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On-chip Kerr frequency comb generation and its effects on the application of optical communications
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On-chip Kerr frequency comb generation and its effects on the application of optical communications
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Content
ON-CHIP KERR FREQUENCY COMB GENERATION AND ITS EFFECTS ON
THE APPLICATION OF OPTICAL COMMUNICATIONS
by
Changjing Bao
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(ELECTRICAL ENGINEERING)
May 2018
Copyright 2018 Changjing Bao
Dedication
To my parents, Bangxie Bao and Ruxiu Liu,
for their selfless love and care,
and to my elder brother Yanyan Bao,
for his support and encouragement.
ii
Acknowledgements
It has been over five years since I started my PhD program in August 2012. I
deeply feel that the PhD period is the most valuable time in my life, up to now,
because I have learned so much from my PhD advisor Prof. Alan E. Willner and also
from my colleagues at Optics Communications Laboratory (OCLab). Without their
help and devotion, it would not be possible for me to complete this dissertation.
First, I would like to deeply thank my PhD advisor Prof. Alan E. Willner who
is always supporting, encouraging, and instructing me. I always feel so lucky
because I have been studying my PhD under the supervision of Prof. Willner at USC
and I have gained invaluable experience and knowledge to study in a well-known lab.
Prof. Willner's knowledge, experience, and vision help me overcome all the
difficulties in my PhD period. I have also learned a lot from him about many
personal lessons, which I believe will be greatly helpful for my future life.
I would also thank Dr. Andrey Matsko from OEwaves Inc., Prof. Tobias
Kippenberg from EPFL, and Prof. Moshe Tur from Tel Aviv University. I cannot
accomplish my research projects without their advice and instructions. I also feel
thankful to our collaborators Martin Pfeiffer, Maxim Karpov, and Arne Kordts from
EPFL for their help with our experiment.
I would also like to thank my colleagues Dr. Lin Zhang, Dr. Yang Yue, Dr.
Jeng-yuan Yang, Dr. Yan Yan, and Peicheng Liao for their insightful discussions and
collaborations in this dissertation. I would also like to thank other current and past
OCLab members, Dr. Hao Huang, Dr. Salman Khaleghi, Dr. Asher Voskoboinik, Dr.
iii
Mohammad Reza Chitgarha, Dr. Yongxiong Ren, Dr. Morteza Ziyadi, Bishara
Shamee, Guodong Xie, Ahmed Almaiman, Amirhossein Mohajerin-Ariaei, Yinwen
Cao, Long Li, Zhe Zhao, Cong Liu, Ahmad Fallahpour, Fatemeh Alishahi, Zhe
Wang, Runzhou Zhang, Kai Pang, Haoqian Song, Kaiheng Zou, and Hao Song for
all their helpful discussions. I am also thankful to other USC friends and roommates
who have helped me in the past five years.
Last but not least, I would like to express my deepest love to my parents, my
elder brother, and my lovely niece who are always supporting and encouraging me
spiritually and providing me with their selfless love !
iv
Table of Contents
Dedication ................................................................................................................... ii
Acknowledgements .................................................................................................... iii
List of Figures ........................................................................................................... vii
Abstract... .................................................................................................................. xv
Chapter 1 Introduction .............................................................................................. 1
1.1 An optical frequency comb ............................................................................. 1
1.2 Application of Microresonator-Based Frequency Comb in Optical
Communications ............................................................................................. 3
1.3 Thesis Outline ................................................................................................. 5
Chapter 2 Increased Bandwidth with Flattened and Low Dispersion in a
Horizontal Double-Slot Silicon Waveguide ........................................... 7
2.1 Introduction .................................................................................................... 7
2.2 Waveguide Design and Modeling .................................................................. 8
2.3 Dispersion Properties and Supercontinuum Generation ............................... 14
2.4 Conclusion .................................................................................................... 18
Chapter 3 Nonlinear Conversion Efficiency in Kerr Frequency Comb
Generation .............................................................................................. 20
3.1 Introduction .................................................................................................. 20
3.2 Definition of Nonlinear Conversion Efficiency ........................................... 21
3.3 Numerical Simulation Results ...................................................................... 22
3.4 Discussion and Conclusion ........................................................................... 29
Chapter 4 Effect of a Breather Soliton in Kerr Frequency Combs on Optical
Communication Systems ....................................................................... 31
4.1 Introduction .................................................................................................. 31
4.2 Waveguide Design and Kerr Comb Simulation ........................................... 32
4.3 Communication System Simulation ............................................................. 36
4.4 Discussion and Conclusion ........................................................................... 40
Chapter 5 High-Order Dispersion in Kerr Comb Oscillators ............................. 41
5.1 Introduction .................................................................................................. 41
5.2 Slot Waveguide Ring Microresonator .......................................................... 42
5.3 Numerical Model .......................................................................................... 43
5.4 Dispersive Wave and Phase Matching Curve ............................................... 44
5.4 Simulation Results for Various GVD Combinations ................................... 46
5.5 Deviation from the Ideal Soliton Due to High-Order GVD ......................... 48
v
5.6 Power Dependence of the Dispersive Wave Spectral Position .................... 51
5.7 Fourth-Order GVD in a Slot Waveguide Ring Resonator ............................ 53
5.8 Conclusion .................................................................................................... 54
Chapter 6 Demonstration of Optical Multicasting Using Kerr Frequency Comb
Lines ........................................................................................................ 55
6.1 Introduction .................................................................................................. 55
6.2 Concept of Optical Multicasting................................................................... 56
6.3 Low-Phase-Noise Comb Generation and Optical Multicasting
Performance .................................................................................................. 58
6.4 Chaotic Comb Generation and Multicasting Performance ........................... 61
6.5 Conclusion .................................................................................................... 63
Chapter 7 Dual-Pump Generation of High-Coherence Primary Kerr Combs
with Multiple Sub-Lines ........................................................................ 64
7.1 Introduction .................................................................................................. 64
7.2 Principle and Experimental Setup ................................................................ 65
7.3 Experimental Results .................................................................................... 67
7.4 Conclusion .................................................................................................... 73
Chapter 8 Tunable Insertion of Multiple Lines into a Kerr Frequency Comb
Using Electro-Optical Modulators........................................................ 74
8.1 Introduction .................................................................................................. 74
8.2 Concept of EO Comb Line Insertion ............................................................ 75
8.3 Experimental Results .................................................................................... 77
8.4 Conclusion .................................................................................................... 81
Chapter 9 Orthogonally Polarized Kerr Frequency Combs ................................ 83
9.1 Introduction .................................................................................................. 83
9.2 Operation Principle ....................................................................................... 84
9.3 Experimental Setup and Experimental Results ............................................ 84
9.4 Numerical Simulation Results ...................................................................... 90
9.5 Discussion ..................................................................................................... 93
References ................................................................................................................. 95
vi
List of Figures
Figure 1.1 An optical frequency comb in the frequency domain (a) and in the time
domain (b). ............................................................................................................ 1
Figure 1.2 (a) A Kerr frequency comb is generated when a pump laser is coupled
into a high Q-factor microresonator. (b) The Kerr frequency comb is
generated by a combination of degenerate FWM and nondegenerate FWM. ...... 3
Figure 1.3 The application of Kerr frequency comb in optical communication
system using wavelength- and polarization-division multiplexing and higher
modulation formats. EDFA, erbium-doped fiber amplifier; MUX,
multiplexer; DEMUX, demultiplexer; and PC, polarization controller. ............... 4
Figure 1.4 The relationship between the comb noise and the constellations and the
BER of high-order modulation QAM signals........................................................ 5
Figure 2.1 Two options of horizontal double-slot silicon waveguide (left and
middle) and single-slot silicon waveguide (right). ................................................ 9
Figure 2.2 Mode distributions of the major electric field component (E
y
) at
different wavelengths in Option 1 (a), Option 2 of the double-slot
waveguide (b), and the single-slot waveguide (c). .............................................. 10
Figure 2.3 Power ratio in the slots with the wavelength in Option 1 (a), Option 2
(b), and single-slot waveguide (c). ...................................................................... 12
Figure 2.4 The chromatic dispersion with four zero-dispersion wavelengths
(ZDWs) in double-slot design (Option 1 and Option 2) and a single-slot
waveguide. ........................................................................................................... 13
Figure 2.5 The chromatic dispersion of Option 1 and single-slot waveguide with
the same height. ................................................................................................... 14
Figure 2.6 Dispersion profiles in Option 2 of double-slot design with the variation
of lower silicon height Hl
1
(a), central silicon height Hl
2
(b), and lower slot
thickness Hs
1
(c). ................................................................................................. 15
Figure 2.7 (a) Silicon waveguide with all-normal dispersion for supercontinuum
generation. (b) The real part of the nonlinear coefficient γ
re
and the effective
mode area A
eff
versus the wavelength. ................................................................ 17
Figure 2.8 Supercontinuum generation in the double-slot waveguide with all-
normal-dispersion at different distances. The insets show the corresponding
temporal waveforms. ........................................................................................... 18
vii
Figure 3.1 (a) A schematic of a microring resonator based on a silicon nitride slot
waveguide coupled to a straight waveguide as well as the corresponding
waveguide cross section. W, Hl, H
u
, and H
s
are the waveguide width, lower
Si
3
N
4
height, upper Si
3
N
4
height, and silica slot height, respectively. (b)
Values of the second-order dispersion, β
2
,
found for the fundamental TM
mode (vertically polarized). The dispersion curve of strip waveguide with
the same height as Slot waveguide 1 is also shown. β
2
is calculated by the
second derivative of the effective index, n
eff
, with respect to the wavelength
(i.e. β
2
=(λ
3
/2πc
2
)·(d
2
n
eff
/dλ
2
)) when the material dispersion is taken into
account according to the corresponding Sellmeier equations, where λ is the
wavelength and c is the speed of light in a vacuum. ........................................... 23
Figure 3.2 (a) The nonlinear conversion efficiency, η, versus the input power P
in
with different dispersion coefficients, β
2
. The solid lines are the analytical
solutions based on Eq. (3), and the dots are the results of numerical
simulations. An example of the intracavity comb spectra corresponding to
Slot waveguide 1 and 2 with P
in
=0.7 W is shown on the right. (b) η is
inversely proportional to the number of comb lines, N, within a 3-dB comb
bandwidth. N increases with input power. The dark yellow, blue, and red
dots stand for the results of the numerical simulations. ...................................... 24
Figure 3.3 (a) The product of the comb lines, N, within a 3-dB comb bandwidth
and the conversion efficiency, η, can be improved by an increase in the
power coupling coefficient, θ. (b) η and the intracavity pulse energy versus θ.
The solid lines come from the analytical expressions and the dots
correspond to numerical simulations (Slot waveguide 1, P
in
=1 W). ................... 26
Figure 3.4 (a) The nonlinear conversion efficiency is varied when the pump
frequency detuning is shifted towards the red side. (Slot waveguide 1,
P
in
=0.7 W) (b) The intracavity power of the comb lines located at 2f
0
/3 and
4f
0
/3. f
0
is the pump frequency. (Slot waveguide 1) ............................................ 28
Figure 3.5 The results of numerical simulations of the conversion efficiency in Slot
waveguide 1 when Raman scattering and self-steepening are considered
separately and together. The minimal considered pump power significantly
exceeds the Raman threshold. ............................................................................. 29
viii
Figure 4.1 (a) Schematic of Kerr comb generation in a microring resonator based
on a Si
3
N
4
waveguide and the waveguide cross section. W and H are the
waveguide width and height, respectively. (b) Dispersion curve β
2
for the
fundamental transverse magnetic mode (vertically polarized). β
2
is
calculated by the second derivative of the effective index, n
eff
, with respect
to the wavelength (i.e., β
2
=(λ
3
/2πc
2
)·(d
2
n
eff
/dλ
2
)) when the material
dispersion is considered, where λ is the wavelength and c is the speed of
light in vacuum. ................................................................................................... 33
Figure 4.2 (a) Comb spectrum in the stable soliton state (left) and spectral
dynamics and peak power of the stable soliton with the slow time (right). (b)
Narrowest (red color) and widest (blue color) frequency spectra when the
breather soliton is excited (left). The spectral dynamics and intracavity peak
power of the breather soliton are shown on the right. ......................................... 34
Figure 4.3 External-cavity spectra of the 89th comb line (f=192.22 THz) away
from the pump in the case of the stable soliton (a) and the breather soliton
(b), respectively. .................................................................................................. 36
Figure 4.4 Simulated communication system based on Kerr comb, which is
modulated with different modulation formats. .................................................... 36
Figure 4.5 (a) Simulated BERs as functions of the OSNR for the 89th comb line
(f=192.22 THz) when the stable soliton and the breather soliton are excited,
respectively. (b) Simulated BERs as functions of the OSNR for the 49th
(f=184.22 THz), 69th (f=188.22 THz), and 89th (f=192.22 THz) comb lines
in the breather soliton state. Also, the respective spectra of these comb lines
with multiple sub-teeth are shown in (c), (d), and (e). ........................................ 38
Figure 4.6 (a) Kerr comb in the breather soliton state with the sub-teeth filtered by
a narrow bandwidth filter (Gaussian) before being modulated. (b) EVM
versus the filter bandwidth for QPSK signals when the OSNR is 11 dB. The
insets show the corresponding constellation diagrams. The respective
spectra of the 89th comb line after being filtered when the filter bandwidth
is 15 (c) and 5 GHz (d). ....................................................................................... 39
Figure 4.7 Simulated BERs versus the filter bandwidth for different OSNR values
(OOK transmission system). ............................................................................... 40
Figure 5.1 (a) A microring resonator based on a Si
3
N
4
slot waveguide is shown
with the waveguide cross section. W, H
l
, H
u
, and H
s
are the waveguide
width, lower section height, upper section height, and silica slot height,
respectively. (b) Two different dispersion types in two microrings for the
fundamental transverse magnetic (TM) mode (vertically polarized). The
second-order GVD β
2
is calculated by the second derivative of the effective
index, n
eff
, with respect to the wavelength (i.e., β
2
=(λ
3
/2πc
2
)·(d
2
n
eff
/dλ
2
))
when the material dispersion is taken into account, and where λ is the
wavelength and c is the speed of light in vacuum. (c) High-order dispersion
values of two dispersion types when β
2
is the same at -32 ps
2
/km. The pump
wavelengths are 1.695 μm for Type-I and 1.6 μm for Type-II. (d) Dispersion
curves considering different high-order dispersion terms for Type-I are
shown by β
2
( ω) = ∑ β
m
( ω − ω
0
)
m − 2
/(m − 2)!
n
m = 2
, where ω
0
is the pump
angular frequency, n is the maximum order of dispersion. ................................. 42
Figure 5.2 The phase-matching curve Δδ and the intracavity comb spectra of Type-
I dispersion when high-order dispersion terms up to the fourth-order
dispersion β
4
are considered (a), up to β
5
are considered (b), up to β
6
are
considered, and when all-order dispersion terms are considered (d). The
input power P
in
is 2 W. All-order dispersion terms are calculated by
∑ β
m
( ω − ω
s
)
m
/m!
+ ∞
m = 2
= β( ω) − β( ω
s
) − β
1
( ω
s
) ∙ ( ω − ω
s
) in the
linear step of the split-step Fourier method. ........................................................ 46
Figure 5.3 The phase-matching curve Δδ and the corresponding intracavity comb
spectra of Type-II dispersion when high-order dispersion terms up to β
4
are
considered (a), up to β
5
are considered (b), up to β
6
are considered, and when
all-order dispersion terms are considered (d). The input power P
in
is 2 W. ........ 48
Figure 5.4 The maximum detuning, 3-dB comb bandwidth, and intracavity soliton
peak power versus the normalized input power X when high-order
dispersion terms up to the fourth-order dispersion β
4
are considered (a) and
up to β
5
are considered (b). The solid lines are the analytical solutions for
Type-I dispersion, and the dashed lines are for Type-II dispersion. The dots
stand for numerical results................................................................................... 49
Figure 5.5 The maximum detuning, 3-dB comb bandwidth, and intracavity soliton
peak power versus the normalized input power X when high-order
dispersion terms up to β
6
are considered (a) and when all-order dispersion
terms are considered (b). The solid lines are the analytical solutions for
Type-I dispersion, and the dashed lines are for Type-II dispersion. The dots
stand for numerical results................................................................................... 51
Figure 5.6 (a) The frequency of dispersive wave ω
d
varies when the normalized
input power X is increased. The solid and dashed lines are from the
analytical expression for the two types of dispersion. The dots are the
numerical results when all-order dispersion terms are considered. The
intracavity comb spectra in Type-I dispersion are shown in (b) and (c),
where the normalized input powers are 33.9 (0.7 W) and 77.4 (1.6 W),
respectively. ......................................................................................................... 53
Figure 5.7 (a) The intracavity soliton peak power and 3-dB comb bandwidth vary
as negative β
4
is further decreased. (b) Both the intracavity pulse energy and
nonlinear conversion efficiency increases as negative β
4
is further decreased.
The phase-matching curves Δδ in (c) and (d) correspond to the intracavity
comb spectra shown in (e) and (f), where β
4
values are -1.28×10
-3
ps
4
/km
and -1.15×10
-1
ps
4
/km, respectively. The normalized input power X is 45.7
(1 W) and the normalized detuning Δ is 45. High-order dispersion terms up
to the fourth-order dispersion β
4
are considered in the simulations. .................... 54
Figure 6.1 (a) Concept diagram of optical multicasting based on Kerr frequency
combs. P1 and P2 represent two pumps. Pump 1 carries phase-modulated
signals and is thus spectrally broadened. SFG, sum frequency generation;
DFG, difference frequency generation; QPM, quasi-phase matching. (b)
Experimental setup for optical multicasting. ECDL, external-cavity diode
laser; TOF, tunable optical filter; PPLN, periodically poled lithium niobate;
EDFA, erbium-doped fiber amplifier; PC, polarization controller. ..................... 57
Figure 6.2 (a) An optical microscope image of an integrated Si
3
N
4
microresonator.
(b) Transmission spectrum around 1552 nm showing a Q factor of 4×10
5
. ........ 58
Figure 6.3 (a) Optical spectrum of Kerr frequency combs in a low-phase-noise
state. (b) RF spectrum of a low-phase-noise state (RBW=10 kHz). ................... 59
Figure 6.4 (a) Optical spectrum at the output of the PPLN waveguide showing
seven multicast copies of 20-Gbaud QPSK signals. Seven Kerr comb lines
with low phase noise are selected and input into the PPLN waveguide. (b)
EVMs versus various multicast channels. The insets show the constellation
diagrams for various channels. (c) BER measurements of seven multicast
channels as functions of the OSNR. .................................................................... 60
Figure 6.5 (a) Optical spectrum for four-fold multicasting of 16-QAM signals. The
input Kerr combs are in a low-phase-noise state. (b) The constellation
diagrams of four multicast copies. ....................................................................... 61
Figure 6.6 (a) Kerr comb spectrum in a high-phase-noise state (chaotic combs). (b)
RF spectrum of a high-phase-noise state (RBW=10 KHz). ................................ 62
Figure 6.7 (a) Optical spectrum at the output of the PPLN waveguide for two-fold
multicasting of 20-Gbaud QPSK signals when two chaotic comb lines are
input. The signal quality of the multicast copy is significantly deteriorated.
The constellation diagram for Channel 1, which indicates high-intensity
noise, is shown in (b). The constellation diagram for the corresponding
chaotic comb is shown in (c) when it is modulated with 20-Gbaud QPSK
signals and directly demodulated. ....................................................................... 63
Figure 7.1 (a) Primary comb generation using dual pumps. (b) Experimental setup
for dual-pump comb generation. P1 and P2 represent the two pumps. TOF,
tunable optical filter; LCoS filter, liquid crystal on silicon programmable
filter; EDFA, erbium-doped fiber amplifier; PC, polarization controller; I/Q,
in-phase/quadrature; QPSK, quadrature phase-shift-keyed. ................................ 66
xi
Figure 7.2 Experimental generation of primary combs by (a) dual pumps and (b) a
single pump. (The total pump power on the chip is P
in
=800 mW). ..................... 68
Figure 7.3 (a) Simulated group velocity dispersion (β
2
) for the fundamental
transverse magnetic mode. β
2
is calculated by the second derivative of the
effective index, n
eff
, with respect to wavelength λ [i.e.,
β
2
=(λ
3
/2πc
2
)·(d
2
n
eff
/dλ
2
)], where c is the speed of light in a vacuum. (b)
Simulated primary comb spectrum based on the LLE (The total pump power
on the chip is P
in
=800 mW). ................................................................................ 68
Figure 7.4 The Kerr comb spectra in Regions A and B of Figure 7.3(b) are shown
in (a) and (b). The inset in (a) shows the RF spectrum of the beat note of the
combs in Region A (resolution bandwidth RBW=30 kHz). ................................ 70
Figure 7.5 The output power of comb line 1 remains practically constant when the
phase of P2 is varied with respect to that of P1. .................................................. 70
Figure 7.6 Experimental generation of primary combs when the total pump power
is decreased to (a) 300 mW and (c) 75 mW. Simulated primary comb
spectra are shown in (b) and (d). The inset of (c) shows a zoom-in view of
the spectrum around the two pumps. ................................................................... 71
Figure 7.7 (a) Measured linewidths versus different wavelengths in two regions.
(b) Linewidth measurements by self-heterodyne detection for two different
wavelengths. ........................................................................................................ 72
Figure 7.8 (a) EVM and output power versus different comb lines. Each comb line
is modulated with 20 Gbaud QPSK signals and demodulated by the coherent
receiver. (b) Example constellation diagrams of four comb lines. ...................... 73
Figure 8.1 (a) Conceptual diagram of an electro-optical (EO) comb insertion into a
Kerr comb using EO modulation. Each EO sideband is generated from an
adjacent Kerr comb line. Both the Kerr and EO combs are then modulated
with data. (b) Experimental setup for inserting the EO comb lines into a
Kerr frequency comb and for the data modulation. The Kerr comb is
generated from a silicon nitride microresonator. ECDL, external-cavity
diode laser; AFG, arbitrary function generator; EDFA, erbium-doped fiber
amplifier; PC, polarization controller; FBG, fiber Bragg grating; TOF,
tunable optical filter; IM, intensity modulator; PM, phase modulator; PS, RF
phase shifter; ESA, electronic spectrum analyzer; I/Q, in-phase/quadrature;
QPSK, quadrature phase-shift-keyed. ................................................................. 76
Figure 8.2 (a) Optical spectrum of a Kerr frequency comb in a single-soliton state
(resolution bandwidth [RBW] = 0.1 nm). The inset shows the five selected
Kerr comb lines in the 1543-1551 nm range. (b) Repetition rate beat note of
the soliton comb in (a) (RBW = 1 kHz). ............................................................. 77
xii
Figure 8.3 Optical spectra of combined Kerr and EO combs after EO modulation
when the modulation frequency f
e
is (a) 1/7, (b) 1/8, and (c) 1/9 of the Kerr
comb spacing (RBW = 0.1 nm). The corresponding RF beat notes are shown
on the right (RBW = 1 kHz). ............................................................................... 78
Figure 8.4 (a) Experimental setup for the linewidth measurement based on the
delayed self-heterodyne method. AOM, acoustic optical modulator; DPO,
digital phosphor oscilloscope. (b) The measured linewidths versus different
wavelengths. The linewidth measurements for the comb line of 1547.24 nm,
with and without EO modulation, are shown in (c). ............................................ 79
Figure 8.5 (a) Optical spectra of the channels in the coherent receiver when
multiple comb lines in the gray-dashed boxes of Figure 8.3 are modulated
with 10 Gbaud QPSK data (RBW = 0.1 nm). The spectrum of the channel
when the Kerr comb line at 1548.76 nm is directly modulated with data is
also shown. (b) Measured EVMs versus different comb lines for different
modulation frequencies of the modulator. As a comparison, the EVM of the
Kerr comb line directly modulated with the data is also shown. ......................... 81
Figure 9.1 Schematic of TE-polarized comb generation from a TM-polarized
soliton comb. A TE-polarized CW with weak power as a "seed" is cross-
phase modulated by a TM-polarized soliton pulse, which results in the
generation of a TE-polarized comb. .................................................................... 84
Figure 9.2 (a) Experimental setup of a TE-polarized comb generation when a TE-
polarized CW signal with low power is cross-phase modulated by a TM-
polarized soliton in the microresonator. The inset shows the intensity
autocorrelation of the single soliton. A TM-polarized soliton pulse is
generated first by a high-power TM-polarized pump, and a TE-polarized
CW signal is then coupled into the cavity as a "seed" to excite the TE-
polarized comb. AFG, arbitrary function generator; ECDL, external-cavity
diode laser; PC, polarization controller; EDFA, erbium-doped fiber
amplifier; FBG, fiber Bragg grating; OSA, optical spectrum analyzer. (b)
Optical microscope image of an Si
3
N
4
microresonator. (c) Measured
transmission spectra of the fundamental TE and TM modes when a
spontaneous emission source is coupled into the microresonator. (d) Optical
spectrum of a single soliton generated by a TM-polarized pump (resolution:
0.1 nm). (e) Repetition rate beat note of the single-soliton comb (resolution
bandwidth [RBW] = 10 kHz). (f) Optical spectrum of the TE- and TM-
polarized combs (resolution: 0.16 pm). The line at 1567.4 nm in the green-
dashed box is generated by the FWM between the TM-polarized pump and
the TE-polarized signal on the bus waveguide. The TE-polarized comb
spectrum is distinguished manually based on the transmission spectra in (c).
(g) Zoom-in spectrum of (f) in a spectral range of 1540 nm to 1550 nm
(resolution: 0.16 pm). (h) Heterodyne beat note of the two TE- and TM-
polarized comb lines in (g) (RBW = 100 kHz). .................................................. 87
xiii
Figure 9.3 (a) Power of the generated first sideband right next to the TE signal
versus the signal power. (b) Sample spectrum of the TE- and TM-polarized
combs when the signal power is only -7 dBm. When the signal power is -0.4
dBm and the signal wavelength is moved to 1549.4 nm and 1564.9 nm, the
optical spectra of the generated TE- and TM-polarized combs are shown in
(c) and (d), respectively (resolution: 0.16 pm). ................................................... 88
Figure 9.4 The optical spectra of two different multisoliton states are shown in (a)
and (d) when only the TM-polarized pump is coupled into the
microresonator. The intensity autocorrelations of two multisoliton states are
shown in (b) and (e). The corresponding spectra of the TE- and TM-
polarized combs are shown in (c) and (f). ........................................................... 89
Figure 9.5 Optical spectrum of two combs, where the TM-polarized comb is
generated by XPM between a TE-polarized soliton pulse and a TM-
polarized signal. The TE-polarized pump power is 740 mW and the power
of the TM-polarized signal is 8.5 mW. ................................................................ 90
Figure 9.6 The simulated spectral and temporal dynamics of the TM- and TE-
polarized combs considering the group-velocity mismatch are shown in (a)
and (b), respectively. The stable comb spectrum and the corresponding
stable temporal waveform are shown in the third and fourth columns,
respectively. ......................................................................................................... 92
Figure 9.7 The simulated spectral and temporal dynamics of the TM- and TE-
polarized combs without considering the group-velocity mismatch are
shown in (a) and (b), respectively. The temporal waveform of the stable TE-
polarized comb corresponds to a dark pulse when the group velocities
between two polarization modes are equal. ......................................................... 93
xiv
Abstract
A large number of precisely spaced spectral lines form an optical frequency
comb, which enables a wide range of applications including frequency metrology,
atomic clocks, precision spectroscopy, and optical communications. In recent years,
compact microresonator-based Kerr frequency combs have attracted much attention
because of their compactness and high repetition rate. Kerr frequency combs in a
low-noise state also have the advantages of high coherence and broadband coverage;
therefore, a Kerr comb source could potentially be used to replace a laser transmitter
in a wavelength-division multiplexing system. Under different pump conditions, the
Kerr comb has several dynamic regimes including different noise properties and
different frequency spectra which could affect its application in optical
communications. Thus, it would be interesting to study the fundamental physics of
Kerr frequency comb generation and further investigate its effects on the
communication system performance.
The first part of this dissertation is the study of Kerr frequency comb
generation. First, a double-slot waveguide with low and flattened dispersion is
designed, which could be potentially beneficial for a broadband comb generation.
Second, nonlinear conversion efficiency in Kerr comb generation has been
analytically and numerically investigated. Third, the effect of a breather soliton comb
on communication system performance has been further numerically studied. Then,
we numerically investigate the effect of high-order dispersion on Kerr comb
generation in microresonators characterized with anomalous dispersion. Last, we
theoretically and experimentally demonstrate that one polarized comb could be used
to create an orthogonally polarized comb via cross-phase modulation effect and these
two combs interact with each other in one microresonator.
The second part of this dissertation is about the experimental demonstration of
Kerr frequency comb in the application of optical communications. First, we
demonstrate optical wavelength multicasting using Kerr frequency comb lines.
xv
Second, two pumps are used to generate high-coherence primary Kerr combs, which
shows that the primary comb state could also be used for multichannel
communication. Last, the combination of Kerr and electro-optical combs has been
demonstrated in a coherent communication link, which indicates the possibility of
obtaining more data channels in the system.
xvi
Chapter 1 Introduction
This chapter will introduce the basic concept of microresonator-based Kerr
frequency comb, followed by the motivations of using on-chip Kerr frequency comb
for optical communications.
1.1 An optical frequency comb
An optical frequency comb consists of evenly spaced spectral lines, which
allow high-precision frequency metrology and are also used in a wide range of
applications including coherent communications, precision spectroscopy, the
photonic generation of spectrally pure radio frequency (RF) signals, and optical
signal processing [1]. The spacing of the comb modes in the frequency domain is f
r
,
which is the inverse of the pulse cycle in the time domain. Each comb line can be
given by f
n
= f
0
+ n f
r
, where n is a large integer index and f
0
is the global offset of the
frequency comb. Thus, measurement and frequency control of these two parameters
f
r
and f
0
provide a link between optical and RF domains, which also enables many
applications.
Figure 1.1 An optical frequency comb in the frequency domain (a) and in the time domain (b).
1
In recently years, there has been increasing interest in microresonator-based
Kerr frequency combs because of their compactness and high repetition rate [2]. An
on-chip Kerr frequency comb can be generated by cascaded four-wave mixing
(FWM) when a coherent continuous wave (CW) is coupled into a high-quality-factor
(Q-factor) microresonator (see Figure 1.2(a)) [3]. The high Q-factor factor of the
microresonator enhances the light intensity and the nonlinear process of parametric
FWM. Several operating regimes with different noise properties, including primary,
chaotic, breather, and stable soliton combs, have been demonstrated under different
pump conditions [4-12]. Of interest are Kerr soliton combs [13], which result from
the balance between the anomalous dispersion and Kerr nonlinearity in the
microresonator, and have the characteristics of low noise [13, 14] and wideband
coverage [11, 12, 15].
In general, the frequency sideband generation can be attributed to two
nonlinear processes [2]. First, the pump photons are converted to secondary
sidebands, which are often spaced multiple free spectral ranges of the cavity. This
degenerate FWM would also result in the generation of equidistant pairwise
sidebands. Second, the generated signal and idler sidebands can serve as seeds for
further parametric frequency conversion. Because these two pump photons have
different frequencies, this is also referred to nondegenerate FWM.
2
Figure 1.2 (a) A Kerr frequency comb is generated when a pump laser is coupled into a high Q-factor
microresonator. (b) The Kerr frequency comb is generated by a combination of degenerate FWM and
nondegenerate FWM.
1.2 Application of Microresonator-Based Frequency Comb in
Optical Communications
Kerr combs in a low-noise state including the soliton state, have the advantages
of high coherence and broadband coverage; therefore, they may present a possible
alternative to individual lasers and a potential source for coherent wavelength-
division multiplexing (WDM) transmission [16, 17]. Figure 1.3 shows a coherent
communication system using the Kerr comb and high-modulation formats. The comb
lines from the microresonator will be first separated by a demultiplexer. Each of
them is modulated by an I/Q modulator to carry signals of advanced modulated
formats. Then, all the channels will be multiplexed and sent to the optical fiber for
transmission. At the receiver, all data channels will be separated by a demultiplexer
and detected by a real-time coherent receiver to measure the error vector magnitude
(EVM) and bit error rate (BER).
3
Figure 1.3 The application of Kerr frequency comb in optical communication system using
wavelength- and polarization-division multiplexing and higher modulation formats. EDFA, erbium-
doped fiber amplifier; MUX, multiplexer; DEMUX, demultiplexer; PC, polarization controller.
Because the Kerr comb in different operating regimes has different noise
properties, it could be interesting to explore the connection between the physics of
the Kerr frequency comb and its applications in coherent communication systems
(see Figure 1.4). When the Kerr comb is used as the source in a communication
system, the physical process of the Kerr frequency comb will determine the noise of
the comb line, and therefore determine the quadrature amplitude modulation (QAM)
order of the modulated signal. If the comb line has narrow linewidth, low amplitude,
and low phase noise, the amplitude and phase level of the modulated signal can be
distinguished by the coherent receiver. Consequently, the bit error rate (BER) will be
low and the constellation of the modulated signal will be clean. On the contrary, if
the noise of the comb line is high, the amplitude and phase of the modulated signal
will not be differentiated, which would therefore result in a high BER and limit the
spectral efficiency and total capacity of the system. In addition to the noise
properties, the number of WDM channels is related to the comb bandwidth and
flatness, which is affected by the dispersion of the microresonator.
4
Figure 1.4 The relationship between the comb noise and the constellations and the BER of high-order
modulation QAM signals.
1.3 Thesis Outline
This dissertation is organized with the following structure: Chapter 2 presents
the design of a horizon double-slot silicon waveguide with low dispersion and broad
bandwidth. The similar structure could be applied in multiple microresonator
platforms which is beneficial for a broadband Kerr frequency comb generation.
Chapter 3 describes the nonlinear conversion efficiency in Kerr frequency comb
generation. Chapter 4 investigates how the Kerr comb in a breather soliton state
affects the communication system performance. Chapter 5 is the study of effect of
high-order dispersion in the micoresonator on the Kerr comb. Chapters 6 presents the
experimental demonstration of Kerr comb in the application of optical multicasting.
5
Chapter 7 shows the experimental results of dual-pump generation of high-coherence
primary combs. Chapter 8 presents the tunable insertion of multiple lines into a Kerr
frequency comb using electro-optical modulators and the use of combined Kerr and
electro-optical combs as light sources in coherent communications has been
experimentally demonstrated. Chapter 9 shows that one comb in one polarization is
generated by an orthogonally polarized soliton comb. Two low-noise, orthogonally
polarized combs interact with each other and exist simultaneously in a single
microresonator.
6
Chapter 2 Increased Bandwidth with Flattened
and Low Dispersion in a Horizontal Double-Slot
Silicon Waveguide
2.1 Introduction
Optical signals passing through on-chip devices and systems can be affected by
waveguide properties, such as optical nonlinearity [18] and dispersion [19]. The
chromatic dispersion is typically produced by a combination of the material
dispersion and the waveguide dispersion. In nonlinear processes, a flat and low
dispersion profile is desirable for phase matching conditions [20-22], conversion
efficiency enhancement [23], octave-spanning spectral broadening [24-28] and comb
generation [29-31].
One traditional method to tailor the dispersion is to change the waveguide
dimensions, i.e., the waveguide dispersion, which shows a parabolic profile with two
zero-dispersion wavelengths (ZDWs) and a narrow bandwidth [32]. Another
approach for tailoring the dispersion is to introduce a slot structure, which has been
shown to provide extra design freedom to tailor the dispersion [25, 28, 33-38].
Chalcogenide glass (As
2
S
3
) slot waveguides show a flat dispersion of 0±170
ps/(km·nm) over a bandwidth of 249 nm and a nonlinear coefficient as high as 160
/(m·w) [33]. Silicon strip/slot hybrid waveguides produce a flat dispersion profile
with three ZDWs from 1562 nm to 2115 nm [34]. Furthermore, a silicon slot
waveguide with four ZDWs from 1461 nm to 2074 nm has been proposed, enabling
octave-spanning supercontinuum generation [25]. A symmetrical vertical-slot silicon
waveguide with two identical slots has also been shown to enlarge low dispersion
bandwidth [28].
The effect of flattened chromatic dispersion over a broad band originates from
the power distribution change of the spatial mode in the slot waveguide at different
7
wavelengths. At shorter wavelengths, the power of the mode mainly stays in the high
index material area, while at longer wavelengths, more power moves into the slot
area [25]. This feature results in the extension of the range of low chromatic
dispersion in a longer wavelength direction.
In this chapter, we study the waveguide with horizontal double slots. Compared
with the vertical slot waveguide, the minimum achievable slot thickness in the
horizontal slot waveguide in the fabrication process is much less [39]. We find that
by introducing an additional horizontal slot layer, the power in the slot area
continues to grow as the wavelength becomes longer and therefore, low waveguide
dispersion is obtained at even longer wavelengths. We investigate two structural
options of horizontal double-slot silicon waveguides to achieve low dispersion over a
broad bandwidth. One option is to use two slots at the bottom of the waveguide,
which shows flattened dispersion from -26 ps/(km·nm) to 21 ps/(km·nm) over the
bandwidth of 802 nm. The second option is to use two slots placed at both sides to
achieve flattened dispersion from -17 ps/(km·nm) to 23 ps/(km·nm) over the
bandwidth of 878 nm. We analyze in details the power ratios of two separate slots in
the double-slot waveguide at different wavelengths and compare the results with the
single-slot waveguide. The comparison further shows that the mode transition from a
strip mode to two slot modes contributes to enlarging the low dispersion bandwidth.
In addition, we find that another way of increasing the bandwidth in the single-slot
waveguide is to increase the waveguide height, but the dispersion fluctuation would
increase simultaneously. Using a double-slot waveguide can overcome this issue and
achieve a larger bandwidth with the same dispersion flatness. Lastly, we show
simulation results on supercontinuum generation in an all-normal-dispersion double-
slot silicon waveguide. The full width at half maximum (FWHM) of the input pulse
is 50 fs and the output pulse has a 3-dB bandwidth of 188 nm after 4 mm of
propagation.
2.2 Waveguide Design and Modeling
8
Double-slot waveguides with different designs are shown in Figure 2.1. Hs
1
,
Hs
2
, Hl
1
, Hl
2
, Hl
3
and W are the lower slot height, upper slot height, bottom silicon
height, central silicon height, upper silicon height, and waveguide width,
respectively. The thickness of the silicon oxide substrate is 2 μm. The upper silicon
height Hl
3
is much larger than other layers in Option 1 while the central silicon
height Hl
2
is the largest layer in Option 2. In the calculation of the chromatic
dispersion, the material dispersion, including Si and SiO
2
, are taken into
consideration using Sellmeier equations. To calculate the dispersion of the quasi-TM
mode (Y-polarized), the effective index n
eff
with respect to the wavelength is
calculated by a full-vector mode solver (COMSOL). The second derivative of n
eff
with respect to the wavelength is equal to the dispersion value, i.e., D=-
(c/λ)·(d
2
n
eff
/dλ
2
), where c and λ are the speed of light and the wavelength in a
vacuum.
Figure 2.1 Two options of horizontal double-slot silicon waveguide (left and middle) and single-slot
silicon waveguide (right).
9
Figure 2.2 Mode distributions of the major electric field component (E
y
) at different wavelengths in
Option 1 (a), Option 2 of the double-slot waveguide (b), and the single-slot waveguide (c).
Figure 2.2 shows the mode distributions at different wavelengths. The mode
transition in single-slot waveguide has two steps in Figure 2.2(a), from a strip mode
to a slot mode as the wavelength increases [25]. An anti-crossing effect due to mode
coupling from a strip mode to a slot mode helps form the flattened dispersion profile.
Similarly, the mode transition process in a double-slot waveguide can be viewed to
have three steps. The mode is concentrated in the strip region at short wavelengths in
the first step, gradually transfers to one slot in the second step and further into the
second slot as the wavelength increases. For Option 1, the mode is mainly
concentrated in the upper silicon layer at short wavelengths (1000 nm) and gradually
transfers to Slot 2 with the wavelength increase (1600 nm) and to Slot 1 with the
further increase of wavelength (2100 nm). The trend of mode transition process is
similar to that in the single-slot waveguide. The difference is that the power extends
10
to Slot 2 at first and then to Slot 1. An additional step of power transfer to Slot 1 is
believed to enlarge the flattened and low dispersion bandwidth. Compared with
Option 1, in which the light is still tightly confined in Slot 2 at long wavelengths,
more power transfers to Slot 1 in Option 2.
For Option 1, the parameters are W=610 nm, Hl
1
=80 nm, Hl
2
=104 nm,
Hl
3
=366 nm, Hs
1
=35 nm, and Hs
2
=25 nm. For Option 2, W=600 nm, Hl
1
=105 nm,
Hl
2
=390 nm, Hl
3
=113 nm, Hs
1
=65 nm, and Hs
2
=42 nm. Figure 2.3 shows the power
ratio in the slot region with the wavelength in the single-slot waveguide and double-
slot waveguides. In the single-slot waveguide, the width W is 610 nm, the upper
silicon height and the lower silicon height are 136 nm and 344 nm, respectively, and
the slot thickness is 40 nm. As seen in Figure 2.3(c), the power ratio in the slot
increases as the wavelength increases in the beginning corresponding to the mode
transition from the strip mode to the slot mode. Then it decreases when the
wavelength is beyond 2.4 μm, which indicates that the mode expands to the cladding.
In double-slot design, the power distribution transits to Slot 2 at first and then to Slot
1, which agrees well with Figure 2.2. Compared with the single-slot waveguide, one
more step in mode transition is believed to enlarge the flattened dispersion
bandwidth. However, the power ratio in Slot 1 of Option 1 remains below 10% even
at long wavelengths when the ratio in Slot 2 decreases and the second slot mode
transition begins. Figure 2.3(a) shows that Slot 2 still holds the power and makes it
difficult to transfer. On the contrary, the power ratio in Slot 1 of Option 2 increases
greatly and even exceeds that in Slot 2 at the wavelength of 2.2 μm. In the mid-
infrared range, Slot 1 begins to work fully so as to complete the mode transition from
Slot 2 to itself.
11
(a)
(b)
(c)
Figure 2.3 Power ratio in the slots with the wavelength in Option 1 (a), Option 2 (b), and single-slot
waveguide (c).
Figure 2.4 shows the chromatic dispersion of TM mode in the single-slot and
double-slot waveguides. Option 1 shows the flattened dispersion from -26
ps/(km·nm) to 21 ps/(km·nm) over the bandwidth of 802 nm, spanning from 1464
nm to 2266 nm. Option 2 shows the flattened dispersion from -17 ps/(km·nm) to 23
12
ps/(km·nm) over the bandwidth of 878 nm from 1498 nm to 2376 nm. The single-
slot waveguide in [25] shows a bandwidth of 667 nm. The results show that the
double-slot waveguide has a much larger bandwidth, and it expands the ultralow
dispersion profile into the middle-infrared region.
Figure 2.4 The chromatic dispersion with four zero-dispersion wavelengths (ZDWs) in double-slot
design (Option 1 and Option 2) and a single-slot waveguide.
Increasing the waveguide dimension would enlarge the bandwidth between the
left-most and right-most ZDWs, but the dispersion fluctuation would increase as well.
Figure 2.5 shows the dispersion of a single-slot waveguide with the same height of
610 nm as a double-slot waveguide in Option 1. The width W is 610 nm, the upper
silicon height Hu and the lower silicon height Hl are 395 nm and 178 nm,
respectively. The slot thickness Hs is 37 nm. The single-slot waveguide with
increased height shows larger dispersion fluctuation from -77 ps/(km·nm) to 93
ps/(km·nm) over the bandwidth of 1118 nm. Thus, one more slot layer should be
introduced as another degree of freedom to better tailor the dispersion and make it
flatter. As shown in Figure 2.4, a double-slot waveguide shows a larger dispersion
bandwidth than an original single-slot waveguide with the same dispersion flatness.
13
Figure 2.5 The chromatic dispersion of Option 1 and single-slot waveguide with the same height.
2.3 Dispersion Properties and Supercontinuum Generation
As seen in Figure 2.6(a), when the lower silicon height Hl
1
increases from 95
nm to 115 nm in Option 2, the dispersion curve moves downward, but the long
wavelength part of the dispersion goes down relatively faster. When the central
silicon height Hl
2
increases from 382 nm to 398 nm, the dispersion moves up from a
normal dispersion to an anomalous dispersion regime and almost maintains its shape
and slope, which shows the dispersion value change of 11.4 ps/(km·nm) per nm.
Moreover, the dispersion slope varies and partially rotates when the short
wavelength part of the dispersion goes up and the long wavelength part goes down,
as Hs
1
increases from 57 nm to 73 nm in Figure 2.6(c). The influence of structural
parameters on the variations of the dispersion indicates a guide for waveguide
dimension design, and that the dispersion properties could be sensitive to fabrication
imperfection [40-42] in high-index-contrast waveguides. The fabrication of such a
double-slot waveguide may require the deposition of amorphous silicon layers in
which optical absorption is high at near-infrared wavelengths.
14
(a)
(b)
(c)
Figure 2.6 Dispersion profiles in Option 2 of double-slot design with the variation of lower silicon
height Hl
1
(a), central silicon height Hl
2
(b), and lower slot thickness Hs
1
(c).
One application of nonlinear waveguide with flat dispersion is the
supercontinuum in the anomalous dispersion regime [25, 28], which can be
explained by the soliton fission and dispersive wave generation. Meanwhile, the
supercontinuum generation with high coherence and flat-top characteristics has been
15
demonstrated in all-normal dispersion photonic crystal fibers. The corresponding
temporal pulse with smooth phase distribution can be recompressed further [24, 43].
It will be interesting to study the supercontinuum generation in the silicon waveguide
with the all-normal-dispersion profile. Flattened and low dispersion is obtained in
Option 2 when W=610 nm, Hl
1
=115 nm, Hl
2
=372 nm, Hl
3
=90 nm, Hs
1
=60 nm and
Hs
2
=40 nm, as shown in Figure 2.7(a). The dispersion varies from -24 ps/(km·nm) to
-13 ps/(km·nm) with the bandwidth of 612 nm, covering from 1550 nm to 2162 nm.
The real part of the nonlinear coefficient γ
re
as a function of wavelength is calculated,
based on a full-vector model [44]. The nonlinear index of silicon is n
2
=5.8×10
−18
m
2
/W and the two-photon absorption (TPA) coefficient is β
TPA
=5.9×10
−12
m/W. As
seen in Figure 2.7(b), the non-monotonic variation of the real part of γ
re
with the
wavelength has a peak of 104.6 /(m·w) at 1760 nm. The effective mode area A
eff
increases monotonously, explaining that the mode expands with the wavelength
increase.
(a)
16
(b)
Figure 2.7 (a) Silicon waveguide with all-normal dispersion for supercontinuum generation. (b) The
real part of the nonlinear coefficient γ
re
and the effective mode area A
eff
versus the wavelength.
We use a nonlinear envelope equation [25] to model the supercontinuum
generation process:
�
∂
∂ z
+
α
2
+ j ∑
( − j)
m
β
m
∂
m
m! ∂ t
m
∞
m = 2
� A = ∑
− jγ
n
n!
�
− j
2
�
n
�1 −
j
ω
0
∂
∂ t
� [A
∗
∂
n
∂ t
n
(A
2
)]
∞
n = 0
, (1)
where A(z, t) is the complex amplitude, and γ
n
is the nth-order dispersion coefficient
of nonlinearity with γ
n
= ω
0
∙ ∂
n
(γ/ω)/ ∂ω
n
. Up to 6th-order term is included
here. α is the propagation loss and ω
0
is the central angular frequency. Raman
scattering cannot occur for TM mode [20].
A 50 fs chirp-free hyperbolic secant pulse with a peak power of 15 W and a
central wavelength of 1.7 μm is launched into such a double-slot waveguide with the
all-normal dispersion profile. The nonlinear coefficient γ is (102+j13.8) /(m·W) at
the wavelength of 1.7 μm, and the linear propagation loss is assumed to be 7 dB/cm.
As seen in Figure 2.8, the spectrum broadening is mainly due to the self-phase
modulation (SPM) and no dispersive wave generation or soliton fission is observed.
After propagating the length of 4 mm, the spectrum broadens and covers from 1.45
μm to 2 μm at the -27 dB level. We can see that the supercontinuum generation in
all-normal-dispersion shows a flat-top profile and the 3-dB bandwidth is increased
from 39 nm to 188 nm. Different from the spectrum broadening in the anomalous
dispersion showed in [25] of which the pulse is significantly compressed, the
17
pulsewidth is increased monotonously here. This is due to the fact that in the SPM-
induced spectrum broadening process, the red-shift on the leading pulse edge travels
faster than the blue-shift on the tailing edge in the normal dispersion, which
contributes to the broadening of the pulsewidth from 50 fs to 88.7 fs. The
corresponding pulse energy is reduced to 0.34 pJ due to the TPA and linear
propagation loss.
Figure 2.8 Supercontinuum generation in the double-slot waveguide with all-normal-dispersion at
different distances. The insets show the corresponding temporal waveforms.
2.4 Conclusion
In this chapter, two options of horizontal double-slot silicon waveguide have been
designed to achieve a low and flattened dispersion, which shows a maximum bandwidth of
887 nm. The dependence of the dispersion profile on the dimensions of the waveguide is
also studied. Furthermore, by designing a waveguide with an all-normal flat-top dispersion
18
profile over a 612 nm bandwidth, the simulated supercontinuum from 1.45 μm to 2 μm is
generated by inputting a femtosecond pulse of 50 fs.
19
Chapter 3 Nonlinear Conversion Efficiency in
Kerr Frequency Comb Generation
3.1 Introduction
Fundamentally, mode-locked Kerr combs, which correspond to a single pulse
confined in the microresonator, can be described analytically with an approximate solution
of the Lugiato-Lefever equation (LLE) [45, 46]. A broad comb spectrum and short optical
pulse can be generated in a nonlinear resonator with low 2nd-order chromatic dispersion,
high input power, and a high Q-factor. The applicability of this analytical model is limited to
comparably long pulses and spectrally narrow combs, because it does not take into account
higher-order chromatic dispersion terms, self-steepening or Raman scattering. These effects,
together with frequency-dependent coupling and loss [30], can be studied through
numerically solving the LLE in order to investigate soliton dynamics, stability and the
coherence of frequency combs in nonlinear resonators [14, 47-53]. For instance, the
dynamic behaviors of cavity solitons, such as cavity soliton formation, evolution, and its
frequency shift, were analyzed. It was found that flattened 2nd-order dispersion is beneficial
to produce wideband comb spectra [30]. It was also shown that the third-order dispersion
helps to stabilize the frequency combs [51], and that the fourth-order dispersion impacts the
comb bandwidth [52].
Kerr frequency combs typically have low nonlinear conversion efficiency. For
instance, the maximum of the observed frequency comb envelope is almost 50 dB below the
CW pump [54]. There has been little reported on the efficiency analysis. It is of particular
interest to analyze and improve the conversion efficiency in octave-spanning Kerr comb
generation.
In this chapter, we analytically and numerically investigate the nonlinear conversion
efficiency of Kerr frequency combs and suggest ways to enhance it. Using a simplified
analytical approach based on the LLE [46], with only the 2nd-order group velocity
dispersion (GVD) considered, we derive an analytical formula for nonlinear efficiency and
20
then verify the analytical results through numerical simulations, taking into account high-
order dispersion, self-steepening and stimulated Raman scattering. As examples, we analyze
nonlinear ring resonators based on both conventional strip waveguide and recently proposed
dispersion-flattened waveguides [25, 30, 55, 56]. The dispersion-flattened cavity is favorable
to obtain broadband combs [30, 55, 56]. Moreover, due to the dispersion flattening, the 2nd-
order chromatic dispersion dominates over the higher-order dispersion terms in a region with
broad wavelengths, making the cavity an ideal “embodiment” of the classical LLE. A
comparison of our numerical and analytical results confirms this prediction. A reasonably
small deviation appears as the pump power increases. Interestingly, the observed deviation
always shows that realistic systems have worse efficiencies than the ideal ones.
3.2 Definition of Nonlinear Conversion Efficiency
In the stable soliton regime, corresponding to the fundamentally mode-locked
Kerr comb, the intracavity electric field amplitude [46] is
E
cs
= �
2α ∆
γ L
sech (
�
2α ∆
�β
2
� L
τ), (2)
which depends on the relative frequency detuning of the pump light and the mode of
the cold resonator. The normalized detuning
Δ
is defined as δ
0
/α; α = (α
i
+ θ)/2; L
is the cavity length; γ is the nonlinear coefficient; δ
0
is the phase detuning of the
pump frequency from the adjacent resonance frequency; β
2
is GVD; α
i
and θ
represent the power loss per round trip and power coupling coefficient, respectively;
and τ is the temporal coordinate of a single round trip. When maximum detuning is
achieved [46] ( ∆= π
2
P
in
γLθ/8α
3
), corresponding to the maximum peak power and
minimum pulsewidth, the nonlinear conversion efficiency, expressed as external
pulse energy divided by the input energy, becomes
η =
6. 3 FS R ⋅θ
1. 5
α
i
+θ
�
�β
2
�
P
in
γ
= ℱ ⋅ FSR ⋅ θ
1. 5
�
�β
2
�
P
in
γ
. (3)
FSR is the free spectral range of the cavity; ℱ = π/α is the cavity finesse.
21
The number of comb lines within the 3-dB comb bandwidth in the case of
maximum detuning is equal to
N =
BW
FS R
=
0. 5 6 1 3
(α
i
+θ) FS R
�
P
in
γθ
�β
2
�
=
0. 0 9ℱ
FS R
�
P
in
γθ
�β
2
�
. (4)
BW is the 3-dB comb bandwidth. The product of the nonlinear conversion efficiency
and the number of comb lines is
Nη = 3.53(
θ
α
i
+θ
)
2
= 0.09(ℱ ⋅ θ)
2
, (5)
which is only related to the power coupling coefficient, θ, and power loss per round
trip, α
i
. For an overcoupled resonator when θ ≫ α
i
, Nη~3.53, which is a constant. As
seen in the above equations, although comb bandwidth (pulsewidth) can be improved
(reduced) with an increase in input power, P
in,
or a decrease in the dispersion
coefficient, β
2
, the conversion efficiency decreases. In terms of the cavity soliton for
the critical coupling θ ≈ α
i
, the product of the number of comb lines and the
conversion efficiency is Nη≈1. Thus, the average comb power is ~P
in
/N and the
average power of a comb line is ~P
in
/N
2
. For P
in
=1 W and N=100, the average power
of a comb line is 100 μW.
3.3 Numerical Simulation Results
(a)
22
(b)
Figure 3.1 (a) A schematic of a microring resonator based on a silicon nitride slot waveguide coupled
to a straight waveguide as well as the corresponding waveguide cross section. W, Hl, H
u
, and H
s
are
the waveguide width, lower Si
3
N
4
height, upper Si
3
N
4
height, and silica slot height, respectively. (b)
Values of the second-order dispersion, β
2
,
found for the fundamental TM mode (vertically polarized).
The dispersion curve of strip waveguide with the same height as Slot waveguide 1 is also shown. β
2
is
calculated by the second derivative of the effective index, n
eff
, with respect to the wavelength (i.e.
β
2
=(λ
3
/2πc
2
)·(d
2
n
eff
/dλ
2
)) when the material dispersion is taken into account according to the
corresponding Sellmeier equations, where λ is the wavelength and c is the speed of light in a vacuum.
We numerically simulate the comb generation in a microring resonator based
on slot waveguide. The generated combs in such a cavity can be approximately
described with the analytical formula, due to the dispersion flattening over a broad
band. A flat β
2
curve of Slot waveguide 1 covering a broad frequency band due to the
anti-crossing effect between the strip and the slot modes [25] is shown in Figure
3.1(b). Two zero-dispersion wavelengths (ZDWs) are located at 1056 nm and 2296
nm. The slot waveguide is bent into a ring resonator, with the radius, R of 114 μm
corresponding to 200 GHz FSR. The dispersion can be increased by decreasing the
upper Si
3
N
4
height, H
u
.
23
(a)
(b)
Figure 3.2 (a) The nonlinear conversion efficiency, η, versus the input power P
in
with different
dispersion coefficients, β
2
. The solid lines are the analytical solutions based on Eq. (3), and the dots
are the results of numerical simulations. An example of the intracavity comb spectra corresponding to
Slot waveguide 1 and 2 with P
in
=0.7 W is shown on the right. (b) η is inversely proportional to the
number of comb lines, N, within a 3-dB comb bandwidth. N increases with input power. The dark
yellow, blue, and red dots stand for the results of the numerical simulations.
We use the LLE [46, 57, 58] to simulate the octave-spanning comb generation.
�τ
0
∂
∂ t
+
α
i
2
+
θ
2
− jδ
0
+ jL ∑
( − j)
m
β
m
m!
∞
m = 2
∂
m
∂τ
m
� E(t, τ) = √θE
in
+ L[K(E) + R(E) (6)
K(E) = −jγ �1 − jτ
s ho c k_ K
∂
∂τ
� E|E|
2
, and
R(E) = −jγ
R
�1 − jτ
sho ck_ R
∂
∂τ
� [E � h
R
(τ − τ
′
)
τ
−∞
|E|
2
dτ
′
]
24
are the nonlinear Kerr and Raman terms respectively; γ
R
is the Raman gain
coefficient and h
R
is the Raman response function. The second part of K(E) indicates
the self-steepening term. The full width at half maximum (FWHM) of the Si
3
N
4
Raman-gain spectrum is 1.72 THz [59]. τ
0
=1/FSR is the roundtrip time; t and τ are
the slow and fast times, respectively; E(t, τ) is intracavity field; and E
in
is the input
field. The pump power is defined as P
in
= |E
in
|
2
; L is the cavity length; γ is the
nonlinear coefficient; and β
m
is the mth-order dispersion coefficient. To derive this
equation, the complex amplitude of intracavity electric field is expressed as A(z,
τ)exp[j(βz-ωτ)] instead of A(z, τ)exp[j(ωτ-βz)], where A(z, τ) is the amplitude and β
is the propagation constant.
The pump wavelengths are 1.695 μm and 1.68 μm for the two selected slot
waveguides and 2.07 μm for the strip waveguide, which are around the center of two
ZDWs. We find that the dispersion coefficients corresponding to the pump
wavelengths are -32 ps
2
/km, -87 ps
2
/km, and -597 ps
2
/km, respectively. The loss (0.2
dB/cm) and the coupling coefficient ( θ = 1.1α
i
corresponding to a slightly
overcoupled resonator) are assumed to be wavelength-independent. The
corresponding Q factors are 8×10
5
for slot waveguides and 6.7×10
5
for the strip
waveguide. The nonlinear coefficients γ are 0.78 /(W·m), 0.83 /(W·m), and 0.54
/(W·m), respectively.
To solve Eq. 6 numerically, we use the split-step Fourier method. The temporal
step is selected to be 1 fs. The pump is initially resonant with a cold cavity mode so
that the four-wave mixing process can be observed. Then, we begin decreasing the
pump frequency with a normalized detuning step,
Δ=6,
every 30 ns and tracking the
waveform of the generated pulses. Eventually, the fundamental soliton is generated.
We continue increasing the detuning until the duration of the soliton reaches its
minimum.
25
(a)
(b)
Figure 3.3 (a) The product of the comb lines, N, within a 3-dB comb bandwidth and the conversion
efficiency, η, can be improved by an increase in the power coupling coefficient, θ. (b) η and the
intracavity pulse energy versus θ. The solid lines come from the analytical expressions and the dots
correspond to numerical simulations (Slot waveguide 1, P
in
=1 W).
Initially, we do not take Raman and self-steepening into account. For Slot
waveguide 1 with relatively small GVD, an increase of input power from 0.54 W to
2.1 W causes the conversion efficiency to decrease from 1.05% to 0.4%, as
illustrated by Figure 3.2(a). The deviation from the analytical solution appears due to
high-order dispersion terms [46] when the input power exceeds 1.2 W. In case of a
higher dispersion waveguide (-87 ps
2
/km), similar behavior is observed. An increase
of input power from 0.7 W to 1.4 W causes the conversion efficiency to decrease
from 1.53% to 0.84%. In accordance with analytical theory, the conversion
efficiency of the lower dispersion resonator is lower than a higher dispersion
26
resonator, while the comb bandwidth is broader. Thus, the conversion efficiency is
highest in the unoptimized resonator. As shown in Figure 3.2(b), the simulation
results show agreement with the analytical expression for β
2
=-32 ps
2
/km when N is
smaller than 125 (P
in
=1.2 W). However, the efficiency is worse than the analytical
prediction for higher pump power. The results for high dispersions of Slot waveguide
2 and Strip waveguide show no deviation from the analytics, even when the input
power reaches 2 W. This can be explained by the fact that the generated frequency
comb is narrower in a larger GVD resonator.
As follows from Eq. (5), an effective way to increase the value of Nη is to
increase the power coupling coefficient, θ, and decrease the power loss, α
i
. The
increase in the power coupling coefficient aids the coupling of more energy in and
out of the resonator. Our simulation confirms the conclusion. As shown in Figure
3.3(a), Nη increases from 0.88 to 2.8 when θ increases from 0.36% to 3.31%. Figure
3.3(b) shows that an increase in the coupling coefficient would improve the
conversion efficiency five times, from 0.74% to 4.3%, although the intracavity pulse
energy is decreased from 10.8 pJ to 6.4 pJ and the corresponding Q factor is reduced
from 8×10
5
to 1.5×10
5
. The 3-dB comb bandwidth is reduced from 23 THz to 13.8
THz.
(a)
27
(b)
Figure 3.4 (a) The nonlinear conversion efficiency is varied when the pump frequency detuning is
shifted towards the red side. (Slot waveguide 1, P
in
=0.7 W) (b) The intracavity power of the comb
lines located at 2f
0
/3 and 4f
0
/3. f
0
is the pump frequency. (Slot waveguide 1)
The multi-soliton state, corresponding to the confinement of multiple weakly
interacting pulses in the nonlinear resonator [13], is also found to increase the
conversion efficiency. Figure 3.4(a) shows how conversion efficiency depends on
the number of cavity solitons. Since the pulses are almost the same shape in the
single-soliton and multi-soliton state, the efficiency is proportional to the number of
pulses in the resonator. Within both states, the efficiency increases with the pump
frequency detuning.
An example of the dependence of the intracavity power of two selected comb
lines on the pump power is shown in Figure 3.4(b). The power of one of the lines
increases monotonously with the pump power, while the other gets saturated. The
dependences are different due to the high-order dispersion of the resonator. Although
the power in the harmonics increases with pump power, higher pump power leads to
lower conversion efficiency.
High power of the frequency comb harmonics is important in various
applications. For instance, to stabilize an octave-spanning frequency comb, one
needs to realize f-2f locking, in which the frequency of the comb harmonic is
doubled at 2f
0
/3 and is locked to the comb harmonic at 4f
0
/3, while f
0
represents the
pump frequency. It is important for the power of those harmonics to be as large as
28
possible to support the high signal-to-noise ratio required for sufficiently tight
locking. Since the power of the harmonics is rather small, a 2f-3f stabilization
configuration can be used instead of the conventional f-2f one for comb self-
referencing.
Finally, we study the impacts of Raman scattering and self-steepening on the
nonlinear conversion efficiency. For Slot waveguide 1, the Kerr shock time, τ
sho ck_ K
,
and Raman shock time, τ
sho ck_ R
, are 1.9 fs and 1.8 fs, respectively. The Raman gain
coefficient, γ
R
,
is 0.11 /(W·m), and the calculated threshold power for Raman
scattering is 17 mW.
Figure 3.5 The results of numerical simulations of the conversion efficiency in Slot waveguide 1
when Raman scattering and self-steepening are considered separately and together. The minimal
considered pump power significantly exceeds the Raman threshold.
We find that self-steepening does not play a significant role in the above
numerical examples, but Raman scattering does. As seen in Figure 3.5, the efficiency
is decreased due to intrapulse Raman amplification accompanied by a reduction in
pulse energy; thus, the value of Nη should also be reduced.
3.4 Discussion and Conclusion
Other factors that impact the generation of ultrabroad Kerr frequency combs
should be further investigated. For instance, the frequency-dependent Q factor
29
should be taken into account. The dependence of the conversion efficiency on the
Raman gain bandwidth should be studied. The interaction among resonator mode
families needs to be understood. All these topics can be further investigated.
In conclusion, we have discussed the nonlinear conversion efficiency in Kerr
frequency combs and the ways to improve it. The efficiency was found to be
inversely proportional to the comb bandwidth. The influences of Raman scattering
and self-steepening on the efficiency were also investigated.
30
Chapter 4 Effect of a Breather Soliton in Kerr
Frequency Combs on Optical Communication
Systems
4.1 Introduction
Kerr frequency combs based on microresonators can be generated via coupling of a
continuous-wave light into a high-quality (Q) factor resonator, which can be potentially
applied in frequency metrology, radio frequency signal generation, and coherent
communications [2, 16, 60, 61]. Kerr combs are capable of covering telecommunication
bands, including S, C, L, and U bands, making them potential sources for data transmission.
For example, 10 and 40 Gbit/s on-off-keyed (OOK) signals have been modulated on each
individual comb line [62-64]. Moreover, a total data capacity of 1.44 Tbit/s has been
demonstrated in coherent communications based on a low-phase noise comb state in which
20 comb lines are modulated with 18 Gbaud polarization multiplexed (pol-mux) quadrature
phase-shift-keyed (QPSK) signals [16]. In theory, one could use a comb line to replace a
laser transmitter in a wavelength-division multiplexing (WDM) system; however, there
might be some issues that arise due to the relationship between comb lines.
The stable temporal soliton in optical microresonators has been generated due to the
balance between Kerr nonlinearity and anomalous group velocity dispersion [9, 15, 65]. The
combs in such a stable state exhibit low phase and amplitude noise [14, 65]. Additionally,
they have the advantage of covering a broad band. Thus, these combs have been used in a 20
Tbit/s WDM transmission covering the full C and L bands [66]. However, a breather soliton,
which is an oscillating soliton with a temporal amplitude and spectrum varying periodically
in time [67], can also be excited in this nonlinear process [68, 69] and has already been
observed in a fiber resonator [70]. To the best of our knowledge, the effect of Kerr combs in
the breather soliton state on the performance of optical communication systems has not been
fully explored.
31
In this chapter, we show the effect of a breather soliton pulse in Kerr combs on
the performance of optical communication systems through numerical simulation.
The breather soliton is found to generate multiple "sub-teeth" (i.e., additional
frequency components) which are close to the main comb line [71]. Through a
comparison of combs in the stable soliton and breather soliton states in a 20-Gbaud
QPSK transmission, we find that the sub-teeth generated from the oscillating soliton
distort QPSK signals. However, we observe that OOK signals are more tolerant to
the sub-teeth and the signal quality might not be affected.
4.2 Waveguide Design and Kerr Comb Simulation
We use the Lugiato–Lefever equation [45, 46, 57, 58] to simulate the octave-spanning
comb generation.
� τ
0
∂
∂t
+
α
i
2
+
θ
2
− j δ
0
+ jL �
( −j)
m
β
m
m!
∞
m = 2
∂
m
∂ τ
m
� E(t, τ)
= √ θE
in
− j γLE(t, τ)|E(t, τ)|
2
. (7)
τ
0
=1/FSR is the roundtrip time, FSR is the free spectral range, t and τ are the slow and fast
times, respectively, E(t, τ) is the intracavity field , and E
in
is the input field. The pump power
is defined as P
in
= |E
in
|
2
, δ
0
is the phase detuning of the pump frequency from the adjacent
resonance frequency, L is the cavity length, and α
i
and θ represent the power loss per round-
trip and power coupling coefficient, respectively. All-order dispersion terms are considered
in the simulation.
32
Figure 4.1 (a) Schematic of Kerr comb generation in a microring resonator based on a Si
3
N
4
waveguide and the waveguide cross section. W and H are the waveguide width and height,
respectively. (b) Dispersion curve β
2
for the fundamental transverse magnetic mode (vertically
polarized). β
2
is calculated by the second derivative of the effective index, n
eff
, with respect to the
wavelength (i.e., β
2
=(λ
3
/2πc
2
)·(d
2
n
eff
/dλ
2
)) when the material dispersion is considered, where λ is the
wavelength and c is the speed of light in vacuum.
The dispersion profile is obtained on the basis of the silicon nitride (Si
3
N
4
) ring
resonator with a waveguide height H of 700 nm and a width W of 2000 nm (see Figure
4.1(a)). The FSR is 200 GHz, and the corresponding cavity length is 114 μm. The pump
wavelength is 1.72 μm (174.42 THz) , which is around the center of two zero-dispersion
wavelengths. The corresponding second-order dispersion β
2
is -112 ps
2
/km. The nonlinear
coefficient γ is 0.89 /( W·m). The propagation loss is assumed to be 0.2 dB/cm and the power
coupling coefficient θ=1.1 α
i
, which are both wavelength independent. The cavity Q factor is
calculated as 7.8×10
5
.
33
The breather soliton and the stable soliton can be excited in different regimes with
different pump power and pump detuning [70, 72]. Figure 4.2(a) (left) shows the spectrum
in the stable soliton state when the pump power P
in
is 1 W, and the normalized pump
detuning Δ (Δ is defined as 2δ
0
/(α
i
+ θ) [73]) is 68. The spectral dynamic and the
intracavity peak power with the slow time show stable characteristics. As a comparison, a
breather soliton is excited when the pump power is increased to 2.2 W, and the normalized
pump detuning Δ is 134. Figure 4.2(b) (left) shows the narrowest (red) and widest (blue)
spectra when the soliton breathes. The corresponding spectrum and peak power of the
soliton show slow temporal oscillation, as depicted in Figure 4.2(b) (right). The comb
spectrum breathes with the period of 345 ps and the 20-dB comb bandwidth changes from
66.1 THz to 68.7 THz. The breathing period of the comb spectrum can be varied by the
pump power and the Q factor [70, 72].
Figure 4.2 (a) Comb spectrum in the stable soliton state (left) and spectral dynamics and peak power
of the stable soliton with the slow time (right). (b) Narrowest (red color) and widest (blue color)
frequency spectra when the breather soliton is excited (left). The spectral dynamics and intracavity
peak power of the breather soliton are shown on the right.
34
The spectrum of the 89th comb line (f=192.22 THz) away from the pump in the stable
soliton can be calculated from a pulse train with 100,000 pulses with the use of Fourier
Transform, as shown in Figure 4.3(a). The narrow linewidth profile of the comb line shown
in Figure 4.3(a) results from the slightly different envelope of each roundtrip pulse in the
simulation. The narrow linewidth profile shows its potential as an optical source for coherent
communication, which can be modulated with high baud rates and different modulation
formats. However, the spectrum of the 89th comb line in the breather soliton has multiple
sub-teeth because of the temporal oscillation of the soliton, as shown in Figure 4.3(b). The
spacing between the first sub-tooth and the main comb line is only 2.9 GHz, which is the
reciprocal of the oscillation period of the soliton with 345 ps. The sub-teeth are so close to
the main comb line that they may degrade the performance of communication systems when
the comb is modulated with signals.
35
Figure 4.3 External-cavity spectra of the 89th comb line (f=192.22 THz) away from the pump in the
case of the stable soliton (a) and the breather soliton (b), respectively.
4.3 Communication System Simulation
Figure 4.4 shows the simulated communication system in which Kerr frequency
combs are sent into the tunable filter and then one selected comb is modulated with QPSK or
OOK signals. At the receiver, the signals are demodulated and the signal quality is measured
through error vector magnitude (EVM) and bit error rate (BER) [74]. A 20-Gbaud rate is
used in both modulation formats.
Figure 4.4 Simulated communication system based on Kerr comb, which is modulated with different
modulation formats.
Figure 4.5(a) shows the comparison between the effects of the stable soliton and the
breather soliton on the QPSK transmission system. As an example, the 89th comb line is
encoded with 20-Gbaud QPSK signals in the transmission. The simulated BER curves as
functions of the optical signal-to-noise ratio (OSNR) in the stable and breather soliton states
are both shown. Simulations indicate that the BER in the case of the breather soliton is much
worse than that of the stable soliton, and the OSNR penalty between two types of solitons is
increased when the BER is low. When the BER decreases from 4.4×10
-3
to 7.9×10
-4
, the
OSNR penalty increases from 1.8 dB to 2.2 dB. The insets show the constellation diagrams
in both states when the OSNR is 11 dB. The constellation of the breather soliton shows a
much higher distortion (34.6% EVM) than that of the stable soliton (28.6% EVM). This
36
result can be explained by the degradation caused by the generated sub-teeth close to the
main comb line from the oscillating comb spectrum.
We also compare the effect of different comb lines in the breather soliton state on the
transmission system. The 49th, 69th, and 89th lines are all encoded with 20-Gbaud QPSK
signals in the simulation. Figure 4.5(c), (d), and (e) depict the spectra of these comb lines.
The simulation shows that the power of the main comb line is reduced when it is further
away from the pump, and the power of the first sub-tooth remains almost the same. The
power difference between the main comb line and the first sub-tooth for the 49th comb line
is 20.3 dB, whereas the difference for the 89th comb line is 16.35 dB. Thus, the degradation
caused by the sub-teeth may become severe because of the reduction in power difference.
Figure 4.5(b), which shows the simulated BERs as functions of the OSNR, illustrates this
trend in which a small power difference results in poor system performance with a high BER.
When we compare the 49th and 89th lines, the BER increases from 8.5×10
-4
to 3.7×10
-3
when the OSNR is maintained at 10 dB. The constellation diagrams also show that the
distortion induced by the sub-teeth becomes high when the power difference is small as the
EVM increases from 31.7% to 37%. Also, high dispersion can improve the nonlinear
conversion efficiency and result in the high power of the comb line [9, 31, 52]. Therefore,
the OSNR per comb line in the stable soliton state can be increased, and the corresponding
signal quality can be improved. As a comparison, how the dispersion affects the sub-teeth in
the breather soliton state and the signal quality in the system can be further studied in other
works.
37
Figure 4.5 (a) Simulated BERs as functions of the OSNR for the 89th comb line (f=192.22 THz) when
the stable soliton and the breather soliton are excited, respectively. (b) Simulated BERs as functions
of the OSNR for the 49th (f=184.22 THz), 69th (f=188.22 THz), and 89th (f=192.22 THz) comb lines
in the breather soliton state. Also, the respective spectra of these comb lines with multiple sub-teeth
are shown in (c), (d), and (e).
Figure 4.6(a) shows that when a narrow bandwidth filter is added before the
transmitter, the power of the sub-teeth in the breather soliton can be reduced, and, thus, the
QPSK transmission performance is expected to be improved. Simulations show that when
the filter bandwidth is reduced from 15 GHz to 5 GHz, the power of the first sub-tooth (the
89th comb line) drops from -31.2 dBm to -34.66 dBm and the EVM of the QPSK signals
improves from 33.8% to 29.7% (see Figure 4.6(b)).
38
Figure 4.6 (a) Kerr comb in the breather soliton state with the sub-teeth filtered by a narrow
bandwidth filter (Gaussian) before being modulated. (b) EVM versus the filter bandwidth for QPSK
signals when the OSNR is 11 dB. The insets show the corresponding constellation diagrams. The
respective spectra of the 89th comb line after being filtered when the filter bandwidth is 15 (c) and 5
GHz (d).
We also simulate the effect of the sub-teeth on the performance of OOK signals. The
89th line is encoded with 20-Gbaud OOK signals in the simulation. Figure 4.7 shows that
the BER is slightly changed from 4.55×10
-6
to 4.1×10
-6
when the filter bandwidth is
decreased from 25 GHz to 5 GHz (the OSNR is 15 dB). The BER almost remains
unchanged as the filter bandwidth is considerably varied with the same OSNR at different
levels of 13, 15, and 17 dB. The observation that the sub-teeth generated from the oscillating
soliton do not affect OOK signal transmission can be explained as follows. The power of the
sub-teeth is not high enough to affect the pulse envelope of OOK signals. Furthermore, the
sub-teeth and the main comb line transmit at the same speed and the pulse envelope
maintains the same shape, because no dispersion is considered in the link; the dispersion can
be compensated by a dispersion compensating fiber. Compared to the QPSK performance in
39
Figure 4.5(a), in which the sub-teeth bring distortion to the QPSK signals with such power,
OOK signals are more tolerant to the generated sub-teeth.
Figure 4.7 Simulated BERs versus the filter bandwidth for different OSNR values (OOK transmission
system).
4.4 Discussion and Conclusion
In conclusion, we study how a breather soliton affects optical communication systems
by simulation. The sub-teeth are generated close to the main comb line when the soliton is
breathing and the comb spectrum is changing periodically. Simulations show that the sub-
teeth can introduce distortion to QPSK signal transmission but have less of an effect on
OOK signal transmission. In the future, the work considering a larger resonator with
different comb spacing, such as 50 GHz ITU channel spacing, can be investigated.
Simulations including the thermal effects [10] can more accurately model the breather
soliton generation.
40
Chapter 5 High-Order Dispersion in Kerr Comb
Oscillators
5.1 Introduction
Kerr frequency combs and associated short temporal pulses have been studied
theoretically [30, 46, 57, 68, 75] and demonstrated experimentally in different resonator
platforms [15, 65, 76-78]. The goal of this contribution is to elucidate several potentially
important effects that occur in microresonator-based Kerr frequency comb generators due to
the presence of high-order GVD. The existence of high-order dispersion terms affects the
pulse dynamics and the frequency spectrum [48, 51, 52, 79]. For instance, it has been shown
that third-order dispersion helps to stabilize the frequency combs [51, 79], and that fourth-
order dispersion affects the comb bandwidth [52]. The comb spectrum becomes asymmetric
and the soliton can emit a dispersive wave (Cherenkov radiation) when perturbed by the
high-order dispersion of a proper type [15, 80]. Such a dispersive wave has been
experimentally shown to broaden the comb spectral bandwidth to two-thirds of an octave or
even an octave. Such a broad frequency comb can be self-referenced and used in frequency
metrology [11, 12, 15]. The 3-dB comb bandwidth, which is related to the temporal pulse
width, is also affected by the dispersive wave [73].
In this chapter, we numerically study several effects associated with Kerr frequency
comb generation in resonators characterized with complex frequency-dependent chromatic
dispersion. Using an example of a slot-waveguide-based silicon nitride (Si
3
N
4
) microring,
we analyze the shapes of the frequency comb envelope and the associated optical pulses. By
adding each dispersion term one by one and simulating the corresponding comb spectrum,
we analyze the roles of different dispersion terms with different signs in generation of the
comb. While the positive fourth-order GVD term initiates and governs the generation of
dispersive waves in the resonator, the negative fourth-order GVD term shifts the dispersive
wave frequency away from the carrier and, eventually, suppresses its generation. We also
consider the power dependence of the dispersive wave spectral position.
41
5.2 Slot Waveguide Ring Microresonator
We study the impact of high-order GVD terms using a particular example of a
realistic slot Si
3
N
4
waveguide microring resonator. The slot waveguide structure
(Figure 5.1[a]), which has been shown to offer more freedom in the waveguide
design [25] than a simple waveguide, is used to tailor the chromatic dispersion of the
resonator to maximize the region with a nearly constant second-order GVD term
(Figure 5.1[b]).
Figure 5.1 (a) A microring resonator based on a Si
3
N
4
slot waveguide is shown with the waveguide
cross section. W, H
l
, H
u
, and H
s
are the waveguide width, lower section height, upper section height,
and silica slot height, respectively. (b) Two different dispersion types in two microrings for the
fundamental transverse magnetic (TM) mode (vertically polarized). The second-order GVD β
2
is
calculated by the second derivative of the effective index, n
eff
, with respect to the wavelength (i.e.,
β
2
=(λ
3
/2πc
2
)·(d
2
n
eff
/dλ
2
)) when the material dispersion is taken into account, and where λ is the
wavelength and c is the speed of light in vacuum. (c) High-order dispersion values of two dispersion
types when β
2
is the same at -32 ps
2
/km. The pump wavelengths are 1.695 μm for Type-I and 1.6 μm
for Type-II. (d) Dispersion curves considering different high-order dispersion terms for Type-I are
42
shown by β
2
( ω) = ∑ β
m
( ω − ω
0
)
m − 2
/(m − 2)!
n
m = 2
, where ω
0
is the pump angular frequency, n is
the maximum order of dispersion.
The slot waveguide with W=1300 nm, H
u
=440 nm, H
l
=920 nm, and H
s
=158
nm shows a flat β
2
curve (Type-I dispersion) covering a bandwidth of 1204 nm
between two zero-dispersion wavelengths (ZDWs). Another slot waveguide with
W=1300 nm, H
u
=300 nm, H
l
=790 nm, and H
s
=151 nm shows a β
2
curve (Type-II
dispersion) with a parabolic shape and the bandwidth between two ZDWs is 830 nm.
The waveguide is bent into a ring resonator with a 200 GHz free spectral range (FSR)
and the corresponding radius is 114 µm.
Looking at the Type-I dispersion shown in Figure 5.1(b), for instance, one can
see that the second-order GVD is anomalous and nearly frequency-independent in a
broad wavelength range. However, Figure 5.1(d) shows that even the dispersion
considering high-order dispersion terms up to the sixth-order β
6
still deviates from
the ideal dispersion in the wavelength range larger than 2 µm and smaller than 1.4
µm. This indicates that to accurately model a broad Kerr comb, we must consider all-
orders of dispersion. Next, to analyze the effect of different high-order dispersion
terms on the broadband comb generation, we add each dispersion term one by one
and numerically simulate the comb generation.
5.3 Numerical Model
We use the Lugiato-Lefever equation (LLE) [45, 46, 57, 58] to simulate the octave-
spanning comb generation,
�τ
∂
∂ t
+
α
i
2
+
θ
2
− iδ
0
+ iL ∑
( − i)
m
β
m
m!
∞
m = 2
∂
m
∂ T
m
� E(t, T) =
√ θE
in
− i γ
n l
LE(t, T)|E(t, T)|
2
. (8)
Here, τ=2π/ω
FSR
is the round-trip time and ω
FSR
is the free spectral range in radians
per second; t and T are the slow and fast times, respectively; E(t, T) is the intracavity
field; and E
in
is the input field. The pump power is defined as P
in
= |E
in
|
2
; δ
0
is the
normalized detuning of the pump frequency from the adjacent resonance frequency;
43
L=2πR is the cavity length for a cavity with radius R ; α
i
and θ represent the relative
power loss per round-trip time and power coupling coefficient, respectively such that
γτ=(α
i
+θ )/2 is the overall loss, γ being the half-width at half-maximum of the
pumped mode; γ
nl
is the nonlinearity coefficient, and β
m
is the mth-order dispersion
coefficient. The fast time coordinate can be replaced with the spatial coordinate,
which is periodic and accurately describes the physical process [45, 57].
For each resonator, the pump frequency is located at an extremum point where
the slope of the β
2
vs. wavelength curve is close to zero, so that the third-order
dispersion β
3
is close to zero. Both β
4
and β
5
in both dispersion types have opposite
signs (Figure 5.1[c]). The cavity loss is assumed to be 0.2 dB/cm and the power
coupling coefficient θ=1.1α
i
. The corresponding cavity Q factors in Type-I and
Type-II resonators are 8×10
5
and 8.5×10
5
, respectively. The nonlinear coefficients γ
nl
are 0.78/(W·m) and 0.93/(W·m), respectively.
The split-step Fourier method is used to solve Eq. 8 numerically. The temporal
resolution for the fast time is selected to be 1 fs. The step for the slow time is τ/20.
The pump is initially resonant with a cold cavity mode, so that the four-wave mixing
process can be observed. Then, the pump frequency is decreased with a normalized
detuning step, Δ = 6 (Δ is defined as 2δ
0
/(α
i
+ θ)) every 30 ns. Eventually, the
fundamental soliton is generated. We continue increasing the detuning until the
duration of the soliton reaches its minimum.
5.4 Dispersive Wave and Phase Matching Curve
Equation 8 can be approximated analytically for the case of pure anomalous second-
order GVD (β
2
<0). The analytical expressions for 3-dB comb bandwidth and the
corresponding pulse peak power [73] are, respectively,
BW = (0.315 1.763 ⁄ ) � 2 γ τ ∆ �β
2
�L ⁄ , and 𝑃 = 2 γ 𝜏 ∆/γ
n l
L. When the maximum
detuning is achieved ∆
m ax
= 𝜋 2
𝑋 /8 (X is defined as 8P
in
γ
n l
Lθ/(α
i
+ θ)
3
) [73], the 3-dB
comb bandwidth can be calculated as
44
BW
m ax
=
0. 3 1 5
1. 7 6 3
�
γ 𝜏 𝜋 2
𝑋 4 �β
2
� L
, (9)
and the corresponding intracavity peak power of the pulse is equal to
𝑃 m ax
= γ 𝜏 𝜋 2
𝑋 /4γ
n l
L . (10)
The presence of high-order dispersive terms complicates the solution and a
numerical analysis is needed to verify how well the simplified analytical model
approximates the parameters of a frequency comb generated in a realistic system.
The presence of high-order GVD results in modifications of the Kerr comb
spectrum. One of the possible modifications is the appearance of a “dispersive wave.”
The wave corresponds to a jump in the power and phase of the Kerr frequency comb.
It is possible to obtain an analytical formula describing the frequency of a dispersive
wave that can be generated in a microresonator due to the presence of high-order
GVD. The dispersive wave represents a convenient feature for visualization of the
impact of high-order dispersion. The spectral position of a dispersive wave ω
d
can be
calculated by the phase-matching condition [81]
∑ β
m
(ω
d
− ω
s
)
m
/m!
+∞
m = 2
− γ
n l
P
s
/2 = 0, (11)
where ω
s
is the soliton carrier frequency and P
s
is the soliton peak power. The
dispersive wave emission is determined by the combination of different orders of
dispersion terms. It can be seen from Eq. 11 that negative even dispersion terms (β
4
,
β
6
, β
8
, etc.) might not satisfy the equation. The presence of both positive and
negative dispersion terms is required to observe dispersive waves. We compare two
types of dispersion curves based on the realistic waveguide designs summarized in
Figure 5.1, one of which can show a negative β
4
, and study how Kerr combs are
affected by the dispersive wave. Also, positive and negative β
4
are referred to as
normal quartic GVD and anomalous quartic GVD, respectively.
In the following section, we utilize the phase-matching parameter defined as
∆δ = ∑ β
m
(ω
d
− ω
s
)
m
/m!
+∞
m = 2
− γP
s
/2, (12)
to characterize the generation of the frequency comb and confirm that this parameter
is more meaningful than the second-order GVD term alone.
45
5.4 Simulation Results for Various GVD Combinations
High-order GVD terms are usually truncated in numerical simulations. Our
goal is to show that this truncation frequently leads to modification of the frequency
comb predicted by the model. We use dispersive waves to illustrate the results.
In order to study the effect of high-order dispersion on the Kerr comb
performance, we show the phase-matching condition as well as the intracavity comb
spectra in the case of maximum pump detuning when different orders of high-order
dispersion terms are taken into account.
Figure 5.2 The phase-matching curve Δδ and the intracavity comb spectra of Type-I dispersion when
high-order dispersion terms up to the fourth-order dispersion β
4
are considered (a), up to β
5
are
considered (b), up to β
6
are considered, and when all-order dispersion terms are considered (d). The
46
input power P
in
is 2 W. All-order dispersion terms are calculated by ∑ β
m
( ω − ω
s
)
m
/m!
+ ∞
m = 2
=
β( ω) − β( ω
s
) − β
1
( ω
s
) ∙ ( ω − ω
s
) in the linear step of the split-step Fourier method.
In the case of Type-I dispersion, where β
4
is negative, Figure 5.2(a) shows that
there is no phase-matching point and no dispersive wave is generated when high-
order dispersion terms up to β
4
are considered. This indicates that negative β
4
could
potentially suppress the emission of the dispersive wave. Further, when β
5
is taken
into account, the dispersive wave is emitted on the blue side of the soliton's carrier
frequency; it is far away from the carrier frequency and its power is low. When the
positive term β
6
is included, the generated dispersive wave is closer to the carrier
frequency, and the power of the dispersive wave becomes higher. When all-order
dispersion terms are considered, however, the dispersive wave is generated on the
red side. This, therefore, could indicate that β
4
and higher dispersion terms are
essential for the simulation of spectrally broad frequency combs. However, the most
important point to note is that the frequency comb simulated for the case of all orders
of GVD taken into account (Figure 5.2d) is significantly different from any other
simplified case.
We also show the simulation results for Type-II dispersion with positive β
4
. When we
compare Figure 5.2(a) and Figure 5.3(a), positive β
4
leads to dispersive wave generation.
Figure 5.3(b) shows the dispersive wave emitted on the red side of the carrier frequency
when β
5
is added, which is different from that emitted on the blue side, as shown in Figure
5.2(b). This is due to the different signs of β
5
for the two dispersion types [82]. The addition
of β
6
even results in the generation of two dispersive waves with high power on both sides of
the soliton's carrier frequency. In the case of all-order dispersion terms, one dispersive wave
is shifted away from the pump and its power is much lower. Again, the frequency comb
observed when all orders of GVD are taken into account differs significantly from the
spectra obtained using the truncated GVD model. We also observe peaks with rather low
power around 380 THz in Figure 5.3(c). The one closer to the pump results from frequency
mixing between the pumped mode and the stronger dispersive wave peaks on either side of
the pump. The weaker peak (also observed in Figure 5.3(b)) may result from a numerical
47
artifact, which needs to be further explored in the future. Another, mathematically more
rigorous way of predicting the dispersive wave, has been given in [79] when high-order
dispersion terms up to β
3
is considered.
Figure 5.3 The phase-matching curve Δδ and the corresponding intracavity comb spectra of Type-II
dispersion when high-order dispersion terms up to β
4
are considered (a), up to β
5
are considered (b),
up to β
6
are considered, and when all-order dispersion terms are considered (d). The input power P
in
is
2 W.
5.5 Deviation from the Ideal Soliton Due to High-Order GVD
Analytical theory accurately predicts the parameters of Kerr frequency comb solitons
in the case of pure anomalous second order GVD. We study how the parameters of the
soliton are impacted by the high-order GVD observed in the Si
3
N
4
microrings (Figure 5.1).
48
Figures 5.4 and 5.5 show the maximum detuning, 3-dB comb bandwidth, and the intracavity
soliton peak power as a variation of normalized input power X when different high-order
dispersion terms are taken into account. No dispersive wave is emitted and no deviations
from the analytical expressions are found in the case of Type-I dispersion and high-order
dispersion up to β
4
, as shown in Figure 5.4(a). The generated dispersive wave is far away
from the soliton carrier frequency when β
5
is considered. No deviations are found between
the numerical and analytical results (see blue dots in Figure 5.4[b]). In the case of Type-II
dispersion, high-order dispersion terms up to β
4
make maximum detuning, 3-dB comb
bandwidth, and intracavity peak power deviate when the normalized input power X is 87
(1.5 W). The consideration of even higher-order dispersion β
5
contributes to the emitted
dispersive wave being located closer to the soliton carrier frequency with high power. Thus,
the deviation appears even when the normalized input power X is only 46 (0.8 W).
Figure 5.4 The maximum detuning, 3-dB comb bandwidth, and intracavity soliton peak power versus
the normalized input power X when high-order dispersion terms up to the fourth-order dispersion β
4
49
are considered (a) and up to β
5
are considered (b). The solid lines are the analytical solutions for Type-
I dispersion, and the dashed lines are for Type-II dispersion. The dots stand for numerical results.
For high-order dispersion terms up to β
6
, Figure 5.5(a) shows that the minimum
normalized input power X that results in the deviation from the analytical values for Type-I
dispersion is 72.5 (1.5 W) while it is 35 (0.6 W) for Type-II dispersion. The all-order
dispersion consideration in Type-I dispersion further decreases the minimum normalized
input power, which leads to the deviation from 72.5 to 59, as shown in Figure 5.5(b). The
impact of the dispersive wave on the comb performance could be explained as follows. It is
known that the input pump power is converted to both the soliton and the dispersive wave.
When the pump power is increased, the power of the emitted dispersive wave also grows
and its proportion to the soliton power is also increased, which makes the soliton's peak
power lower than the ideal predicted value. Thus, the maximal nonlinear phase shift of the
pump light that overlaps with the soliton per roundtrip decreases. As the maximum pump
detuning is proportional to the maximal nonlinear phase shift [65], the pump detuning would
be limited and would deviate from the analytical prediction. In some case, when the pump
power conversion becomes more complex, a more accurate explanation could be given.
50
Figure 5.5 The maximum detuning, 3-dB comb bandwidth, and intracavity soliton peak power versus
the normalized input power X when high-order dispersion terms up to β
6
are considered (a) and when
all-order dispersion terms are considered (b). The solid lines are the analytical solutions for Type-I
dispersion, and the dashed lines are for Type-II dispersion. The dots stand for numerical results.
5.6 Power Dependence of the Dispersive Wave Spectral Position
The repetition rate of an ideal Kerr frequency comb (with second-order GVD only)
does not depend on the pump power. However, this is no longer the case in presence of
high-order GVD, because the third and higher-order dispersion terms result in the transfer of
pump fluctuation to the repetition rate [83]. This effect could be understood based on
dispersive waves. The dispersive wave is one of the possible consequences of high-order
GVD and it is a coherent part of the frequency comb whose spectral position depends on the
pump power. The dispersive wave radiation is accompanied by the spectral shift of the peak
of the comb spectrum envelope (i.e., soliton recoil) and will hence affect the comb repetition
rate. Consequently, the pump fluctuation will be transferred to the comb repetition rate
51
through the dispersive wave. On one hand, the dispersive wave allows the comb repetition
rate to be tuned by changing the optical power. On the other hand, it destabilizes the
frequency comb if the pump is unstable. This property can be important for several practical
applications of Kerr frequency combs such as optical clocks.
Figure 5.6(a) shows that the frequency of dispersive wave ω
d
is shifted towards longer
wavelengths as the input power X is increased. We can see that the analytical solution from
Eq. 11 also shows this trend. For Type-I dispersion, when the normalized input power X is
increased from 33.9 to 77.4, the dispersive wave frequency shifts from 98.477 THz to
97.077 THz (Figure 5.6[b, c]).
Note that the power of the dispersive wave increases as the input power grows. The
power growth of the dispersive wave shifts the center of the comb spectrum away from the
pump by 5 THz in the simulation, as shown in Figure 5.6(c). This soliton recoil [84] can be
understood by looking at soliton momentum [85], which is defined as 𝑝 = ∫
i
2
𝐸 𝜕 𝑇 𝐸 ∗
𝜋 − 𝜋 +
c. c. , where “c. c.” stands for complex conjugate. Momentum plays the role of the center of
mass for the frequency comb spectrum ( 𝑝 ∝ ∑ 𝜔 𝑚 � 𝐸 �
( 𝜔 𝑚 ) �
2
+ 𝑚 − 𝑚 , where 𝐸 �
( 𝜔 𝑚 ) is the
comb spectral component at frequency ω
m
and m is an integer denoting mode number with
respect to the pumped mode). Momentum of a steady-state comb is zero when losses are
nonzero [86, 87] or the Raman effect is absent [69]. As a result, the appearance of a
dispersive wave on one side of the spectrum tends to shift its center (peak of the spectrum)
away from the pump and to the opposite side. This effect is less pronounced for a weaker
dispersive wave (see Figure 5.6(b)).
52
Figure 5.6 (a) The frequency of dispersive wave ω
d
varies when the normalized input power X is
increased. The solid and dashed lines are from the analytical expression for the two types of
dispersion. The dots are the numerical results when all-order dispersion terms are considered. The
intracavity comb spectra in Type-I dispersion are shown in (b) and (c), where the normalized input
powers are 33.9 (0.7 W) and 77.4 (1.6 W), respectively.
5.7 Fourth-Order GVD in a Slot Waveguide Ring Resonator
We consider realistic waveguide designs in which negative β
4
is dominant while β
2
and
β
3
are small. We consider a slot waveguide with W=1300 nm, H
u
=480 nm, H
l
=920 nm, and
H
s
=156 nm [30], of which the dispersion profile is Type-I and both β
2
and β
3
are small.
When the pump wavelength is 1.72 µm, the first three dispersion terms are β
2
= -6.8 ps
2
/km,
β
3
= -6.3×10
-3
ps
3
/km, β
4
= -6.4×10
-4
ps
4
/km. When the high-order dispersion terms up to β
4
are considered, no dispersive wave is emitted as expected due to the contribution of the
negative β
4
. Yet, as shown in Figure 5.7(a), when the absolute value of β
4
is further increased
from 1.28×10
-4
ps
4
/km to 1.15×10
-1
ps
4
/km, the intracavity soliton peak power decreases
slightly and the 3-dB comb bandwidth drops noticeably from 39.9 THz to 19.4 THz.
Meanwhile, the intracavity pulse energy grows and the corresponding conversion efficiency
[31] increases from 0.48% to 1.05%. This indicates that a negative β
4
of larger magnitude
can result in increased pulse energy and, consequently, conversion efficiency improvement.
53
Figure 5.7 (a) The intracavity soliton peak power and 3-dB comb bandwidth vary as negative β
4
is
further decreased. (b) Both the intracavity pulse energy and nonlinear conversion efficiency increases
as negative β
4
is further decreased. The phase-matching curves Δδ in (c) and (d) correspond to the
intracavity comb spectra shown in (e) and (f), where β
4
values are -1.28×10
-3
ps
4
/km and -1.15×10
-1
ps
4
/km, respectively. The normalized input power X is 45.7 (1 W) and the normalized detuning Δ is
45. High-order dispersion terms up to the fourth-order dispersion β
4
are considered in the simulations.
5.8 Conclusion
We have studied how high-order chromatic dispersion affects Kerr frequency
comb as well as dispersive wave generation by adding high-order dispersion terms
one by one and performing numerical simulations for each particular case. The
approach confirms that to optimize resonator dispersion for a broad Kerr comb
generation, one should account for all orders of GVD. Our results highlight the
potential influence of comb power on the spectral position of a generated dispersive
wave.
54
Chapter 6 Demonstration of Optical Multicasting
Using Kerr Frequency Comb Lines
6.1 Introduction
Primary combs with low noise are initiated by modulation instability of the
pump [75]. Chaotic combs are characteristically highly noisy due to multiple comb
lines oscillating within one resonance [54]. Stable soliton combs with low noise
result from the balance between Kerr nonlinearity and anomalous group velocity in
the resonant [9, 15, 65, 88-90]. Of interest are low-phase-noise Kerr combs, which
have been experimentally demonstrated in multiple microresonator platforms [9, 15,
65, 88-91]. They may present a possible alternative to individual lasers and potential
source for coherent wavelength-division multiplexing (WDM) transmission due to
the high coherence of the frequency comb. Specifically, a total data capacity of 1.44
Tbit/s using low-phase-noise combs has been demonstrated in coherent
communications [16]. A 20 Tbit/s WDM transmission covering the full C and L
bands has also been demonstrated based on soliton combs [90]. Optical wavelength
multicasting is a potentially important networking function, which copies the same
data onto many different output signals for efficient data distribution and parallel
processing [92-98]. It appears that microresonator-based Kerr combs could be also a
potential tool for wavelength multicasting. However, it remains unclear how Kerr
combs with different noise properties affect optical multicasting.
In this chapter, we describe the experimental investigation of the effect of Kerr combs
with different noise properties on the multicasting performance. First, based on the low-
phase-noise combs, we demonstrate seven-fold multicasting of a 20-Gbaud quadrature
phase-shift-keyed (QPSK) signal to 71.7-GHz spaced copies. Seven generated comb lines
are used as sources and input into a periodically poled lithium niobate (PPLN) waveguide. A
constellation diagram with error vector magnitude (EVM) of 16% is achieved when the
multicast copies are demodulated. The effect of high noise chaotic combs on the
55
multicasting performance is also studied. It is observed that although wavelength conversion
is taking place in the PPLN waveguide for the high-noise comb, the constellation diagrams
of the multicast copies are dominated by high-intensity noise.
6.2 Concept of Optical Multicasting
Figure 6.1(a) shows the concept of optical multicasting based on Kerr frequency
combs. To generate N signal copies, two pumps (ω
p1
and ω
p2
) are placed symmetrically
around the quasi-phase matching (QPM) wavelength of a PPLN waveguide with N Kerr
comb lines as the inputs. Pump 1 is encoded with phase-modulated signals with the
amplitude E
p1
(t). Two pumps are used to preserve the phase information of the modulated
signal. First, due to the sum frequency generation (SFG) in the χ
(2)
nonlinear process, a new
frequency component (ω
SFG =
ω
p1
+ ω
p2
) is produced, of which the amplitude E
SFG
(t) is
proportional to E
p1
(t)·E
p2
. Simultaneously, each frequency line of the Kerr comb, ω
comb
,
causes the generation of a converted signal (ω
signal =
ω
SFG
- ω
comb
) through difference
frequency generation (DFG). The field amplitude of the converted signal E
signal
(t) is
proportional to E
SFG
(t)·E*
comb
. The cascaded SFG and DFG are similar to non-degenerate
FWM and have been used to achieve quite a few advanced all-optical signal processing
functions [99]. Due to the efficient SFG+DFG in the PPLN waveguide [100], multiple
copies of the signal are simultaneously created, symmetrically located with respect to the
input combs.
56
Figure 6.1 (a) Concept diagram of optical multicasting based on Kerr frequency combs. P1 and P2
represent two pumps. Pump 1 carries phase-modulated signals and is thus spectrally broadened. SFG,
sum frequency generation; DFG, difference frequency generation; QPM, quasi-phase matching. (b)
Experimental setup for optical multicasting. ECDL, external-cavity diode laser; TOF, tunable optical
filter; PPLN, periodically poled lithium niobate; EDFA, erbium-doped fiber amplifier; PC,
polarization controller.
Figure 6.1(b) shows the experimental setup for optical multicasting. A
continuous-wave (CW) pump laser is amplified by a high-power erbium-doped fiber
amplifier (EDFA) to a value of 1.6 W and then coupled into a comb generating
microresonator. A tunable optical filter (TOF) is used to select various numbers of
comb lines, which are then amplified by EDFA 1 before entering the PPLN
waveguide. A 1547.5 nm laser (P1) is encoded with 20-Gbaud QPSK or 10-Gbaud
16-quadrature amplitude modulation (16-QAM) signals. Another laser, having a
wavelength of 1553.5 nm, is amplified by EDFA 2, and serves as Pump 2. A 13-nm-
wide bandpass filter (BPF) selects the phase-modulated multicast copies of interest,
and finally, a coherent receiver is used to demodulate the copied signals.
57
6.3 Low-Phase-Noise Comb Generation and Optical Multicasting
Performance
The microresonator used in our work is based on a silicon nitride (Si
3
N
4
)
waveguide (see Figure 6.2[a]). The coupling loss is ~3 dB per facet. The Si
3
N
4
waveguide has a height of 0.9 µm and a width of 1.5 µm. The measured loaded Q-
factor of the pump mode in the cavity is ~4×10
5
(see Figure 6.2[b]).
Figure 6.2 (a) An optical microscope image of an integrated Si
3
N
4
microresonator. (b) Transmission
spectrum around 1552 nm showing a Q factor of 4×10
5
.
When the microresonator pump is tuned to resonance, the primary combs are
generated due to the modulation instability. Then, the pump wavelength is tuned toward
longer wavelengths until a broadband comb is observed, which normally corresponds to a
chaotic comb [54]. Furthermore, a low-phase-noise comb state can be obtained by further
tuning the pump frequency in a short step (5 MHz in our experiment; see Figure 6.3(a)). A
high-speed photodetector is used to characterize the RF beat note of Kerr combs after the
pump is suppressed by a fiber Bragg grating (FBG). An RF spectrum (resolution bandwidth
(RBW) of 10 KHz) with a single line peak, as shown in Figure 6.3(b), demonstrates the low-
phase-noise characteristics of Kerr combs. The measured comb spacing is ~71.7 GHz. Note
that another low-phase-noise state, the soliton combs, can be obtained by sweeping the
pump wavelength or modulating the pump power using a two-step "power kicking"
technique [9, 15, 65].
58
Figure 6.3 (a) Optical spectrum of Kerr frequency combs in a low-phase-noise state. (b) RF spectrum
of a low-phase-noise state (RBW=10 kHz).
To demonstrate seven-fold optical multicasting, seven Kerr comb lines of the
generated low-phase-noise comb state are selected by the TOF and then sent into the
PPLN waveguide. Figure 6.4(a) shows the PPLN output spectrum for the seven-fold
multicasting of a 20-Gbaud QPSK signal. The multicast copies of 20-Gbaud QPSK
signals (from Channel 1 to Channel 7) are located at wavelengths from 1564.56 nm
to 1568.16 nm, respectively. The conversion efficiency of the PPLN waveguide in
our experiment, which is the power difference between the comb line and the
multicast copy, is -15.8 dB. Figure 6.4(b) shows the measured EVMs for each
multicast copy from Channel 1 to Channel 7. The figure shows that the EVM is 16%
for Channel 1 and 22% for Channel 4 and that all the multicast signals have error-
free performance. The EVM differences among these channels can be explained by
the optical signal-to-noise ratio (OSNR) difference for each channel at the receiver.
Bit error rate (BER) measurements are obtained for all seven multicast QPSK
channels, as shown in Figure 6.4(c). The BER performances for the seven channels
59
are similar. The back-to-back BER is also measured when Pump 1 bypasses the
PPLN waveguide and is directly demodulated by the receiver, and an OSNR penalty
of around 0.5 dB is found between the back-to-back and multicast copies.
Figure 6.4 (a) Optical spectrum at the output of the PPLN waveguide showing seven multicast copies
of 20-Gbaud QPSK signals. Seven Kerr comb lines with low phase noise are selected and input into
the PPLN waveguide. (b) EVMs versus various multicast channels. The insets show the constellation
diagrams for various channels. (c) BER measurements of seven multicast channels as functions of the
OSNR.
Next, we demonstrate the four-fold multicasting of 10-Gbaud 16-QAM signals
with four low-phase-noise comb lines (see Figure 6.5[a]). The constellation diagrams
60
in Figure 6.5(b) show that the EVMs (BERs) are 10.3% (4.5 × 10
-4
), 10.8% (7.4 ×
10
-4
), 10.9% (8.2 × 10
-4
), and 11.3% (1.2 × 10
-3
) for the four channels.
Figure 6.5 (a) Optical spectrum for four-fold multicasting of 16-QAM signals. The input Kerr combs
are in a low-phase-noise state. (b) The constellation diagrams of four multicast copies.
6.4 Chaotic Comb Generation and Multicasting Performance
It might be interesting to study how another Kerr comb state with different
noise properties affects optical multicasting performance. Chaotic combs with high
intensity and phase noise have been shown to result from multiple comb lines
oscillating within one resonance [5, 54]. Chaotic combs covering a broad band are
shown in Figure 6.6(a). The larger beat note in the RF spectrum relative to the pump
laser linewidth in Figure 6.6(b) indicates a high-phase-noise comb state.
61
Figure 6.6 (a) Kerr comb spectrum in a high-phase-noise state (chaotic combs). (b) RF spectrum of a
high-phase-noise state (RBW=10 KHz).
Two comb lines in a high-phase-noise comb state are filtered by the TOF and
sent to the PPLN waveguide. Pump 1 is also encoded with 20-Gbaud QPSK signals.
Figure 6.7(a) shows that there is still wavelength conversion because two multicast
copies are still generated at the longer wavelengths. The conversion efficiency of the
PPLN waveguide is comparable to that of the low-phase-noise combs. However,
when the multicast copy of Channel 1 is demodulated, we find that the signal quality
is rather poor. The constellation diagram in Figure 6.7(b) indicates that the signal is
deteriorated by phase noise and intensity noise and high-intensity noise is dominant
as compared to the constellation diagrams shown in Figure 6.4(b). The EVM (BER)
is as high as 43% (1.1 × 10
-2
). When the corresponding chaotic comb is modulated
with 20-Gbaud QPSK signals and directly demodulated without being sent into the
PPLN waveguide, we find that the constellation diagram of the modulated chaotic
comb is similar to that of the multicast copy (see Figures 6.7[b] and 6.7[c]). This
62
indicates that the noise properties of the chaotic comb is the same as that of the
multicast copy after the wavelength conversion.
Figure 6.7 (a) Optical spectrum at the output of the PPLN waveguide for two-fold multicasting of 20-
Gbaud QPSK signals when two chaotic comb lines are input. The signal quality of the multicast copy
is significantly deteriorated. The constellation diagram for Channel 1, which indicates high-intensity
noise, is shown in (b). The constellation diagram for the corresponding chaotic comb is shown in (c)
when it is modulated with 20-Gbaud QPSK signals and directly demodulated.
6.5 Conclusion
In conclusion, we experimentally demonstrate seven-fold multicasting of 20-
Gbaud QPSK signals using low-phase-noise Kerr frequency combs. The results show
that error-free performance can be achieved following demodulation. For
comparison, chaotic combs are also tried, resulting in wavelength conversion,
including the transfer of noise. However, error free data transmission could not be
achieved.
63
Chapter 7 Dual-Pump Generation of High-
Coherence Primary Kerr Combs with Multiple
Sub-Lines
7.1 Introduction
Kerr combs, which are capable of covering multiple telecommunication bands
including C and L bands, can potentially replace multiple individual lasers in a
wavelength-division multiplexing (WDM) system and enable high-quality data
transmission [16, 17]. By changing the pump conditions, several dynamical regimes,
including Turing patterns, chaos, breathing solitons, and stable solitons, have been
found both theoretically and experimentally [5, 7, 9, 50, 65, 68, 75, 88, 89, 101-104].
Turing patterns, which result from the modulation instability (MI) of a pump [75],
are also referred to as primary combs in the frequency domain. They are of interest
because of their potential high coherence and robustness to external perturbations [14,
50, 104]. As an example, such high coherence can enable a high capacity of 144
Gbit/s for a single primary comb line [104].
Nonetheless, a limited number of primary comb lines may restrict the number
of multiplexed channels in the WDM system, as have been demonstrated using low-
phase-noise combs [16, 17]. A goal, therefore, is to identify an effective method of
increasing the number of high-coherence frequency lines in the primary comb state.
This may be possibly achieved with the use of two pumps that enables
nondegenerate FWM [105-110]. This approach could also serve as another route to
multichannel communications.
In this chapter, we experimentally demonstrate the generation of high-
coherence primary combs with multiple sub-lines by using dual pumps in a silicon
nitride (Si
3
N
4
) microresonator. Compared with employing a single pump, which
generates only one comb line within the MI gain spectrum, using dual pumps enables
64
the generation of more than 10 highly coherent comb lines within the same localized
spectrum. The RF spectrum of the beat note from these Kerr combs and the linewidth
measurement demonstrate the high coherence of the combs. We also conduct a
coherent communication experiment, wherein 10 selected comb lines are encoded
with 20 Gbaud quadrature phase-shift-keyed (QPSK) signals.
7.2 Principle and Experimental Setup
Figure 7.1(a) illustrates the generation of the primary combs when the two
pumps are in resonance. The optical frequency comb is initiated by the MI in which
the gain spectrum overlaps with at least one resonance. Two MI gain spectra are
symmetrically located with respect to pump frequency. First, two main comb lines
are produced within the left/right MI gain spectrum because of energy transfer from
two pump modes. Then, more adjacent sub-lines are generated because of
nondegenerate FWM with the two main lines, which is a thresholdless process [106].
Some lines around the two pumps can also be generated given nondegenerate FWM
with the two pumps. Thus, the primary comb generation for which dual pumps are
used differs from single-pump generation, wherein normally only one line is
produced in the MI gain spectrum.
65
Figure 7.1 (a) Primary comb generation using dual pumps. (b) Experimental setup for dual-pump
comb generation. P1 and P2 represent the two pumps. TOF, tunable optical filter; LCoS filter, liquid
crystal on silicon programmable filter; EDFA, erbium-doped fiber amplifier; PC, polarization
controller; I/Q, in-phase/quadrature; QPSK, quadrature phase-shift-keyed.
Figure 7.1(b) shows the experimental setup. An integrated Si
3
N
4
microresonator (top
right, Figure 7.1[b]) comprises a 900 nm high and 1500 nm wide waveguide. The dual
pumps for Kerr comb generation are derived from a CW laser which is modulated by an
intensity modulator driven by a sinusoidal signal with a frequency of 35.976 GHz. The two
generated tones are spaced at 71.952 GHz and input into a liquid crystal on silicon (LCoS)
filter along with the central carrier. The LCoS filter is used to suppress the central carrier,
and the two other tones are amplified by a pre-amplifier to satisfy the minimum input power
requirement of a high-power erbium-doped fiber amplifier (EDFA). Two pumps are coupled
into the silicon chip to excite two resonances at 1551.66 and 1552.23 nm with loaded Q-
factors of 4×10
5
and 3.3×10
5
, respectively. With a combined power of 1.6 W and around 3
dB of coupling loss per facet, an on-chip power of 800 mW can excite primary combs.
High-bit-rate optical communication systems perform through the use of phase
encoding and coherent receivers. These systems also use narrow-linewidth coherent sources
66
in their transmitters. Therefore, a practical and informative approach to assess the coherence
of resultant primary combs is to study their adequacy as sources in a coherent optical
communication system. To this end, we select individual comb lines by using a tunable
optical filter (TOF), which is subsequently amplified and encoded with a 20 Gbaud QPSK
modulated data stream. Finally, the data is processed by a coherent receiver, which analyzes
its quality on the basis of constellation diagrams and corresponding error vector magnitudes
(EVMs).
7.3 Experimental Results
Figure 7.2(a) also shows the experimentally obtained optical spectrum of the primary
lines generated by the dual pumps. For comparison, a primary comb generated by a single
pump is also displayed with the same pump power input (Figure 7.2[b]). The first sideband,
which depends on the pump power and the resonator features [54], is located 44 free spectral
ranges (FSRs, FSR≈72 GHz) away from the single pump. The spectral locations of the
primary lines generated by the dual pumps are close to those of the single pump. Many sub-
lines are generated by the dual pumps.
67
Figure 7.2 Experimental generation of primary combs by (a) dual pumps and (b) a single pump. (The
total pump power on the chip is P
in
=800 mW).
Using the Lugiato–Lefever equation (LLE) [45], we numerically simulate the
expected primary comb spectrum. Figure 7.3(a) presents the simulated dispersion profile
based on waveguide dimensions. The material dispersion, including Si
3
N
4
and SiO
2
, are
taken into consideration using Sellmeier equations. The group velocity dispersion, β
2
, is -130
ps
2
/km. High-order dispersion terms are also taken into account. The calculated nonlinear
coefficient, γ, is 0.87/(W·m). For dual -pump excitation, we assume a total input field of the
form E
in
[1+exp(jω
f
t)], where ω
f
is the angular modulation frequency of the intensity
modulator, t denotes fast time in the LLE, and E
in
and E
in
·exp(jω
f
t) represent the input fields
for two pumps, respectively. The total pump power (P
in
=2| 𝐸 𝑖 𝑛 |
2
) is 800 mW, which
corresponds to the estimated power on the chip in the experiment. Figure 7.3(b) shows the
simulation of the primary comb spectrum with the generation of multiple sub-lines. The
difference between the simulation and experimental results may be attributed to several
factors. One may be that the two pumps are assumed to have the same cavity detuning in the
simulation. Another factor is that the simulated dispersion deviates from actual waveguide
dispersion when there is a sidewall inclination in the microresonator.
Figure 7.3 (a) Simulated group velocity dispersion (β
2
) for the fundamental transverse magnetic mode.
β
2
is calculated by the second derivative of the effective index, n
eff
, with respect to wavelength λ [i.e.,
β
2
=(λ
3
/2πc
2
)·(d
2
n
eff
/dλ
2
)], where c is the speed of light in a vacuum. (b) Simulated primary comb
spectrum based on the LLE (The total pump power on the chip is P
in
=800 mW).
68
Figure 7.4(a) provides an expanded view of the spectral section of the primary combs
generated by the dual pumps in the 1515 to 1535 nm range. More than 10 comb lines are
generated by the dual pumps, whereas only one line around 1525 nm is produced by the
single pump (Figure 7.2[b]). This result can be explained as follows. The dual pumps
produce two MI-induced spectral lines around 1525 nm, thereby generating, through
nondegenerate FWM interaction, a multitude of sub-lines (Figure 7.4[a]). The inset of Figure
7.4(a) shows the beat note of the comb lines in Region A, which is characterized by a high-
speed photodetector. The RF spectrum (resolution bandwidth (RBW)=30 kHz) with a
narrow single peak indicates that these combs are phase locked. Figure 7.4(b) shows the
spectral region from 1540 to 1560 nm, in which the comb lines are generated from
nondegenerate FWM between the two pumps. The comb lines generated at longer
wavelengths have higher power than those at shorter wavelengths. This could be ascribed to
the different cavity detunings of the two pumps in the experiment. The high noise level in
Figure 7.4(b) is caused by the pre-amplifier, which brings the amplified spontaneous
emission (ASE) noise into the high-power EDFA. The low-power line between the two
pumps is the central carrier that is inadequately suppressed by the LCoS filter.
69
Figure 7.4 The Kerr comb spectra in Regions A and B of Figure 7.3(b) are shown in (a) and (b). The
inset in (a) shows the RF spectrum of the beat note of the combs in Region A (resolution bandwidth
RBW=30 kHz).
Degenerate FWM is a phase-sensitive process when the signal and the idler have the
same wavelength [111]. Here, it might be interesting to study the effect of the phase
difference between the two closely placed pumps on the generation of primary combs.
When the phase of P2 is varied by the LCoS filter, both the spectral structure of the primary
comb lines and their power levels remain practically unchanged. Figure 7.5 indicates that the
power of comb line 1 remains almost constant under a varying P2 phase. This result is
attributed to the fact that the power conversion mediated by nondegenerate FWM is
independent of the phase difference between the two pumps [81].
Figure 7.5 The output power of comb line 1 remains practically constant when the phase of P2 is
varied with respect to that of P1.
We further study the dependence of the primary comb generation on pump power.
When the total power decreases from 800 to 300 mW, the primary comb spectrum in Figure
7.6(a) remains similar to that in Figure 7.3(b). However, the primary lines around the MI
gain spectrum becomes closer to the location of the dual pumps. This is because the reduced
pump power causes the peak of the MI gain spectrum to move towards the pumps. When
the total pump power is further reduced to 75 mW, the MI threshold is not reached, no
energy transfer takes place and no primary comb lines are generated away from the dual
70
pumps. Nonetheless, sub-lines that originate from the dual pumps are still generated via
nondegenerate FWM. These experimental results are verified by numerical simulations, as
shown in Figures 7.6(b) and (d). Note that because the decreased pump power is associated
with a change in comb spacing [91], a necessary requirement is to adjust the modulation
frequency of the modulator to ensure that the two pumps are still in resonance. In addition,
the slight spacing change between two pumps could vary the cavity detunings of two pumps
and the spacing of generated primary lines. Thus, the modulation frequency of the modulator
can be used to tune the comb spacing.
Figure 7.6 Experimental generation of primary combs when the total pump power is decreased to (a)
300 mW and (c) 75 mW. Simulated primary comb spectra are shown in (b) and (d). The inset of (c)
shows a zoom-in view of the spectrum around the two pumps.
To experimentally demonstrate the high coherence of the primary comb lines
generated by the dual pumps, we measure the linewidth of each line by the delayed self-
heterodyne interferometric method [112]. The linewidth measurements for different comb
lines in Regions A and B are shown in Figure 7.7(a). The linewidth of the generated primary
71
line is around 10 kHz, which is comparable to that of the two pumps. This indicates that the
performance of the generated primary combs depends on that of the two pumps.
Figure 7.7 (a) Measured linewidths versus different wavelengths in two regions. (b) Linewidth
measurements by self-heterodyne detection for two different wavelengths.
To reinforce the demonstration of high coherence, 10 of the generated primary comb
lines (labeled 1 to 10, Figures 7.4[a] and 7.4[b]) are each encoded with 20 Gbaud QPSK
signals and demodulated using the coherent receiver. Figure 7.8(a) shows the EVM of each
comb line, indicating the signal quality of each channel. Each line that carries signals can be
demodulated with a low bit error rate (BER) and thus demonstrated to have adequately high
coherence. Specifically, comb line 3 exhibits the lowest EVM (12.8%), which is attributed to
its high optical signal-to-noise ratio (OSNR). Comb line 10 has the highest EVM (27.3%)
with a corresponding BER of 8.3×10
-4
. It is also observed that although the output power of
comb line 7 is higher than that of comb line 3, the signal quality of the former is lower. This
is because the noise level of comb line 7 is much higher than that of comb line 3, which
decreases the OSNR and deteriorates the signal quality.
72
Figure 7.8 (a) EVM and output power versus different comb lines. Each comb line is modulated with
20 Gbaud QPSK signals and demodulated by the coherent receiver. (b) Example constellation
diagrams of four comb lines.
7.4 Conclusion
To conclude, to overcome the limited number of lines in the primary comb state, we
have experimentally generated a number of primary lines with high coherence by using dual
pumps. We have also demonstrated the application of the primary comb state in a WDM
system. Compared with a single pump, with which only one primary comb is generated
within the MI gain spectrum, dual pumps can produce more than 10 primary lines in the
same spectral region. The high coherence of the lines has been demonstrated through the
linewidth measurement and the modulation of each line with QPSK signals.
73
Chapter 8 Tunable Insertion of Multiple Lines
into a Kerr Frequency Comb Using Electro-
Optical Modulators
8.1 Introduction
Of interest are Kerr soliton combs [7, 9, 15, 60, 88], which result from the
balance between anomalous dispersion and Kerr nonlinearity, and have been
demonstrated to cover an octave [11, 12]. Recently, based on a soliton comb with a
spacing of 100 GHz, a 30.1 Tbit/s wavelength-division multiplexing transmission
experiment covering the full C and L bands was demonstrated [17].
However, Kerr comb spacing depends on the circumference of the
microresonator [3], and it is a challenge to change the comb spacing significantly
after device fabrication. The number of comb lines within the desired wavelength
bands is therefore limited when the comb line spacing is quite large (e.g., >200 GHz).
This limited number could limit the applications of Kerr comb, such as in coherent
communications and spectroscopy. One way to increase the number of comb lines is
to generate a second comb from another microresonator and to interleave the two
combs [17]. However, these two combs are not coherent to each other. Another way
is to redesign and fabricate a larger microresonator with a smaller comb spacing (e.g.,
25 GHz ITU spacing). Such a large microresonator would require a much higher
pump power because of the larger mode volume [113] and the conversion efficiency
from the pump to the comb is relatively lower [31]. Moreover, this approach is
neither cost effective nor tunable. Overcoming this challenge in a flexible and
tunable manner presents an interesting problem.
Electro-optical (EO) modulation can also be used to generate comb lines [114].
Such EO-generated combs have been previously inserted into Kerr combs to either
measure the repetition rate of the Kerr comb [15, 115] or to enable the referencing of
74
a Kerr comb spacing of >100 GHz to a microwave frequency standard [116].
Carrying this insertion idea one step further could result in an effective way to
increase the number of lines in the Kerr comb, which could then overcome the
above-mentioned challenge and be potentially advantageous for many applications.
In addition, further demonstrating the tunable insertion of multiple lines into a Kerr
frequency comb would be interesting.
In this chapter, we experimentally demonstrate the flexibility of inserting
multiple EO comb lines into a Kerr soliton comb via EO modulation. Different
numbers of lines can be inserted by changing the modulation frequency. When
measured, the linewidth of the newly generated EO comb is comparable to that of the
Kerr comb. As an example, we also show the advantage of EO comb insertion in the
application of coherent communications. The combination of Kerr and EO combs is
demonstrated in a coherent communication system when one Kerr and five EO lines
are encoded with 10 Gbaud quadrature phase-shift-keyed (QPSK) signals. Thus, the
number of data channels is increased, and the spectral efficiency of the
communication system could be potentially improved.
8.2 Concept of EO Comb Line Insertion
Figure 8.1(a) shows the conceptual diagram of the EO line insertion into a Kerr
frequency comb that is used for data transmission. When a Kerr comb is sent into an
optical modulator with a modulation frequency of f
e
, each Kerr comb line generates
several electro-optic sidebands spaced f
e
apart. Thus, the comb at the output of the
modulator comprises a combination of a Kerr comb and an EO comb. The comb
spacing in the newly formed comb is equal to f
e
, which is determined by the radio
frequency (RF) modulation frequency. Ideally, an evenly distributed channel
spectrum for optical communications can be formed when the Kerr comb spacing is
an integer multiple of the modulation frequency f
e
, and a flexible number of EO
comb lines can be inserted into the Kerr comb when the RF modulation frequency f
e
is changed.
75
Figure 8.1(b) shows the experimental setup for inserting EO lines into a Kerr
comb by using EO modulation. A continuous-wave light is amplified by a high-
power erbium-doped fiber amplifier and coupled into a silicon nitride microresonator
to generate a Kerr frequency comb. An arbitrary function generator controls the laser
wavelength, and a soliton comb can be generated when the wavelength is swept from
shorter to longer wavelengths, stopping at a step region of the resonance [15]. The
pump wavelength is 1555.9 nm, and the measured loaded Q-factor of the pump mode
is 1.3×10
6
. The waveguide in the microresonator has a height of 900 nm and a width
of 1500 nm. The coupling loss is around 3 dB per facet. The high pump power at the
output of the microresonator is suppressed by a fiber Bragg grating. A tunable
optical filter is used to select several Kerr comb lines, and these Kerr lines are then
sent into cascaded intensity and phase modulators, which are used to generate a flat
EO comb [114]. By changing the RF modulation frequency, a flexible number of EO
sidebands are generated around the Kerr comb lines. The newly formed comb splits
into two paths: one is used to measure the comb spacing, and the other is used to
send the selected comb lines into a transmitter. The comb line is modulated with 10
Gbaud QPSK signals and demodulated by a coherent receiver.
Figure 8.1 (a) Conceptual diagram of an electro-optical (EO) comb insertion into a Kerr comb using
EO modulation. Each EO sideband is generated from an adjacent Kerr comb line. Both the Kerr and
EO combs are then modulated with data. (b) Experimental setup for inserting the EO comb lines into
a Kerr frequency comb and for the data modulation. The Kerr comb is generated from a silicon nitride
76
microresonator. ECDL, external-cavity diode laser; AFG, arbitrary function generator; EDFA,
erbium-doped fiber amplifier; PC, polarization controller; FBG, fiber Bragg grating; TOF, tunable
optical filter; IM, intensity modulator; PM, phase modulator; PS, RF phase shifter; ESA, electronic
spectrum analyzer; I/Q, in-phase/quadrature; QPSK, quadrature phase-shift-keyed.
8.3 Experimental Results
Figure 8.2(a) shows the spectrum of a soliton comb that is generated in the
microresonator. The spectral envelope exhibits a sech
2
shape with a 3 dB bandwidth of 5
THz. The repetition rate of the soliton comb is measured using the EO down-conversion
technique [15, 115]. An RF signal of 36 GHz drives both EO modulators, which generate
EO sidebands. The difference between the two adjacent Kerr comb lines is reduced to
48.002 GHz based on the detection of the spacing between their second EO sidebands by a
high-speed photodetector (PD). Thus, the Kerr comb spacing is measured as 192.0020 GHz
(see Figure 8.2[b]). Note that the Kerr comb can be stabilized via phase-locking to a known
optical reference and a microwave synthesizer [15, 91]. The inset of Figure 8.2(a) shows the
five Kerr comb lines, which are of almost equal power and which are selected for later use in
EO comb insertion.
Figure 8.2 (a) Optical spectrum of a Kerr frequency comb in a single-soliton state (resolution
bandwidth [RBW] = 0.1 nm). The inset shows the five selected Kerr comb lines in the 1543-1551 nm
range. (b) Repetition rate beat note of the soliton comb in (a) (RBW = 1 kHz).
Figure 8.3 shows the comb spectra at the output of the cascaded modulators
(left) and the corresponding RF beat notes of the comb lines (right) when the RF
77
modulation frequency f
e
is changed to 1/7, 1/8, and 1/9 of the original Kerr comb
spacing. The RF phase shifter and the bias voltage of the intensity modulator are
optimized to flatten the newly generated EO lines. Figure 8.3(c) shows that when the
RF modulation frequency is set as 21.3336 GHz, eight EO peaks are generated
between two adjacent Kerr lines in the optical spectrum analyzer, indicating that the
newly generated comb lines could be used as carriers for data modulation. Note that
for high-order EO sidebands between two Kerr lines, two closely spaced EO
sidebands of relatively low power are generated from the left and right Kerr lines
[117]. Therefore, we only investigate the EO lines close to the Kerr lines and encode
them with high-order modulation format signals.
Figure 8.3 Optical spectra of combined Kerr and EO combs after EO modulation when the
modulation frequency f
e
is (a) 1/7, (b) 1/8, and (c) 1/9 of the Kerr comb spacing (RBW = 0.1 nm). The
corresponding RF beat notes are shown on the right (RBW = 1 kHz).
78
To study the effect of EO modulation on the Kerr comb and also demonstrate that the
newly inserted EO comb lines are suitable for data modulation, we measure the linewidths
of both the Kerr and EO comb lines on the basis of the delayed self-heterodyne method
[112]. Figure 8.4(a) shows the linewidth measurement setup for the Kerr and EO comb lines.
The selected comb line is amplified before being divided into two paths. One path contains
an acoustic optical modulator (AOM), which is used to shift the frequency of the comb line
by 590 MHz. The other path contains a 20.3 km fiber delay, which is longer than the
coherence length of the input comb line. Then, the combined beam is sent to a PD after
being amplified, and the RF spectrum of the beat note is recorded by a digital phosphor
oscilloscope. Figure 8.4(b) shows the linewidth measurements for different comb lines. The
linewidth of the generated EO comb line falls in the range of 20-30 kHz, which is
comparable to that of the Kerr comb line. In addition, the measured linewidth of the Kerr
line remains nearly the same after EO modulation (see Figure 8.4[c]).
Figure 8.4 (a) Experimental setup for the linewidth measurement based on the delayed self-
heterodyne method. AOM, acoustic optical modulator; DPO, digital phosphor oscilloscope. (b) The
measured linewidths versus different wavelengths. The linewidth measurements for the comb line of
1547.24 nm, with and without EO modulation, are shown in (c).
79
To further demonstrate that this insertion technique provides high-quality
communication-grade optical lines, we select one Kerr comb line and several surrounding
EO lines and modulate them with data. Multiple comb lines in the 1547.9-1549.5 nm range
are filtered (see the lines in the gray-dashed boxes of Figure 8.3) with different comb
spacings in the case of different modulation frequencies. Figure 8.5(a) shows the spectrum of
multiple data channels in the coherent receiver when those comb lines are encoded with 10
Gbaud QPSK signals. The error vector magnitude (EVM) performance in Figure 8.5(b)
shows that these data channels are error free, demonstrating that the new inserted EO lines
are suitable for data transmission. Thus, we are able to increase the channel number to six
with only one channel from a Kerr comb line. The EVM difference between these data
channels can be explained by the optical signal-to-noise ratio (OSNR) difference in the
receiver. In addition, we also show the case when the Kerr comb line at 1548.76 nm has not
been EO modulated and is directly modulated with 10 Gbaud QPSK. We see that the signal
power of the corresponding data channel is higher than those of data channels in the case of
EO modulation, and the noise level is lower. The resulting higher OSNR makes the EVM
lower and the signal quality better (see Figure 8.5(b)). This phenomenon can be attributed to
the following reason. In the case of EO comb line insertion, EO modulators can bring loss to
the input Kerr comb, and the amplified spontaneous emission noise level of each comb line
may increase when it is amplified. The amplified signal power is also limited. Thus, the
OSNR of the corresponding data channel would then decrease, and the signal quality could
be affected.
80
Figure 8.5 (a) Optical spectra of the channels in the coherent receiver when multiple comb lines in the
gray-dashed boxes of Figure 8.3 are modulated with 10 Gbaud QPSK data (RBW = 0.1 nm). The
spectrum of the channel when the Kerr comb line at 1548.76 nm is directly modulated with data is
also shown. (b) Measured EVMs versus different comb lines for different modulation frequencies of
the modulator. As a comparison, the EVM of the Kerr comb line directly modulated with the data is
also shown.
8.4 Conclusion
In conclusion, the number of available comb lines can be increased when the Kerr
comb is input into EO modulators, which proves useful in potentially improving the spectral
efficiency of optical communications systems. The linewidths of the inserted EO comb lines
are found to be comparable to those of the input Kerr comb lines, indicating that the inserted
EO lines have coherence similar to that of their Kerr comb counterparts. A coherent
communication experiment confirms the high coherence of Kerr and EO combs when six
comb lines are encoded with 10 Gbaud QPSK signals. We also find that the EO modulation
can bring loss to the Kerr comb and affect the signal quality when the comb is modulated
81
with data. Thus, using EO modulation to increase the number of comb lines is preferred for
cases in which many data channels are needed and OSNR requirements are not too stringent.
Furthermore, EO comb insertion could be beneficial for other applications. For example, the
increased number of comb lines could improve the number of taps in a tapped-delay-line [99]
used for all-optical signal processing.
82
Chapter 9 Orthogonally Polarized Kerr
Frequency Combs
9.1 Introduction
There are multiple transverse mode families inside an optical resonator, each of which
can theoretically be exploited to generate a frequency comb. Previous theoretical studies
have shown that mode coupling exists between different mode families, in both the fiber ring
cavity and the microresonator [118-120]. Effective mode coupling between different modes
potentially indicates that one Kerr frequency comb could create another Kerr comb, and both
combs could interact with each other in one microresonator. Recently, a Stokes soliton comb
in a long-wavelength range has been experimentally demonstrated when a primary soliton
comb creates both an effective potential well via the Kerr effect and local Raman
amplification [121]. Additionally, considering that the microresonator has two orthogonally
polarized modes (i.e., transverse electric [TE] mode and transverse magnetic [TM] mode), a
laudable goal would be to further demonstrate that one polarized Kerr comb could be used to
generate an orthogonally polarized comb via nonlinear interaction and these two overlapping
combs in the frequency domain could exist simultaneously in a single resonator.
In this chapter, we demonstrate that one comb in one polarization can help
generate an orthogonally polarized comb via cross-phase modulation (XPM), and
that two combs in orthogonal polarization states can exist simultaneously in a single
resonator. Specifically, the interaction effect between two polarized modes caused by
XPM can alleviate the power requirement for one polarized pump light. Moreover,
after the two combs are generated, the existence of the second comb is dependent on
the other vertically polarized comb, and strong interaction still exists between two
orthogonally polarized frequency combs. In the experiment, we pump the microring
resonator first with a TM-polarized CW light which leads to the generation of a
soliton comb. Then, after the introduction of a much weaker TE-polarized CW light
as a "seed", a TE-polarized frequency comb is excited. Numerical simulations that
83
consider the group–velocity mismatch between two polarization modes also confirm
the experimental observation. Thus, not only do we observe that one comb is
generated from an orthogonally polarized soliton comb via XPM, we also find that it
is possible to achieve two combs in different polarizations in a single microresonator.
9.2 Operation Principle
Figure 9.1 shows the schematic of TE- and TM- polarized comb generation in
a microresonator. After a TM-polarized pump is coupled into the microresonator to
generate a soliton comb, a weak TE-polarized CW interacts with the TM-polarized
comb via XPM and the TE-polarized comb is created.
Figure 9.1 Schematic of TE-polarized comb generation from a TM-polarized soliton comb. A TE-
polarized CW with weak power as a "seed" is cross-phase modulated by a TM-polarized soliton pulse,
which results in the generation of a TE-polarized comb.
9.3 Experimental Setup and Experimental Results
Figure 9.2(a) shows our experimental setup. The microresonator in the setup is
based on a 1,500-nm height and 900-nm wide silicon nitride waveguide. The radius
of the ring resonator is ~119 µm. The TM and TE mode families of the
microresonator are characterized by 192.1 GHz and 190.3 GHz free spectral ranges
(FSRs), respectively. The microresonator is pumped with the emissions of two
external cavity diode lasers (ECDLs) amplified by erbium-doped fiber amplifiers
(EDFAs). In the first step, a single-soliton pulse is generated when only a high-
power TM-polarized pump laser is coupled into the microresonator. The on-chip
pump power is ~600 mW. The pump wavelength is controlled by a function
84
generator, and the comb regime is approached by sweeping the laser from shorter to
longer wavelengths across the resonance and stopping at a soliton-existence region
[5, 65]. Subsequent laser wavelength backward tuning turns the multi-soliton state
into a single soliton state [10] after the observation of a multi-soliton state. In the
second step, a TE-polarized CW signal light is also coupled into the cavity. Its on-
chip power is ~6.3 mW, a hundred times lower than the power of the first pump. In
fact, the power of the TE-polarized signal is lower than the threshold power of the
comb generation [122]. Interestingly, we can observe the generation of a TE-
polarized comb in the microresonator. The second comb is generated because of the
XPM effect between the TE and TM mode families. We also observe that the
existence of the second comb relies on the orthogonally polarized soliton comb and
the disappearance of the TM-polarized soliton comb also results in the disappearance
of the TE-polarized comb.
We use an incoherent optical source to characterize the cavity spectrum. Figure
9.2(c) shows the measured transmission curves for the fundamental TE and TM
modes when a spontaneous-emission source in a single polarization state is coupled
into the microresonator. We distinguish these two polarization states by the FSR
difference between TE and TM modes which also agrees with the calculation. The
wavelengths of the TE-polarized signal and TM-polarized pump are 1544.6 nm and
1555.9 nm, respectively, and the corresponding Q factors of the resonant modes are
7.8×10
5
and 1.3×10
6
, respectively. We can see that the TE and TM modes do not
overlap in frequency. Figure 9.2(d) shows the single-soliton comb spectrum when
only the TM-polarized pump laser is coupled into the cavity. The comb envelope
shows a sech
2
shape and the 3 dB comb bandwidth is ~5 THz. The repetition rate is
~191.8 GHz, as seen in Figure 9.2(e). The repetition rate of the comb is measured
with electro-optical down conversion. The comb is sent into the cascaded optical
modulators [115], and the spacing between adjacent electro-optical sidebands is
measured.
85
Figure 9.2(f) shows the spectrum of both TE- and TM-polarized combs. The
generated TE- and TM-polarized combs can be determined considering that they are red-
shifted with respect to the resonant wavelengths of two polarization modes in Figure 9.2(c)
because of Kerr effect and thermal effect. In addition to the TE-polarized comb, which is
generated because of the XPM effect between the TE-polarized CW and the TM-polarized
soliton pulse, we can also see a frequency harmonic generated at 1567.4 nm. This line is
created because of the degenerate four-wave mixing (FWM) effect between the TE-
polarized signal and the TM-polarized pump in the bus waveguide [123]. The wavelength of
this line changes when we tune the wavelength of the TE-polarized signal. This line is
generated even when the TE-polarized signal is not resonant and the TE-polarized comb is
not generated. Figure 9.2(g) shows the zoom-in spectrum of (f) and we find that the
repetition rates of the TE- and TM-polarized combs are identical within the measurement
error. Our microresonator has a single-mode "filtering" section [124, 125], which is intended
to suppress higher-order modes. That is why the TM-polarized frequency comb spectrum in
Figure 9.2(d) is smooth. The cross-polarization-based XPM interaction occurs only between
the fundamental TM and TE modes, and interaction between the fundamental mode and
high-order mode families [126, 127] is avoided.
To demonstrate the low-phase-noise characteristics of the newly generated TE-
polarized comb, we use a narrow-band filter to select the first comb line right next to the TE-
polarized signal and the adjacent TM-polarized comb line (see the black-dashed box in
Figure 9.2[g]). The beat note of these two selected comb lines is measured when they are
amplified by an EDFA and sent to a photodetector (PD). Note that these two comb lines are
not orthogonally polarized before the PD because of the polarization change in the fiber and
then we can measure the beat note. A single line peak in the radiofrequency (RF) spectrum
of Figure 9.2(h) demonstrates that the TE-polarized comb is also in a low-noise state, as is
the TM-polarized soliton comb [65].
86
Figure 9.2 (a) Experimental setup of a TE-polarized comb generation when a TE-polarized CW signal
with low power is cross-phase modulated by a TM-polarized soliton in the microresonator. The inset
shows the intensity autocorrelation of the single soliton. A TM-polarized soliton pulse is generated
first by a high-power TM-polarized pump, and a TE-polarized CW signal is then coupled into the
cavity as a "seed" to excite the TE-polarized comb. AFG, arbitrary function generator; ECDL,
external-cavity diode laser; PC, polarization controller; EDFA, erbium-doped fiber amplifier; FBG,
fiber Bragg grating; OSA, optical spectrum analyzer. (b) Optical microscope image of an Si
3
N
4
microresonator. (c) Measured transmission spectra of the fundamental TE and TM modes when a
spontaneous emission source is coupled into the microresonator. (d) Optical spectrum of a single
soliton generated by a TM-polarized pump (resolution: 0.1 nm). (e) Repetition rate beat note of the
single-soliton comb (resolution bandwidth [RBW] = 10 kHz). (f) Optical spectrum of the TE- and
TM-polarized combs (resolution: 0.16 pm). The line at 1567.4 nm in the green-dashed box is
generated by the FWM between the TM-polarized pump and the TE-polarized signal on the bus
waveguide. The TE-polarized comb spectrum is distinguished manually based on the transmission
87
spectra in (c). (g) Zoom-in spectrum of (f) in a spectral range of 1540 nm to 1550 nm (resolution: 0.16
pm). (h) Heterodyne beat note of the two TE- and TM-polarized comb lines in (g) (RBW = 100 kHz).
Figure 9.3(a) shows the relationship between the TE-polarized signal power and the
power of the generated first sideband right next to the signal. Even at -9.4 dBm (0.1 mW),
the TE light generates the second comb, and the corresponding first sideband comb's relative
power is -71.3 dB. The ability of the low signal power to excite a frequency comb is a result
of the assistance from the TM-polarized soliton pulse and the high peak power of the soliton
pulse in the cavity. We also observe the generation of the TE-polarized comb, even when the
signal wavelength is moved to other resonant wavelengths. Figures 9.3(c) and (d) show two
example spectra when the signal wavelength is moved to 1549.4 nm and to 1564.9 nm,
respectively. Simultaneously, the line directly generated from the FWM between the TE-
polarized signal and the TM-polarized pump in the bus waveguide has moved to 1562.5 nm
and 1547.0 nm, respectively.
Figure 9.3 (a) Power of the generated first sideband right next to the TE signal versus the signal
power. (b) Sample spectrum of the TE- and TM-polarized combs when the signal power is only -7
dBm. When the signal power is -0.4 dBm and the signal wavelength is moved to 1549.4 nm and
88
1564.9 nm, the optical spectra of the generated TE- and TM-polarized combs are shown in (c) and (d),
respectively (resolution: 0.16 pm).
The above results experimentally demonstrate that a TE-polarized CW can be
modulated by a single-soliton pulse in the microresonator. Moreover, the excitation of
multiple solitons at the TM polarization (see Figures 9.4[a] and [d]) modifies the TE-
polarized comb. This observation indicates that a different number of solitons in the
microresonator can assist in generating different TE-polarized comb states. The TM-
polarized multi-soliton comb envelope results in the irregularity of the TE-polarized comb
envelope.
Figure 9.4 The optical spectra of two different multisoliton states are shown in (a) and (d) when only
the TM-polarized pump is coupled into the microresonator. The intensity autocorrelations of two
89
multisoliton states are shown in (b) and (e). The corresponding spectra of the TE- and TM-polarized
combs are shown in (c) and (f).
A TE-polarized soliton pulse can excite a TM-polarized comb in the same way a TM-
polarized pulse excites a TE-polarized comb. Figure 9.5 shows the spectrum of two combs
when the TE-polarized pump wavelength is 1557.1 nm and the TM-polarized signal
wavelength is 1545.4 nm. This indicates that a soliton pulse in any polarization state can
modulate a CW light in an orthogonal state.
Figure 9.5 Optical spectrum of two combs, where the TM-polarized comb is generated by XPM
between a TE-polarized soliton pulse and a TM-polarized signal. The TE-polarized pump power is
740 mW and the power of the TM-polarized signal is 8.5 mW.
9.4 Numerical Simulation Results
We have numerically simulated TE-polarized comb generation via XPM between a
TM-polarized soliton pulse and a TE-polarized CW. We consider the XPM effect and the
group-velocity mismatch between the TE and TM modes in the modeling. Two coupled
Lugiato-Lefever equations [45, 46, 57, 58] have been used in the simulation. The second
term on the right-hand side of the two equations describes the XPM effect.
�τ
0
TM
∂
∂ t
+
α
i
TM
2
+
θ
TM
2
− jδ
0
TM
+ jL ∑
( − j)
m
β
m
TM
m!
∞
m = 2
∂
m
∂τ
m
� E
TM
(t, τ) =
�
θ
TM
E
in
TM
−
jγ
TM
L ∙ E
TM
(t, τ)(|E
TM
(t, τ)|
2
+
2 � E
TE
( t,τ) �
2
3
). (13)
90
�τ
0
TE
∂
∂ t
+
α
i
TE
2
+
θ
TE
2
− jδ
0
TE
+ jL ∑
( − j)
m
β
m
TE
m!
∞
m = 2
∂
m
∂τ
m
+ L(
1
v
g
TM
−
1vgTE)∂∂τETEt,τ=θTEEinTE−jγTEL∙ETEt,τ(ETEt,τ2+2ETMt,τ23). (14)
Here, τ
0
TM
and τ
0
TE
are the round-trip times for the TM and TE modes, respectively; t and τ
are the slow and fast times, respectively; E
TM
(t, τ) and E
TE
(t, τ) represent the intracavity
fields for the two polarization modes, respectively; E
in
TM
and E
in
TE
are the input fields for the
two modes, respectively; α
i
TM
and α
i
TE
represent the power loss per round trip; θ
TM
and θ
TE
represent the power coupling coefficient; δ
0
TM
and δ
0
TE
are the phase detunings of the pumps
from the adjacent resonance frequencies, respectively; β
m
TM
and β
m
TE
are the mth-order
dispersion coefficients; v
g
TM
and v
g
TE
are the group velocities; L is cavity length; and γ
TM
and γ
TE
are nonlinearity coefficients. In the simulation, the calculated dispersion coefficients
for the TM and TE modes are -129 ps
2
/km and -105 ps
2
/km, respectively. The nonlinear
coefficients are 0.86/(W·m) and 0.98/(W·m), respectively. The numerical simulations
involve two steps to follow the experimental process. In the first step, we use only Eq. 13, in
which we ignore the XPM effect to simulate TM-polarized soliton generation. After the
soliton becomes stable in the simulation, we then use Eqs. 13 and 14 to simulate both TE-
polarized comb generation and TM-polarized comb evolution. We assume that the phase
detuning δ
0
TE
for the TE mode is zero in the second step.
Figure 9.6 shows the spectral and temporal dynamics of the TM-polarized comb after
the TE-polarized CW signal is turned on. The TM-polarized soliton comb evidently exhibits
almost no change (see Figure 9.6[a], Column 1) because the TE-polarized signal power is
much lower than the TM-polarized soliton power, and the XPM effect on the soliton pulse
from the TE-polarized signal is consequently weak. On the other hand, we see that the TE-
polarized light is modulated, and the corresponding comb is generated. The first column of
Figure 9.6(b) shows that the TE-polarized comb is generated and it becomes stable within 2
ns. The stable TE-polarized comb spectrum shown in the third column of Figure 9.6(b) has a
concave envelope, which agrees with the experimental results illustrated in Figure 9.2(f).
91
The corresponding temporal waveform in the fourth column of Figure 9.6(b) has a dip in the
center and does not exhibit a pulse shape. In the pulse-assisted comb generation, several
initial comb lines close to the TE-polarized signal are created first, and the comb bandwidth
grows until the comb reaches a stable state (see Figure 9.6[b], Column 1). We also observe
that the temporal evolution trace for the TE-polarized comb has the same slope as that for the
TM-polarized comb (see Figure 9.6, Column 2), which indicates that the repetition rates of
the TE- and TM-polarized combs are the same, although the FSR difference for the TE and
TM modes is ~1.8 GHz. The TE-polarized comb envelope could be explained as follows.
The repetition rate of the TE-polarized comb is different from the FSR of the TE mode. The
generation of higher-order harmonics can be suppressed because the further the harmonics
are from the TE-polarized CW signal, the larger the frequency mismatch between the
stimulated harmonics and the TE mode that accommodates this harmonics.
Figure 9.6 The simulated spectral and temporal dynamics of the TM- and TE-polarized combs
considering the group-velocity mismatch are shown in (a) and (b), respectively. The stable comb
spectrum and the corresponding stable temporal waveform are shown in the third and fourth columns,
respectively.
We also simulate TE-polarized comb generation when the group-velocity mismatch
between the TE and TM modes is ignored by assuming v
g
TE
= v
g
TM
in Eq. 14. The TM-
polarized soliton comb evolution shown in the first column of Figure 9.7(a) is similar to that
shown in Figure 9.6(a). However, the TE-polarized comb exhibits different characteristics
92
(see Figure 9.7[b]). As the third and fourth columns of Figure 9.7(b) show, the TE-polarized
comb spectrum is broader, and the temporal waveform corresponds to a dark pulse [78]. The
pulsewidth is much broader than that of the bright soliton in Figure 9.6(b), Column 4. Note
that this dark pulse is not a standard dark soliton [81], and some oscillation occurs at the
pulse edge. A TE-polarized dark pulse can be formed in the anomalous dispersion regime as
a result of the strong XPM effect on the signal CW, which counteracts the temporal
spreading of the dark pulse induced by the anomalous dispersion [128]. By comparing the
TE-polarized comb in Figures 9.6(b) and 9.7(b), we find that even when the group velocities
for two modes are close, the small group-velocity mismatch could affect comb generation
when the CW is influenced by the soliton pulse.
Figure 9.7 The simulated spectral and temporal dynamics of the TM- and TE-polarized combs without
considering the group-velocity mismatch are shown in (a) and (b), respectively. The temporal
waveform of the stable TE-polarized comb corresponds to a dark pulse when the group velocities
between two polarization modes are equal.
9.5 Discussion
We have experimentally demonstrated the generation of one polarized frequency
comb from an orthogonally polarized soliton comb. This results from the strong XPM effect
when a CW signal in one polarization state is modulated by a soliton pulse in an orthogonal
polarization state. Because of the high peak power of the soliton pulse in the cavity, even a
weak CW signal with low power can excite a comb with the assistance of a soliton. This
93
study may provide another method of generating two combs [129-137] in a single
microresonator. Specifically, intracavity phase interferometry is one potential application of
two combs with the same repetition rate [138]. Several methods could be used to increase
the power of the second comb in the present work. One approach is to increase the input
signal power, which results in an increase in the power of the second comb. It should also be
noted that the high input power of the signal could disturb the stability of the soliton pulse
due to the thermal effect. This could be partially overcome by tuning the pump wavelength
of the soliton comb when the signal power is gradually increased and the soliton state is still
maintained. We could also tune the group velocity difference between two polarization
modes by designing the waveguide structure so that the second comb is optimized. In
addition, dispersive wave generation [15] could be used to increase the power and broaden
the bandwidth of the second comb. Furthermore, these two polarized combs could be
separated by placing a polarization beam splitter at the output of the microresonator so that
their corresponding temporal waveforms can be experimentally characterized.
Moreover, because there is a nonlinear interaction between two orthogonally
polarized frequency combs, the state of one comb could be finely tuned when the
other comb is controlled externally. In this chapter, we use a soliton comb as an
example to generate another comb, and we demonstrate the interaction between two
orthogonally polarized Kerr combs. Thus, Kerr combs in other states, such as
primary combs and breathing soliton combs, could possibly generate different combs
in the orthogonally polarization and interact with the orthogonally polarized combs.
94
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Abstract (if available)
Abstract
A large number of precisely spaced spectral lines form an optical frequency comb, which enables a wide range of applications including frequency metrology, atomic clocks, precision spectroscopy, and optical communications. In recent years, compact microresonator-based Kerr frequency combs have attracted much attention because of their compactness and high repetition rate. Kerr frequency combs in a low-noise state also have the advantages of high coherence and broadband coverage
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Asset Metadata
Creator
Bao, Changjing
(author)
Core Title
On-chip Kerr frequency comb generation and its effects on the application of optical communications
School
Viterbi School of Engineering
Degree
Doctor of Philosophy
Degree Program
Electrical Engineering
Publication Date
01/30/2018
Defense Date
12/11/2017
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
Kerr frequency comb,microresonator,nonlinear optics,OAI-PMH Harvest,optical communications,soliton
Language
English
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Electronically uploaded by the author
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Advisor
Willner, Alan (
committee chair
), Armani, Andrea (
committee member
), Wu, Wei (
committee member
)
Creator Email
baochangjing@gmail.com,changjib@usc.edu
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-c40-466636
Unique identifier
UC11265721
Identifier
etd-BaoChangji-5975.pdf (filename),usctheses-c40-466636 (legacy record id)
Legacy Identifier
etd-BaoChangji-5975.pdf
Dmrecord
466636
Document Type
Dissertation
Rights
Bao, Changjing
Type
texts
Source
University of Southern California
(contributing entity),
University of Southern California Dissertations and Theses
(collection)
Access Conditions
The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...
Repository Name
University of Southern California Digital Library
Repository Location
USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA
Tags
Kerr frequency comb
microresonator
nonlinear optics
optical communications
soliton