherical harmonics corresponds to specific
symmetry of the oscillation of the charge in the dielectric of the DBB that produces
them- these oscillations are referred to multipoles. The symmetry in the displacement
current can be divided into the radial order (n in eqn. (A.2) and (A.3)), angular
periodicity (m in eqn. (A.2) and (A.3)). Under spherical boundary condition, the
contribution from the radial order and the angular order can be separated- leading to the
vector spherical harmonics defined as: [A.3-A.6]
𝑚 ̄
𝑛 ,𝑚 = 𝛻 ̄
× (𝑟 ̄ 𝜓 𝑛 ,𝑚 ) (A.2)
and,
𝑛 ̄
𝑛 ,𝑚 = 𝛻 ̄
× (𝛻 ̄
× (𝑟 ̄ 𝜓 𝑛 ,𝑚 )) (A.3)
272
where 𝜓 𝑛 ,𝑚 = √𝑛 (𝑛 + 1)𝑧 𝑛 (𝛽𝑟 )𝑌 𝑛 ,𝑚 (𝜃 ,𝜙 ) , 𝑧 𝑛 (𝛽𝑟 ) being the appropriate spherical
Bessel function. The 𝑚 ̄
𝑛 ,𝑚 and 𝑛 ̄
𝑛 ,𝑚 complementarily describe the electric and
magnetic field components of a propagating spherical wave for the two types of
multipole modes, named TER (Transverse electric field to the radial direction) and TMR
(Transverse magnetic field to the radial direction) modes, also known as magnetic and
electric multipole modes respectively. These two types of multipole modes can be
mathematically expressed as,
𝐸 ̄
𝑇 𝐸 𝑞 ,𝑝 = 𝑚 ̄
𝑞 ,𝑝 ,𝐻 ̄
𝑇 𝐸 𝑞 ,𝑝 =
1
𝑖𝜂
𝑛 ̄
𝑞 ,𝑝 (A.5)
and
𝐸 ̄
𝑇 𝑀 𝑞 ,𝑝 = 𝑛 ̄
𝑞 ,𝑝 ,𝐻 ̄
𝑇 𝑀 𝑞 ,𝑝 =
1
𝑖𝜂
𝑚 ̄
𝑞 ,𝑝 (A.6)
where 𝜂 represents the impedance of the surrounding medium.
From equation (A.5) and (A.6), the exact expressions for the E-field and H-field can be
obtained as
𝐸 ̅
𝑇 𝐸 𝑛𝑚
(𝑟 ̅ )= 𝜂 𝐻 ̅
𝑇 𝑀 𝑛𝑚
(𝑟 ̅ )
=
1
√𝑛 (𝑛 + 1)
(𝑖𝑛 𝑧 𝑛 (𝛽𝑟 )
𝑌 𝑛 ,𝑚 (𝜃 ,𝜙 )
sin(𝜃 )
𝜃 ̂
− 𝑧 𝑛 (𝛽𝑟 )
𝜕 𝑌 𝑛 ,𝑚 (𝜃 ,𝜙 )
∂𝜃 𝜙 ̂
) (A.7)
And,
273
𝐸 ̅
𝑇 𝑀 𝑛𝑚
(𝑟 ̅ )= 𝜂 𝐻 ̅
𝑇 𝐸 𝑛𝑚
(𝑟 ̅ )
=
1
√𝑛 (𝑛 + 1)
(
𝑛 (𝑛 + 1)𝑧 𝑛 (𝛽𝑟 )
𝛽𝑟
𝑌 𝑛 ,𝑚 (𝜃 ,𝜙 )𝑟 ̂+
𝜕 (𝛽𝑟 𝑧 𝑛 (𝛽𝑟 ))
𝜕 (𝛽𝑟 ) 𝛽𝑟
𝜕 𝑌 𝑛 ,𝑚 (𝜃 ,𝜙 )
∂𝜃 𝜃 ̂
+ 𝑖𝑛
𝜕 (𝛽𝑟 𝑧 𝑛 (𝛽𝑟 ))
𝜕 (𝛽𝑟 ) 𝛽𝑟
𝑌 𝑛 ,𝑚 (𝜃 ,𝜙 )
sin (𝜃 )
𝜙 ̂
) (A.8)
The type of this Bessel function is chosen for 𝑧 𝑛 is according to the nature of the wave.
For scattered wave, the outward radiating nature is represented by the spherical Hankel
function of type 1 [A.3, A.6]. On the other side, for representing incident wave on any
sphere, the spherical Bessel function of type J is used [A.3, A.6].
For an N particle assembly with the particle center positions denoted by the
vectors {𝑟 ̄
𝑖 }
𝑖 =1
𝑁 such as shown in Fig. A.1. , the expansion of the scattered electric and
magnetic fields under the multipole mode basis of individual DNPs takes the form
,
𝐸 ̄
𝑆𝑐𝑎𝑡𝑡𝑒𝑟𝑒𝑑 (𝑟 ̄ )= ∑ ∑ ∑ (𝑎 𝑛 ,𝑚 (𝑖 )
𝐸 ̄
𝑇 𝐸 𝑛 ,𝑚 (𝑟 ̄− 𝑟 ̄
𝑖 )+ 𝑏 𝑛 ,𝑚 (𝑖 )
𝐸 ̄
𝑇 𝑀 𝑛 ,𝑚 (𝑟 ̄
𝑛 𝑚 =−𝑛 𝑛 𝑚𝑎𝑥 𝑛 =1
𝑁 𝑖 =1
− 𝑟 ̄
𝑖 )) (𝐴 .9)
𝐻 ̄
𝑆𝑐𝑎𝑡𝑡𝑒𝑟𝑒𝑑 (𝑟 ̄ )= ∑ ∑ ∑ (𝑎 𝑛 ,𝑚 (𝑖 )
𝐻 ̄
𝑇 𝐸 𝑛 ,𝑚 (𝑟 ̄− 𝑟 ̄
𝑖 )
𝑛 𝑚 =−𝑛 𝑛 𝑚𝑎𝑥 𝑛 =1
𝑁 𝑖 =1
+ 𝑏 𝑛 ,𝑚 (𝑖 )
𝐻 ̄
𝑇 𝑀 𝑛 ,𝑚 (𝑟 ̄− 𝑟 ̄
𝑖 )) (𝐴 .10)
274
The series is terminated by setting a 𝑛 𝑚𝑎𝑥
as the maximum order of the multipole
modes. We show later that the series quickly converges.
To form a self-consistent equation to solve for the scattered field, we need to consider
two processes- Translation of the spherical vector harmonics and scattering event by a
single DBB.
Spherical Vector Translation:
The scattered multipole modes comprise of the spherical Hankel function of type (1)
and represent radially outward emitted wave. However, the radially outward wave
originating from ith sphere can be also broken down in the multipole basis around the
sphere j, leading to spherical vector harmonics with the spherical Bessel function J as
the radial component. These J-type Bessel function vector harmonics represent the
incident wave on the jth sphere that is caused by the scattered wave from ith sphere, in
the multipole basis centered on DBB j. This is aided by the spherical vector translation
that is illustrated in Figure 2.
275
Figure A. 2. Translation of the multipole modes from radiated (scattered) wave from
sphere i to incident wave on sphere j. The superscript J on the multipole modes signify
that they comprise of the J-type spherical Bessel functions.
𝐸 ̅
𝑇𝐸𝑛𝑚 (𝑟 − 𝑟 𝑖 )= ∑(𝜁 𝑞 ,𝑝 ,𝑛 ,𝑚 𝑖 ,𝑗 𝐸 𝑇𝐸𝑞𝑝 (𝐽 )
(𝑟 − 𝑟 𝑗 )+ 𝜏 𝑞 ,𝑝 ,𝑛 ,𝑚 𝑖 ,𝑗 𝐸 𝑇𝑀𝑞𝑝 (𝐽 )
(𝑟 − 𝑟 𝑗 ))
𝑞 ,𝑝 (A.11)
𝐸 ̅
𝑇𝑀𝑛𝑚 (𝑟 − 𝑟 𝑖 )= ∑(𝜁 𝑞 ,𝑝 ,𝑛 ,𝑚 𝑖 ,𝑗 𝐸 𝑇𝑀𝑞𝑝 (𝐽 )
(𝑟 − 𝑟 𝑗 )+ 𝜏 𝑞 ,𝑝 ,𝑛 ,𝑚 𝑖 ,𝑗 𝐸 𝑇𝐸𝑞𝑝 (𝐽 )
(𝑟 − 𝑟 𝑗 ))
𝑞 ,𝑝 (A.12)
The analytical expressions for 𝜁 and 𝜏 require use of the translation of the spherical
vector harmonics and can be found in [A.3, A.6].
Scattering by a Single Sphere:
Owing to the assumed spherical symmetry of the scatterers, an incident wave of the
nature TEnm
(J)
will always lead to a TEnm type scattered wave and same applies for the
TM modes as well. The conversion from the incident wave to the scattered wave can be
described by a factor that is only dependent on the radius of the sphere (R), refractive
276
index (ni) of the sphere and the wavelength of light. These are expressed as 𝜒 𝑇 𝐸 𝑛 ,𝑚 and
𝜒 𝑇 𝑀 𝑛 ,𝑚 .
𝜒 𝑇 𝐸 𝑛 ,𝑚 =
−𝑛 𝑖 𝑗 ̂
𝑛 (𝛽𝑅 )𝑗 ̂
𝑛 ′
(𝛽 𝑑 𝑅 )+ 𝑛 𝑜𝑢𝑡 𝑗 ̂
𝑛 ′
(𝛽𝑅 )𝑗 ̂
𝑛 (𝛽 𝑑 𝑅 )
𝑛 𝑖 ℎ
̂
𝑛 (1)
(𝛽𝑅 )𝑗 ̂
𝑛 ′
(𝛽 𝑑 𝑅 )− 𝑛 𝑜𝑢𝑡 ℎ
̂
′
𝑛 (1)
(𝛽𝑅 )𝑗 ̂
𝑛 (𝛽 𝑑 𝑅 )
(A.13)
𝜒 𝑇 𝑀 𝑛 ,𝑚 =
−𝑛 𝑖 𝑗 ̂
𝑛 ′
(𝛽𝑅 )𝑗 ̂
𝑛 (𝛽 𝑑 𝑅 )+ 𝑛 𝑜𝑢𝑡 𝑗 ̂
𝑛 (𝛽𝑅 )𝑗 ̂
𝑛 ′
(𝛽 𝑑 𝑅 )
𝑛 𝑖 ℎ
̂
′
𝑛 (1)
(𝛽𝑅 )𝑗 ̂
𝑛 (𝛽 𝑑 𝑅 )− 𝑛 𝑜𝑢𝑡 ℎ
̂
𝑛 (1)
(𝛽𝑅 )𝑗 ̂
𝑛 ′
(𝛽 𝑑 𝑅 )
(A.14)
Here 𝛽 =
2𝜋 𝜆 and 𝛽 = 𝑛 𝑖 𝛽 , i.e. wavevector inside the dielectric.
With these, now the general Matrix equation is formulated for Mie scattering by array
of spherical DBBs.
Formulation for a Numerical Solution: Analytical Method to Matrix Equation
The truncated multipole decomposition of the scattered wave, as shown in equation
(A.7) and (A.8) is constructed in the form of a vector, 𝑉 𝑂𝑢𝑡 , as
277
[𝑉 𝑜𝑢𝑡 ] =
[
[𝑎 𝑛 ,𝑚 (1)
]
1×𝑛 𝑑𝑖𝑚
[𝑏 𝑛 ,𝑚 (1)
]
1×𝑛 𝑑𝑖𝑚 [𝑎 𝑛 ,𝑚 (2)
]
1×𝑛 𝑑𝑖𝑚 [𝑏 𝑛 ,𝑚 (2)
]
1×𝑛 𝑑𝑖𝑚 .
.
.
[𝑎 𝑛 ,𝑚 (𝑁 )
]
1×𝑛 𝑑𝑖𝑚 [𝑏 𝑛 ,𝑚 (𝑁 )
]
1×𝑛 𝑑𝑖𝑚 ]
𝑁 𝑑𝑖𝑚 ×𝑁 𝑑𝑖𝑚 .......................( A.15)
Here 𝑛 𝑑𝑖𝑚 = (𝑛 𝑚𝑎𝑥
2
+ 2𝑛 𝑚𝑎𝑥
) represents the number of TE or number of TM modes
per DBB. Thus, [𝑉 𝑜𝑢𝑡 ], the matrix representing the state of the whole system comprising N
DBBs has a dimension 𝑁 𝑑𝑖𝑚 × 𝑁 𝑑𝑖𝑚 where 𝑁 𝑑𝑖𝑚 = 2𝑁 × 𝑛 𝑑𝑖𝑚 .
With this Matrix representation, the translation of scattered wave from one sphere to the
incident wave on the other sphere (Figure A. 2) is expressed as a translation matrix T:
Thus [𝑉 𝑖𝑛𝑐𝑖𝑑𝑒𝑛𝑡 ] = [𝑇 ]
𝑁 𝑑𝑖𝑚 ×𝑁 𝑑𝑖𝑚 [𝑉 𝑜𝑢𝑡 ]+ [𝑉 𝑆𝑜𝑢𝑟𝑐𝑒 ] (A.16)
Where,
[𝑇 ]
=
[
0 [
[𝜁 𝑞 ,𝑝 ,𝑛 ,𝑚 2,1
]
𝑛 𝑑𝑖𝑚 ×𝑛 𝑑𝑖𝑚 [𝜏 𝑞 ,𝑝 ,𝑛 ,𝑚 2,1
]
𝑛 𝑑𝑖𝑚 ×𝑛 𝑑𝑖𝑚 [𝜏 𝑞 ,𝑝 ,𝑛 ,𝑚 2,1
]
𝑛 𝑑𝑖𝑚 ×𝑛 𝑑𝑖𝑚 [𝜁 𝑞 ,𝑝 ,𝑛 ,𝑚 2,1
]
𝑛 𝑑𝑖𝑚 ×𝑛 𝑑𝑖𝑚 ] .
[
[𝜁 𝑞 ,𝑝 ,𝑛 ,𝑚 1,2
]
𝑛 𝑑𝑖𝑚 ×𝑛 𝑑𝑖𝑚 [𝜏 𝑞 ,𝑝 ,𝑛 ,𝑚 1,2
]
𝑛 𝑑𝑖𝑚 ×𝑛 𝑑𝑖𝑚 [𝜏 𝑞 ,𝑝 ,𝑛 ,𝑚 1,2
]
𝑛 𝑑𝑖𝑚 ×𝑛 𝑑𝑖𝑚 [𝜁 𝑞 ,𝑝 ,𝑛 ,𝑚 1,2
]
𝑛 𝑑𝑖𝑚 ×𝑛 𝑑𝑖𝑚 ] . .
.
.
.
. [
[𝜁 𝑞 ,𝑝 ,𝑛 ,𝑚 𝑁 ,𝑁 ]
𝑛 𝑑𝑖𝑚 ×𝑛 𝑑𝑖𝑚 [𝜏 𝑞 ,𝑝 ,𝑛 ,𝑚 𝑁 ,𝑁 ]
𝑛 𝑑𝑖𝑚 ×𝑛 𝑑𝑖𝑚 [𝜏 𝑞 ,𝑝 ,𝑛 ,𝑚 𝑁 ,𝑁 ]
𝑛 𝑑𝑖𝑚 ×𝑛 𝑑𝑖𝑚 [𝜁 𝑞 ,𝑝 ,𝑛 ,𝑚 𝑁 ,𝑁 ]
𝑛 𝑑𝑖𝑚 ×𝑛 𝑑𝑖𝑚 ]
]
𝑁 𝑑𝑖𝑚 ×𝑁 𝑑𝑖𝑚
..( A.17)
278
Also, the scattering of the incident wave on each sphere to the scattered wave is
represented in matrix form:
[𝑉 𝑜𝑢𝑡 ] = [χ]
𝑁 𝑑𝑖𝑚 ×𝑁 𝑑𝑖𝑚 [𝑉 𝑖𝑛𝑐𝑖𝑑𝑒𝑛𝑡 ] (A.18)
Where,
[𝜒 ] =
[
[
[𝜒 𝑇𝐸
(1)
]
𝑛 𝑑𝑖𝑚
×𝑛 𝑑𝑖𝑚
0
0 [𝜒 𝑇𝑀
(1)
]
𝑛 𝑑𝑖𝑚
×𝑛 𝑑𝑖𝑚
] 0 0
0 [
[𝜒 𝑇𝐸
(2)
]
𝑛 𝑑𝑖𝑚
×𝑛 𝑑𝑖𝑚
0
0 [𝜒 𝑇𝑀
(2)
]
𝑛 𝑑𝑖𝑚
×𝑛 𝑑𝑖𝑚
] 0
0
0
0
0
0
[
[𝜒 𝑇𝐸
(𝑁 )
]
𝑛 𝑑𝑖𝑚
×𝑛 𝑑𝑖𝑚
0
0 [𝜒 𝑇𝑀
(𝑁 )
]
𝑛 𝑑𝑖𝑚
×𝑛 𝑑𝑖𝑚
]
]
𝑁 𝑑𝑖𝑚
×𝑁 𝑑𝑖𝑚
(A.19)
We note that 𝜒 is a diagonal matrix as long as the scatterers have spherical ssymmetry.
By combining equation (A.16) and (A.18), we get a self-consistent matrix equation for
out
V as follows:
[𝑉 𝑜𝑢𝑡 ] = [𝜒 ][𝑇 ][𝑉 𝑜𝑢𝑡 ]+ [𝜒 ][𝑉 𝑆𝑜𝑢𝑟𝑐𝑒 ] (A.20)
Or,
(𝕀 − [𝜒 ][𝑇 ])[𝑉 𝑜𝑢𝑡 ] = [𝜒 ][𝑉 𝑆𝑜𝑢𝑟𝑐𝑒 ] (A.21)
Or,
[𝑉 𝑜𝑢𝑡 ] = (𝕀 − [𝜒 ][𝑇 ])
−1
[𝜒 ][𝑉 𝑆𝑜𝑢𝑟𝑐𝑒 ] (A.22)
279
This provides the solution to the scattered wave defined fully by [𝑉 𝑜𝑢𝑡 ] for any arbitrary
source, defined by [𝑉 𝑆𝑜𝑢𝑟𝑐𝑒 ].
Figure A. 3. magnetic dipole mode coefficients as a function of mode order for
scattering of a plane wave by a single spherical nanoparticle.
The dimension of the matrix equation in equation (A.22) is 𝑁 𝑑𝑖𝑚 = 2(𝑛 𝑚𝑎𝑥
2
+
2𝑛 𝑚𝑎𝑥
)𝑁 . To show the convergence of the solution with increasing nmax, we consider
the physical situation shown in Fig. A.3, same as system shown in Fig. 2.16 in Chapter
2 in the results of the scattering cross section of the Mie resonances. In Chapter 2 we
established that the magnetic dipole mode (represented by the coefficient 𝑎 1,1
) is the
dominant mode. Thus here, in Fig. A.3 we show the 𝑎 𝑛 ,1
coefficients (representing the
magnetic dipole, magnetic quadrupole, and higher order terms) as a function of the
radial order n. An exponential trend is observed in the reduction of the coefficients for
the higher order multipole modes. This indicates that the series in equation (A.9) and
280
(A.10) can be truncated. We estimate that choosing 𝑛 𝑚𝑎𝑥
= 4 results in a computational
error <1e-7. This is far better than the computational accuracy of finite difference
method based numerical computational tool (error ~1%) and thus is suited for our
purpose. Also, this choice of 𝑛 𝑚𝑎𝑥
= 4 results in the dimension of the Matrix equation
𝑁 𝑑𝑖𝑚 = 48 × 𝑁 -- negligible compared to the few million degrees of freedom for a
numerical calculation.
References:
[A. 1] H. C. Van-de-Hulst, “Light Scattering by Small Particles”, Dover Publications
Inc., New York, 1981.
[A. 2] G. Mie, "Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen".
Annalen der Physik. 330 (3): 377–445 (1908).
[A. 3] J. M. Gerardy and M. Ausloos, “Absorption spectrum of clusters of spheres from
the general solution of Maxwell’s equations: II. Optical properties of aggregated metal
spheres”, Phys Rev. B 25, 6 (1982)
[A. 4] S. J. Adams, “Electromagnetic theory”. McGrawhill Book Company, New York
and London, 1941.
[A. 5] J. D. Jackson, “Classical Electrodynamics”, Wiley student edition, 1999.
[A. 6] M. Abramowitz and I. A. Stegun, “handbook of mathematical functions with
formulas, graphs, and mathematical tables”, National Bureau of Standards Applied
Mathematics Series. 55(1964).
281
Appendix B. Green function and Local Density of States
In the field of quantum optics, one of the most useful tools is photonic Green function
[B.1, B.2, B.3, B.4]. The Green functions are the classical solution of the E-field of
Maxwell equation for an impulse response- which in this context means a point source.
Thus, the Green function capture the E-field produced by a point dipole source.
Photonic Green function has the unique applicability to quantum physics. Since
the description of light in Maxwell equation is already via electromagnetic waves,
Maxwell equation can represent exactly evolution of a single photon state. Therefore,
although classical, the Green function from Maxwell equation directly correspond to the
photon density of states that is responsible for radiative decay of any emitter- a
phenomenon that is best captured by the Fermi golden rule:
Γ =
𝜋𝜔
3ℏ𝜖 𝑝 2
𝜌 𝐿𝐷𝑂𝑆 (𝑟 ̅
𝑆𝑃𝑆 , 𝜔 𝑆𝑃𝑆 ) (𝐵 . 1)
Thus, the photonic Green function finds huge applicability in any category of
optical light manipulating circuits- both in the strong and weak coupling limit.
In this dissertation work we have applied the concepts of photon Green function
and density of states extensively. In this appendix we capture the key mathematical
formulation used.
282
§B.1. Definition of the Photon Green Function:
The starting equation for this formulation is the Maxwell equation of the Electric
Field as a function of position and frequency, resulted by an arbitrary current density
field. If the E- and H-fields as a function of position 𝑟 ̅ and angular frequency 𝜔 is given
as 𝐸 ̅
(𝑟 ̅ , 𝜔 ) and 𝐻 ̅
(𝑟 ̅ , 𝜔 ), the from Maxwell equation we have,
∇
̅
× 𝐸 ̅
(𝑟 ̅ , 𝜔 ) = 𝑖𝜔𝜇 𝐻 ̅
(𝑟 ̅ , 𝜔 ) (𝐵 . 2)
∇
̅
× 𝐻 ̅
(𝑟 ̅ , 𝜔 ) = −𝑖𝜔𝜖 𝐸 ̅
(𝑟 ̅ , 𝜔 ) + 𝐽 ̅
(𝑟 ̅ , 𝜔 ) (𝐵 . 3)
Combining equation (B.1) and (B.2), we get,
∇
̅
× ∇
̅
× 𝐸 ̅
(𝑟 ̅ , 𝜔 ) = 𝜔 2
𝜇𝜖 𝐸 ̅
(𝑟 ̅ , 𝜔 ) + 𝑖𝜔𝜇 𝐽 ̅
(𝑟 ̅ , 𝜔 ) (𝐵 . 4)
Now, the current density 𝐽 ̅
(𝑟 ̅ , 𝜔 ) is related to the polarization of the dielectric medium
by
𝐽 ̅
(𝑟 ̅ , 𝜔 ) = −𝑖𝜔 𝑃 ̅
(𝑟 ̅ , 𝜔 ) (𝐵 . 5)
Which leads to,
∇
̅
× ∇
̅
× 𝐸 ̅
(𝑟 ̅ , 𝜔 ) = 𝜔 2
𝜇𝜖 𝐸 ̅
(𝑟 ̅ , 𝜔 ) + 𝜔 2
𝜇 𝑃 ̅
(𝑟 ̅ , 𝜔 ) (𝐵 . 6)
=> ∇
̅
× ∇
̅
× 𝐸 ̅
(𝑟 ̅ , 𝜔 ) − 𝛽 2
𝐸 ̅
(𝑟 ̅ , 𝜔 ) =
𝛽 0
2
𝜖 0
𝑃 ̅
(𝑟 ̅ , 𝜔 ) (𝐵 . 7)
where 𝛽 2
= 𝜔 2
𝜇𝜖 is the wave-vector at any point of the dielectric medium. Equation
(B.7) now provides the notion of a Green function that describes the E-field distribution
resulting from a point-like dipole source. We thus define the Green function as,
283
∇
̅
× ∇
̅
× 𝐺 ̿
(𝑟 ̅ , 𝑟 ̅ ′, 𝜔 ) − 𝛽 2
𝐺 ̿
(𝑟 ̅ , 𝑟 ̅ ′, 𝜔 ) =
𝛽 0
2
𝜖 0
𝐼 ̿
𝛿 (𝑟 ̅ − 𝑟 ̅
′
) (𝐵 . 8)
where 𝐼 ̿
is the unit Dyadic and is given by 𝑥 ̂𝑥 ̂ + 𝑦 ̂𝑦 ̂ + 𝑧 ̂ 𝑧 ̂ . Thus, from any arbitrary
source-polarization field 𝑃 ̅
𝑆𝑜𝑢𝑟𝑐𝑒 (𝑟 ̅ , 𝜔 ), the resultant Electric field can now be expressed
as
𝐸 ̅
(𝑟 ̅ , 𝜔 ) = ∫ 𝐺 ̿
(𝑟 ̅ , 𝑟 ̅ ′, 𝜔 ) ⋅ 𝑃 ̅
𝑆𝑜𝑢𝑟𝑐𝑒 (𝑟 ̅ ′, 𝜔 )𝑑 3
𝑟 ′
(𝐵 . 9)
For an exciton recombination process in a quantum dot, the resultant polarization field
can be expressed as
𝑃 ̅
𝑆𝑜𝑢𝑟𝑐𝑒 (𝑟 ̅ , 𝜔 ) = 𝜂 𝑒 (𝑟 ̅ )𝜂 ℎ
(𝑟 ̅ ) < 𝜒 𝑒 |𝑒 𝑟 ̂ |𝜒 ℎ
> (𝐵 . 10)
where 𝜂 𝑒 (𝑟 ̅ ), 𝜂 ℎ
(𝑟 ̅ ) are the electron and hole envelop function and 𝜒 𝑒 , and 𝜒 ℎ
are the
corresponding orbital functions. [B.5]. Typically, the resultant 𝑃 ̅
𝑆𝑜𝑢𝑟𝑐𝑒 (𝑟 ̅ , 𝜔 ) can be
expanded as a sum of different orders of multipoles for states that are more spread out
[B.6]. For our case, the spatial extent of the QD electronic state is ignored in comparison
to the wavelength of the emitted photons. Thus, we rewrite equation (B.10) as
𝑃 ̅
𝑆 𝑜𝑢𝑟𝑐𝑒 (𝑟 ̅ , 𝜔 ) = 𝑝 ̅ (𝜔 )𝛿 (𝑟 ̅ − 𝑟 ̅
𝑆𝑜𝑢𝑟𝑐𝑒 ) (𝐵 . 11)
where 𝑝 ̅ (𝜔 ) represents the dipole moment corresponding to the ground level exciton
recombination of the SPS, and 𝑟 ̅
𝑆𝑃𝑆 represents its position in space. The total electric
field produced by the ensemble of the emitters is now,
𝐸 ̅
(𝑟 ̅ , 𝜔 ) = 𝐺 ̿
(𝑟 ̅ , 𝑟 ̅
𝑆𝑃𝑆 , 𝜔 ) ⋅ 𝑝 ̅ (𝜔 ) (𝐵 . 12)
284
At this point it must be noted that 𝐺 ̿
in equation (B.11) represents the electromagnetic
propagator for the E-field vector for an arbitrary point dipole source- under the
assumption that the dipole source itself does not scatter the light/ interact with the
photon emitted by itself. We refer this Green function hereby onwards as the zeroth-
order Green function, and schematically represent it in as Fig. B.1.
Figure B.1. The zeroth order Green function is defined as the E-field produced by a unit
dipole source in uniform medium- as indicated in the left panel. The right panel
schematically defines such Green function.
§B.2. Green function and relation with radiative decay rate
Note, however, that this zeroth-order Green function does not represent reality
as there is always a chance that the photon emitted from the SPS can get back to the
SPS and be scattered again. This process is physically represented in the left panel of
Fig. B.2. To accommodate this self-interaction process, a corrected Green function is
needed. We represent this Green function as the Green function of first order, or, 𝐺 ̿
(1)
.
285
Figure B.2. A conceptual schematic showing that in the presence of a non-uniform
medium, such as a nanoantenna, there a chance that the emitted photon is scattered by
the same source dipole where it came from. This results in the first order Green function
as indicated in the right panel. Note- this multiple scattering of the emitted photon is the
origin of Purcell enhancement.
In general, any scattering process can be thought of as absorption and reemission
of an incident photon by the scatterer, all within the timescale determined by the
uncertainty relationship. Classically we can express this scattering process by a linear
phenomenon, where the E-field of the incident photon induces an oscillating dipole
moment at the location of the QD.
𝑝 ̅
𝑖𝑛𝑑𝑢𝑐𝑒𝑑 (𝜔 ) = 𝛼 ̿(𝜔 ) ⋅ 𝐸 ̅
𝑖𝑛𝑐 (𝑟 ̅
𝑆𝑃𝑆 , 𝜔 ) (𝐵 . 13)
Here 𝛼 ̿(𝜔 ) is a dyadic that corresponds to the linear process of induction of the
oscillating dipole for a given incident E-field. Here, we assume that the direction of the
dipole moment is fixed for the emitter by the unit vector 𝑝 ̂ . Under this assumption, the
scattering dyadic 𝛼 ̿(𝜔 ) can be simplified as
𝛼 ̿(𝜔 ) = 𝑝 ̂ 𝑉 (𝜔 )𝑝 ̂ (𝐵 . 14)
where 𝑉 (𝜔 ) is a scalar response function. This response function for a zero-linewidth
Lorentz-like scatterer can be derived as [B.2]
286
𝑉 (𝜔 ) =
𝑝 1
ℏ
2𝜔 𝑆𝑃𝑆 𝜔 2
− 𝜔 𝑆𝑃𝑆 2
(𝐵 . 15)
Combining the zeroth order Green function and the scattering response dyadic
𝛼 ̿(𝜔 ) by the emitter as defined above, it allows us to solve for the first-order Green
dyadic in the form of a Dyson expansion, as follows:
𝐺 ̿
(1)
(𝑟 ̅ , 𝑟 ̅
𝑆𝑃𝑆 , 𝜔 ) = 𝐺 ̿
(𝑟 ̅ , 𝑟 ̅
𝑆𝑃𝑆 , 𝜔 ) + 𝐺 ̿
(𝑟 ̅ , 𝑟 ̅
𝑆𝑃𝑆 , 𝜔 ) ⋅ 𝛼 ̿(𝜔 ) ⋅ 𝐺 ̿
(𝑟 ̅
𝑆𝑃𝑆 , 𝑟 ̅
𝑆𝑃𝑆 , 𝜔 )
+𝐺 ̿
(𝑟 ̅ , 𝑟 ̅
𝑆𝑃 𝑆 , 𝜔 ) ⋅ 𝛼 ̿(𝜔 ) ⋅ 𝐺 ̿
(𝑟 ̅
𝑆𝑃𝑆 , 𝑟 ̅
𝑆𝑃𝑆 , 𝜔 ) ⋅ 𝛼 ̿(𝜔 ) ⋅ 𝐺 ̿
(𝑟 ̅
𝑆𝑃𝑆 , 𝑟 ̅
𝑆𝑃𝑆 , 𝜔 ) + ⋯ (𝐵 . 16)
This series can be summed exactly in the form:
𝐺 ̿
(1)
(𝑟 ̅ , 𝑟 ̅
𝑆𝑃𝑆 , 𝜔 ) =
𝐺 ̿
(𝑟 ̅ , 𝑟 ̅
𝑆𝑃𝑆 , 𝜔 )
1 − 𝑝 ̂ ⋅ 𝛼 ̿(𝜔 ) ⋅ 𝐺 ̿
(𝑟 ̅
𝑆𝑃𝑆 , 𝑟 ̅
𝑆𝑃𝑆 , 𝜔 ) ⋅ 𝑝 ̂
(𝐵 . 17)
Importantly, the complex poles of this modified Green function 𝐺 ̿
(1)
contains the critical
information on the time-dynamics [B.2, B.3]—the imaginary roots representing the
decay constants and the real part of the root representing the oscillatory behavior, i.e.
emission energy and Lamb shift. Let us now solve for the poles of the first order Green
dyadic as shown in equation (B.16). By replacing the form for 𝛼 ̿(𝜔 ), we get,
𝐺 ̿
(1)
(𝑟 ̅ , 𝑟 ̅
𝑆𝑃𝑆 , 𝜔 ) =
(𝜔 2
− 𝜔 𝑆𝑃𝑆 2
)𝐺 ̿
(𝑟 ̅ , 𝑟 ̅
𝑆𝑃𝑆 , 𝜔 )
𝜔 2
− 𝜔 𝑆𝑃𝑆 2
− 2 𝜔 𝑆𝑃𝑆 𝑝 2
ℏ
𝑝 ̂⋅ 𝐺 ̿
(𝑟 ̅
𝑆𝑃𝑆 , 𝑟 ̅
𝑆𝑃𝑆 , 𝜔 ) ⋅ 𝑝 ̂
(𝐵 . 18)
We define,
𝑋 (𝜔 ) =
𝑝 2
ℏ
(𝑝 ̂ ⋅ 𝐺 ̿
(𝑟 ̅
𝑆𝑃𝑆 , 𝑟 ̅
𝑆𝑃𝑆 , 𝜔 ) ⋅ 𝑝 ̂ ) (𝐵 . 19)
287
Thus, the poles of the Green function are the roots of the equation
𝜔 2
− 𝜔 𝑆𝑃𝑆 2
− 2𝜔 𝑆𝑃𝑆 𝑋 (𝜔 ) = 0 (𝐵 . 20)
Here we make two critical assumptions:
First, |𝑋 (𝜔 )| ≪ 𝜔 𝑄𝐷
. Intuitively this approximation can be understood if the
interaction between the emitter and the photons is weak. This criterion is the formal
definition of the week coupling [Refs] limit in this context.
Second, the surrounding dielectric medium does not affect the emission frequency of
the emitter significantly. In such case, the only value of 𝑋 (𝜔 ) that actually matters is at
𝜔 = 𝜔 𝑆𝑃𝑆 . Thus, we can replace the function 𝑋 (𝜔 ) by a constant 𝑋 (𝜔 𝑆𝑃𝑆 ).
With these two assumptions above, the equation (B.19) now leads to
𝜔 2
− 𝜔 𝑆𝑃𝑆 2
− 2𝜔 𝑆𝑃𝑆 𝑋 (𝜔 𝑆𝑃𝑆 ) = 0 (𝐵 . 21)
⇒ 𝜔 ≈ (𝜔 𝑆𝑃𝑆 + 𝑋 (𝜔 𝑆𝑃𝑆 )) (𝐵 . 22)
288
Figure B.3. Shift of the complex poles of the E-field owing to the multiple scattering by
the emitter results in Purcell effect.
Let us take a look what this means on a complex plane representing the poles as
𝑠 𝑝𝑜𝑙𝑒𝑠 = −Γ
𝑆𝑃𝑆 + 𝑖 (𝜔 𝑆𝑃𝑆 + Δ
𝑆𝑃𝑆 ) (𝐵 . 23)
where Γ
𝑄𝐷
= 𝐼𝑚 (𝑋 (𝜔 𝑆𝑃𝑆 )), and Δ
𝑄𝐷
= 𝑅𝑒 (𝑋 (𝜔 𝑆𝑃𝑆 )). Needless to say, Γ
𝑆𝑃𝑆 represents
the radiative decay rate of the emitter in the uniform medium, and Δ
1
represents the
corresponding Lamb shift. Thus, the radiative decay rate of the emitter can be expressed
in terms of the zeroth order Green function as,
Γ = 2𝐼𝑚 (𝑋 (𝜔 𝑆𝑃𝑆 )) =
2𝑝 2
ℏ
𝐼𝑚 (𝑝 ̂ ⋅ 𝐺 ̿
(𝑟 ̅
𝑆𝑃𝑆 , 𝑟 ̅
𝑆𝑃𝑆 , 𝜔 ) ⋅ 𝑝 ̂ ) (𝐵 . 24)
289
i.e. the imaginary component of the trace of the Green dyadic results in Purcell
enhancement. While, the real part results in shift in frequency known as Lamb shift.
This is captured in Fig.B.3 showing the effect of the multiple scattering phenomenon
by the nanoantenna on the complex poles of the electric field.
Now we can compare equation (B.24) with equation (B.1) to derive an expression
for the photon local density of state in the weak coupling limit as
𝜌 𝐿𝐷𝑂𝑆
(𝑟 , 𝜔 𝑆𝑃𝑆 ) =
6𝜖 0
𝜋𝜔
𝐼𝑚 (𝑝 ̂ ⋅ 𝐺 ̿
(𝑟 ̅
𝑆𝑃𝑆 , 𝑟 ̅
𝑆𝑃𝑆 , 𝜔 ) ⋅ 𝑝 ̂ ) (𝐵 . 25)
This equation constitutes the fundamental relation between the classical Green function
of electromagnetic wave and density of states of photons and forms the most general
rule governing decay of a single emitter in any arbitrary environment so long as the
weak coupling approximation is satisfied.
290
§B.3. Two Emitters: Second Order Green Function
When the number of emitters is increased from one, the Green function can be
appropriately modified to capture the behavior of a single photon (classical) emission
and propagation via an arbitrary dielectric environment including the effect of scattering
by multiple emitters. We have exploited this in Chapter 4 to derive the sub-radiant and
super-radiant decay rate of two coupled SPS in the back to back nanoantenna waveguide
system. In this section, the derivation of the Green function is presented.
Let us assume a scenario similar to what is shown in figure B.4, i.e. two radiating
dipoles, 𝑝 1
(𝜔 ) and 𝑝 2
(𝜔 ) at position 𝑟 ̅ = 𝑟 ̅
1
and 𝑟 ̅ = 𝑟 ̅
2
, respectively. Compared to Fig.
B.2, now a single photon can be scattered by either of the two emitters. Thus, the Green
function needs to be further modified to capture this effect. Let us symbolically denote
the modified Green function by 𝐺 ̿
(2)
in this case, where the subscript (2) now represents
two point-scatterers involved in the multiple scattering process. The resultant E-field
vector at an arbitrary point can be expressed as,
𝐸 ̅
(𝑟 ̅ , 𝜔 ) = 𝐺 ̿
(2)
(𝑟 ̅ , 𝑟 ̅
1
, 𝜔 ) ⋅ 𝑝 ̅
1
(𝜔 ) + 𝐺 ̿
(2)
(𝑟 ̅ , 𝑟 ̅
2
, 𝜔 ) ⋅ 𝑝 ̅
2
(𝜔 ) (𝐵 . 26)
Schematically this multiple scattering process is now represented in the following:
291
Figure B.4. Representing the second-order Green function when the both the emitters
interact with the emitted photon.
In Figure B.4, the red wavy arrows represent the zeroth order propagator 𝐺 ̿
. The diagram
can be further simplified by using the propagators based on 𝐺 ̿
(1)
as show in Fig.B.5.
Figure B.5. Representing the second-order Green function in terms of the first order
Green function.
The derivation of the exact expression of 𝐺 ̿
(2)
(𝑟 ̅ , 𝑟 ̅
1
, 𝜔 ) from Fig. B.5 straightforward.
To assist, we now break down the diagram as shown in Fig. B.6
292
Figure B.6. Breakdown of 𝐺 ̿
(2)
(𝑟 ̅ , 𝑟 ̅
1
, 𝜔 ) from the schematic shown in Fig. B.5.
From Fig. B.6, now the second order modified Green function for two point-scatterers
can be expressed as
𝐺 ̿
(2)
(𝑟 ̅ , 𝑟 ̅
1
, 𝜔 ) =
𝐺 ̿
(1)
(𝑟 ̅ , 𝑟 ̅
1
, 𝜔 )
1 − 𝑝 ̂
1
⋅ 𝛼 ̿
1
(𝜔 ) ⋅ 𝐺 ̿
(1)
(𝑟 ̅
1
, 𝑟 ̅
2
, 𝜔 ) ⋅ 𝛼 ̿
2
(𝜔 ) ⋅ 𝐺 ̿
(1)
(𝑟 ̅
2
, 𝑟 ̅
1
, 𝜔 ) ⋅ 𝑝 ̂
1
+
𝐺 ̿
(1)
(𝑟 ̅ , 𝑟 ̅
2
, 𝜔 ) ⋅ 𝛼 ̿
2
(𝜔 ) ⋅ 𝐺 ̿
(1)
(𝑟 ̅
2
, 𝑟 ̅
1
, 𝜔 )
1 − 𝑝 ̂
1
⋅ 𝛼 ̿
1
(𝜔 ) ⋅ 𝐺 ̿
(1)
(𝑟 ̅
1
, 𝑟 ̅
2
, 𝜔 ) ⋅ 𝛼 ̿
2
(𝜔 ) ⋅ 𝐺 ̿
(1)
(𝑟 ̅
2
, 𝑟 ̅
1
, 𝜔 ) ⋅ 𝑝 ̂
1
(𝐵 . 27)
Similar expression can be derived for 𝐺 ̿
(2)
(𝑟 ̅ , 𝑟 ̅
2
, 𝜔 ). We rewrite equation (B.27) as,
𝐺 ̿
(2)
(𝑟 ̅ , 𝑟 ̅
1
, 𝜔 ) =
𝐺 ̿
(1)
(𝑟 ̅ , 𝑟 ̅
1
, 𝜔 ) + 𝐺 ̿
(1)
(𝑟 ̅ , 𝑟 ̅
2
, 𝜔 ) ⋅ 𝛼 ̿
2
(𝜔 ) ⋅ 𝐺 ̿
(1)
(𝑟 ̅
2
, 𝑟 ̅
1
, 𝜔 )
1 − 𝑉 1
(𝜔 ) (𝑝 ̂
1
⋅ 𝐺 ̿
(1)
(𝑟 ̅
1
, 𝑟 ̅
2
, 𝜔 ) ⋅ 𝑝 ̂
2
) 𝑉 2
(𝜔 ) (𝑝 ̂
2
⋅ 𝐺 ̿
(1)
(𝑟 ̅
2
, 𝑟 ̅
1
, 𝜔 ) ⋅ 𝑝 ̂
1
)
(𝐵 . 28)
293
⇒ 𝐺 ̿
(2)
(𝑟 ̅ , 𝑟 ̅
1
, 𝜔 ) =
𝐺 ̿
(𝑟 ̅ , 𝑟 ̅
1
, 𝜔 )
1 − 𝑉 1
(𝜔 )𝐺 1,1
(𝜔 )
+
𝐺 ̿
(𝑟 ̅ , 𝑟 ̅
2
, 𝜔 ) ⋅ 𝑝 ̂
2
𝑉 2
(𝜔 )𝑝 ̂
2
⋅ 𝐺 ̿
(𝑟 ̅
2
, 𝑟 ̅
1
, 𝜔 )
(1 − 𝑉 2
(𝜔 )𝐺 2,2
(𝜔 )) (1 − 𝑉 1
(𝜔 )𝐺 1,1
(𝜔 ))
1 − 𝑉 1
(𝜔 )
𝐺 1,2
(𝜔 )
(1 − 𝑉 2
(𝜔 )𝐺 2,2
(𝜔 ))
𝑉 2
(𝜔 )
𝐺 2,1
(𝜔 )
(1 − 𝑉 1
(𝜔 )𝐺 1,1
(𝜔 ))
(𝐵 . 29)
where we have defined the shorthand notation:
𝐺 𝑖 ,𝑗 (𝜔 ) = 𝑝 ̂
𝑖 ⋅ 𝐺 ̿
(𝑟 ̅
𝑖 , 𝑟 ̅
𝑗 , 𝜔 ) ⋅ 𝑝 ̂
𝑗 (𝐵 . 30)
Simplifying equation (B.29) further, we get,
𝐺 ̿
(2)
(𝑟 ̅ , 𝑟 ̅
1
, 𝜔 )
=
𝐺 ̿
(𝑟 ̅ , 𝑟 ̅
1
, 𝜔 )(𝜔 2
− 𝜔 1
2
)
𝜔 2
− 𝜔 1
2
− 2𝜔 1
𝑋 1
(𝜔 )
+
(𝜔 2
− 𝜔 1
2
)𝐺 ̿
(𝑟 ̅ , 𝑟 ̅
2
, 𝜔 ) ⋅
𝑝 ̂
2
2𝜔 2
𝑝 2
2
ℏ
𝐺 2,1
(𝜔 ) 𝑝 ̂
1
(𝜔 2
− 𝜔 1
2
− 2𝜔 1
𝑋 1
(𝜔 ))(𝜔 2
− 𝜔 2
2
− 2𝜔 2
𝑋 2
(𝜔 ))
1 −
2𝑝 1
2
𝜔 1
ℏ(𝜔 2
− 𝜔 1
2
)
𝐺 1,2
(𝜔 ) (𝜔 2
− 𝜔 2
2
)
(𝜔 2
− 𝜔 2
2
− 2𝜔 2
𝑋 2
(𝜔 ))
2𝑝 2
2
𝜔 2
ℏ(𝜔 2
− 𝜔 2
2
)
𝐺 2,1
(𝜔 ) (𝜔 2
− 𝜔 1
2
)
(𝜔 2
− 𝜔 1
2
− 2𝜔 1
𝑋 1
(𝜔 ))
(𝐵 . 31)
=
(𝜔 2
− 𝜔 1
2
)[𝐺 ̿
(𝑟 ̅ , 𝑟 ̅
1
, 𝜔 )(𝜔 2
− 𝜔 2
2
− 2𝜔 2
𝑋 2
(𝜔 )) + 𝐺 ̿
(𝑟 ̅ , 𝑟 ̅
2
, 𝜔 ) (𝑝 ̂
2
𝑝 ̂
1
)
𝑝 2
𝑝 1
2𝜔 2
𝐽 2,1
(𝜔 )]
(𝜔 2
− 𝜔 2
2
− 2𝜔 2
𝑋 2
(𝜔 ))(𝜔 2
− 𝜔 1
2
− 2𝜔 1
𝑋 1
(𝜔 )) − 4𝜔 1
𝜔 2
𝐽 2,1
(𝜔 )𝐽 1,2
(𝜔 )
(𝐵 . 32)
where
𝑋 𝑖 (𝜔 ) =
𝑝 𝑖 2
ℏ
𝐺 𝑖 ,𝑖 (𝜔 ) =
𝑝 𝑖 2
ℏ
𝑝 ̂
𝑖 ⋅ 𝐺 ̿
(𝑟 ̅
𝑖 , 𝑟 ̅
𝑖 , 𝜔 ) ⋅ 𝑝 ̂
𝑖 (𝐵 . 33)
𝐽 𝑖 ,𝑗
(𝜔 ) =
𝑝 𝑖 𝑝 𝑗 ℏ
𝐺 𝑖 ,𝑗 (𝜔 ) =
𝑝 𝑖 𝑝 𝑗 ℏ
𝑝 ̂
𝑖 ⋅ 𝐺 ̿
(𝑟 ̅
𝑖 , 𝑟 ̅
𝑗 , 𝜔 ) ⋅ 𝑝 ̂
𝑗 (𝐵 . 34)
294
Note, that by Reciprocity theorem of electromagnetism [B.7], we have 𝐽 1,2
(𝜔 ) =
𝐽 2,1
(𝜔 ) = 𝐽 (𝜔 ), say.
Equation (𝐵 . 32) represents the time-evolution of the E-field under scattering by two
emitters. The poles of this Green function thus represent the characteristic time-scales-
both oscillatory and lossy. This is used in Chapter 4.2 to derive the transition rates for
the super-radiant and sub-radiant emission from the coupled SPSs.
§B.4. Green function from Mode Decomposition
Equation (B.8) defines the notion of a classical Green function for electromagnetic wave
for arbitrary dielectric environment. However, for situations, where the modes of the
dielectric environment can be solved directly, the Green function can be expressed in
terms of the modes [B.1, B.2] of the dielectric environment. This is exploited to solve
for the Green function and density of states for the spherical DBB shown in Chapter 2.
Here we first define the electromagnetic modes as
∇
̅
× ∇
̅
× ℇ
̅
𝑖
(𝑟 ̅ ) − 𝛽 2
ℇ
̅
𝑖
(𝑟 ̅ ) = 0 (𝐵 . 35)
Here we only consider the transverse modes for which ∇ ⋅ (𝜖 ℇ
̅
𝑖
) = 0 which is in general
true for any arbitrary dielectric environment. The index i indicates the ith mode of the
system.
295
The Green function can be expressed in terms of the known modes of the environment
as
𝐺 ̿
(𝑟 ̅ , 𝑟 ̅ ′, 𝜔 ) =
𝜔 2
𝜖 0
∑
ℇ
̅
𝑖
(𝑟 ̅ )⨂ℇ
̅
𝑖
∗
(𝑟 ̅ ′)
(𝜔 2
− 𝜔 𝑖 2
)
𝑎𝑙𝑙 𝑖 (𝐵 . 36)
As an illustration, let us consider a single spherical DBB. The electromagnetic modes
around the DBB is well described using the TEn,m and TMn,m modes as discussed in
Appendix A. Thus the mode expansion with the spherical harmonics leads to the
analytical estimate of the Green dyadic and thus the density of states for the spherical
DBB [B.1].
§B.5. References:
[B.1] Cole. P. Van Vlack. “Dyadic Green function and their applications in classical
and quantum nanophotonics”, Dissertation, Queen’s University 2012.
[B.2] M. Wubs, L. G. Suttorp, and A. Lagendijk, “Multiple-scattering approach to
interatomic interactions and superradiance in inhomogenelos dielectrics”, Phys. Rev. A
70, 053823 (2004).
[B.3] E. N. Economou, “Green’s functions in quantum physics”, Springer 2006.
[B.4] C. Carlson, D. Dalacu, C. Gustin, S. Haffouz, X.Wu, J. Lapointe, R L. Williams,
P. J. Poole, and S. Hughes, “Theory and experiments of coherent photon coupling in
semiconductor nanowire waveguides with quantum dot molecules”, Phys. Rev. B, 99,
085311(2019).
296
[B.5] M. Minkov and V. Savona, “Radiative coupling of quantum dots in photonic
crystal structures”, Phys. Rev. B. 87, 125306, (2013).
[B.6] Chapter 5, “The mesoscopic nature of quantum dots in photon emission”, P.
Michler (Ed.), “Quantum Dots for Quantum Information Technologies”, Springer 2017.
[B.7] R. F. Harrington, “Time-harmonic electromagnetic fields”, IEEE Press 1961.
297
Appendix C:
Instrumentation: Optical Measurement on DBB Array
In this appendix, we discuss in details the optical measurement configurations that were
built and used for the experimental studies on propagation via fabricated DBB array as
reported in Chapter 3 section 3.3.2.a and Section 3.3.2.b..
The measurements are divided into two stages—(1) measuring the spectral position of
the collective magnetic dipole mode of the fabricated DBB array waveguide using
vertical excitation/vertical detection (Section 3.3.2.a) and (2) measuring the
transmission via DBB array waveguide in the horizontal geometry (Section 3.3.2.b).
The measurement set-up for these are discussed in the next:
§C.1. Instrumentation for Vertical Excitation/Vertical Detection
The vertical excitation-vertical detection measurement (geometry shown in Fig. C.1(a))
on the fabricated DBB array is discussed in Section 3.3.2.a. Figure C.1 (b) shows the
schematic of the instrumentation corresponding to this set-up.
As the backbone of the measurement set-up, an inverted microscope from
Olympus (Olympus IX71) is used for sample mounting and imaging purpose. The
sample was mounted on a MS-2000 microscope stage allowing translation of the sample
298
with respect to the excitation and detection branch with a precision of ~40nm. The
vertical excitation and vertical detection branch are built with the following
specifications to allow the target measurement:
Excitation Beam: A supercontinuum light source (SuperK Compact from NKT
photonics) is used as excitation light source in the wavelength range of 400nm-2200nm-
well-encompassing the needed wavelength corresponding to the Mie modes of the
DBBs. The excitation beam is brought in via a collimator and focused on the sample
surface via a 60x objective lens. The excitation beam focused on the sample surface is
approximately a gaussian beam with ~2𝜇𝑚 waist diameter and 10
0
beam divergence
angle. This closely mimicks a parallel beam of light exciting one end of the DBB array.
The excitation-beam collimator is mounted on XYZ translation stage and rotation stage
allowing control on both the angle of incidence and point of incidence of the excitation
laser on the sample surface for ready alignment with the DBB array.
Detection: The photons for the other end of the DBB array are collected via the same
60x objective using a confocal geometry- thus limiting the collection area ~2𝜇𝑚 around
the end of the DBB array. A NIR optimized achromatic pair of lens system is used to
focus the detected photons to the Action 300i spectrometer slit. A 1200/mm grating is
used for spectrally resolving the photons with a resolution of ~0.6nm and thus are able
to resolve photons from the Mie resonance of ~10nm bandwidth with no problem. The
spectrally resolved photons are detected by a LN2-cooled Si CCD detector that allows
299
detection in the 500nm-1100nm wavelength range, suited for observing the collective
electric and magnetic dipole modes.
Figure C.1. (a) Vertical excitation/vertical detection experimental geometry on DBB
array waveguide and (b) Instrumentation schematic.
The NA1.41 numerical aperture of the objective of the microscope allows
detection in an angular spread of ±70
0
around the surface normal. As shown, the light
collected by the Objective is transformed into a parallel beam in-between the two
300
doublet lenses. Thus, placing a pinhole aperture in that parallel beam allows only beam
that corresponds to a particular detection angle—and adjusting position of this pinhole
allows changing the angle of detection. A pinhole of diameter 500𝜇 m that results into
an angular resolution of ~30
0
was chosen. For scanning the Angular distribution, the
pinhole is moved in steps of 100𝜇 m which corresponds to ~6
0
step in the detection
angle- thus allowing measurement of the spectrally resolved angular distribution of the
light coming out of the DBB array waveguide as reported in Section 3.3.2.a.
§C.2. Horizontal Excitation/ Horizontal Detection
The Horizontal excitation/detection measurement (Fig. C.2(a)) set-up is built
around the BX53 upright microscope from Olympus. Figure C.2 (b) schematically
captures the instrumentation. Two 10x and 100x objective are used with the upright
optical microscope for imaging and alignment purpose. The DBB array sample is
mounted a translation stage (Newport Piezometer stage- allowing ~20nm step in X, Y,
and Z) and also rotation stage allowing rotation around all three axes with ~0.1
0
degree
precision allowing easy alignment of the DBB array (DBBs of size ~200nm) with the
excitation and detection branch. The excitation and detection branch are built around
these with the following specifications:
Horizontal Excitation Branch: Similar to the vertical excitation/detection set-up in the
last section, the same NKT-SuperK Compact supercontinuum is used as the external
light source providing photons of wavelength range of ~400nm-~2000nm- safely
301
encompassing the needed wavelength range ~900 nm for the Mie resonance. The
excitation supercontinuum laser is brought in via a collimator and focused using a NIR
Figure C.2. (a) Horizontal measurement geometry on the DBB array and (b) schematic
of the corresponding optical instrumentation.
optimized Mitutoyo 50x objective to a Gaussian beam of beam diameter at waist being
~5𝜇 m and beam divergence angle ~5
0
. This focused excitation beam serves to excite on
302
one end of the DBB array in our measurement. The excitation branch is mounted on a
XYZ translation and rotation (around Z) stage to allow course movement of the
Figure C.3. An optical microscope image showing the DBB array on the edge of the
sample surface with lensed optical fiber aligned with one of the DBB arrays. The
303
excitation laser, coming in at an angle of ~30
0
with respect to the array is shown
schematically.
excitation beam with a precision ~1𝜇𝑚 . For fine alignment, the piezo-motor translation
of the sample stage (step of ~20nm) is used.
Horizontal Detection Branch: For the detection, we simply bring in a lensed optical
fiber (Thorlab HP780, Focal spot beam diameter ~2𝜇𝑚 , focal length from the tip of the
fiber ~10𝜇𝑚 ) horizontally, mounted on a high precision MP225 micromanipulator
allowing ~40nm precision in X, Y, and Z translation movements. Figure C.3 shows an
actual optical microscope image on the DBB array substrate showing the lensed optical
fiber aligned with one of the DBB arrays and the excitation beam schematically shown
on the same image. The collected photons are then focused into the entrance slit of the
Action300i spectrometer using a pair of achromatic doublet lenses and resolved
spectrally using a 1200/mm grating with ~0.6nm spectral resolution. The detection is
dione via LN2 cooled Si CCD (same as the vertical setup in the previous section)
allowing photon detection in the wavelength range of ~500nm-1100nm.
Detection Efficiency with the Lensed Optical Fiber:
With the detection fiber configured as shown in Fig. C.3, the collection
efficiency of the Mie mode into the optical fiber mode is calculated using the mode
overlap [C.1] between the mode of the DBB array and the mode of the lensed optical
fiber (Gaussian mode with 2𝜇𝑚 diameter at the beam waist) to represent the collection
efficiency of a photon already in the collective Mie mode of the DBB array into the
304
mode of the collection optical fiber. If 𝐸 ̅
𝑓 and 𝐻 ̅
𝑓 represents the TEM00 gaussian mode
of the lensed optical fiber and 𝐸 ̅
and 𝐻 ̅
represents the E- and H-field of the collective
Mie resonance, then the collection efficiency is expressed as the mode overlap:
𝜂 =
𝑅𝑒 {∬ 𝐸 ̅
𝑓 × 𝐻 ̅
∗
⋅ 𝑧 ̂𝑑𝑆 𝑆 ∬ 𝐸 ̅
× 𝐻 ̅
𝑓 ∗
⋅ 𝑧 ̂𝑑𝑆 𝑆 }
𝑅𝑒 {∬ 𝐸 ̅
𝑓 × 𝐻 ̅
𝑓 ∗
⋅ 𝑧 ̂𝑑𝑆 𝑆 } 𝑅𝑒 {∬ 𝐸 ̅
× 𝐻 ̅
∗
⋅ 𝑧 ̂𝑑𝑆 𝑆 }
(𝐶 . 1)
Figure C.4. Estimated collection efficiency of photons from the collective Mie mode of
the DBB array into the mode of the single mode lensed optical fiber as a function of the
wavelength.
Figure C.4. shows the estimated collection efficiency from the Mie mode of the
DBB into the lensed optical fiber for two orthogonal (Horizontal and Vertical)
polarizations. Note that, for the magnetic dipole mode ~930nm, a collection efficiency
of ~20-25% is estimated.
305
§C.3. References:
[C.1] M. Davanco, J. Liu, L. Sapienza, C-Z. Zhang, J. V. D. M. Cardoso, V. verma, R.
Mirin, S. W. Nam, L. Liu, and K. Srinivasan, “Heterogeneous integration for on-chip
quantum photonic circuits with single quantum dot devices”, Nature Communications
8, 889 (2017).
Abstract (if available)
Abstract
This dissertation proposes and explores a new and novel class of on-chip optical circuits aimed at optical quantum information processing (QIP). Realization of on-chip optical QIP systems demands conceiving, fabricating, and examining material structures that exploit designed modulation in the refractive index (in the wavelength regime of interest) to manipulate on-chip generated photons from their source through their pathway to the detector where they are finally detected having performed the many desired information carrying and transfer functions on the way. Such light manipulating functional metastructures made of appropriate material combinations laid out in planar architectures to mediate photon interactions with other photons and material entities (typically electrons) are dubbed optical circuits (OCs). Such OCs are to embed on-chip single photon sources (SPSs) for manipulation of the emitted photons to enable on-chip photon interference and establish communication between distinct SPSs for quantum entanglement as a resource for quantum information processing. Generically the optical circuits are required to provide such functions as (1) enhancement of emission rate of the SPS (2) enhancement of emission directionality, (3) state-preserving propagation of emitted photons, (4) splitting and (5) combining to enable interference between photons emitted from pre-determined distinct on-chip SPSs. In the conventional approaches to on-chip photon manipulation, these functions are implemented using distinct functional structures, categorized as discrete components such as cavity, waveguide, beam splitter etc.—using existing platforms such as 2D photonic crystal and ridge waveguides and couplers. However, this modular approach also demands mode-matching between the distinct functional components such as cavity, waveguide, etc. This has always been an obstacle towards scalability. As a result, an optical circuit that provides all the above-mentioned five functions eludes realization to this day. ❧ In contrast, we envision and pursue an approach fundamentally different from this conventional way of thinking of an optical circuit as a collection of mode-matched “components”. The physics we exploit is also new—the collective Mie resonance of arrays (metastructures) of subwavelength scale dielectric building blocks (DBBs) in contrast to Bragg scattering in photonic crystal. Engineering the collective Mie resonance of a DBB based optical metastructure allows simultaneous control over the E-field spatial distribution and dispersion characteristics of the collective mode, which, in turn, allows to implement all the above noted needed five light manipulating functions using the same collective Mie mode. Thus, on-chip integration of SPS arrays with DBB metastructures opens a new paradigm for quantum information processing applications. The conceptualization, theoretical (classical and quantum) modelling, and numerical examination of this new paradigm constitutes the core of this dissertation. ❧ Our conceptualization and thus modelling of the DBB metastructure optical circuits and judicious classical and quantum theoretical studies are guided by the recently established mesa-top single quantum dots (MTSQDs) in spatially regular arrays as on-chip scalable SPSs with considerable potential for providing sufficiently spectrally uniform emission (< 2nm standard deviation) from arrays distributed over 1000µm² areas to enable proven local tuning methodologies to bring to resonance photons emitted from different known emitters in the array to enable controlled interference and entanglement. Thus the simulation studies undertaken are of DBB units built around each MTSQD SPS of the array to provide—using the same collective Mie-mode of the interconnected units forming the metastructure dubbed the optical circuit—all the needed functions: enhancement of SPS emission rate and directionality, propagation, splitting, and recombining aimed at photon interference and entanglement. As all light manipulating functions are provided by the nature of the electric and magnetic field distribution of the same collective Mie mode in different spatial regions of the unit as part of the larger metastructure, there is no concept of “components” and therefore no issue of “impedance matching” between them. In this new paradigm based upon collective Mie resonance of the whole system (i.e. the metastructure that is the optical circuit itself) the same mode of the whole system provides the different needed “functions” in pre-specified spatial regions as part of the co-design of the optical circuit (i.e. the metastructure) to include even the DBB that contains inside the effectively point-like photon emitting source. ❧ To expose the fundamental physics of the nature of the Mie resonances in high refractive index dielectric blocks of subwavelength size for optical wavelengths, we have carried out analytical and numerical (finite element method) studies for spherical and cubic DBBs respectively, exploiting the Mie theory to establish the nature of the Mie resonance of the DBBs. The various magnetic and electric Mie resonance are identified by studying the different symmetries associated with the oscillation of displacement current within an individual dielectric block and it is revealed that the interference between these oscillation of different symmetries provide the fundamental basis behind light manipulation functions. The dominant magnetic resonance of the dielectric was identified to result in enhanced density of photon states and, for cubic DBBs of linear dimension 200nm with nearest wall-to-wall separation ~50nm, provide Purcell enhancement of ~5 to 10 in the spontaneous emission rate of the source modelled as a dipole-driven two-level system emitting at 980nm. The interference between the magnetic and electric dipole modes controls the degree of directionality in photon emission with ~0.5 coupling efficiency of the emitted photon to the collective Mie resonance. Supporting the extensive theoretical studies, we also pursued experimental work with limited scope of fabrication array of DBB array in the Silicon on Insulator platform and via far-field scattering spectroscopy measurement, demonstrating the propagating collective Mie resonance of such DBB array validating our approach. ❧ Complementing the above examined phenomena of single photon generation to interference between photons, the direct coupling between two or more SPSs mediated by the Mie mode is of significance to QIP. To examine this coupling, we exploited a Dyson sequence of the classical Green dyadic description to account for the radiative rate enhancement of coupled SPSs. We found that the collective Mie resonance of the DBB array is capable of inducing on-chip coupling between distinct SPSs over large on-chip separations of ~10 to 100μm. Moreover, theoretically a super-radiant emission rate, approximately 1.7 times the single emitter is demonstrated. However, as a classical description cannot account for two photon states, to account for the coherent and incoherent decay processes that lead to such super-radiant coupled SPS state, we exploit Von-Neumann Lindblad approach and provide a detailed quantum field theoretic study of the evolution of the density matrix of SPSs coupled via the collective Mie resonance to show that the collective Mie resonance is responsible for the emergence of coherence and entanglement between the coupled SPSs over time. The reported study lays the foundation for further exploration and study on this new paradigm of “component-less” optical circuits based on common Mie resonance of DBB metastructures providing spatial location dependent photon manipulation functions for optical classical and quantum information processing.
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University of Southern California Dissertations and Theses
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Asset Metadata
Creator
Chattaraj, Swarnabha
(author)
Core Title
Investigations of Mie resonance-mediated all dielectric functional metastructures as component-less on-chip classical and quantum optical circuits
School
Viterbi School of Engineering
Degree
Doctor of Philosophy
Degree Program
Electrical Engineering
Degree Conferral Date
2021-08
Publication Date
05/11/2021
Defense Date
10/29/2020
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
dielectric,Mie resonance,OAI-PMH Harvest,optical quantum information processing,quantum photonics
Format
theses
(aat)
Language
English
Contributor
Electronically uploaded by the author
(provenance)
Advisor
Madhukar, Anupam (
committee chair
), Hashemi, Hossein (
committee member
), Willner, Alan (
committee member
), Zanardi, Paolo (
committee member
)
Creator Email
chattara@usc.edu
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https://doi.org/10.25549/usctheses-oUC112720151
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UC112720151
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etd-ChattarajS-9628.pdf (filename)
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Chattaraj, Swarnabha
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Tags
dielectric
Mie resonance
optical quantum information processing
quantum photonics