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Analysis of photonic crystal double heterostructure resonant cavities

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Analysis of photonic crystal double heterostructure resonant cavities

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Content ANALYSISOFPHOTONICCRYSTALDOUBLEHETEROSTRUCTURE
RESONANTCAVITIES
by
AdamMock
ADissertation Presentedtothe
FACULTYOFTHEGRADUATESCHOOL
UNIVERSITYOFSOUTHERNCALIFORNIA
InPartialFulfillmentofthe
RequirementsfortheDegree
DOCTOROFPHILOSOPHY
(ELECTRICALENGINEERING)
August2009
Copyright 2009 AdamMock
Dedication
Tomymom.
ii
Epigraph
iamascientist-iseektounderstandme
iamanincurableandnothingelsebehaveslikeme
FromiamascientistbyRobertPollard,GuidedbyVoices
iii
Acknowledgments
Iacknowledgefiveyearsofsteadyfinancialsupportfrommyresearchadvisor,John
O’Brien, which allowed me ample time to educate myself on subjects of interest
and to pursue research unencombered by financial constraints. I appreciate his
suggestiontostudythephotoniccrystaldoubleheterostructurecavitywhichmakes
up much of this Thesis. Our technical conversations often resulted in a clarified
viewoftheresearchprojectathand.
I appreciate the training and mentoring I received during my first year in the
Microphotonic Device Group from Wan Kuang. Without his help I would never
I have learned how to use the finite-difference time-domain method to the extent
thatIdotoday. HesetastandardofexcellenceforwhichIcontinuetostrive. Iam
thankfulforthecontinuedadvisingandconversations thatwecontinuetohave.
I thank Stephen Farrell for providing a different and relatable perspective on
graduate school. IthankLingLuforbeingasuperb colleague–Iamgladwewere
abletocollaborateonsomeinterestingprojects.
I would like to thank Dr. William Steier and Dr. Aiichiro Nakano for being
on my qualifying exam and thesis defense committees. I appreciate Dr. Aluizo
Prata and Dr. Steve Cronin for being on my qualifying exam committee. I am
thankfulforallofthejobsearchadvicethatIgotfromDr. MichellePovinelli. Her
perspectiveonbeingayoungresearcherwasinvaluable.
iv
Iamappreciativeofhavingunfetteredaccesstothehighperformancecomput-
ing and communications resource at USC. Almost every numerical computation
presentedinthisworkwasdoneusingHPCC.
Finally,Iwouldliketothankmyfriendsandfamilyfortheirsupport. Iextend
thanks to my parents for helping to pay some of the bills and for always being
supportive. IthankBenCollinsforbeingagoodpartner-in-sciencesincefreshman
year of college. I thank Sayaka Nasu for making me take a break every once in a
while.
v
TableofContents
Dedication ii
Epigraph iii
Acknowledgments iv
ListofFigures viii
Abstract xiv
Chapter1: OverviewandMotivation 1
1.1 Photonicintegratedcircuits . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Two-dimensionalphotonic crystals . . . . . . . . . . . . . . . . . . . . 3
1.3 Photoniccrystal integratedcircuits . . . . . . . . . . . . . . . . . . . . 7
1.4 Numericalanalysisofphotonic crystals . . . . . . . . . . . . . . . . . 10
1.5 Thesisoverview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Chapter2: Electromagnetic resonantcavityanalysis 21
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 Temporalbehavioroffields,energyandpowerinaresonantcavity . 22
2.3 Derivationoflaserthresholdcondition . . . . . . . . . . . . . . . . . . 25
Chapter3: QualityFactorEstimationUsingPad´ eInterpolation 32
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2 EstimatingtheQfactorfromfinite-differencetime-domainsimulations 32
3.3 DirectextractionapproachtoPad´ einterpolation . . . . . . . . . . . . 39
3.4 Pad´ einterpolation forgeneralizedtransferfunctionanalysis . . . . . 50
Chapter4: Photonic CrystalDoubleHeterostructure ResonantCavities 55
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.2 Spectral properties of photonic crystal double heterostructure reso-
nantcavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
vi
4.3 Modal properties of photonic crystal double heterostructure bound
states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.4 Higherorderboundstatesinphotoniccrystaldoubleheterostructures 73
Chapter5: PhotonicCrystalDoubleHeterostructureCavitiesWithDielectric
LowerSubstrates 84
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.2 Substratelossinphotoniccrystaldoubleheterostructure cavities . . . 86
5.3 Strategyforreducingtheout-of-planeradiation . . . . . . . . . . . . 89
5.4 Towardelectricallyinjectedphotoniccrystal lasers . . . . . . . . . . . 98
Chapter 6: Finite-Difference Time-Domain Analysis With a Material Gain
Model 107
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.2 Dispersive gain modeling using the finite-difference time-domain
method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
6.3 Interactionbetweenspatialgaindistribution andcavitymode . . . . 110
6.4 Futuredirections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
References 119
vii
ListofFigures
1.1 Schematicdiagramofaphotonicintegratedcircuit[Koc08]. . . . . . . 2
1.2 PhotographoftheBlazarproductfromLuxteraInc. . . . . . . . . . . 3
1.3 Schematicdiagramofatwo-dimensionalphotoniccrystaldefinedin
asemiconductorslab. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Photonic band diagram for a two-dimensional photonic crystal cal-
culated using the three-dimensional finite-difference time-domain
method. c denotes free space speed of light. a denotes the lattice
cosntant. The hole radius to lattice constant ratio is r/a = 0.29 and
the slab thickness to lattice constant ratio is d/a = 0.6. The index of
the slab was set to n = 3.4. The diagram depicts the lowest three
bands for the TE-like modes of the slab. The beige shaded regions
denotethelightconeprojectionontothecorrespondingpropagation
directions. The inset shows the region of the first Brillouin zone
describedbythedispersiondiagram. . . . . . . . . . . . . . . . . . . . 5
1.5 (a) Photograph of a peacock. (b) Scanning electron micrograph of a
peacock wing showing microstructure with two-dimensional peri-
odicity. Adaptedfrom[ZYL
+
03]. . . . . . . . . . . . . . . . . . . . . . 6
1.6 H
z
(x,y,z = 0) field distribution of a single line defect photonic
crystal waveguideobtained usingthefinite-difference time-domain
method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.7 H
z
(x,y,z= 0) fielddistributions for the sixmodes associated with a
photoniccrystalcavityformedbyremovingthreeneighboringholes
obtainedusingthefinite-differencetime-domainmethod. . . . . . . . 8
1.8 Schematic diagram illustrating a photonic crystal based photonic
integratedcircuit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
viii
1.9 Qualityfactorfordifferenttwo-dimensionalphotoniccrystalcavities
as a function of time. Data points and figures come from [PLS
+
99,
RKP
+
02,AASN03,ZQ04,NB06,AASN05,SNAA05,ASN06,TAN08] 11
1.10 Illustrationofdiscretesamplingofafunctionu(x). . . . . . . . . . . . 12
1.11 Illustration of the spatial arrangement of the electric and magetic
fields at a single spatial point in the finite-difference time-domain
method. ThisarrangementisknownasaYeecell. . . . . . . . . . . . 15
3.1 Time sequence obtained from a three-dimensional finte-difference
time-domainsimulationofaphotoniccrystaldoubleheterostructure
resonantcavity. Thesequenceisobtainedbystoringthediscretetime
evolutionofH
z
(t)atapointnearthecenterofthecavity. . . . . . . . 33
3.2 DiscreteFouriertransformofthetimesequenceinFig.3.1. cdenotes
freespacespeedoflight. adenotesthelatticecosntant. . . . . . . . . 35
3.3 (a) User generated time sequence of the form Eq. 2.5. (b) Discrete
Fouriertransform ofthesequencein(a). . . . . . . . . . . . . . . . . . 36
3.4 Diagramshowingdiscretesamplingpointsassociatedwithatypical
resonancepeak. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.5 QfactorconvergencestudyofthePad´ emethodusingaP(0,1)Pad´ e
function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.6 QfactorconvergencestudyofthePad´ emethodusingaP(0,1)anda
P(1,1)Pad´ efunction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.7 DiscreteFourierspectrumoftworesonancepeaksseparatedby0.005
innormalizedfrequencyunits. . . . . . . . . . . . . . . . . . . . . . . 45
3.8 Convergence study of the Q factor extraction by the Pad´ e method
whentworesonancesarebroughtclosetogetherinfrequency. . . . . 47
3.9 ConvergencestudyoftheQfactorextractionwhenthePad´ emethod
is applied to data obtained from three-dimensional finite-difference
time-domainsimulations. (a)-(c)depicttheconvergenceofthemethod
withincreasinglyhigherorderPad´ efunctions. . . . . . . . . . . . . . 49
3.10 Convergence study of Q factor extraction using Pad´ e interpolation
andfilterdiagonalization. . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.11 Systemleveldiagramsillustrating(a)feedbackand(b)feedforward. 53
ix
4.1 Schematicdiagramsofphotoniccrystaldoubleheterostructurereso-
nantcavitiesformedinuniformsinglelinedefectwaveguidesby(a)
increasingand(b)decreasingthelatticeconstantalongthex-direction. 58
4.2 Close up view of a photonic crystal waveguide dispersion diagram
illustrating the candidate frequencies for a bound state resonance
associatedwithaheterostructure cavity. . . . . . . . . . . . . . . . . . 59
4.3 Left: photonic crystal waveguide dispersion diagram. Black lines
correspond to the photonic crystal waveguide dispersion bands.
Blue regions denote photonic crystal cladding modes. The beige
region denotes the light cone projection. Right: photonic crystal
double heterostructure resonant spectra for defects resulting from
an increase and a decrease in the lattice constant. Dashed lines
illustrate correspondance between heterostructure bound state fre-
quenciesandwaveguidedispersionextrema. . . . . . . . . . . . . . . 61
4.4 Schematic diagram showing the regions of the cavity supporting
Fabry-Perotresonances. . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.5 Group index extracted from frequency spacing in numerically cal-
culated resonance spectra associated with photonic crystal double
heterostructures. The back curves correspond to the group index
obtained from the slope of the photonic crystal waveguide disper-
sion diagram. (a) Photonic crystal waveguide band spanning the
normalized frequency range 0.26-0.34. (b) Photonic crystal waveg-
uidebandspanningthenormalizedfrequencyrange0.28-0.30. . . . . 63
4.6 Left: H
z
(x,y,z= 0)forthefivemodessummarizedinTable4.1. Right:
spatial Fourier transform of each mode illustrating the distribution
ofspatialwavevectors makingupthedifferentresonantmodes. . . . 66
4.7 (a) Diagram illustrating the directional radiation of Mode 2 from
Fig. 4.6. (b) First Brillouin zone of a two-dimensional triangular
lattice. Projection oftheMpointontotheΓ−Kdirectionatβ
x
= π/a
isindicated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.8 Q factor versus perturbation depth for the first, second and third
orderboundstates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.9 DirectionalQfactorbreakdownversusperturbation depth. . . . . . . 75
x
4.10 Left: H
z
(x,y,z= 0) field distributions for the first, second and third
orderbound states. Right: field envelopes extracted from|H
z
(x,y=
0,z= 0). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.11 (a) Nominal heterostructure cavity with a 10% lattice constant per-
turbation. (b) Modified cavity with two extra holes to suppress the
secondorderboundstate. (c)Modifiedcavitywithoneextraholeto
suppressthefirstorderboundstate. . . . . . . . . . . . . . . . . . . . 78
4.12 ResonancespectraassociatedwiththethreecavitiesshowninFig.4.11.
Topcurve: cavityinFig.4.11(a). Middlecurve: cavityinFig.4.11(b).
Bottomcurve: cavityinFig.4.11(c). Thephotoniccrystalwaveguide
bandedge is denoted. Any features in the resonance spectrum at
frequencieslargerthanthephotoniccrystalbandedgeareassociated
with resonances in the uniform waveguide sections of the structure
asdenotedinFig.4.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.13 Q factor of the first order bound state as a function of the distance
fromthecenterthatthetwoextraholesareplaced. . . . . . . . . . . . 80
4.14 DiagramillustratingtheoverlapofthetwoextraholeswithE
y
(x,y=
0,z= 0)forthefirstandsecondorderboundstatesfortheoptimum
placementof2.4afromthecenterofthestructurealongthex-direction. 81
4.15 (a)-(c)Lasingspectraofthreedouble-heterostructurelaserswith10%
perturbation(a
x
= 1.10a
x
). TheirSEMimages,shownasinsets,were
taken at the same experimental conditions. (d) is the light-in–light-
outcurvesofthelasersin(a)-(c). . . . . . . . . . . . . . . . . . . . . . 83
5.1 Scanning electron micrograph of a fabricated photonic crystal het-
erostructurecavitythathasbeendamagedbyhighcontinuouswave
pumppower. CourtesyofLingLu. . . . . . . . . . . . . . . . . . . . . 85
5.2 Schematicrenderingofamembranephotoniccrystalheterostructure
cavity(lavender)bondedorgrown onalowersubstrate(green). . . . 87
5.3 Directional Q factor as a function of index of lower substrate for a
typeAphotonic crystalheterostructure cavity. . . . . . . . . . . . . . 88
5.4 Illustration of radiation directions for a substrate bonded photonic
crystalheterostructure cavity. . . . . . . . . . . . . . . . . . . . . . . . 88
xi
5.5 SchematicdiagramofatypeBphotoniccrystalwaveguide. Thetop
photonic crystal claddingisshiftedonehalflattice periodalongthe
x-directionwithrespecttothebottom cladding.. . . . . . . . . . . . . 90
5.6 Photonic crystal waveguide dispersion diagram for propagation
alongtheΓ−Kdirection. Blueregionscorrespondtophotoniccrys-
tal cladding modes. The straigh black line is the light line. The
various colored curves are waveguide dispersion bands for waveg-
uideswhosephotoniccrystalcladdingarraysareshiftedbyvarying
degrees along the x-direction. A lattice shift of 0.0a corresponds to
a type A photonic crystal waveguide. A lattice shift of 0.5a corre-
spondstoatypeBphotoniccrystalwaveguide. . . . . . . . . . . . . . 91
5.7 DoubleheterostructurecavityformedfromatypeBphotoniccrystal
waveguide. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.8 H
z
(x,y,z= 0)atthemidplaneoftheslabforatypeBhetetostructure
cavitywith awaveguide widthof 0.8w. Theresonance frequencyis
0.2631449. Thephotoniccrystalgeometryissuperimposed. . . . . . 95
5.9 H
z
(x,y,z= 0)atthemidplaneoftheslabforatypeBhetetostructure
cavity with a waveguide width of 0.8w and tapered perturbation.
Theresonancefrequencyis0.2617761. Thephotoniccrystalgeometry
issuperimposed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.10 Directional Q factor as a function of index of lower substrate for a
typeBphotoniccrystalheterostructure cavity. . . . . . . . . . . . . . . 98
5.11 Comparison of the Q factors as a function of the index of the lower
claddingfortypeAandtypeBhetetrostructure cavities. . . . . . . . 99
5.12 DirectionalQfactorasafunctionofindexoftopandbottomcladding
foratypeAphotoniccrystal heterostructure cavity. . . . . . . . . . . 101
5.13 Schematicdiagramdepictingtheverticalslabstructureforaphotonic
crystalheterostructure withatopandbottomverticalcladding. . . . 101
5.14 Comparison of the Q factors as a function of the index of the top
andbottomverticalcladdingfortypeAandtypeBhetetrostructure
cavities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
xii
5.15 Qfactorversustopandbottomcladdingindexwithvaryingamounts
of hole penetration into the vertical cladding for a type B photonic
crystal heterostructure cavity. The hole depth corresponds to the
depth into both the top and the bottom cladding. These structures
maintainverticalmirrorsymmetryaboutthemidplaneoftheslab. . 103
5.16 Q factor versus top and bottom cladding index for different top
cladding thicknesses for a type B photonic crystal heterostructure
cavity. Insetdepictstheverticalcladdinggeometry. . . . . . . . . . . 104
5.17 Q factor versus top and bottom cladding index for different high-
index slab thicknesses for a type B photonic crystal heterostructure
cavity. The vertical slab structure is shown in the inset of Fig. 5.16
witha1.0μmetchdepthinthebottomcladdingand0.5μmperforated
topcladdinglayer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.1 1/Q
tot
versus peakmaterial gain where the spatial gain distribution
is varied. The gain is uniform in the plane of the slab but absent in
the air-holes. The three curves correspond to different fractions of
theverticalslabprovidinggain. . . . . . . . . . . . . . . . . . . . . . . 110
6.2 Illustration of field evolution versus time for (a) 1/Q
tot
> 0, (b)
1/Q
tot
= 0and(c)1/Q
tot
< 0. . . . . . . . . . . . . . . . . . . . . . . . . 113
6.3 (a)Two-dimensionalGaussiangaindistribution superimposedona
photoniccrystal heterostructure cavity. (b)1/Q
tot
versus peakmate-
rialgainfortwodifferentpumpspotsizes. . . . . . . . . . . . . . . . 115
xiii
Abstract
Two-dimensional photonic crystals represent a versatile technology platform for
constructing photonic integrated circuits. Low-loss and small footprint waveg-
uides and cavities can be combined to make delay lines, modulators, filters and
lasers for efficient optical signal processing. However, this diverse functionality
comesattheexpenseofhighercomplexityinboththefabricationandthemodeling
of these devices. This Thesis discusses the finite-difference time-domain numer-
ical modeling of large quality factor photonic crystal cavities for chip-scale laser
applications.
In Chapter 2 the role of the quality factor in estimating laser threshold is
derived starting from Maxwell’s equations. Expressions for modal loss and gain
are derived. Chapter 3 discusses methods for extracting the quality factor from
finite-difference time-domain simulations. Evenwith large-scale parallelcomput-
ing, only a short record of the time evolution of the fields can be recorded. Toget
around this issue, Pad´ e functions are fitted to the available data in the frequency
domain.
Oncetheanalysistoolshavebeendescribedanddemonstrated,theyareapplied
to the photonic crystal double heterostructure cavity which has been shown to
have quality factors in excess of one million and mode volumes on the order of
a cubic wavelength. A detailed description of the spectral and modal properties
xiv
of heterostructure cavities is presented, and a method for mode discrimination is
discussed.
The effect of heat sinking dielectric lower substrates on the optical loss of the
heterostructure cavity is investigated, and it is seen that the quality factor is sig-
nificantly reduced as the index of the lower substrate is increased. A modified
heterostructure cavity with glide plane symmetry is shown to have significantly
reduced out-of-plane leakage. An optimized design is proposed for continuous
waveedge-emittinglaseroperation.
Finally,anovelapproachforlasersimulationisintroducedinwhichamaterial
gainmodelisincludedinthefinite-differencetime-domainsimulation. Theeffects
of spatially varying gain distributions are investigated and agree well with the
modalgainderivationinChapter2.
xv
Chapter1
OverviewandMotivation
1.1 Photonicintegratedcircuits
This work on photonic crystal devices is motivated by their use in photonic inte-
gratedcircuits. Photonicintegratrationisanalogoustotheintegrationofelectronic
devicesonasiliconchip. Typicalmicroprocessors containontheorderof10
9
tran-
sistors inanareaontheorderof1cm
2
. Suchdensedeviceintegration hasresulted
in microprocessors with exceptional functionality. And because the devices share
a common substrate and metal wiring network, they can be mass produced with
limited overhead costs. Due to its exceptional functionality coming at a low cost,
theelectronicintegratedcircuithasbecomeaubiquitoustechnologicalcomponent.
Similar to electronic integrated circuits, photonic integrated circuits are useful
for any application in which a large number of devices need to be contained in a
confined space. Photonic integrated circuits haveavarietyof applications includ-
ing telecommunications, sensing and imaging. In this Thesis, I will be discussing
devices to be used primarily in telecommunication systems where photonic inte-
grated circuits havethe potential for lowercost systems, improved reliability and
improvedbandwidth. Fig.1.1showsaschematicdiagramofaphotonicintegrated
circuit that includes on-chip sources, modulators, filters and detectors integrated
on a single chip. Although these devices currently exist as discrete components,
1
Figure1.1: Schematicdiagramofaphotonicintegratedcircuit[Koc08].
buildingthemonasinglechipcanreducepackagingcosts andpreventalignment
errors.
Anotherapplicationofphotonicintegratedcircuitsincludeopticalbusesinmul-
ticore computerarchitectures. Becausetransistor scalingisbeginningtoapproach
its physical limits, microprocessor performance improvement is being achieved
by including multiple processor cores which communicate with each other and
the off-chip memory through a bus made up of electrical transmission lines. For
applicationswithheavymemoryaccess,thebusbandwidthcanbeaperformance
bottleneck. Optics has the potential to improve memory access bandwidth due
in part to its ability to transmit signals at multiple wavelengths through a single
waveguide. It also has the advantage of operating at a lower temperature due to
absenceofresistiveheating.
Photonic integrated circuit technology has only recently began to emerge in
industrial settings. One problem that has hampered progress in integrating pho-
tonic devices isthe lackof asingle devicethatcanbeusedtobuild morecomplex
functionality the way that the transistor has been used in electronic circuits. This
has made reaching a consensus on a particular material system or fabrication
technology difficult, as each device may be optimized with a certain fabrication
2
Figure1.2: Photograph oftheBlazarproductfromLuxteraInc.
procedure. Oneexampleofthisissueistheproblemofintegratingalasermadeof
activematerialwithalowloss waveguidemadeofpassivematerial. Ontheother
hand, Luxtera Inc. of Carlsbad, California recently won the 2009 Best in Show
PrismAwardforPhotonic InnovationwithitsBlazarproduct. TheBlazarconsists
ofafiberoptictranseiverandaCMOSchipenablingfiber-to-the-chipconnectivity
tobeusedinconnectingmassivelyparallelcomputersystems. Withtheincreasing
commericaldeploymentoffiber-to-the-home technologyfordataandmultimedia
communication, integrated photonic microsystems are sure to play an increasing
roleinreliablelowcostandhighbandwidthapplications.
1.2 Two-dimensionalphotoniccrystals
One solution to the problem of lacking a common basic building block for pho-
tonicintegrationisthephotoniccrystal. Atwo-dimensionalphotoniccrystalisany
structure possessing two-dimensional periodicityin itsoptical properties. Fig.1.3
displays a schematic diagram of a semiconductor slab perforated with a two-
dimensionalperiodicarrayofairholes. Lightincidentonthisstructure willexpe-
rience Bragg reflection from the material interfaces. Fig. 1.4displays the photonic
bandstructureassociatedwiththegeometryinFig.1.3forTE-likepolarization. In
3
Figure 1.3: Schematic diagram of atwo-dimensional photonic crystal definedin a
semiconductorslab.
Fig. 1.4therightvertical axis isinnormalizedfrequencyunits. Whenconsidering
lineardielectricmaterials,Maxwell’sequationsarelengthscaleinvariantmeaning
thatMaxwell’sequationsyieldthesamesolutionwhetherworkingonananometer
lengthscaleoronakilometerscalesolongasthewavelengthsandfrequenciesare
scaled appropriately (assuming the material properties are not dispersive). This
allows one to work with normalized length units. In Fig. 1.4, the frequency is
normalizedtothelatticeconstanta. Therightverticalaxishasbeendenormalized
usingalatticeconstantofa= 400nm.
In Fig. 1.4 there is a range of normalized frequencies in which there are no
propagating modes for any in-plane wavevector. This is known as a photonic
bandgap and corresponds to the range of frequencies that will be reflected by
the lattice. It is apparent that for a = 400nm, the low loss (λ = 1.5μm) and
low dispersion (λ = 1.3μm) fiber transmission wavelengths fall into the photonic
bandgap.
4
Figure 1.4: Photonic band diagram for a two-dimensional photonic crystal calcu-
latedusingthethree-dimensionalfinite-differencetime-domainmethod. cdenotes
free space speed of light. a denotes the lattice cosntant. The hole radius to lat-
tice constant ratio is r/a = 0.29 and the slab thickness to lattice constant ratio is
d/a= 0.6. Theindexoftheslabwassetton= 3.4. Thediagramdepictsthelowest
three bands for the TE-like modes of the slab. The beige shaded regions denote
thelightconeprojection ontothecorrespondingpropagationdirections. Theinset
showstheregionofthefirstBrillouinzonedescribedbythedispersiondiagram.
5
Figure 1.5: (a) Photograph of a peacock. (b) Scanning electron micrograph of a
peacockwingshowingmicrostructurewithtwo-dimensionalperiodicity. Adapted
from[ZYL
+
03].
Before discussing the technology aspects of photonic crystal structures, it is
interestingtopointoutthatnaturehastakenadvantageofthereflectiveproperties
ofphotoniccrystals. Fig.1.5showsthatpartsofpeacockwingshavemicrostructure
withtwodimensionalperiodicity[ZYL
+
03]. Thephotonic crystal hereisactingas
a frequency selective mirror, reflecting only those wavelengths that fall in the
bandgap of the lattice. The directional dependence of the reflectivity lends an
irridescenteffecttothecolorationwhichhastheusualbiologicalpurposesofscaring
awaypredatorsandattractingmatingpartners.
6
Figure 1.6: H
z
(x,y,z = 0) field distribution of a single line defect photonic crystal
waveguideobtainedusingthefinite-differencetime-domainmethod.
Figures1.6and1.7displaylinearandpointdefectsintroducedintotheperiodic
lattice in Fig. 1.3. Also shown are the H
z
(x,y) components of the electromagnetic
eigenstates supported by the various defects. It is clear in both figures that the
fieldamplitudeisconfinedtothedefectregionduetothepresenceofthebandgap
introduced by the surrounding periodic lattice. In Fig. 1.6, the linear defect acts
as a waveguide that can transmit and guide optical signals in plane. In addition
tosimplytransmittingopticalsignals,photoniccrystalwaveguidescanbeusedin
modulators and slow wave structures. In Fig. 1.7, the three missing hole defect
behaves as an optical cavity confining the light at the resonant frequency. Such a
structure canbeusedtobuildlasers,filtersandopticalbuffers.
1.3 Photoniccrystalintegratedcircuits
Photoniccrystalsofferaversatileplatformforrealizingavarietyofdifferentdevices
by local rearrangements of the hole pattern as shown in Fig. 1.6 and 1.7. It is this
7
Figure 1.7: H
z
(x,y,z = 0) field distributions for the six modes associated with
a photonic crystal cavity formed by removing three neighboring holes obtained
usingthefinite-differencetime-domainmethod.
8
Figure 1.8: Schematic diagram illustrating a photonic crystal based photonic inte-
gratedcircuit.
flexibilitythatmakesphotoniccrystalsaviablesolutiontoestablishingacommon
technologyforphotonicintegration. Furthermore,devicesbasedonphotoniccrys-
talstendtooccupysmallerspacesandhavemoreusefuldispersioncharacteristics
thandevicesbasedonindexguiding.
Figure 1.8 is a schematic diagram of a phototonic crystal based photonic inte-
grated circuit. This particular structure consists of a bus waveguide passing from
left to right carrying modulated optical signals at wavelengths λ
1
and λ
2
. First,
thesignalsencounterfrequencyselectivefilterswhichcouplethefilteredsignalto
an optical detector. On-chip lasers operating at λ
1
and λ
2
generate a new carrier
beam which is modulated and rerouted to the bus waveguide. The input and
output ports could lead to other on-chip processing or to coupled optical fibers.
Fig.1.8suggests thatphotoniccrystalbaseddevicesareapromisingcandidatefor
a photonic integrated circuit technology due to their ability to realize a variety of
functionalityinasmallspace.
9
Much of the research on photonic crystal based photonic integrated circuits is
still on the individual device level. In this Thesis, I will be discussing resonant
cavities tobe used in on-chip lasersources. In addition to theirability to realizea
varietyofdifferentdevicesbysimplerearrangementsoftheholepatterns,photonic
crystals offer a high degree of freedom in designing any one device. The cavity
in Fig. 1.7, which consists of removing three neighboring holes from a uniform
triangular lattice, represents just one geometry for realizing an optical resonator.
A photonic crystal cavity can be realized just as well by removing only one hole
or five holes. Fig. 1.9 displays four cavity designs as well as the evolution of
their quality (Q) factors over the passed decade. The Q factor is a figure of merit
usedtoquantifytheradiationlossesofanopticalresonatorandwillbethesubject
of subsequent Chapters. Early photonic crystal cavities were formed by remov-
ing a single hole from a uniform lattice [PLS
+
99, RKP
+
02]. More recently, linear
defects have been shown to have higher Q factors than single missing hole cav-
ities [AASN03, AASN05], and the photonic crystal double heterostructure cavity
hasbeenshowntohavethelargestQfactoramongtwo-dimensionalphotoniccrys-
talcavities[SNAA05,TAN08]. BecauseofitsexceptionallyhighQfactorandsmall
mode volume, the photonic crystal double heterostructure has beenthe subject of
intense research for building efficient chip scale optical sourcs in our group and
willbethesubjectofChapters4and5inthisThesis.
1.4 Numericalanalysisofphotoniccrystals
Althoughthehugedegreeoffreedomindesigningphotoniccrystaldevicesmakes
them versatile components for photonic integration, it comes with an increased
difficulty in predicting the theoretical behavior of a given structure. Maxwell’s
10
Figure1.9: Qualityfactorfordifferenttwo-dimensionalphotoniccrystalcavitiesas
afunctionoftime. Datapointsandfigurescomefrom[PLS
+
99,RKP
+
02,AASN03,
ZQ04,NB06,AASN05,SNAA05,ASN06,TAN08]
11
Figure1.10: Illustration ofdiscretesamplingofafunctionu(x).
equations govern the electromagnetic properties of a material geometry and can
be solved in closed form only for a small set of problems. The complicated and
often subtle electromagnetic properties associated with the two-dimensional hole
patterndemandanumericalapproachtoobtainingtheirelectromagneticfeatures.
Fig. 1.7 is an example of the complex electromagnetic behavior associated with
photonic crystal cavities. The cavity shown in Fig. 1.7 supports six modes whose
frequencies lie in the bandgap of the surrounding lattice. It would not have been
possibletoobtainthenumberofmodes,theirresonantfrequenciesortheirspatial
fielddistribution withoutanumericalsolution.
The field distributions shown in Fig. 1.6 and Fig. 1.7 were calculated using
thefinite-difference time-domainmethod forthesolution toMaxwell’s equations.
SeveralnumericalmethodsforsolvingMaxwell’sequationsexist. Someexamples
include the finite-element method [Kim04], transmission line method [BSB
+
05],
scatterning based methods [PRM98, Sad00] and plane wave expansion methods
for periodic structures [JMW95, Sak01]. In this work, we will be using the finite-
differencetime-domain method due to its generality, simplicity andlinearscaling
withproblemsize.
The finite-difference time-domain method is an explicit discretization of
Maxwell’scurlequations[Yee66,TH00]whichareshowninEqs.1.1and1.2.
12
∂
~
B
∂t
= −∇×
~
E (1.1)
∂
~
D
∂t
=∇×
~
H−
~
J (1.2)
∇·
~
D= 0 (1.3)
∇·
~
B= 0 (1.4)
Fig. 1.10 illustrates the discretization scheme. The continuous field distributions
are sampled on a discrete grid. The first order spatial derivatives resulting from
the curl operation are approximated by finite differences. In the finite-difference
time-domain method, second order accuracy is achieved by using a three point
differencing scheme. Eq. 1.5 and Eq. 1.6 represent second order Taylor series for
obtainingthedistributionofsomefieldcomponent(labeledu)atx
i
±Δ. Theresult
of subtracting Eq. 1.6 from Eq. 1.5 is shown in Eq. 1.7. Eq. 1.8 shows the discrete
approximationofthecontinuousfirstorderderivativethatresultsinsecondorder
accuracy. Eq. 1.8 indicates that the first order spatial derivative located halfway
betweentwopointsonthegridcanbeevaluatedtosecondorderaccuracybyusing
theimmediatelyneighboringpoints.
u(x
i
+Δ)= u(x
i
)+
du
dx
Δ+
d
2
u
dx
2
Δ
2
2
+O(Δ
3
) (1.5)
u(x
i
−Δ)= u(x
i
)−
du
dx
Δ+
d
2
u
dx
2
Δ
2
2
+O(Δ
3
) (1.6)
13
u(x
i
+Δ)−u(x
i
−Δ)= 2
du
dx
Δ+O(Δ
3
) (1.7)
du
dx
=
u(x
i
+Δ)−u(x
i
−Δ)
2Δ
+O(Δ
2
) (1.8)
TheresultofapplyingthesecondorderdifferencingschemetoMaxwell’sequa-
tions in three dimensions is shown in Fig. 1.11and is known as a Yee cell [Yee66].
The electric and magnetic fields are spatially staggered in three dimensions. In
addition to illustrating the second order accuracy in the spatial derivatives, the
results of Fig. 1.11 must be taken into account when assigning a spatially vary-
ing geometry to the three dimensional grid. In particular, if a boundary between
twodifferentmaterialscutsthrough acell,thefieldcomponents mustbeassigned
proper material coefficients. This is especially important in working with curved
boundariesasisthecasewithphotoniccrystal structures.
OncethespatialderivativesassociatedwiththecurloperationinEqs.1.1and1.2
have been evaluated according to Eq. 1.8, the electric and magnetic fields are
temporally updated. Inspection of Eqs. 1.1 and 1.2 indicates that the magnetic
fields are updated using spatial derivatives of the electric fields, and the electric
fieldsareupdatedusingspatialderivativesofthemagneticfields. Thisrelationship
imparts an ordering to the temporal update. This is illustrated in Eq. 1.9 which
is equivalent to Eq. 1.8 except the derivative is taken with respect to time instead
of space. In Eq. 1.9 the left side is obtained from the curl operator in accordance
withMaxwell’sequations. Ifurepresentsanelectricfieldcomponent,thentheleft
side of Eq. 1.9 is expressed in terms of the appropriate component of the curl of
the magnetic field and vice versa. u(t
n
−Δ) corresponds to the field sample at the
previous time step, and u(t
n
+Δ) is the desired updated field value. From Eq. 1.9
14
Figure 1.11: Illustration of the spatial arrangement of the electric and magetic
fields at a single spatial point in the finite-difference time-domain method. This
arrangementisknownasaYeecell.
it is clear that the electric and magnetic fields are evaluated at staggered points
in time. This results in second order accuracy in the first order finite difference
approximationtothetemporalderivativesinMaxwell’sequations.
du
dt
=
u(t
n
+Δ)−u(t
n
−Δ)
2Δ
+O(Δ
2
) (1.9)
Thefinite-differencetime-domainmethodisageneraltoolfortheexplicitsolu-
tion to Maxwell’s curl equations. However, the discretized spatial and temporal
derivatives do result in unintended consequences that go beyond simple inaccu-
racies due to finite sampling of the geometry. The discretized grid introduces
numerical dispersion and anisotropy which is not a result of the geometry under
investigation [TH00]. The dispersion arises from the fact that plane waves with
different frequencies (and thus wavelengths) have a different number of spatial
15
samples per wavelength. This results in a different amount of error being associ-
atedwitheachfrequencycausingphasevelocitydispersion. Theanisotropyresults
from the fact that plane waves which propagate along the three Cartesian axes of
the computational domain experience a discretization of Δ whereas plane waves
that propagate along a diagonal experience a discretization of
√
3Δ. In this work,
thespatialdiscretizationissufficientlyfinethatissuesofnumericaldispersionand
anisotropyarenotimmediatelyapparent.
In the analysis of photonic crystal structures, the default grid density is set to
20pointsperlatticeconstant. Usuallyourtargetoperatingwavelengthis1500nm,
and a lattice constant of around 400nm is required to align the relevant spectral
features of ourdeviceswiththe targetwavelength. This griddensitycorresponds
to 75 points per wavelength in free space and 22 points per wavelength inside
the semiconductor slab with refractive index n = 3.4 which is sufficiently fine to
minimizenumericaldispersionandanisotropy. Becausephotoniccrystalstructures
employ subwavelength semiconductor features, the grid density is more often
dictated bythe smallest spatial feature in the device. A related issue concerns the
mappingofthecurvedsurfacesofthephotoniccrystalholesontoacarteseangrid.
Typicallythecurvedsurfaceisapproximatedbyastaircasingpatternresultingfrom
assigning a refractive index of 1.0 if the pixel lies inside the hole and a refractive
index of 3.4 if it lies outside of the hole. For coarse grids, the staircasing effect
can severely distort the shape of the hole and introduce unintended scattering.
One method to alleviate this issue is to employ averaging at material interfaces.
Insteadofassigningarefractiveindexvalueofeither1.0or3.4foracellthatfallson
theboundarybetweenairandsemiconductor, aproperlyweightedaveragecould
be employed to help smoothen the transition. In our code, we employ a simple
16
spatialaveragetoassignaboundarycellanintermediaterefractiveindexnumber;
although,moresophisticated schemesexist[FRR
+
06].
In order to avoid the possibility of superluminal propagation, the time step
cannotexceedthetimerequired foraplanewavetotravel from oneedgeofaYee
cell to the other. In one dimension this condition is given by Δt < Δx/c. If this
conditionisnotfulfilled,unstableandunphysicalsolutionstoMaxwell’sequations
aregenerated. Althoughusingasufficientlyfinegridcanreduceerrorsassociated
with finite spatial differences, it restricts the time step to correspondingly small
values. This often requires the calculation of many more time steps per oscilla-
tion period than is necessary for spectral analysis. An alternative formulation of
the finite-difference time-domain algorithm which casts Maxwell’s equations in
terms of aunitary propagator matrixhas beenproposed to overcome the stability
requirement[DRMKF03]. However,theoverallcomputationalperformanceofthe
algorithmremainscomparabletothestandardmethod. Anotheralternativeformu-
lation,basedonthealternating-direction-implicittechnique,hasbeenshowntobe
unconditionallystable andhasdemonstrated modestcomputational performance
improvementsoverthestandardmethod[ZCZ00].
One of the most important elements in the solution of any partial differential
equationisthespecificationoftheboundaryconditions. WhensolvingMaxwell’s
equations in dielectric structures, typical boundary conditions demand that the
fields decay to zero at infinity or are radiating with an appropriate spatial decay
rate. In the case of the finite-difference time-domain method, the computational
domain is made as small as possible to reduce the computer effort required to
update Maxwell’s equations. This implies that computational boundaries can be
withinafewwavelengthsofthestructure. Inordertomimicaninfinitelyextended
medium, absorbing boundary conditions are introduced. Electromagnetic fields
17
thatreachtheboundaryareabsorbedasiftheyhaveleftthemedium. Thisapproach
iscomplicatedbythefactthatperfectabsorptioncanonlybeachievedforincidence
at a certain angle and at a specified wavelength. This issue was overcome by
the use of perfectly matched layer absorbing boundary conditions [Ber94] which
theoretically completely absorb electromagnetic fields incident at any angle and
at any frequency. However, implementing the perfectly matched layer absorbing
boundaryconditionsonadiscretizedgridintroduceserrors. Inourcodeagraded
material profile is introduced to overcome these issues and 15 perfectly matched
layersareused. Powerreflectivitiesassmallas10
−4
aretypical.
Because the finite-difference time-domain method is a direct solution to
Maxwell’sequations,itcanbeappliedtoanyarbitrarygeometry. Theonlyapprox-
imationcomesfromthefinite-differenceapproximationtothespatialandtemporal
derivatives. The generality of the method makes it a good candidate for explor-
ingthe electromagnetic behavior of differentphotonic crystal structures. Another
advantage of the finite-difference time-domain method is that it scales linearly in
the problem size. Because the algorithm consists of a serial update of the dis-
cretized Maxwell’s equations, enlarging the compuational domain by a factor of
tworequirestwiceasmanymicroprocessoroperations. However,itshouldbekept
inmind thatalarger domainis often associated with phenomenahappeningon a
longertimescale,sothesimulationmayrequiremoretimestepsaswell. Theserial
update scheme also makes the finite-difference time-domain amenable to straight
forwardparallelizationusingasimpleCarteseandomaindecomposition.
Afinaladvantageofthefinite-differencetimedomainmethodisthatfrequency
domain information is often simply extracted from the time domain results. One
of the primary uses of the finite-difference time domain method will be to extract
the complex frequencies associated with optical resonators. The real part of the
18
complex frequency yields the temporal oscillation period. The imaginary part is
realated to the optical loss of the cavity. Explicit methods for extracting complex
frequencyfromtime-domainsimulationwillbethesubjectofChapter3.
1.5 Thesisoverview
In this Chapter, I have discussed the technological applications of the work to
be described in this Thesis. Specifically I introduced two-dimensional photonic
crystals as a technology platform for photonic integrated circuits. The modeling
of such structures is difficult, andthe finite-difference time-domain approach was
introducedasournumericalmethodofchoice. InthenextChapterIwilldefineand
derive the quantities of interest when doing electromagnetic analysis of dielectric
resonators. TheQfactorwillbedefinedandinterpretedonanintuitiveandphysical
level. Anovelderivationofthelaserthresholdwillbeprovided.
InChapter3Idiscussmethodsofspectralanalysisoftime-sequencesobtained
from finite-difference time-domain simulations. The discrete Fourier transform
is used to convert the time-domain data to the frequency domain. Interpolation
methods are introduced that allow for accurate spectral analysis in the presence
of limitedfrequencyresolution. Adirectextraction approach isintroduced which
allowsfortheaccurateanalysisofspectrawithcloselyspacedresonances.
Chapter4discussesthephotoniccrystaldoubleheterostructureresonantcavity
which is among the highest Q factor cavities with a mode volume on the order
of a cubic wavelength. Their numerically calculated resonance spectra are dis-
cussedandthree-dimensionalelectromagneticmodeprofilesarepresented. These
cavities are shown to support higher order bound states, and methods for mode
discriminationaresuggested. Onedrawbackoftwo-dimensionalphotonic crystal
19
structures is their susceptibility to out-of-plane loss. This issue is investigated in
Chapter5. The feasibility of growing or bonding active material to a heatsinking
lowersubstrateisdiscussed,andanewheterostructuredesignbasedonglide-plane
symmetryissuggestedtoreduceout-of-plane radiation.
Finally in Chapter 6 I present first steps toward building an electromagnetic
simulationtoolthatincorporatesamaterialgainmodel. Arudimentarydispersive
gain model is introduced, and the laser threshold is discerned by monitoring the
rate of growth or loss of the electromagnetic fields. It is shown that the direct
extraction approach developed in Chapter 3 allows for the efficient extraction of
the temporal growth and loss coefficients. The threshold gain is estimated and
compared to the analytical result derived in Chapter 2. They differ by less than
1%.
20
Chapter2
Electromagneticresonantcavity
analysis
2.1 Introduction
In the previous chapter, an overview of the finite-difference time-domain method
was presented. In this chapter, we focus on the specific problem of analyzing
electromagnetic resonant cavities and answer the question, given the ability to
solveMaxwell’sequationsnumerically,whatquantitiesareimportanttocalculate?
Because our primary interest is to use these resonant cavities in semiconductor
lasers, the relevent laser design parameters will be derived. In particular the
quality(Q)factorwillbedefinedasametricforcharacterizingtheelectromagnetic
lossofacavity,andtheopticalconfinementfactorwillbederivedwhichdescribes
theinteractionbetweentheopticalmodeandthegainmedium.
AnovelderivationfortheconditionforlaserthresholdintermsoftheQfactor
andopticalconfinementfactorwillbederivedstartingfrom Maxwell’sequations.
The first theoretical discussion of lasers stated that oscillation threshold could
be reached when the optical losses of the lasing mode were compensated by an
external energy source [ST58]. In [ST58] it was stated that the basic threshold
condition could be derived from Maxwell’s equations; however, the steps of this
derivation have yet to be presented. In addition to its theoretical elegance, this
21
derivation has a pragmatic motivation which is to obtain the correct method for
calculatingthecollectionefficiencyandopticalconfinementfactorusingtheresults
fromthree-dimensionalfinite-differencetime-domainsimulations.
2.2 Temporalbehaviorof fields, energyandpower in
aresonantcavity
TheinstantaneousPoyntingtheoremwillformthebasisofmuchofthediscussion
in this Chapter and is displayed in Eq. 2.1. It is a direct result of Maxwell’s
equations[Jac99]. InEq.2.1
~
S=
~
E×
~
HisthePoyntingvectorandisintegratedover
aclosedsurfacedenotedbyAthatenclosestheresonantcavityofinterest. Uisthe
total electromagnetic energycontained in thevolume enclosedbythesurface and
isgivenexplicitlyinEq.2.2. P
a
representspowerlosttoabsorptioninthematerial,
andP
s
representssuppliedpower.
I
~
S·d
~
A=−
∂U
∂t
−P
a
+P
s
(2.1)
U=
Z

1
2
ǫ
~
E·
~
E+
1
2
μ
~
H·
~
H

dV (2.2)
TheQfactorisacommonlyusedmetricthatdescribesthetemporallossrateof
aresonantcavity. Itisdefinedby
Q= ω
0
U
−
dU
dt
. (2.3)
where U is given by Eq. 2.2. Eq. 2.3 can be considered a first order ordinary
differentialequationinU. Itssolution is
22
U(t)= U(0)exp(−ω
0
t/Q). (2.4)
Equation 2.4 tells us that the temporal decayof the electromagnetic energy stored
in a large Q factor cavity will be slower than that of a low Q factor cavity. From
Eq. 2.4 a photon lifetime can be defined as τ
p
= Q/ω
0
. This quantity is related to
the average length of time that a photon spends inside a cavity. It is clear that a
largeQfactorresultsinalongphotonlifetime.
If Eq. 2.4 describes the temporal behavior of the electromagnetic energy in
the cavity, then the temporal behavior of the individual field components will be
describedby
f(t)= f
0
exp(−ω
o
t/2Q)cos(ω
0
t). (2.5)
Theindividualfieldcomponentswillexhibitatemporaldecayathalftherateofthe
decay in Eq. 2.4 and will oscillate at the electromagnetic resonance frequency. In
Eq.2.4thetotalenergyhasatemporalresponsedescribedbyadampedexponential
function. However,fromEq.2.2onewouldexpectanoscillatorytemporalbehavior
attwicetheopticalfrequency. Ifweconsiderahypotheticalpassivecavityinwhich
thereisnosuppliedorabsorbedpower,thenEq.2.1becomes
I
~
S·d
~
A=−
∂U
∂t
. (2.6)
Inapurelypassivecavity,theenergydecreaseasafunctionoftimeisduetoradia-
tionleakagethrough theclosedsurface. Theelectromagnetic energyisradiatedin
theformofplanewavespropagatingoutwardawayfromthecavity. Theradiating
plane waves that pass through the closed surface are oscillating at the resonance
23
frequency. This indicates that the rate of energy leakage is not simply a damped
exponential but varies in relation to the oscillation frequency. However, it is true
that the rate of energy leakage will be the same at times separated by one optical
period. Thissuggeststhatperformingaoneopticalperiodtemporalaverageofthe
relaventenergyquantitieswouldremovetheeffectsofoscillationattheresonance
frequency. Toillustratetheimportanceoftime-averagingEq.2.6inordertoobtain
meaningful results, imagine evaluating the Poynting vector when either the tem-
poral response of the electric field or magnetic field passes through zero. At this
instant the Poynting vector is zero, but this does not imply the absense of optical
powerexitingthecavity. Thetruequantityofinterestisthetime-averagePoynting
vector. This quantity will not be zero unless zero energy is radiating through the
closedsurface.
FortheremainderofthisChapterwewillworkwiththetime-averagedversions
ofEqs.2.1and2.3. Specifically,
I
h
~
Si·d
~
A= −h
∂U
∂t
i−hP
a
i+hP
s
i (2.7)
Q= ω
0
hUi
−h
dU
dt
i
. (2.8)
where
hf(t)i=
1
T
Z
T
0
f(t
′
)dt
′
(2.9)
whereT corresponds toanintegermultipleofopticalperiods.
24
2.3 Derivationoflaserthresholdcondition
If we consider a hypothetical passive cavity in which there are no material losses
orgain, thenwe cansethP
a
i= hP
s
i= 0in Eq.2.7andsubstitute Eq.2.8forh
∂U
∂t
i in
Eq.2.7resultingin
I
h
~
S
m
i·d
~
A= ω
0
hU
m
i
Q
p
. (2.10)
Equation 2.7 is a general relationship that applies to an arbitrary electromagnetic
field. However, Eq. 2.8 is true for a particular resonant mode m with frequency
ω
0
andqualityfactorQ.WhereasEq.2.8expressestherelationshipbetweentheQ
factor and the temporal energy decay, Eq. 2.10 expresses the relationship between
the Q factor and the spatial radiation properties of the optical mode. The Q factor
in Eq. 2.10 has been labeled with a subscript p to distinguish it as the passive Q
factor.
WhenmaterialabsorptionisincludedinEq.2.7,onecanuseEq.2.10toobtain
ω
0
hU
m
i
Q
p
=h
∂U
∂t
i−hP
a
i. (2.11)
WecanuseEq.2.8forh
∂U
∂t
iinEq.2.11exceptwewilllabelQasQ
tot
todistinguishitas
atheQfactorcharacterizingthelossesarisingfrombothradiationandabsorption.
ω
0
hU
m
i
Q
p
= ω
0
hU
m
i
Q
tot
−hP
a
i. (2.12)
If we define Q
a
according to ω
0
hU
m
i
Q
a
= hP
a
i, then ω
0
and U
m
cancel from both sides
of Eq. 2.12 showing that Q
p
and Q
a
contribute to Q
tot
of the cavity according to
1
Q
tot
=
1
Q
p
+
1
Q
a
. ThetotalQfactorofacavityiscomposedofQfactorscharacterizing
25
different loss mechanisms in the cavity. The dominant loss mechanism will have
thelowestQfactorandwilllimitthetotalQoftheresonator.
Asdiscussedin[ST58],thefundamentalprinciplebehindlaserthresholdisthat
theopticallossesshouldbeoffsetbyasuppliedenergysource. Whenthiscondition
is met, the energy inside the cavity oscillates with an amplitude constant in time
andh
∂U
∂t
i= 0. FromEq.2.7,Eq.2.10andEq.2.12oneobtains
ω
0
hU
m
i
Q
tot
= hP
s
i (2.13)
which says that in order to reach laser threshold the total rate of energy loss
defined by Q
tot
should be matched by a supplied power source. It is apparent
thatindesigningalasercavity,thecavityQfactorcharacterizingthelossesshould
be made sufficiently large so as to reduce the amount of power required to reach
threshold.
Ifamaterialpossessesaconductivity,thenwhenanelectromagneticplanewave
passes through the material, the electric field will generate a current according to
Ohm’s law,
~
J = σ(ω)
~
E. The power associated with the current generation is given
by
P
a
=
Z
~
J·
~
EdV=
Z
σ
~
E·
~
EdV. (2.14)
TheplanewavelossandpropagationconstantsaregiveninEq.2.15andEq.2.16
whereaplanewavesolutionoftheforme
jβx
e
−αx
isassumed.
α=
σ
√
2cǫ
0

ǫ
r
+
r
ǫ
2
r
+(
σ
ωǫ
0
)
2

−
1
2
(2.15)
26
β=
ω
√
2c

ǫ
r
+
r
ǫ
2
r
+(
σ
ωǫ
0
)
2

1
2
(2.16)
In order to simplify the following steps, we assume that the material is low
loss[HJT96],sothat
σ
ωǫ
0
≪ ǫ
r
issatisfied. InthiscaseEq.2.15andEq.2.16simplify
to
α≈
σ
2cǫ
0
n
(2.17)
β≈
ω
c
n (2.18)
where n =
√
ǫ
r
. One important aspect of these results is that the plane wave loss
coefficientαisproportional toσ. Thisallowsustowrite
P
a
=
Z
σ
~
E·
~
EdV=
Z
2cǫ
0
nα
~
E·
~
EdV. (2.19)
Equation 2.19 indicates that if the amplitude loss coefficient (α) of a material is
known, then we can calculate the absorbed power by integrating the portion of
the electric field spatial energy distribution that overlaps the absorbing material.
Insemiconductor laserdevices,thelosscoefficientα canbespatiallyvarying, and
Eq. 2.19 tells us that significant absorption occurs only where there is significant
electricenergydensityandlargematerialabsorptioninthesamespatiallocation.
Substituting Eq.2.12fortheleftsideofEq.2.13resultsin
ω
0
hU
m
i
Q
p
+hP
a
i=hP
s
i (2.20)
27
which states that the supplied power must compensate for radiation losses char-
acterized by Q
p
and for loss due to material absorption described by P
a
. It is
interesting to note that P
a
and P
s
contribute to Eq. 2.20 with different signs. Cer-
tainlythisresultsfromthefactthatP
a
representspowerlost,whereasP
s
represents
power supplied. The interesting point is that P
s
can also be attributed to current
generatedinthematerial;however,nowthematerial,nottheelectromagneticfield,
generates the current which then imparts energy to the field. Following Eq. 2.19
wemaywriteP
s
as
P
s
=
Z
σ
~
E·
~
EdV=
Z
2cǫ
0
ng
~
E·
~
EdV. (2.21)
where a gain coefficient g has been introduced and will have the same effect on
planewavesolutionsasthelosscoefficientαexceptwithoppositesign,sothatthe
fieldswillgrowalongthex-directionaccordingtoe
jβx
e
gx
. Intermsoftheproperties
ofthematerial,becauseP
s
andP
a
haveoppositesignsinEq.2.20,theconductivity
σ appearing in Eq. 2.21 is negative in value [And65]. At this point, assigning a
negative value for the conductivity is only an intermediate step for introducing
a gain coefficient g. However, later in Chapter six we will see that gain can
be introduced into finite-difference time-domain simulations through a negative-
valuedconductivity[HJT96]. SubstitutingEq.2.19andEq.2.21intoEq.2.20yields
thethresholdcondition
ω
0
Q
p
+
R
(
c
n
)2α(x,y,z)ǫh
~
E·
~
EidV
R

1
2
ǫh
~
E·
~
Ei+
1
2
μh
~
H·
~
Hi

dV
=
R
(
c
n
)2g(x,y,z)ǫh
~
E·
~
EidV
R

1
2
ǫh
~
E·
~
Ei+
1
2
μh
~
H·
~
Hi

dV
. (2.22)
28
In Eq. 2.22, the left term represents radiation loss per unit time. The middle term
representsabsorption. Thefactorc/nconvertstheunitsofthelosscoefficientfrom
perunitlengthtoperunittime. Therighttermrepresents gain,andthefactorc/n
convertstheunitsofthegaincoefficientfromperunitlengthtoperunittime. The
factorsoftwothatmultiplyαandgintherighttwotermsresultfromthedefinition
of α and g as amplitude loss and gain coefficients as opposed to power loss and
gaincoefficients.
Equation 2.22 is the laser threshold condition. It states that the material gain
shouldhavesufficientmagnitudeandoverlapwiththeelectricfieldenergydistri-
bution tocompensate forradiative andabsorption losses sufferedbythe resonant
mode. The middle term is known as the modal absorption loss, and the right
term is known as modal gain. These two terms respresent the specific gain and
loss that a particular cavity mode will experience in the presence of a given spa-
tial loss and gain distribution. There are two interesting features of Eq. 2.22 that
haveimmediateconsequencesfornumericalanalysis. First,alloftheenergyquan-
tities appear as time-averages. Evaluating Eq. 2.22 using field profiles from a
finite-difference time-domain simulation taken at an arbitrary time step will give
erroneous results. The time-average is crucial for reasons discssed in the previ-
ous section. Second, the denominators in the right two terms in Eq. 2.22 include
both the electric andmagnetic field energies–notjust theelectric fieldenergyas is
oftenreported[And65,HPW96,VBDL97,RKC02,RPPL08]. However,becausethe
electric field energy in the numerators and denominators of the right two terms
in Eq. 2.22 have the same temporal behavior, one could use the alternative form
shown in Eq. 2.23. In this case the time functions in the numerator and denomi-
natorcancel,andtheissuesrelatedtotemporalbehaviordiscussedintheprevious
Section are avoided. Although Eq. 2.23 represents a logistically simpler method
29
of evaluatingthe threshold condition, evaluating the fields when the electric field
timefunctionpasseszeroispossibleresultinginanindeterminantform. Although
in a numerical calculation it is unlikely to evaluate the fields exactly at zero, the
possibility of evaluating them close to zero could introduce significant errors due
tolimitednumericalprecision.
ω
0
Q
p
+
R
(
c
n
)2α(x,y,z)ǫ
~
E·
~
EdV
R
ǫ
~
E·
~
EdV
=
R
(
c
n
)2g(x,y,z)ǫ
~
E·
~
EdV
R
ǫ
~
E·
~
EdV
. (2.23)
Fundamentally Eq. 2.22 is not a new result. The idea that that in order to
reachlaserthresholdtheopticallossesmustbecompensatedbyanexternalenergy
source was stated in [ST58]. The utility of our derivation is twofold. First, we
present the explicit steps to obtain the laser threshold condition starting directly
from Maxwell’s equations. Second, Eq. 2.22 tells us exactly what parameters we
needtoestimatelaserthresholdfromnumericalsolutionstoMaxwell’sequations.
It should be noted that the threshold condition in Eq. 2.22 takes into account
material gain in the form of stimulated emission only. In a real gain medium,
emission into the lasing mode originates from both stimulated and spontaneous
transitions. A more complete model would include a second energy source on
the right hand side of Eq. 2.22 representing spontaneous photon emission. The
magnitude of this term depends on several factors including the spontaneous
lifetime of the material, the volume of the active region, the optical mode volume
andtheQfactor. Forthequantumwellmaterialthatisusedinourgroup tobuild
photonic crystal lasers, the effect of spontaneous emission is insignificant for Q
factors less than about 10,000. Although much of this Thesis will be concerned
with photonic crystal cavities with Q factors significantly larger than 10,000, in a
realtechnologyapplication,theQfactorwillbeintentionallylowered(or“loaded”)
30
inordertoextractsufficientoutputpowerinapreferreddirection. LoadedQfactors
ontheorderofafewthousandaretypical[LMY
+
09].
31
Chapter3
QualityFactorEstimationUsingPad´ e
Interpolation
3.1 Introduction
In the previous Chapter the role of the quality (Q) factor in characterizing the
radiation loss was derived. The Q factor is one of the primary metrics used in
assessingwhetheranovelresonantcavityissuitableforlasinginasemiconductor
gain medium. In this Chapter we discuss methods for estimating the Q factor
fromfinite-differencetime-domainsimulations. InparticularwefocusonthePad´ e
interpolation method and discuss a novel technique for expressing the Q factors
andresonantfrequenciesintermsofthePad´ efittingparameters. Thismodification
allowsfortheextraction ofboth polesandzerosdistributed anywhereinthetwo-
dimensionalcomplexfrequencyplane.
3.2 EstimatingtheQfactorfromfinite-differencetime-
domainsimulations
One of the primary data sources resulting from a finite-difference time-domain
simulation is a time sequence representing the discrete time evolution of a rep-
resentative field component at one or more locations of interest in the structure
32
Figure3.1: Timesequenceobtainedfromathree-dimensionalfinte-differencetime-
domain simulation of a photonic crystal double heterostructure resonant cavity.
The sequence is obtained by storing the discrete time evolution of H
z
(t) at a point
nearthecenterofthecavity.
under analysis. Fig. 3.1 displays a typical time sequence obtained from a three-
dimensional finite-difference time-domain simulation of a photonic crystal res-
onant cavity. This time sequence resulted from a broad band initial condition
initiatedbyassigninganarbitrarystartingamplitudetoaspatiallyrandomdistri-
bution of cells in the computational domain. The goal is to extract the resonant
frequenciesandQfactorsofthecavityresonancesfromthisdata.
In Fig. 3.1 the bottom axis indicates the number of time steps obtained in
this simulation which is typical for the simulations that will be discussed in the
remainder of this Thesis. The top axis displays the corresponding real time axis
corresponding to these time steps. For optical frequencies in the near infrared
region,thissimulationcorresponds toapproximiately1,000opticalperiods.
33
In the previous Chapter it was determined that the temporal evolution of the
electromagneticfieldscorrespondingtoacavityresonantmodewithfrequencyω
0
and quality factor Q was given by Eq. 2.5. Inspection of Fig. 3.1 shows that the
generaltrendofthefieldamplitudeisqualitativelysimilartoadampedsinusoid. It
shouldbepointedoutthatthe“noisy”appearanceofthefieldamplitudeinFig.3.1
is not noise, as this simulation is completely deterministic. Because we excite the
modesofthecavity withabroadbandinitialcondition, modeswith awiderange
offrequenciesareexcitedandaresuperposedtoformthetimesequenceofFig.3.1.
The“noise”inFig.3.1issimplythecontribution fromhighfrequencymodes.
GiventhatthedatainFig.3.1consistsofasuperpositionofdampedsinusoidsof
theformEq.2.5,oneapproachtoextractingtheQfactorsandresonantfrequencies
ofthevariousmodeswouldbetofitaseriesofdampedsinusoidstothisdata. One
populartechniqueforperformingthisfitisknownasfilterdiagonalization[WN95,
MT97a,MT97b,CG99].
Alternatively, one can work with the discrete Fourier transform of the data in
Fig. 3.1 and note that the Fourier transform of a damped sinusoid is a Lorentzian
function. Specifically, if Eq. 2.5 represents the time sequence associated with a
typicalcavityresonance,thenitsFouriertransform willhavetheform
F(ω)=
f
0
/2
ω
0
2Q
−i(ω−ω
0
)
. (3.1)
One may then extract the Q factor by measuring the full width at half maximum
of each resonant peakaccording to Q = ω
0
/Δω
fwhm
[CH86, ZZX
+
08]. The discrete
FouriertransformofthedatafromFig.3.1calculatedbyapplyingtheFastFourier
Transform method is shown in Fig. 3.2. Using the discrete Fourier transform
has the advantage that the resonance peaks are clearly visible from the raw data,
34
Figure 3.2: Discrete Fourier transform of the time sequence in Fig. 3.1. c denotes
freespacespeedoflight. adenotesthelatticecosntant.
whereas inspecting Fig. 3.1 does not provide any information from the outset.
Furthermore,thefittingusingthePad´ emethodtendstobesimplerandfasterthan
filter diagonalization, and we will show that our implementation of Pad´ e fitting
reliablyextractstheQfactorwithfewertimestepsthandoesfilterdiagonalization.
Whether working in the time-domain or in the frquency domain, one of the
mainchallengeswithreliablyextractingtheQfactorsisthatthetimesequencehas
a finite duration. In the time-domain this results in having access to only a small
windowoftimeinwhichtofitafunctionthat,inprinciple,hasinfiniteduration. In
thefrequencydomain, this results in limitedresolution, sothatreliablyextracting
the true width of a resonance peak becomes difficult. In particular the frequency
spacing between adjacent frequency samples is equal to the inverse of the total
time length Δf = 1/NΔt. To illustrate these issues Fig. 3.3(a) illustrates a user
35
Figure 3.3: (a) User generated time sequence of the form Eq. 2.5. (b) Discrete
Fouriertransform ofthesequencein(a).
36
generated time sequence of the form of Eq. 2.5 with a Δt and ω
0
consistent with
what would be analyzedin a typical finite-difference time-domain calculation. In
Fig. 3.3(a) we have used a low Q factor of 50, so that the energy decay is clear.
Fig.3.3(b)displaysthediscreteFouriertransformofthetimesequenceinFig.3.3(a)
for varying lengths of the time sequence. It is clear that increasing the length of
the timesequenceresults in improved resolution inthe Fouriertransform. Ifonly
the first 2,500 time steps were used to compute the discrete Fourier transform,
then estimating the full width at half maximum would yield an inaccurate value.
IncreasingthenumberoftimestepsusedinthediscreteFouriertransformto10,000
improvestheresolutionbyafactorof4,andusingtheavailablefrequencysamples
to measure the full width at half maximum would yield a reasonable measure of
theQfactor.
A 10,000 time step sequence will yield a normalized frequency spacing of
4×10
−3
. AresonancewithaQfactorof50willhaveafullwidthathalfmaximum
of5×10
−3
. Thissuggeststhattheminimumfrequencyspacingrequiredtoresolve
a given full width at half maximum should be no larger than the full width at
halfmaximum. Thephotoniccrystaldoubleheterostructurecavitytobediscussed
in later chapters has been shown to have a Q factor in excess of 10
5
. In order
to resolve a Q factor of 10
5
from the raw data, one would need a resolution of
2.5×10
−6
which would require 2×10
7
time steps. As discussed earlier, currently
weuseontheorderof10
5
timestepsinourcalculationswhichalreadyrequireover
one hundred parallel processors running for ten to twenty hours. Attempting to
runourcalculationsfor2×10
7
timestepsisnotpractical.
37
One way to get around this issue is to use an interpolation scheme to obtain
information about the fully evolved spectrum from the available frequency sam-
ples. A popular method for performing this task is known as Pad´ e interpola-
tion [DM98, GLH01, Qiu05]. Pad´ e interpolation fits a ratio of polynomials to the
rawfrequencysamples.
F(ω
i
)=
α
0
+α
1
ω
i
+···+α
M
ω
M
i
β
0
+β
1
ω
i
+···+β
M
ω
N
i
= P(M,N) (3.2)
The ratio of polynomials is shown in Eq. 3.2 where F(ω
i
) is the ith frequency
sample, and the α
j
and β
j
are to be determined. P(M,N) labels a Pad´ e function
with a Mth order polynomial in the numerator and a Nth order polynomial in
the denominator. The fitting procedure starts by observing that Eq. 3.2 can be
multiplied by an arbitrary constant without affecting the fit. This allows us to
multiplythetopandbottomby1/β
0
. Afterrelabelingthedenominatorcoefficients
β
j
/β
0
asβ
j
onegets
F(ω
i
)=
α
0
+α
1
ω
i
+···+α
M
ω
M
i
1+β
1
ω
i
+···+β
M
ω
N
i
. (3.3)
ThenextstepmovesthedenominatorpolynomialtotheleftsideofEq.3.3resulting
in
(1+β
1
ω
i
+···+β
M
ω
N
i
)F(ω
i
)= α
0
+α
1
ω
i
+···+α
M
ω
M
i
(3.4)
andaftersomesimplerearrangementonegets
α
0
+α
1
ω
i
+···+α
M
ω
M
i
−(1+β
1
ω
i
+···+β
M
ω
N
i
)F(ω
i
)= F(ω
i
). (3.5)
38
Figure 3.4: Diagram showing discrete sampling points associated with a typical
resonancepeak.
Equation 3.5 is a linear equation in the α
j
and β
j
coefficients. There are a total
of M+ N+ 1 unknown coefficients, and to determine them uniquely, one needs
M+N+1equationsoftheformEq.3.5. ThisrequiresM+N+1frequencysamples.
Typically the M+ N+ 1 frequency samples are distributed symmetrically about
the resonance peak of interest. Fig. 3.4 displays the frequency samples that may
beincorporated intoaPad´ e fit. Once the M+N+1frequency sampleshavebeen
obtained, the α
j
and β
j
coefficients are obtained through matrix inversion. With
the Pad´ e expansion coefficients, Eq. 3.3 can be replotted with a high resolution
frequency array, and the full width at half maximum can be measured with the
desiredaccuracy.
3.3 DirectextractionapproachtoPad´ einterpolation
Although using the extracted α
j
and β
j
coefficients to replot Eq. 3.3 works well to
obtaintheQfactorandresonancefrequencyformostresonancepeaksofinterest,it
39
doessufferfromanumberofdrawbacks. First, byreplottingEq.3.3withahigher
resolutionfrequencyaxis,theusermustdeterminejusthowfinetheresolutionmust
be. Often thisrequires iteratingthrough increasinglyhighresolution spectrauntil
convergence is obtained which can be time consuming and error prone. Second,
if there are two closely spaced resonance peaks, it becomes difficult to explicitly
measurethefullwidthandhalfmaxduetotheoverlapofthetwospectralfeatures.
Finally, the Pad´ e method as described above is incapable of extracting the sign of
the Q factor. The Q factor may be positive or negative with a positive Q factor
correspondingtoopticallossandanegativeQfactorcorrespondingtoagrowthin
the optical energy that may result from optical gain. The importance of this final
pointwillbediscussedinalaterchapter.
An alternative approach [MO08a, MO09a] that addresses these shortcomings
starts by observing that a Lorentzian spectrum corresponds to a specific Pad´ e
functionconsistingofaratioofazerothorderpolynomialtoafirstorderpolynomial
as shown in Eq. 3.1. This observation motivates the use of a Pad´ e function of the
form
P(0,1)=
α
0
1+β
1
ω
=
−iα
0
/β
1
−i/β
1
−iω
=
−iα
0
/β
1
−i/β
1
−iω
0
−i(ω−ω
0
)
(3.6)
wherethePad´ efunctionhasbeenalgebraicallyrearrangedtobringitintotheform
ofEq.3.1.
Comparison between Eq. 3.1 and Eq. 3.6 suggests that one may express the Q
factorandresonancefrequencydirectlyintermsofthePad´ eexpansioncoefficients
accordingto
−i/β
1
−iω
0
=
ω
0
2Q
(3.7)
40
Eq. 3.7 is a single equation for the Q factor and resonance frequency. Rewriting
Eq.3.7as
1/β
1
+ω
0
= i
ω
0
2Q
(3.8)
indicatesthatQandω
0
maybeobtainedbyequatingtherealandimaginaryparts
ofEq.3.8accordingto
ω
0
=−Re{1/β
1
} (3.9)
ω
0
2Q
= Im{1/β
1
} (3.10)
Q=−
Re{1/β
1
}
2Im{β
1
}
. (3.11)
Equations 3.9 and 3.11 represent a method for the direct extraction of the Q
factorandresonantfrequencyfromthePad´ eexpansioncoefficientswhichremoves
the need to replot Eq. 3.3 with a high resolution frequency axis. One method for
evaluating the quality of a parameter extraction method is by determining the
minimum number of time steps required to extract a parameter set that does not
fluctuate as the time sequence is lengthened. As discussed earlier, as the time
sequence is lengthened, the spectrum more closely samples the true Lorentzian
function, andlessinformation isrequired from theinterpolation procedure. Once
the extracted paramter set ceases to fluctuate as the time sequence is lengthened,
theparametersethasconvergedtoastablevalue.
Figure 3.5 displays the convergence of the Q factor as the time steps used in
the discrete Fourier transform range from 5,000 to 200,000 for a discrete Fourier
41
Figure 3.5: Q factor convergence study of the Pad´ e method using a P(0,1) Pad´ e
function.
transform of ausergenerateddampedsinusoid timesequence. Table3.1displays
the convergence of the normalized resonant frequency where Δ stands for the
difference between the user defined frequency and the extracted frequency. The
userdefinednormalisedfrequencywassetto0.26495797forthisstudy. Theresults
inTable3.1showthatthenormalizedfrequencyisalreadyaccurate tosixdecimal
places after only 5,000 time steps which is sufficiently accurate for comparison
to experimental lasing spectra. It is apparent that the Q factor stabilizes much
moreslowlythantheresonantfrequencyandisthusthelimitingparameterwhen
determiningconvergenceoftheoverallextractionprocedure. FromFig.3.5wesee
that as the Q factor increases, more time steps are required to obtain a stable Q
factor which is expected due to the narrowing of the Lorentzian peak width with
increasingQ.ForaQfactorof10
6
,theQfactorhasnotcompletelystabilizedeven
at200,000timesteps.
ItshouldbenotedthatinfittingthePad´ efunctioninEq.3.6totherawdiscrete
Fourier transform data one uses only M+N+1 = 0+1+1 = 2 discrete Fourier
42
TimeSteps Q= 10
4
Q= 10
5
Q= 10
6
(×1000) Δ Δ Δ
5 −9.4×10
−6
9.5×10
−6
−8.9×10
−6
10 −4.8×10
−7
−5.8×10
−7
−5.9×10
−7
20 1.9×10
−6
1.9×10
−6
1.9×10
−6
30 1.8×10
−7
1.6×10
−7
1.6×10
−7
40 −6.4×10
−8
−7.2×10
−8
−7.3×10
−8
50 −5.5×10
−8
−6.8×10
−8
−6.9×10
−8
60 2.1×10
−7
1.9×10
−7
1.9×10
−7
70 7.2×10
−8
6.6×10
−8
6.5×10
−8
80 −2.4×10
−8
−3.0×10
−8
−3.1×10
−8
90 −2.2×10
−8
−2.8×10
−8
−2.9×10
−8
100 5.7×10
−8
−1.3×10
−7
5.5×10
−8
150 2.9×10
−8
2.8×10
−8
2.8×10
−8
200 1.3×10
−9
3.6×10
−9
3.9×10
−9
Table 3.1: Difference between the user defined frequency set at 0.26495797 and
the frequency extracted using the Pad´ e interpolation for resonances with Q factor
values10
4
,10
5
and10
6
. Δ= n.f.
user
−n.f.
extracted
.
transform samples. Considering Fig. 3.4, this implies that only the samples F(ω
j
)
and F(ω
j+1
) are used to determine the Q factor and center frequency. It is rather
remarkable that the Pad´ e method can successfully extract any information at all
fromsuchasmallamountofinformationaboutthetotalspectrum. Thisobservation
suggests that one way to improve the convergence properties of the method is to
includemoresamplesintothefittingprocedure.
One method for including more discrete Fourier transform samples into the
fitting procedure is to increase the order of the polynomials making up the Pad´ e
function. Forexample
P(1,1)=
α
0
+α
1
ω
1+β
1
ω
=
α
1
β
1
+
α
0
−α
1
/β
1
1+β
1
ω
(3.12)
43
Figure 3.6: Q factor convergence study of the Pad´ e method using a P(0,1) and a
P(1,1)Pad´ efunction.
could be used instead of Eq. 3.6 where we have rewritten the ratio of first order
polynomials as a constant term added to a ratio of a zeroth order polynomial to
a first order polynomial. Rewriting Eq. 3.6 illustrates that although we are not
starting with a form consistent with the Lorentzian function, it can be simply
manipulatedtoresembletheLorentzianformandEqs.3.9and3.11canbeapplied
to extract the Q factor and resonance frequency. Near the center frequency, the
rightmostterminEq.3.12isontheorderofQlargerthantheconstanttermα
1
/β
1
suggestingthatthedeviationfromthepreciseLorentzianformresultsinnegligible
error. Fig. 3.6 displays the convergence of the Q factor extraction using both the
P(0,1)andthe P(1,1)Pad´ e functions. The improvementresulting from increasing
thenumberofsamplesusedinthefit(byone)isapparent.
Before applying this method to time sequences generated from a finite-
differencetime-domainsimulation,theabilityofthemethodtoextracttherelavent
parameters in the presence of multiple resonances is explored. Fig. 3.7 displays a
discrete Fourier transform of a user generated time sequence that is comprised of
44
Figure 3.7: Discrete Fourier spectrum of two resonance peaks separated by 0.005
innormalizedfrequencyunits.
two damped sinusoids. One resonance is centered near 0.265 and has a Q factor
of 10
6
. The higher frequency resonance has a Q factor of 10
5
, and its resonance
frequencyis0.005largerthanthefirstresonancepeak.
Whenthe resonator system underanalysis supports more than oneresonance,
thenthefrequencyresponsewillconsistofasumofLorentzianfunctionseachone
corresponding to a resonance of the system. It should be noted that this is valid
onlyforstructures madefromlinearmaterials. Forthecaseoftworesonances,the
two Lorenztian functions will add to give a ratio of a first order polynomial to a
secondorderpolynomial. ThisisillustratedinEq.3.13.
c
1
ω
1
2Q
1
−i(ω−ω
1
)
+
c
2
ω
2
2Q
2
−i(ω−ω
2
)
=
(c
1
+c
2
)[d
0
−i(ω−d
1
)]
−ω
2
+k
1
ω+k
0
(3.13)
where
d
0
=
c
1
ω
1
2Q
2
+
c
2
ω
2
2Q
1
c
1
+c
2
(3.14)
45
d
1
=
c
1
ω
2
+c
2
ω
1
c
1
+c
2
(3.15)
k
0
=
ω
1
2Q
1
ω
2
2Q
2
−ω
1
ω
2
+iω
2
ω
1
2Q
1
+iω
1
ω
2
2Q
2
(3.16)
k
1
= (ω
1
+ω
2
)−i(
ω
1
2Q
1
+
ω
2
2Q
2
) (3.17)
This analysis suggests thatthe properPad´ efunction forfitting twoLorentzian
functions would be P(1,2). In what follows we will use a P(2,2) function so that
we can include an extra frequency sample into the fit to improve convergence at
the expense of deviating slightly from the proper functional form. Eq. 3.18shows
a Pad´ e P(2,2) function and the algebraic manipulation to express it as a constant
termaddedtoaratioofafirstorderpolynomialtoasecondorderpolynomial. The
denominator polynomial in the right most term in Eq. 3.18 can be factored, and
a partial fraction expansion can be used to write it as the sum of two Lorenztian
functions. Eqs. 3.9 and 3.11 can be used on each of the resulting Lorentzians to
extracttheQfactorsandresonantfrequenciesofthetworesonances.
P(2,2)=
α
0
+α
1
ω+α
2
ω
2
1+β
1
ω+β
2
ω
2
=
α
2
β
2
+
α
0
+α
1
ω−α
2
/β
2
−α
2
β
1
/β
2
ω
1+β
1
ω+β
2
ω
2
(3.18)
In Fig. 3.8 we explore the ability of the Pad´ e method to extract the Q factor of
the resonance near 0.265 as the frequency separation between the two resonances
is decreased. The bottom axis in Fig. 3.8 corresponds to the frequency separation
between the two resonances. It is apparent that doubling the length of the time
sequence allows for the extraction of the Q factor when the two resonance peaks
46
Figure3.8: ConvergencestudyoftheQfactorextractionbythePad´ emethodwhen
tworesonancesarebroughtclosetogetherinfrequency.
are more closely spaced. This is anatural result of the improved ability toresolve
spectral features as the frequency resolution increases. For the specific situation
shown in Fig. 3.8,itappearsthat thePad´ emethod successfully extracts a Qfactor
of10
6
whenthetworesonancepeakshaveafrequencyseparationnosmallerthan
theresolution inthediscreteFouriertransform.
In Fig. 3.9 we display the convergence of the direct extraction approach to
Pad´ e interpolation applied to a discrete Fourier transform obtained from a three-
dimensionalfinite-differencetime-domainsimulationofaphotoniccrystaldouble
heterostructurecavity. Thiscavitypresentsaparticularlychallengingspectrumfor
accurateparameterextraction duetoitshighmodedensityandlargeQfactorres-
onances. Becauseofthelargespectraldensity,weusehigherorderPad´ efunctions
to accomodate as manyneighboring modes as possible. It should be noted that it
is our practice to discard the first 10,000 time steps, so as to remove any transient
elementsinthetimesequence. InFig.3.9,thebottomaxiscorrespondstothenum-
beroftimestepsusedinthediscreteFouriertransformwhichwasperformedafter
47
neglectingthefirst10,000timesteps. FromFig.3.9weseethatusingtheP(2,2)and
P(2,3)functionsresultsinconvergenceafter60,000timesteps,andusingtheP(3,3),
P(3,4)andP(4,4)shows convergenceafterabout50,000timesteps. Wedonotuse
anorderhigherthan4inthepolynomialsduetothelackofclosedformexpressions
for their roots. This method may be extended to higher order polynomials, but a
numericalrootfindingroutinewillberequired.
Figure3.9displaystheconvergenceoftheQfactorasthenumberoftimesteps
isincreasedandsuggeststhatourinterpolationschemewillconsistentlyreportthe
same number once convergence is achieved. What remains is to verify that the
number we get from the Pad´ e interpolation is correct. To perform an alternative
calculation of the Q factor we used Eq. 2.5. Because we are dealing with large
Q factor cavities, the denominator in Eq. 2.5 will be small, and care should be
excercised in its evaluation. In order to obtain an accurate evaluation of the Q
factorbasedonEq.2.5,weusedthetime-averageversionofEq.2.5
Q= ω
0
hUi
−h
dU
dt
i
= ω
0
1
T
R
T
0
Udt
1
T
(U(T)−U(0))
. (3.19)
where T corresponds to one optical period. Because the electric and magnetic
fields are recorded at one half time step apart, we employed a small time step of
1/50000 of an optical period to reduce errors associated with this effect. The Q
factor obtained through Pad´ e interpolation was 336,700. The Q factor obtained
from Eq. 3.19 was 329,300. They differby less than 4% and the discrepency could
beattributedtoinaccuraciesinevaluatingEq.3.19,asnoconvergenceanalysishas
beendoneonthisvalue.
AsmentionedintheintroductiontothisChapter,theotherpopularmethodfor
extractingtheQfactorfromfinite-differencetime-domaintimesequencedataisto
48
Figure 3.9: Convergence study of the Q factor extraction when the Pad´ e method
is applied to data obtained from three-dimensional finite-difference time-domain
simulations. (a)-(c)depicttheconvergenceofthemethodwithincreasinglyhigher
orderPad´ efunctions.
49
Figure3.10: ConvergencestudyofQfactorextractionusingPad´ einterpolationand
filterdiagonalization.
applythe filterdiagonalization method. Fig. 3.10displays aQfactor convergence
comparison between Pad´ e interpolation and filter diagonalization. It is apparent
thethePad´ emethodconverges muchmorequicklythanthefilterdiagonalization
method.
3.4 Pad´ e interpolation for generalized transfer func-
tionanalysis
The electromagnetic fields associated with agiven geometry constructed of linear
materials are obtained from Maxwell’s equations which are linear vector partial
differential equations. This implies that the fields resuling from an excitation
(
~
f(~ x,t))canbefoundviaadyadicGreen’sfunction(G(~ x
′
,~ x;t
′
,t))accordingto
~ u(~ x,t)=
Z
V
d~ x
′
Z
dt
′
G(~ x
′
,~ x;t
′
,t)
~
f(~ r
′
,t
′
) (3.20)
50
where ~ u(~ x,t)referstotheelectricormagneticfield.
Ourinitialconditions consistofspatio-temporalimpulsefunctionsoftheform
~
f(~ x,t)= ˆ uδ[~ x−~ x
0
]δ[t−t
0
] (3.21)
where ˆ u refers to the polarization of the source. Plugging Eq. 3.21 into Eq. 3.20
yields
~ u(~ x,t)= G(~ x
0
,~ x;t
0
,t) (3.22)
whichshowsthattheresulting~ u(~ x,t)isthespatio-temporalimpulseresponseofthe
system. The permittivity and permeability that describe the material geometries
are constant in time but spatially varying. This implies that the impulse response
of the structure is time-invariant but not spatially invariant. For time-invariant
impulse responses, the relationship between the input time (t
0
) and the fields at
time t is not a function of t or t
0
but of their difference t−t
0
. Eq. 3.20 can then be
writtenas
~ u(~ x,t)=
Z
V
d~ x
′
Z
dt
′
G(~ x
′
,~ x;t−t
0
)
~
f(~ r
′
,t
′
) (3.23)
The temporal integral on the right sideof Eq. 3.23is a convolution integral which
meansthattheFouriertransform ofthetemporalimpulsereponseyieldsthetem-
poraltransferfunction. Theremainingspatialintegralyieldinformationaboutthe
overlap of the impulse at x = x
0
and the spatial Green’s function. For instance, if
x
0
corresponds toafieldnull,thenG(~ x
0
,~ x;t−t
′
)= 0.
The temporal electromagnetic behavior of the dielectric structures that will be
thefocusintheremainingpartsofthisThesiscanbeanalyzedwithinthecontextof
51
lineartime-invariantsystemtheory. FortherestofthisSection, Iwillusetheterm
“transfer function” to describe the Fourier transform of the temporal response of
the Green’s function. In order to avoid issues associated with the spatial aspects
of the excitation, low symmetry points are chosen to avoid preferentially exciting
modes of a given symmetry. One benefit of this theory is the expression of the
transfer function as a series of poles and zeros located in the two-dimensional
complexplane. Atypicaltransferfunctionwithrzerosandspoleswouldhavethe
form
H(z)=
(z−z
1
)(z−z
2
)···(z−z
r
)
(z−p
1
)(z−p
2
)···(z−p
s
)
. (3.24)
Comparing Eq. 3.24 to Eq. 3.1 shows that the Lorentzian spectral response corre-
sponds to a transfer function with a single pole at p
1
= ω
0
−i
ω
0
2Q
. From a system
theoryperspective,aresonatorisafeedbacksystemwhoseblockdiagramisshown
inFig.3.11(a). Tounderstandthecorrespondancebetweenanopticalresonatorand
afeedbacksystem,consideraFabry-Perot resonator. Thereflectionofelectromag-
neticenergybythemirrors actstofeedenergybackintothesystem.
Inspection of Eq. 3.2indicatesthat thenumerator anddenominator polynomi-
als can be factored into a product of monomial terms to correspond to the form
of Eq. 3.24. This implies that the Pad´ e function is an ideal form for extracting
the pole-zero structure of a general transfer function. It should be emphasized
thatthedirectextraction methodofPad´ einterpolation iscapableofextractingthe
complex frequencies of poles and zeros located anywhere in the complex plane.
Pad´ e interpolation as previously reported [DM98] is able to extract the complex
frequencyonlyofsinglepoletransferfunctions. Futhermore itisunabletodistin-
guish between a positive and negative imaginary part of the complex frequency.
52
Figure3.11: Systemleveldiagramsillustrating(a)feedbackand(b)feedforward.
Filterdiagonalizationislimitedtotransferfunctionsconsistingofpolesonly,butit
candistinguishbetweenpositiveandnegativeimaginarypartsofthecomplexfre-
quency. ThedirectextractionmethodofPad´ einterpolationcanextractinformation
aboutgeneraltransferfunctions.
AlthoughtheremainderofthisThesiswillbeconcernedwithanalyzingsingle
pole transfer functions, the role of zeros in optical transfer functions should be
discussed. Fig. 3.11(b) displays the system theory view of a zero in the transfer
function. From an optics standpoint a transfer function zero is realized by feed-
ing a portion of the input forward and combining it with the output to produce
destructive interferenceasinaMach-Zehnderinterferometer.
Thereisanotherwaythatazerocanappearinthefrequencyresponseofares-
onator. ConsiderEq.3.13which describes a structure supporting two resonances.
Writing the sum of the two Lorenztians as a single ratio of polynomials results in
a first order polynomial divided by a second order polynomial. The first order
53
polynomialinthenumeratorrespresentsazero. Anytimeasystemsupportsmul-
tipleresonancestherewillautomaticallybezerosofthistype. Fundamentallythis
zeroisnotthe samezeroasdiscussed intheprevious paragraphinrelation tothe
systemstructureinFig.3.11(b). Thelocationofthezeroresultingfromtwoormore
resonances is not fixed by the system. Instead its location in the complex plane
is dependent on the spectral amplitude of the two associated resonance peaks as
seen by the appearance of the amplitudes c
1
and c
2
in Eqs. 3.14 and 3.15. This
explainssharpdipsintheresonancespectraassociatedwithfinite-differencetime-
domain simulations as can be seen in Fig. 3.2 in the normalized frequency range
0.28−0.29. Althoughpresentinnumericallycalculatedresonancespectra,oberving
thesekindsofzerosinexperimentaldataisunlikely. First,thepositionofthezeros
will bespatially dependentinaccordance with the spatial field distribution of the
two modes responsible for the presence of the zero. Depending on the particular
geometry, the effectof the zero will likelybe manifestedby aspatial average over
two orthogonal field profiles nullifying the interference effect. Second, in laser
devices, spontaneous emission noise will smear out to some degree the temporal
phaserelationshipbetweenthetwomodes.
Whenanalyzingthespectraresultingfromtimedomainsimulation,itisimpor-
tant to be able to distinguish between real zeros of the system and zeros arising
from twoLorentzian resonances. Becausethecomplexfrequencyof azeroarising
from two Lorentzian resonances is uniquely defined by the properties of the two
resonance peaks, comparing the frequency of a zero with the results in Eqs. 3.13-
3.17 will determine whether the zero is a true system zero or the results of two
Lorentzianresonances. Alternatively,onemayperformmultiplesimulationswith
varying initial conditions to determine whether the complex frequency of a zero
changeswhichwouldindicatethatitisnotatruesystemzero.
54
Chapter4
PhotonicCrystalDouble
HeterostructureResonantCavities
4.1 Introduction
In this Chapter, I discuss the analysis of photonic crystal double heterostcture
cavitiesusingthefinite-differencetime-domainmethod. TheChapterbeginswitha
summaryofthepublishedworkrelatedtoheterostructurecavities. Thisisfollowed
by an analysis of the spectral and modal properties of the cavities. Higher order
boundstateswillbediscussedandcavitydesignsforsinglemodeoperationwillbe
presented. The Chapter concludes with experimental lasing data demonstrating
thefunctionalityofthesecavities.
Devicesbasedoncombiningtwoormorephotoniccrystallatticeswithdifferent
attributes werefirstproposedandanalyzedforthepurposeofenlargingphotonic
bandgaps[ZWZ98,SYM98,ZQW00]. Ahybridsquarelattice–triangularlatticehet-
erostructure wasproposed toimprovethetotal throughput ofaY-branch[SSP02],
and the potential to use heterostructures to form cavities and waveguides was
discussedandanalyzed[IS02,ILCS02].
Thetransmissionandreflectionofphotoniccrystalwaveguideheterostructures
formedbyconcatinatingtwowaveguideswithdifferentlatticeconstantswasinves-
tigatedexperimentallyandconfirmedmanyofthepreviouslyreportedtheoretical
55
analyses[SAA
+
04]. Acompactbeamsplittingdevicewasdemonstratedthatincor-
porated a superprism and heterostructure photonic crystal to enhance angular
separation [WMGK05]. Polarization beam splitters with 20 dB extinction ratios
havealsobeendemonstratedusingphotoniccrystal heterostructures [SWP
+
06].
In 2005 Song et al. showed that a two-dimensional photonic crystal waveg-
uide with a small, localized perturbation can form an ultra high quality (Q)
factor greater than 10
5
with a mode volume on the order of one cubic wave-
length [SNAA05, SAN07]. Further experimental work has demonstrated devices
with passive Q factors as large as 10
6
[ASN06]. Since these initial reports, sev-
eralgroupshavereportedforminghighQphotoniccrystaldoubleheterostructure
cavities through a variety of methods including local modulation of a photonic
crystal line defect width [KNM
+
06], local air-hole infiltration [SWL
+
07], photo-
sensitive materials [THSdSM07], effective index change through micro-fiber cou-
pling [KHSL07] and local modulation of the hole radii [KSKF08b]. A numerical
analysis showed that Q factors as high as 10
9
are possible with a tapered pertur-
bation [TAN08]. Recentstudies haveinvestigated theeffectofaddingaslotalong
thecenterofthephotoniccrystal waveguide[KSKF08a,YNT
+
08].
The ultra high Q factors and cubic wavelength mode volumes along with the
waveguide-like shape of the cavities have made them attractive for a variety of
applicationsincludingchemicalsensing[KSKF08a],slowlight[TNK
+
07,THY
+
07],
elements of coupled resonator optical waveguides [OSK
+
07] and edge-emitting
lasers[SKM
+
06,SMH
+
06,YLM
+
07,LYM
+
07,LMB
+
08,LMY
+
09].
56
4.2 Spectralpropertiesofphotoniccrystaldoublehet-
erostructureresonantcavities
Figure 4.1 is a schematic illustration of two photonic crystal double heterostruc-
turecavitiesinwhichthelatticeconstantofthelightcolored holeshaseitherbeen
increased (a) or decreased (b) along the x-direction. Confinement of the resonant
modealongthey-directionoccursthroughphotonicbandgapguiding,andconfine-
ment along the z-direction (out-of-plane) is through index guiding. Confinement
alongthex-directionoccursduetothepresenceoftheperturbationwhichcreatesa
one-dimensionalphotonic wellas illustrated inFig.4.1. Whenthe latticeconstant
isincreased(decreased),itshiftsthefrequenciesofthewaveguidebandassociated
withtheperturbedregiontolower(higher)frequencies. Theboundstatewilloscil-
latenearfrequenciesoftheperturbedwaveguidesectionthatfallintothemodegap
oftheuniformwaveguidesections. Whentheperturbationisformedbyincreasing
(decreasing) the lattice constant, the bound states will form near photonic crystal
waveguidedispersionminima(maxima). Onlybelowandabove(forpositiveand
negativedefects,respectively)theextremaofthedispersionrelationintheuniform
photonic crystal waveguideregionsisthereapossibilityforamodetoexistinthe
central region withoutthepossibility of theresimultaneously beingamodeinthe
claddingatthesamefrequencyasmalldistanceinwavevectoraway. Fig.4.2illus-
tratestherangeoffrequenciesinwhichtheboundstatefrequencywillfall. Inother
words,onlyinthesecasesistherenomodethatisnearby(inthewavevectorsense)
in the cladding at the same frequency [MKS
+
06, MLO08]. This mode formation
is analogous to the formation of bound states in electronic heterostructures at the
extremaoftheelectronicdispersionrelations.
57
Figure4.1: Schematicdiagramsofphotoniccrystaldoubleheterostructureresonant
cavitiesformedinuniformsinglelinedefectwaveguidesby(a)increasingand(b)
decreasingthelatticeconstantalongthex-direction.
58
Figure 4.2: Close up view of a photonic crystal waveguide dispersion diagram
illustrating the candidate frequencies for a bound state resonance associated with
aheterostructure cavity.
59
Figure 4.3 is a comparison between the spectral features of photonic crystal
double heterostructure resonance spectra and the frequency axis of the photonic
crystal waveguide dispersion diagram corresponding to the underlying straight
waveguide. The red (green) dotted lines illustrate that when the heterostructure
cavity is formed by a local lattice constant increase (decrease), the bound state
resonancefrequenciesoccurjustbelow(above)thewaveguidedispersionminima
(maxima).
Theresonance spectra were obtained bytaking adiscrete Frourier transforma-
tion of a 2 × 10
5
element time sequence. The time sequence was calculated via
the three-dimensional finite-difference time-domain method. The computational
domain included 20 uniform photonic crystal cladding periods on either side of
the central defectregion along the x-direction and 8photonic crystal layers above
and below the central waveguide core along the y-direction. This geometry was
discretizedwith950×340×200discretizationpointsalongthex×y×zdirections
andparallelizedusing11×4×3processors(132totalprocessors)alongthex×y×z
directions. The geometry was discretized using 20 points per lattice constant (a).
Theholeradiustolatticeconstantratiowassettor/a= 0.29andtheslabthickness
to lattice constant ratio was set to d/a = 0.6. The lattice constant of the perturbed
regionwaseitherincreasedordecreasdby5%alongthex-direction.
Table 4.1 summarizes the resonance frequencies, Q factors and associated
waveguide band edges for the five modes highlighted in Fig. 4.3. Resonance
frequencies and Q factors were obtained from the frequency spectrum using the
Pad´ e method described in Chapter 3. It is clear that the bound state resonance
frequencies fall just below (above) the corresponding photonic crystal waveguide
band edge when the heterostructure is formed by increasing (decreasing) the lat-
tice constant. It should be pointed out that the resonance frequency of mode 1 in
60
Figure 4.3: Left: photonic crystal waveguide dispersion diagram. Black lines
correspond to the photonic crystal waveguide dispersion bands. Blue regions
denote photonic crystal cladding modes. The beige region denotes the light cone
projection. Right: photonic crystal double heterostructure resonant spectra for
defects resulting from an increase and a decrease in the lattice constant. Dashed
lines illustrate correspondance between heterostructure bound state frequencies
andwaveguidedispersionextrema.
Table 4.1 is in fact higher than the band edge frequency of the lowest waveguide
band even though it appears in Fig. 4.3 that the resonance peak of this mode falls
belowthewaveguidebandedge. Thediscrepencyarisesduetothefinitesampling
intervalassociatedwiththediscreteresonancespectrum. Thismakesitdifficultto
determine the exact center frequency of a given resonance peak by inspection of
therawdataalone.
InadditiontotheboundstateresonancessummarizedinTable4.1,therearesev-
eralothersharpresonancepeaksthatappearintheresonancespectraofFig.4.3. In
particularthereareclustersofpeaksinthefrequencyranges0.263-0.270,0.283-0.292
and 0.323-0.330. These spectral features correspond to Fabry-Perot resonances in
thestraightwaveguidesectionsoftheheterostructurecavityasdepictedinFig.4.4.
The disruption of the uniform photonic crystal waveguide by the lattice constant
61
Figure4.4: SchematicdiagramshowingtheregionsofthecavitysupportingFabry-
Perotresonances.
Mode Resonance PCWGBand Quality Pertur-
Frequency EdgeFrequency Factor bation
1 0.2100608 0.2098001 1080 -
2 0.2605721 0.2629375 336701 +
3 0.2800121 0.2823920 10802 +
4 0.2949732 0.2928421 357 -
5 0.3184359 0.3227484 8254 +
Table4.1: Summaryofresonancefrequencies,associatedbandedgefrequenciesand
Qfactors forthefiveboundstateresonancesappearinginthespectrainFig.4.3
.
62
Figure 4.5: Group index extracted from frequency spacing in numerically calcu-
lated resonance spectra associated with photonic crystal double heterostructures.
The back curves correspond to the group index obtained from the slope of the
photonic crystal waveguide dispersion diagram. (a) Photonic crystal waveguide
band spanning the normalized frequency range 0.26-0.34. (b) Photonic crystal
waveguidebandspanningthenormalizedfrequencyrange0.28-0.30.
63
perturbationatoneendandtheterminationofthephotoniccrystalwaveguidelat-
ticeattheotherendintroducesascatteringmechanismforthestraightwaveguide
modes. Aportion of the scattered modeis reflectedbackintothe waveguidethus
forming a Fabry-Perot cavity. The free spectral range in a Fabry-Perot cavity is
givenby
Δf =
c
2n
g
L
(4.1)
orinnormalizedfrequencyunits
Δ(
fa
c
)=
1
2n
g
(L/a)
. (4.2)
wheren
g
isthegroupindexofthephotoniccrystalwaveguide,andListhelength
oftheFabry-PerotcavitylabeledinFig.4.4.
InordertoconfirmthatthesefeaturesdocorrespondtoFabry-Perotresonances,
we measured their free spectral range and calculated the group index. We then
compared this group index to what one would obtain by measuring the slope of
photoniccrystalwaveguidedispersiondiagram. TheresultsareshowninFig.4.5.
Fig. 4.5(a) shows the group index obtained from spectra corresponding to het-
erostructurecavitiesformedbybothincreasinganddecreasingthelatticeconstant
forthelowestfrequencyphotoniccrystalwaveguidebandinsidethephotoniccrys-
tal bandgap in Fig. 4.3. Fig. 4.5(b) shows the same comparison for the photonic
crystalwaveguidebandspanningthenormalizedfrequencyrange0.280-0.295. The
agreementisgood,andweattributetheslightscatterintheheterostructureresults
tolimitedabilitytoextractanexactcenterfrequencyinthepresenceofhighmode
density.
64
Inadditiontobeingabletoexplainthefeaturesinnumericallycalculatedreso-
nancespectra, identifyingthe Fabry-Perot modes associated with theheterostruc-
ture cavity is important for mode idintification in performing spectroscopic mea-
surement of fabricated laser devices. The Q factors of the Fabry-Perot modes can
beaslargeas10
4
whichishighenoughtoreachthresholdinsemiconductoractive
materialsifthespatialarrangementoftheinvertedcarrierpopulationissuchasto
favorlasingoverthe higherQbound state resonance. Anotherinterestingfeature
ofthisstudy, isthatitcanbeusedtoextractthemodalreflectivityassociatedwith
thephotoniccrystalwaveguideterminationandtheinhomogeneityintroducedby
thelatticeconstantperturbation. OncetheQfactorsoftheFabry-Perotresonances
isknown,themirrorreflectivitiescanbecalculatedaccordingto
r
1
r
2
= exp(−
Lω
0
n
g
c
0
Q
). (4.3)
Applying Eq. 4.3 to the Fabry-Perot resonance peaks resulting from a double het-
erostructurewillonlyyieldtheproductofthereflectivitiesarisingfromtwodiffer-
ent mirrors. In order to unambiguously analyze the modal reflectivity associated
witheitheroneoftheinterfaces,aseparatecalculationwouldberequiredinwhich
a finite length photonic crystal waveguide is used with the same termination on
bothends. ItshouldalsobenotedthatextractingtheQfactorfromtheveryclosely
spacedFabry-Perot resonancepeaksnearthebandedgewasmadepossiblebythe
directextraction approachtoPad´ einterpolation discussedinChapter3.
65
Figure 4.6: Left: H
z
(x,y,z = 0) for the five modes summarized in Table 4.1.
Right: spatialFouriertransformofeachmodeillustratingthedistributionofspatial
wavevectors makingupthedifferentresonantmodes.
66
4.3 Modal properties of photonic crystal double het-
erostructureboundstates
Figure4.6depictsthespatialmodeprofilesofthefiveboundstateresonanceslisted
in Table 4.1 at the midplane of the semiconductor slab. These single frequency
mode profiles were extracted from the multi-mode finite-difference time-domain
simulation by applying a discrete-time filter centered at the frequency of inter-
est [OSB99]. For the TE-like modes of the slab, only the E
x
, E
y
and H
z
fields are
nonzeroatthemidplane,andH
z
isdisplayedduetoitsscalarnature. Thesemode
profilesmaybeinterpretedasconsistingofthewaveguidemodeoftheunderlying
straight waveguide multiplied by a confining envelope function centered at the
perturbation.
Tothe right of eachH
z
(x,y,z= 0)mode profile is the spatialFourier transform
of each mode. Specifically, log(|FT(E
x
)|
2
+ |FT(E
y
)|
2
) is plotted where FT stands
for Fourier transform. The two-dimensional spatial Fourier transform yields the
spatialwavevectorcomponentsthatmakeuptheboundstateresonance. Formodes
(a), (b), (c) and (e) the spatial wavevector distribution is centered at β
x
= ±π/a.
Thisobservationisconsistentwithourearlierdiscussionofthespectralproperties
of the bound state resonances: bound states resonance frequencies occur near the
waveguidedispersionextremawhichoftenoccurneartheBrillouinzoneboundary
at β
x
= ±π/a. The spatial wavevector distribution of mode (d) is centered near
β
x
= ±0.6 × π/a. Again, this is consistent with the location of the waveguide
dispersion maxima near β
x
= ±0.6×π/a in Fig. 4.3. Also evident in Fig. 4.6(d) is
thepresenceof peaksatβ
x
= 0.6×π/a−2π/aandβ
x
= −0.6×π/a+2π/awhichis
thespatialwavevectordistribution inthesecondBrillouinzone.
67
Figure4.7: (a)DiagramillustratingthedirectionalradiationofMode2fromFig.4.6.
(b)FirstBrillouinzoneofatwo-dimensionaltriangularlattice. Projection oftheM
pointontotheΓ−Kdirectionatβ
x
= π/aisindicated.
68
ItisinterestingthattheQfactorvalueslistedinTable4.1forthedifferentbound
states differ so drastically from mode to mode. This suggests that each mode has
differentdominantloss mechanisms. Inordertounderstandthe loss mechanisms
of the heterostructure cavity, the time-averaged Poynting vector of each bound
state resonance mode was calculated and integrated over a surface enclosing the
cavity. Time-averagingthePoyntingvectorremovesthetemporaloscillationattwo
timestheresonancefrequencyandextractsthepurepowerflowingoutwardaway
from thecavity asdiscussed in Chapter2. The amountof energythatleaks outof
the cavity during one optical period is proportional to 1/Q, so for high Q factor
cavities,onlyaverysmallamoundofenergyleaksoutofthestructure perperiod.
Inordertoobtainanaccuratetime-average,theFDTDtimestepwasdecreased,so
that one period consisted of as many as 50,000 time steps. In order to assess the
accuracyofthehighresolutiontime-averageweturntotheinstantaneousPoynting
theorem4.4.
I
~
S·d
~
A= −
∂U
∂t
(4.4)
In Eq. 4.4,
~
S =
~
E ×
~
H is the Poynting vector and U is the stored energy. If we
integrate both sides of Eq. 4.4 with respect to time over one optical period, T one
gets
1
T
Z
T
0

I
~
S·d
~
A

dt= U(0)−U(T) (4.5)
Equation4.5statesthatthedifferencebetweentheenergyinthecavityattimet= 0
and the energy in the cavity one optical period later must equal the amount of
energy that radiated out of the cavity. When the time-averaged Poynting vector
is calculated, the initial stored energy, U(0) and the final stored energy, U(T) are
69
Mode P
z
/(P
x
+P
y
) P
x
/P
y
1 0.25 23.7
2 1.80 0.19
3 1.19 42.6
4 106.5 0.71
5 0.43 0.014
Table 4.2: Power flow ratios between the x, y and z directions for the five modes
displayedinFig.4.6.
recorded,sothatEq.4.5canbeverified. TheleftandrightsidesofEq.4.5typically
differbynomorethan1%.
ThedirectionallosspropertiesaredeterminedbyintegratingthePoyntingvec-
toroverthesurfacethroughwhichtheopticalenergyisradiating. Asdiscussedin
theprevioussection,theopticalmodeinaphotoniccrystaldoubleheterostructure
is confined via three different mechanisms: photonic well confinement along the
x-direction,photonicbandgapconfinementalongthe y-directionandtotalinternal
reflectionalongthez-direction. Table4.2summarizestheratiosofpowerradiating
along the three loss directions. Because we are analyzing a passive cavity, the
amplitudeofthePoyntingvectordoesnothaveanyphysicalsignificance,andthe
powerflowratiosuniquelydeterminethelosspropertiesofthecavity.
Table 4.2 indicates that Mode 1 is dominated by in-plane loss along the x-
direction. ThisisconsistentwiththequalitativefeaturesoftheH
z
(x,y,z= 0)mode
distribution associated with Mode 1 shown in Fig. 4.6. It is clear that the field is
not well confined along the x-direction. For Modes 2 and 3 the in-plane loss and
out-of-planelossesarenearlyequal. ForMode3thein-planelossisdominatedby
lossalongthex-direction. Thereasoningbehindthiswillbediscussedlater. Mode
4isheavilydominatedbyout-of-planelosswhichweattributetotheproximityof
themode’sFourier spacedistribution totheradiation cone. Thisisduetothefact
70
Mode Q
x
Q
y
Q
z
Q
tot
1 1400 33100 5470 1080
2 5962000 1120000 525000 337000
3 24200 1031000 19900 10800
4 92600 65500 360 357
5 83900 12100 27400 8250
Table 4.3: Directional Q factors for the x, y and z directions for the five modes
displayedinFig.4.6.
thatMode4isassociatedwithawaveguidebandmaximumnearthecenterofthe
Brillouin zone. Mode 5 is dominated byin-plane loss along the y-direction which
due to the small frequency difference between the bound state resonance and the
photoniccrystal claddingmodes.
Table 4.3 summarizes the directional Q factors for the five bound state reso-
nances. The directional Q factors were obtained according to the following rela-
tionships
Q
in-plane
Q
z
=
P
z
P
x
+P
y
(4.6)
1
Q
=
1
Q
in-plane
+
1
Q
z
(4.7)
1
Q
in-plane
=
1
Q
x
+
1
Q
y
(4.8)
Q
y
Q
x
=
P
x
P
y
(4.9)
Eqs.4.6through 4.9representthreeindependentequations (Eq.4.8is adefinition,
not an independent equation) for the three unknowns Q
x
, Q
y
and Q
z
. Expressing
71
thedirectionallosspropertiesviatheQfactorallowsforaquantitativecomparison
of the loss channels between different modes. Table 4.3 indicates that Mode 2 has
the largest directional Qfactor for eachof the x, yand z-directions. However, itis
interesting to note that Q
y
of Mode 3 is almost equal to Q
y
of Mode 2, suggesting
that Modes 2 and 3 experience similar confinement along the y-direction. This
helps understand the large P
x
/P
y
ratio for Mode 3 in Table 4.2: Mode 3 is not
sufferingfromenhancedleakagealongthex-directionsomuchasitisexperiencing
enhancedconfinementalongthey-direction. Weattributetheenhancedy-direction
confinement to the central location of the resonance frequency in the photonic
crystalbandgap.
AnotherinterestingfeatureofTable4.3isthatthein-planeconfinementofModes
2and5islimitedbyradiationinthe y-direction. Fig.4.7(a)depictsthatdirectional
dependenceofthisradiationandshowsthatthefieldprofileofMode2extendsinto
thephotoniccrystalcladdingatthirtydegreeangleswithrespecttothex-axis. The
reasoningbehindthisdirectionalradiationisillustratedinFig.4.7(b)whichshows
theoutlineofthefirstBrillouinzoneassociatedwithatwo-dimensionaltriangular
lattice. The photonic crystal waveguide is formed along theΓ−K direction. The
Brillouin zone boundary for the photonic crystal waveguide occurs at β
x
= π/a.
The way that the photonic crystal cladding modes (shown in Fig. 1.4) map onto
the photonic crystal waveguide dispersion diagram (shown on the leftin Fig. 4.3)
results in an M point getting mapped to β
x
= π/a. Because the two-dimensional
spatial Fourier transform for Mode 2 shows that the mode is made up of a distri-
bution ofwavevectors concentrated nearβ
x
= π/a, itisnotsuprisingthatin-plane
leakage along the M direciton (thirty degrees from the x-axis) is dominant. The
same reasoning explains the in-plane leakage properties of Mode 5. Its proximity
72
to the photonic crystal cladding modes results in significant leakage along the y-
direction,andthepropagationdirectionofthephasefrontsassociatedwiththefield
penetrationshowninFig.4.6havethesamethirtydegreeorientation withrespect
to the x-axis. This observation may be exploited to optimize the in-plane cou-
pling between the resonant cavity and output waveguides. Although intuitively,
onewouldsuggestcouplinglightoutofaphotoniccrsytal doubleheterostructure
resonant cavity along the waveguide direction, based on the directional leakage
results presented here it maybe more efficientto extract light via the leakage into
thephotoniccrystal claddingforMode2.
4.4 Higherorderboundstatesinphotoniccrystaldou-
bleheterostructures
Inthissectionweexplorenovelproperties ofphotonic crystal doubleheterostruc-
turecavitiesfocusingprimarilyontheobservationofhigherorderboundstatesin
analogy with bound states in a finite quantum well. From Table 4.1, we see that
mode 2 has the largest Q factor of all the bound states by an order of magnitude,
sothediscussioninthissectionwillfocusonmode2.
Figure 4.8 illustrates two interesting features of photonic crystal heterostruc-
turecavities. First,thebluecurveindicatesthatasthelatticeconstantperturbation
is increased, the Q factor of the bound state decreases approximately exponen-
tially. In order to explain this trend, the total Q factor was broken down into
directional Q factors as was done in the previous section. The results are dis-
played in Fig. 4.9 which shows that as the perturbation is increased from 2.5%
to 15.0%, each directional Q factor decreases. The limiting Q factor is Q
z
for
73
Figure4.8: Qfactorversusperturbationdepthforthefirst,secondandthirdorder
boundstates.
perturbations of 2.5%, 5.0% and 10.0% and Q
y
for a 15.0% perturbation. As the
perturbation is increased, the interface between the uniform waveguide sections
andtheperturbation region ismademore abrupt. Qualitatively, abruptinterfaces
in photonic crystal structures tend to introduce high spatial frequencies into the
mode, and the high frequency spatial wavevectors tend to radiate out of plane
which explains Q
z
being a limiting loss channel. This idea has been discussed by
several authors [SP02, VLMS02, AASN03], and the ultra high theoretical Q factor
of10
9
discussedintheintroductionwasobtainedbyusingataperedphotonicwell
whichsmoothedouttheinterfacebetweentheuniformwaveguidesectionsandthe
perturbationregion[TAN08]. Between10.0%and15.0%perturbation,thelimiting
Qfactor changes from Q
z
toQ
y
. This trend is explainedusingthe samereasoning
discussed in the previous section. As the perturbation is increased, the well is
deepened along the x-direction. However, the confinement along the y-direction
isweakenedduetothereductioninfrequencyseparationbetweentheboundstate
mode and the photonic crystal cladding modes. In principle, the leakage in the
74
Figure 4.9: DirectionalQfactorbreakdownversusperturbation depth.
y-direction can always be mitigated by increasing the claddingperiods along this
direction,andthefundamentallossmechanismremainstheverticaldirection.
ThesecondinterestingfeatureshowninFig.4.8istheemergenceofhigherorder
boundstates[MLH
+
09]. Ataperturbationof7.5%,thesecondorderboundstateis
supported,andataperturbationof20.0%thethirdorderboundstateissupported.
It is interesting to note that for a given perturbation, the highest Q factor mode is
thehighestsupportedboundstate. Thisisinterestingasitallowsonetoobtainhigh
Q factor cavities without having to resort exclusively to very small perturbations.
Fig. 4.10 illustrates the mode profile of the first, second and third order bound
statesatthemidplaneoftheslab. Totherightofeachmodeprofileistheenvelope
function extracted bymeasuring the maxima of|H(x,y= 0,z= 0)|. The first order
boundstatehasonelobe,thesecondorderboundstatehastwolobesandthethird
orderboundstatehasthreelobeswhichisconsistentwiththefirstthreeelectronic
boundstatesofaquantumwell.
75
Figure 4.10: Left: H
z
(x,y,z = 0) field distributions for the first, second and third
orderboundstates. Right: fieldenvelopesextractedfrom|H
z
(x,y= 0,z= 0).
76
From an application perspective, if one chooses to work with a cavity that
supports multiple bound states, then it is useful to be able to enhance one mode
relative to the others. The benefits of mode discrimination include the ability to
select a given mode even if the pumping conditions favor lasing in a different
mode and enhanced side mode suppression. The strategy used to perform mode
discrimination is to place extra holes in the cavity near the maxima of the electric
fieldcorrespondingtothemodewewishtosuppresswhichenhancesout-of-plane
radiationandlowerstheQfactor[KCC
+
05]. Fig.4.11displaysthemodifiedcavities.
Fig.4.11(a)isacavitywitha10.0%perturbationwhichsupportsboththefirstorder
andthesecondorderboundstates. Fig.4.11(b)illustratesacavitywithholesplaced
atx= ±2.4atosuppressthesecondorderboundstate,andFig.4.11(c)illustratesa
cavitywithaholeplacedatx= 0tosuppressthefirstorderboundstate.
Figure 4.12 displays the numerically calculated resonance spectra correspond-
ing to the three cavities shown in Fig. 4.11. In the blue curve resonance peaks
corresponding to the first order and second order bound states are visible. The
photonic crystal waveguide band edge is also labeled. The green curve is the res-
onancespectrum corresponding tothecavity shown inFig. 4.11(b). Itis clearthat
thesecondorderboundstateresonancepeakisnolongervisible,andthefirstorder
bound state resonance peak is unaffected. The converse is true for the red curve
in Fig. 4.12 which corresponds to the cavity shown in Fig. 4.11(c): the first order
boundstateresonancepeakisnolongervisible,andthesecondorderboundstate
resonancepeakisunaffected.
The results of Fig. 4.12 suggest that our cavity modification is successful in
suppressing the unwanted bound state modes. The next issue is whether the
cavity modficiation affects the featured mode. Fig. 4.13 displays the Q factor of
the first order bound state as a function of the distance from the origin of the two
77
Figure 4.11: (a) Nominal heterostructure cavity with a 10% lattice constant per-
turbation. (b) Modified cavity with two extra holes to suppress the second order
bound state. (c) Modified cavity with one extra hole to suppress the first order
boundstate.
78
Figure4.12: ResonancespectraassociatedwiththethreecavitiesshowninFig.4.11.
Topcurve: cavityinFig.4.11(a). Middlecurve: cavityinFig.4.11(b). Bottomcurve:
cavity in Fig. 4.11(c). The photonic crystal waveguide bandedge is denoted. Any
features in the resonance spectrum at frequencies larger than the photonic crystal
bandedgeareassociatedwithresonancesintheuniformwaveguidesectionsofthe
structure asdenotedinFig.4.4
79
Figure4.13: Qfactorofthefirstorderboundstateasafunctionofthedistancefrom
thecenterthatthetwoextraholesareplaced.
extra holes added to suppress the second order bound state (see Fig. 4.11(b)). For
eachdisplacementshowninFig.4.13,thesecondorderboundstateissuppressed;
however,thereisanoptimumintheQfactorforthefirstorderboundstateatahole
placementofx= ±2.4a. Thisisexplainedbyconsideringtheextra holeplacement
relativetotheelectricfieldofthefirstorderboundstatemode. Fig.4.14illustrates
the overlap of the extra holes with the electric fields (E
y
(x,y = 0)) of the first and
second order bound states. It can be seen that the Q of the first order bound state
isoptimumwhentheextraholesoverlapwithanodeofitselectricfield.
The proposed modified photonic crystal double-heterostructure cavities for
preferential selection betweenthe first andsecond bound states, were experimen-
tally verified by fabricating a set of three double-heterostructure laser cavities
in a 240-nm-thick suspended InGaAsP membrane containing four compressively
strainedquantumwells. ThiswasdonebyLingLuincollaborationwithDr. Dap-
kus’slabatUSC.Thesemiconductordry-etchwasdoneinaninductivelycoupled
80
Figure 4.14: Diagram illustrating the overlap of the two extra holes with E
y
(x,y=
0,z= 0)forthe first andsecond orderbound states for theoptimum placementof
2.4afromthecenterofthestructure alongthex-direction.
plasma etcher using BCl
3
chemistry at 165
◦
C. The rest of the fabrication pro-
cesses are the same as those in [SKM
+
06]. Scanning electron microscope images
of the nal devices are shown as insets in Fig. 4.15(a)-(c), where (a) is the regular
double-heterostructure cavity, (b) has two extra holes 2.4a away from the device
center along the waveguide core, and (c) has one extra hole placed at the center
of the device. The lattice constant (a) is 405 nm and the perturbed lattice constant
(a
′
) is 10% larger than a along the waveguide direction in all three devices. The
devices are optically pumped at room temperature by an 850 nm diode laser at
normalincidencewithan8nspulsewidthand1%dutycycle. Thesizeofthepump
spot is about 2 μm in diameter. The lower spectrum in Fig. 4.15(a) is the single-
modelasingspectrumoperatingintherstboundstate,whiletheuppermultimode
lasing spectrum shows the existence of the second bound state approximately 20
nm away from the first one when the pump spot is slightly moved off the device
center along the waveguide core. This wavelength separation is consistent with
the predicted spacing of 25 nm from the 3-D FDTD calculation. The two modied
81
structures in Fig. 4.15(b) and (c) are both stable in single-mode operation against
thepumpoffsets. Theirlasingwavelengthslineupwiththerstandsecondbound
states lasing in the nonmodied structure. All four lasing spectra were taken at
the peak incident power of 1.7 mW. The broad resonance peak between 1.40 and
1.45 μm corresponds to the second waveguide dispersion band shown in Fig. 2.
Fig.4.15(d)depictsthelight-in–light-out (L-L) curvesof thethreelasersdescribed
be-fore. Theyhavealmostidenticalthresholdsbutdifferentslopes,indicatingthe
same amount of total optical loss but different portions of collected laser power.
Devices(a)and(b)havethesamelasingmodesandthresholds, butdevice(b)has
ahigherslopeefciency. Thisislikelyduetotheincreasedout-of-planeopticalloss
introducedbythetwoextraholes,which,inthefabricateddevice,haveradiiabout
30%largerthanwhatwasanalyzednumericallybefore.
In this Chapter I have presented a detailed description of the spectral and
modal properties of the photonic crystal double heterostructure. It was shown
that the heterostructure cavity supports an ultra high Q factor mode with a mode
volumeontheorderofacubicwavelength. Itsdirectionalradiationpropertieswere
discussed,andIcommentedontheuseofthismodeforin-planecoupling. Higher
orderboundstateswereinvestigatedandmethodsforsidemodesuppressionwere
presented. Experimentallasingresultssupportedthespectralpropertiesassociated
withselecteddesignspresentedinthisChapter.
82
pump off center
a’ a a
1μm 1kV
(a)
(b)
(c)
(d)
1st
2nd
1st
1st
2nd
Figure 4.15: (a)-(c) Lasingspectra of three double-heterostructure lasers with 10%
perturbation (a
x
= 1.10a
x
). Their SEM images, shown as insets, were taken at the
same experimental conditions. (d) is the light-in–light-out curves of the lasers in
(a)-(c).
83
Chapter5
PhotonicCrystalDouble
HeterostructureCavitiesWith
DielectricLowerSubstrates
5.1 Introduction
In the previous chapter we saw that the photonic crystal double heterostructure
cavitysupportedahighQfactormodewithaQfactorof337,000. Therehavebeen
severaldemonstrationsoflasinginthismodeincavitiesformedfromfree-standing
air-clad membranes[SKM
+
06,YLM
+
07,LYM
+
07,LMB
+
08,LMY
+
09]. Theselasers
were pumped optically under pulsed conditions. In order for these lasers to be
a candidate for chip-scale optical sources, they must be able to operate under
continuouswave(CW)conditions. CWoperationischallenginginthesestructures
becauseheatbuildsupintheactiveregioninresponsetothepumpexcitationand
degradesthegainproperties. Thephotoniccrystallatticereducestheabilityofthe
material to dissipate heat in-plane, and heat is isolated vertically due to the finite
extentofthesemiconductormembrane. Itisevenpossibletodamagethematerial
by attempting to pump at CW powers required to reach threshold [NIW
+
06]. An
exampleofdamageresultingfromhighCWpumppowersisshowninFig.5.1.
84
Figure 5.1: Scanning electron micrograph of a fabricated photonic crystal het-
erostructurecavitythathasbeendamagedbyhighcontinuouswavepumppower.
CourtesyofLingLu.
Because of the problem of heating in photonic crystal membrane lasers, there
havebeenrelatively few reports of CWlasinginphotonic defectcavities [NKB07,
NIW
+
06, HRS
+
00, CKW
+
05, SKY
+
06a]. Although CW lasing was achieved in the
small foot print point shift cavity owing to the reduced mode volume and high
Q [NKB07], using a thermally conductive lower substrate has advantages which
include better thermal conduction for the entire chip, better CMOS compatibility
andimprovedmechanicalstability[HRS
+
00,CKW
+
05,SKY
+
06a]. Inaddition,CW
lasingwithdielectriclowersubstratesrepresentsafirststeptowarddemonstrating
electricallyinjectedphotoniccrystallasersthatutilizeasemiconductorverticalslab
structurewithlowrefractiveindexcontrast. Theadvantagesofthelowersubstrate
come at the cost of enhanced optical leakage into the substrate [KKS
+
02, Qiu05,
TAHN06,MO08b,MO09b].
Inthischapter,wequantifythesubstrateleakageinphotoniccrystaldoublehet-
erostructure resonantcavitiesusingthefinite-differencetime-domainmethod. We
then propose an alternative cavity design that helps mitigate out-of-plane optical
85
Material RefractiveIndex ThermalConductivity(W/cm-K)
Air 1.003 0.00024
SiliconDioxide 1.46 0.014
Sapphire 1.75 0.34
Table5.1: Candidatematerialsforheatsinkinglowersubstratesandtheirrefractive
indiciesandthermalconductivities.
leakage. Finallyweinvestigatedeeplyetchedstructuresanddiscussthepossibility
oftheiruseaselectricallyinjectedphotoniccrystallasers.
5.2 Substrate loss in photonic crystal double het-
erostructurecavities
Figure 5.2 is an illustration of a photonic crystal double heterostructure cavity
that has been defined in a semiconductor slab which has been either bonded or
grownonalowersubstrate. Asdiscussedintheintroduction,themainbenefitfor
introducingthelowersubstrate istoaidintheheatdissipationintheout-of-plane
direction. Possible substrates include silicon dioxide (SiO
2
) and sapphire (Al
2
O
3
)
whoseopticalandthermalpropertiesaresummarizedinTable5.1. Itisinteresting
to note that among the materials considered in Table 5.1, sapphire has the largest
thermal conductivity but also has the highest refractive index which is going to
introducethemostsubstrate leakageofthethreematerialslistedinTable5.1.
Figure5.3plotstheQfactorofthephotoniccrystaldoubleheterostructurecavity
discussedinthepreviouschapterasafunctionoftheindexofthelowersubstrate.
Asubstraterefractiveindexof1.0correspondstoasuspendedmembranestructure
withoutasubstrate. Asdiscussedinthepreviouschapter,thehighQfactormode
86
Figure 5.2: Schematic rendering of a membrane photonic crystal heterostructure
cavity(lavender)bondedorgrownonalowersubstrate (green).
ofthesuspendedmembranecavityhasaQfactorof337,000. Forthiscavity,theQ
factordropsapproximatelyexponentiallyasthesubstrateindexisincreased.
AlsoshowninFig.5.3aretheQfactorscharacterizingtheopticallossalongthe
waveguide (WG), photonic crystal (PC), substrate and air directions as a function
oftheindexofthelowersubstrate. ThesedirectionsarelabelledinFig.5.4. Because
thedirectionalQfactorsaddasinverses,thesmallestdirectionalQfactorwillbethe
limitingQfactor. From theresults ofFig. 5.3weseethatthelimitingloss direcion
is into the substrate. From the previous chapter, we found that the dominant loss
direction for the high Q photonic crystal heterostructure cavity mode was out-
of-plane, so it is not surprising that the dominant loss direction is into the lower
substrate.
It is interesting to note that the in-plane loss is comparable to the out-of-plane
loss for substrate refractive indicies between 1.2 and 1.4. The enhanced in-plane
loss is due to the vertical symmetry breaking as the index of the lower substrate
is increased. For an air-clad structure the modes of the slab can be characterized
87
Figure5.3: DirectionalQfactorasafunction ofindexoflowersubstrate foratype
Aphotonic crystalheterostructure cavity.
Figure 5.4: Illustration of radiation directions for a substrate bonded photonic
crystalheterostructure cavity.
88
as even (TE-like) and odd (TM-like). A photonic crystal structure consisting of a
hexagonalarrayofairholessupportsabandgapfortheevenmodesbutnotforthe
odd modes for normalized frequencies below 0.5. For a symmetric air-clad struc-
ture, the even and odd modes are rigorously orthogonal, so no coupling between
them is expected. However, as the vertical symmetry is broken, the slab modes
are no longer purely even or purely odd, so some coupling is expected. Because
the odd modes are not confined by a bandgap, coupling of the even modes to
the odd modes results in in-plane leakage. There have been reports investigat-
ing the role of in-plane leakage in photonic crystal cavities formed in silicon-on-
insulator [TAHN04, TAHN06] and InGaAsP quantum well material bonded to a
sapphire substrate [KKS
+
02] andhavecome tosimilarconclusions. From Fig. 5.4,
one sees that for the photonic crystal double heterostructure cavity, in-plane leak-
age along the waveguide direction is present, but the substrate leakage remains
dominant.
5.3 Strategyforreducingtheout-of-planeradiation
One interesting feature of photonic crystal double heterstructure cavities is that
theirpropertiesarerelatedtothepropertiesoftheunderlyingstraightwaveguide.
Thishasalreadybeenpointedoutwithregardtotheelectromagneticfieldprofiles
displayedin4.6. Therefore,wecanconsideralternativephotoniccrystalwaveguide
structuresthathavebeendesignedtoreducetheout-of-planeloss. Inparticularthe
typeBphotoniccrystalwaveguidedesignhasbeenshowntohavereducedout-of-
planelossformodesabovethelightlineinverticalslabgeometriesconsistentwith
epitaxiallygrownsemiconductorlayers[KO04]. InthisSection,weinvestigatethe
89
Figure 5.5: Schematic diagram of a type B photonic crystal waveguide. The top
photonic crystal cladding is shifted one half lattice period along the x-direction
withrespecttothebottom cladding.
possiblityofformingaheterostructurecavityfromsuchawaveguideconfiguration
anddiscussitsQfactorandradiationloss[MO08b,MO09b].
A type B photonic crystal waveguide configuration is shown in Fig. 5.6. We
willrefertoanominalmissinglinedefectphotonic crystal waveguideasatypeA
waveguide. ThetypeBwaveguideisformedfromatypeAwaveguidebyshifting
one side of the cladding by one half lattice constant (a/2) along the waveguide
direction(x-direction). Fromaqualitativestandpoint,thehalflatticeconstantshift
introduces a π-phase shift between the fields on the upper half of the figure and
those on the lower half. On-axis, the out-of-phase fields interfere destructively
reducing the power radiating out-of-plane. A different perspective considers the
spatialFouriertransformofthestructureinFig.5.5whichshowsthattheamplitude
of the spatial Fourier coefficient in the first Brillouin zone goes to zero at β
y
=
0[KO04]. Becausethewaveguidemodecanbeexpandedinaseriesofwavevectors
thatmakeupthelattice, reducedamplitudeinthefirstBrillouin zoneimpliesless
90
Figure 5.6: Photonic crystal waveguidedispersiondiagramforpropagation along
theΓ−K direction. Blue regions correspond to photonic crystal cladding modes.
The straigh black line is the light line. The various colored curves are waveg-
uidedispersionbandsforwaveguideswhosephotoniccrystalcladdingarraysare
shiftedbyvaryingdegreesalongthex-direction. Alatticeshiftof0.0acorresponds
toatypeAphotoniccrystalwaveguide. Alatticeshiftof0.5acorrespondstoatype
Bphotoniccrystalwaveguide.
fieldamplitudeinthefirstBrillouinzone. Becausethelightlineaffectsthephotonic
crystal waveguide band in only the first Brillouin zone for the frequency range of
interest,oneexpectlesscouplingtotheradiationmodes[KO04].
To understand the waveguide dispersion diagram associated with a type B
photonic crystal waveguide, it is instructive to monitor the frequency evolution
of the waveguide bands as one side of the photonic crystal lattice is shifted with
respect to the other side. Fig. 5.6 displays the waveguide dispersion bands for
six differentwaveguides. The red curves represent the dispersion of the standard
91
typeAwaveguideinwhichthephotoniccrystalcladdingmakingupthetwosides
of the waveguide have not been shifted. The purple curves correspond to the
dispersionofthetypeBwaveguideinwhichthephotoniccrystalcladdingmaking
up the two sides of the waveguide have been shifted by one half lattice constant.
The remaining curves correspond to the dispersion of waveguides with varying
degreesoflatticeshift.
One striking feature of Fig. 5.6 is the transition from a band anti-crossing to a
bandcrossingnearβa/π= 0.44andωa/2πc= 0.288. Thetwobandsthateithercross
oranti-crossatthispointhavethree-dimensionalfielddistributionswhoseoverlap
integrals equate to zero (
R
~
E
1
·
~
E
2
d~ r = 0). However, coupled mode theory tells
us that the coupling constants characterizing the interaction between waveguide
modesinvolvetransversespatialoverlapintegralsoftheform
κ ∝
"
E
1
(y,z)E
2
(y,z)ǫ(y,z)dydz (5.1)
where E
i
(y,z) is a tranverse electric field component of waveguide mode i = {1,2}
andǫ(x,z)representsthetransversedielectricdistributionofthewaveguidestruc-
ture. The main point is that when E
1
(y,z) and E
2
(y,z) and ǫ(y,z) are odd about
the center of the waveguide, then κ = 0. In the geometries considered in Fig. 5.6,
the only structure thatis symmetric about the center of the waveguide is the type
A waveguide. Therefore, the various waveguide bands do not interact with each
other, andbandcrossings areallowed. Assoon as thephotonic crystal lattices are
shiftedwithrespecttoeachother,κ, 0,andanti-crossingisexpected.
Another interesting feature of Fig. 5.6 is the degeneracy of the type B waveg-
uide bands at the Brillouin zone boundary (βa/π = 1.0). This degeneracy results
92
from the symmetry of the type B structure which is known as a glide plane sym-
metry. The band structure implications of glide planes have been discussed in
the context of solid state physics which predicts degeneracy at the Brillouin zone
boundaries[Hei60].
Returningtothetaskofdesigningphotoniccrystaldoubleheterostructurecav-
ities with reduced substrate loss, Fig. 5.6 provides useful information about the
spectralpropertiesoftheboundstatemodes. First,thedegeneracyattheBrillouin
zoneboundarywillresultintwoboundstatemodesspacedverycloseinfrequency.
Second,thelowestorderwaveguidemodeinthephotoniccrystalbandgapisclose
tothephotoniccrystalcladdingmodes. Basedontheresultsofthepreviouschap-
ter, one expects significant leakage into the cladding due to this small frequency
separation. Thewaveguidemodecanbemovedtohigher(lower)frequencies rel-
ative to the photonic crystal cladding modes by reducing (increasing) the width
of the photonic crystal waveguide core [NSY
+
02]. Another approach would be to
increase(decrease)theinnermostrow ofholes inthephotonic crystal claddingto
shift the photonic crystal waveguide mode to higher (lower) frequencies. In both
casesthefrequencyofthewaveguidebandshiftsasaresultofeitherdecreasingor
increasingtheeffectiveindexofthestructure.
Figure 5.7 is a schematic diagram of a photonic crystal double heterostructure
cavityformedfromatypeBphotoniccrystalwaveguide. Thewidthofthephotonic
crystalwaveguidecoreregionislabeledas1.0wwhichcorrespondstoasinglerow
of holes removed to form the line defect. Table 5.2 displays the Q factors and
resonant frequencies of the type B photonic crystal heterostructure bound state
corresponding to the lowest waveguide band in the bandgap. Also included in
Table 5.2 are the Q factors and resonant frequencies for heterostructure modes
corresponding to structures with reduced waveguide widths. From Table 5.2 one
93
Figure 5.7: Double heterostructure cavity formed from a type B photonic crystal
waveguide.
Waveguide Resonance Quality
Width Frequency Factor
1.0w 0.2454558 659
0.2454691 733
0.9w 0.2517232 1153
0.2517905 2216
0.8w 0.2631054 915
0.2631449 1493
0.7w 0.2792621 365
0.2793310 2480
Table 5.2: The resonance frequencies and Q factors for bound state resonances
associatedwithtypeBheterostructure cavityfordifferentwaveguidewidths.
sees that there are two resonance modes closely spaced in frequency for each
structure. Thesetwomodesarisefromthetwodegeneratewaveguidebandsatthe
Brillouinzoneboundaryasdiscussedabove.
Table 5.2 also shows that the resonance frequency increases as the waveguide
width is decreased which is consistent with the basic design principle discussed
above. An improvement in the Q factor is seen from decreasing the waveguide
widthfromitsnominalvalueof1.0w. However,thereisnotaclearoptimumvalue
oncethewaveguidebandhasbeensufficientlyseparatedfromthephotoniccrystal
94
Figure 5.8: H
z
(x,y,z = 0) at the midplane of the slab for a type B hetetostructure
cavitywithawaveguidewidthof0.8w. Theresonancefrequencyis0.2631449. The
photoniccrystal geometryissuperimposed.
modes. For the remainder of this Chapter, we will focus on a waveguide width
of 0.8was this places the resonance frequency of the bound state resonances close
tothatof thecorresponding typeAstructure. This facilitatescomparingtheeffect
of radiation into the substrate as both the type A mode and the type B mode will
have the same frequency location relative to the light line in the photonic crystal
waveguidedispersiondiagram.
Fromthepreviouschapter,wesawthatthetypeAphotoniccrystalheterostruc-
ture supported a bound state resonance with a Q factor of 337,000. The Q factor
of the type B photonic crystal heterostructure modes listed in Table 5.2 are signif-
icantly smaller. Inspection of the of electromagnetic field profile at the midplane
of the slab shown in Fig. 5.8 shows significant field amplitude distributed in the
uniformwaveguidesections ofthedevice. Thissuggests thattheprimaryleakage
direction of this cavity is along the x-direction. The enhanced leakage into the
uniform waveguide can be explained by the presence of the two closely spaced
waveguidebandsshowninpurpleinFig.5.6. ForauniformtypeBphotoniccrys-
talwaveguide, thesetwobandsareorthogonal, andnocouplingbetweenthetwo
bands is expected. However, introducing the perturbation breaks the glide plane
95
Figure 5.9: H
z
(x,y,z = 0) at the midplane of the slab for a type B hetetostructure
cavity with a waveguide width of 0.8w and tapered perturbation. The resonance
frequencyis0.2617761. Thephotoniccrystalgeometryissuperimposed.
symmetry in the structure, and coupling from the bound state mode to a nearby
waveguidemodebecomespossibleandexplainstheleakageintothewaveguide.
Inordertoreducethisleakage,weintroducedataperedperturbation. Usually
these taperings function to reduce the high spatial frequency components of the
mode which in turn reduce overlap with the light cone and reduce out-of-plane
radiation [SNAA05, TAN08]. However, in this case the tapering is designed to
softentheglideplanesymmetrybreaking,sothatthein-planecouplingcausedby
theperturbationisreduced. Thetaperedstructureandcorrespondingfieldprofile
are shown in Fig. 5.9. Thecentral perturbation region consists of a 5% lattice con-
stantincreasealongthex-directionasbefore;however,thiscentralperturbation is
sandwitched between perturbations of 2.5% lattice constant increase. This inter-
mediateperturbationservesasatransitionregionbetweentheuniformwaveguide
section and the central perturbation region. In Fig. 5.9 one can see that the field
amplitudeintheuniformwaveguidesectionsisgreatlyreduced. Table5.3summa-
rizestheresonancefrequenciesandQfactorsforthetypeBdoubleheterostructure
96
Waveguide Resonance Quality
Width Frequency Factor
0.8w 0.2617761 21085
0.2620127 9869
Table 5.3: The resonance frequencies and Q factors for bound state resonances
associated with a type B heterostructure cavity with a waveguide width of 0.8w
andataperedperturbation.
withthetaperedperturbation. ImprovementintheQfactorbyanorderofmagni-
tude isapparent. Inwhatfollows wewill focus on the mode with thehigherQin
Table5.3
Figure5.10displaysthebreakdownofthedirectionalQfactorsastheindexofthe
lowercladdingisincreased. Fig.5.10showsthatforsubstrateindicesbetweenn=
1.0andn= 1.5thetypeBtotalQfactorisdominatedbylossalongthewaveguide
direction. Itisinterestingtonotethatintroducingthetaperedperturbationgreatly
reduced the coupling to the uniform waveguide; however, this leakage direction
remains dominantforlow indexsubstrates. Once the substrate indexis increased
aboven= 1.5,substratelossesdominateaswasthecaseforthetypeAstructure.
Figure 5.11 shows a comparison between the total Q factors of a type A het-
erostructure cavityandatypeBheterostrucrture cavity. OneseesthattheQfactor
of the nominally air-clad type B cavity is an order of magnitude smaller than the
air-cladtypeAheterostructure cavity. However,becausethetypeBcavity’slosses
are not dominated by out-of-plane radiation for low index substrates, its total Q
factor isflatbetweensubstrate indicesbetweenn= 1.0andn= 1.4. The Qfactors
of the type A and the type B cavities become equal at around n = 1.3, and the
type B heterostructure cavity has a larger Q factor by as much as a factor of 5 for
substrate indices n > 1.4. In particular, the type B cavity has a larger Q factor
when the indexof the lower substrate is consistent with silicon dioxide (n = 1.45)
97
Figure5.10: DirectionalQfactorasafunctionofindexoflowersubstrateforatype
Bphotoniccrystalheterostructure cavity.
andsapphire(n= 1.75). Experimentalstudiesofsapphirebondedphotoniccrystal
lasers has suggested that a cavity requires a Q factor of at least 1000 in order to
achieve CW lasing [SKY
+
06b]. It should be noted that the type B heterostructure
hasaQfactorgreater than1000forasubstrate indexofn= 1.75whereasthe type
A cavity does not. We believe this cavity is a promising design for achieving CW
lasing in a photonic crystal double heterostructure cavity with a dielectric lower
substrateandiscurrentlythefocusofexperimentaleffortsinourgroup.
5.4 Towardelectricallyinjectedphotoniccrystallasers
One strategy that may lead to an electrically addressable photonic crystal laser
is to define the vertical slab structure epitaxially using semiconductor deposition
techniques. For alloys employing InGaAsP materials the index contrast between
98
Figure 5.11: Comparison of the Q factors as a function of the index of the lower
claddingfortypeAandtypeBhetetrostructure cavities.
99
the core and vertical cladding layers is small. Realistic values for the refractive
indicesforthecoreandcladdingwouldbe3.4and3.2,respectively[BL84]. Onthe
other hand, it becomes possible to etch through the entire vertical slab structure
which has been shown to reduce vertical radiation loss. Another advantage of
usinganepitaxiallydefinedslabstructureistheabilitytointroduceatopcladding
layer. The advantage of including a top cladding layer is shown in Fig. 5.12.
Fig.5.12isadirectionalQfactorstudycorrespondingtotheverticalslabstructure
depictedin Fig. 5.13fora type Aheterostructure cavity. There are twointeresting
features of this data: (1) The total Q of the symmetrically clad structure is larger
than the structure in Fig. 5.3 in which the semiconductor slab has only a lower
claddingeven though intuitively, one would expectthe top claddingto introduce
moreleakage. (2)ThesubstrateQisalmostthreetimeslargerforthesymmetrically
cladstructure thanfortheasymmetricallycladstructure. Oneexplanationforthis
effectis that the slab mode remains in the center of the slab for the symmetrically
cladstructure,whereasthemodeisshiftedtowardthesubstratesideinthebottom
cladstructure whichincreasestheevanescentfieldpenetrationintothesubstrate.
It is also interesting to note that the in-plane Q factors (PC and WG) are as
muchasanorderofmagnitudelargerforthesymmetricallycladstructurethanfor
thethe structure with aonesidedcladding. This illustrates the degreeofin-plane
leakagetobeexpectedduetovertical symmetrybreaking. Although out-of-plane
lossesdominate,maintainingverticalsymmetrycanreducein-planeleakagebyan
orderofmagnitude.
Figure 5.14displays acomparison betweenthe Qfactors of the typeAandthe
typeBheterostructurecavitiesversuslowersubstrateindexforsymmetricvertical
claddings. ThebasictrendofFig.5.11isreproduced,buttheQfactors decreaseat
a slower rate. Because the type B heterostructure exhibits a higher Q factor than
100
Figure5.12: DirectionalQfactorasafunctionofindexoftopandbottomcladding
foratypeAphotonic crystalheterostructure cavity.
Figure5.13: Schematicdiagramdepictingtheverticalslabstructure foraphotonic
crystalheterostructure withatopandbottom verticalcladding.
101
Figure 5.14: Comparison of the Q factors as a function of the indexof the top and
bottom verticalcladdingfortypeAandtypeBhetetrostructure cavities.
102
Figure5.15: Qfactorversustopandbottomcladdingindexwithvaryingamounts
of hole penetration into the vertical cladding for a type B photonic crystal het-
erostructure cavity. Theholedepthcorrespondstothedepthintoboththetopand
the bottom cladding. These structures maintain vertical mirror symmetry about
themidplaneoftheslab.
the type A structure as the substrate index is increased, we will focus on the type
Bstructure fortheremainderofthissection.
Figure 5.15 shows the dependence of the Q factor on the substrate index of
the vertical claddings as the air holes that perforate the semiconductor slab are
extendedintothecladding. SignificantimprovementintheQfactorcanbeachieved
byextendingtheairholesintheverticaldirection. TheQfactorincreasesnegligibly
forholedepthsgreaterthan1.0μm. ThedatainFig.5.15correspondstostructures
withinfinitetopandbottom claddinglayers. Inarealdevicetheverticalcladding
layers would have a finite extent. Current fabrication procedures use surface
patterning techniques, so the prominant user defined features of the device are
103
Figure 5.16: Q factor versus top and bottom cladding index for different top
cladding thicknesses for a type B photonic crystal heterostructure cavity. Inset
depictstheverticalcladdinggeometry.
confined to the top 1 − 2μm of the device. Fig. 5.16 shows the Q factor versus
claddingindexforastructurewithashortenedtopcladdinglayerwhichisdepicted
intheinset. Itisinterestingtonotethatreducingthetopcladdingto0.5μmresults
inahigherQfactorthanatopcladdinglayerof1.0μm. Althoughwesawthatthe
Q factor tends to increase as the air holes penetrate more deeply into the vertical
claddings, by shortening the cladding layer and replacing it with a lower index
material (air), the effective index of the top cladding decreases and improves the
confinementtothehighindexslabregion.
Figure 5.17 explores the dependence of the type B heterostructure Q factor on
the cladding index for a structure with varying high index slab thickness. The
structureinvestigatedinFig.5.17hasa0.5μmtopperforatedcladdinganda1.0μm
104
Figure5.17: Qfactorversustopandbottomcladdingindexfordifferenthigh-index
slab thicknesses for a type B photonic crystal heterostructure cavity. The vertical
slab structure is shown in the inset of Fig. 5.16 with a 1.0μm etch depth in the
bottom claddingand0.5μmperforatedtopcladdinglayer.
bottomperforatedcladdingfollowedbyanunpatternedsubstrateofinfiniteextent.
Asthethicknessofthehighindexslabisincreased,itsmodeconfinementimproves
and reduced out-of-plane radiation is expected. The improvement in the Q factor
as the slab thickness is increased is exhibited in Fig. 5.17, but the improvement is
modest.
In this Chapter, a strategy based on glide-plane symmetry was introduced to
reduce the out-of-plane losses associated with the photonic crystal double het-
erostructure cavity. The dependence of the Q factor on the substrate was investi-
gated,andouranalysispredictedthataQfactergreaterthan1,000wasachievable
forthethermallyconductivesubstratessapphireandsilicondioxide.
105
It was found that etching through a thick epitaxially defined slab improved
the Q facter significantly. Q factors of 1,000 were obtained for a structure with a
1.9μmtotal etchdepthandacladdingindexof 2.3. Althoughanindexcontrast of
2.3 to 3.4 is not achievable using different alloys in the InGaAsP material system,
this represents a significant improvement over simply using a shallow etch type
A geometry. Furthermore, AlGaN materials have refractive indices in the range
2.0-2.3[BAB
+
97]andSiChasarefractiveindexof2.6[Har95]suggestingthatinitial
demonstrations could be carried out by bonding an InGaAsP quantum well layer
toanAlGaNorSiCsubstrateandpatterningthestructureaccordingtotheoptimal
typeBdeeplyetchedgeometry.
106
Chapter6
Finite-DifferenceTime-Domain
AnalysisWithaMaterialGainModel
6.1 Introduction
In the preceding Chapters, the finite-difference time-domain method was applied
topassivecavitystructures madeofmaterialsthatexhibitedneithergainnorloss.
In this Chapter we present a novel method for analyzing the interaction between
resonant cavities and materials exhibiting gain and absorption. We also discuss
methods for introducing dispersive materials into finite-difference time-domain
simulations.
The motivation for this work is to investigate the results derived in Chapter 2
astheyareappliedtophotoniccrystalmicrocavities. Specificallywewouldliketo
verifythelaserthresholdconditionEq.2.22forthecaseofaphotoniccrystaldouble
heterostructure cavity. We will take advantage of the direct extraction version of
Pad´ e interpolation and its ability to extract the sign of the Q factor. This will
allowustoanalyzethetemporalbehavioroftheenergyconfinedtothecavityand
determinewhetheritisgrowing, decayingorremainingconstantintime.
107
6.2 Dispersive gain modeling using the finite-
differencetime-domainmethod
Dispersive material gain and absorption will be introduced through a frequency
dependent conductivity. As discussed in Chapter 2, gain and absorption can be
modeledthroughmaterialconductivitieswithoppositesigns[And65,HJT96]. Gain
would be assigned a negative conductivity, and absorption would be assigned a
positive conductivity. The relationship between the gain and absorption coeffi-
cientsandtheirassociatedmaterialconductivities isgiveninEqs.2.17and2.21.
Although not explicitly required for the present analysis, we also introduce
dispersion into the material conductivity. This is motivated bya general desire to
makeournumericaltoolset asgeneralas possible. Italsoopensupthepossibility
ofanalyzingtheeffectofdispersiononthepropertiesofthelasingmodeaslongas
thedispersionisconsistentwiththepropertiesofarealgainmedium[PPG07].
Because the finite-difference time-domain method works in the time domain,
frequencydependentmaterialsmustbehandledviaappropriateFouriertransform
relationships. Specifically,Ohm’slawinthefrequencydomainreadsas
~
J(ω)= σ(ω)
~
E(ω) (6.1)
whereas inthe timedomain
~
J(t) and
~
E(t) are related bya convolution as shown in
Eq.6.2.
~
J(t)=
Z
t
−∞
σ(t−t
′
)
~
E(t
′
)dt
′
(6.2)
One approach for including a frequency dependent conductivity is known as
piecewise linear recursive convolution which starts with Eq. 6.2 and implements
108
the convolution integral via a discretized Riemannsum [TH00]. Asimplerandin
somecasesmoreefficientapproachisknownastheauxilliarydifferentialequation
(ADE) method. The ADE method begins with the specific form of the frequency
dependence of the conductivity. In this case we will use Lorentzian pairs placed
symmetrically about zero. This frequency dependence yields a real-valued fre-
quency response that is consistent with the Kramers-Kronig relationship. Eq. 6.3
displays the frequencyprofile of the conductivity. The gainis peakedat±ω
0
with
a magnitude of σ
0
/2 and full width athalf maximum of 2/T. Asstated earlier, we
introducedispersionintoourgainmediumtomakeournumericalanalysistoolas
general as possible. However, because we are focussing here on a specific cavity
mode in a narrow spectral window, the shape of the gain spectrum is not crucial
fortheanalysisthatfollows.
σ(ω)=
σ
0
/2
1+ j(ω−ω
0
)T
+
σ
0
/2
1+ j(ω+ω
0
)T
(6.3)
The next step in the ADE method is to insert Eq. 6.3 into Eq. 6.1 and rearrange to
yield
(1+ j(ω−ω
0
)T)(1+ j(ω+ω
0
)T)J(ω)= σ
0
(1+ jωT)E(ω) (6.4)
Convertingfromthetimeharmonicregimetotheinstantaneousregimeconsistsof
substituting
∂
∂t
for jω. Theresultingtimedomainequationis
(1+ω
2
0
T
2
)J+2T
2
∂J
∂t
+T
2
∂
2
J
∂t
2
= σ
0
E+σ
0
T
∂E
∂t
. (6.5)
Only first order time derivatives appear in the finite-difference time-domain
method. However, a second order time derivative appears in Eq. 6.5. Eq. 6.5
109
Figure 6.1: 1/Q
tot
versus peak material gain where the spatial gain distribution is
varied. Thegainisuniformintheplaneoftheslabbutabsentintheair-holes. The
threecurvescorrespond todifferentfractions oftheverticalslabprovidinggain.
can be modfied by introducing an auxilary variable F =
∂J
∂t
, so that the algorithm
works exclusively with first order time derivatives. This comes at the expense of
including extra variables and equations that must be sequentially updated along
withtheMaxwellcurlequations[TH00].
6.3 Interaction between spatial gain distribution and
cavitymode
In the previous Section we discussed methods for including a dispersive material
gainmodelintothefinite-differencetime-domainmethod. InthisSectionwevary
themagnitudeandspatialdistributionofthematerialgain. Thisisdonebymaking
σ
0
spatiallyvarying.
110
Figure 6.1 displays the results of analyzingthe interaction between a photonic
crystaldoubleheterostructurecavitymodeandamaterialwithaspatiallyvarying
gain distribution. The cavity mode under investigation is Mode 2 from Table 4.1
whichisthehighestQmodeofthiscavity. Thegaindistributionisconfinedtothe
semiconductor slab region and only exists in the semiconductor material. The air
holesmakingupthephotoniccrystallatticedonotprovidegain. Aconstantvalue
of gain was assigned to the entire in-plane distribution. Fig. 6.1 displays three
curveseachcorrespondingtodifferentverticalportionsoftheslabsupplyinggain
tothemode. Fig.6.1plots1/Q
tot
versusmaterialgain. ThequantityQ
tot
isdefined
inEq.6.6.
ω
0
Q
tot
=
ω
0
Q
p
−
R
(
c
n
)2g(x,y,z)ǫh
~
E·
~
EidV
R

1
2
ǫh
~
E·
~
Ei+
1
2
μh
~
H·
~
Hi

dV
. (6.6)
Equation 6.6 follows directly from Eq. 2.22. In this case there is noabsorption,
so the two terms remaining on the right side are the passive Q factor and the
modal gain. In the present analysis, the user controls the magnitude and spatial
distribution of g(x,y,z). In Fig. 6.1, the bottom axis starts at zero gain. When
g(x,y,z) = 0, then 1/Q
tot
= 1/Q
p
. In this case 1/Q
p
= 1/336700 = 2.97 × 10
−6
which is the intercept with the vertical axis for all three curves. As g(x,y,z) is
increased from zero, 1/Q
tot
decreases consistent with Eq. 6.6. At the point where
1/Q
tot
= 0, the modal gain is exactly equal to the radiative losses. Because there
is no absorption in this case, this corresponds to the threshold condition Eq. 2.22.
Because the magnitude of the gain is completely user controled, the gain can be
increased beyond the threshold point. In this case the modal gain ovetakes the
radiativelossterminEq.6.6resultinginanegative1/Q
tot
.
111
In order to extract the Q factors appearing in Fig. 6.1 we employed the same
procedureusedtoanalyzethepassivecavities. Thecavitywasexcitedwithaspa-
tiallyrandombroadbandinitialcondition,andthedirectextractionPad´ eapproach
was used to measure the Q factor. It should be noted that the ability to extract a
negativevaluedQfactorismadepossiblebythedirectextractionapproachtoPad´ e
interpolation.
Figure 6.2 is an intuitive description of positive, negative and zero values for
1/Q
tot
. Fig.6.2(a)showsthedampedsinusoidbehaviorofatypicalelectromagnetic
field component experiencing a net loss characterized by 1/Q
tot
> 0. Fig. 6.2(b)
showsthepurelysinusoidalbehaviorofatypicalelectromagneticfieldcomponent
experiencing no netloss characterized by 1/Q
tot
= 0. Finally, Fig. 6.2(c) shows the
growing sinusoidal behavior of a typical electromagnetic field component expe-
riencing net gain characterized by 1/Q
tot
< 0. In this case the field will increase
indefinitelyandrepresentsanunstablesituation. Physicallythisisimplausibledue
togainsaturationwhichcausesthegaintoclampatitsthresholdvalue.
The results displayed in Fig. 6.1 show two interesting features. First, the three
curves each exhibit a linear dependence of 1/Q
tot
on g(x,y,z). To explain this
behavior,Eq.6.6isrewrittenas
ω
0
Q
tot
=
ω
0
Q
p
−
2cg
max
R
(
1
n
)
2g(x,y,z)
2g
max
ǫh
~
E·
~
EidV
R

1
2
ǫh
~
E·
~
Ei+
1
2
μh
~
H·
~
Hi

dV
=
ω
0
Q
p
−2cg
max
Γ (6.7)
where
Γ=
R
g(x,y,z)
ng
max
ǫh
~
E·
~
EidV
R

1
2
ǫh
~
E·
~
Ei+
1
2
μh
~
H·
~
Hi

dV
(6.8)
112
Figure6.2: Illustrationoffieldevolutionversustimefor(a)1/Q
tot
> 0,(b)1/Q
tot
= 0
and(c)1/Q
tot
< 0.
113
isknownastheopticalconfinementfactorandg
max
isthemaximumofthefunction
g(x,y,z). Foragivenspatialgaindistribution,Γisaconstant. InFig.6.1thebottom
gainaxiscorrespondstog
max
. Therightsideof6.7showsthat1/Q
tot
shouldbehave
linearlyon g
max
whichisconsistentwithwhatisdisplayedinFig.6.1.
ThesecondinterestingfeatureinFig.6.1isthatasthesizeofgainregioninthe
slabdecreases,theamoungofgainrequiredtoreach1/Q
tot
= 0increases. Againthis
isconsistentwiththeanalysisshowninEqs.6.7and6.8. Asthegainregionshrinks,
the numerator in Eq. 6.8 decreases while the denominator remains unchanged
resulting in a smaller confinement factor and a smaller slope in Fig. 6.1. For gain
materialutilizingquantumwells,onlyverythinslicesoftheslabactuallyprovide
gain. Comparingthethresholdgainrequiredtoreachthresholdobtainedfromthe
finite-difference time-domain simulation to what would result from performing
thespatialoverlapintegralinEq.2.22usingpassivefieldprofilesyieldsagreement
towithin1%. WeconcludeFig.6.1representsasuccessfulverificationofEq.2.22.
Now we consider a spatial gain distribution that would arise from an optical
pump spot incident from the top of the structure. Fig 6.3(a) illustrates the gain
distributionsuperimposedonthephotoniccrystalheterostructurecavitystructure.
Inthiscasethefullthicknessoftheslabprovidesgainintheverticaldirection,while
the in-plane distribution is described by a two-dimensional gaussian function.
Fig.6.3(b)displaysthe1/Q
tot
versusmaterialgainfortwodifferentspotsizes. The
trends are qualitatively similar to the results of Fig. 6.1 and the finite-difference
time-domain simulation yields a threshold gain to within 1% of what Eq. 2.22
wouldpredict. Theseresults supporttheresultshowninEq.2.22.
114
Figure 6.3: (a) Two-dimensional Gaussian gain distribution superimposed on a
photoniccrystalheterostructurecavity. (b)1/Q
tot
versuspeakmaterialgainfortwo
differentpumpspotsizes.
115
6.4 Futuredirections
Recently,therehavebeenseveralreportsofincorporatinggainintofinite-difference
time-domain simulations for the analysis of laser cavities and amplifiers. In most
cases, gain saturation is introduced either phenomenalogically [HJT96, KM04] or
through some version of a carrier rate equation [NY98, SKH02, SAZ04, CT04,
BRPO04, BPO05, HH06, SSIKL05, PPG07, SP07]. These reports focus on two fea-
tures of active devices: gain dynamics and laser threshold. Laser threshold is
determined by gradually increasing the pump source and monitoring the output
powerofthedevice. Becausegainsaturationisincludedinthemodel,thesimula-
tion mustreach equilibriumbetween thegain dynamics andthesteadystate field
amplitudewhichcanrequirelongsimulationtimes.
In the simulations reported in this Chapter, we use interpolation techniques
to extract the steady state behavior of the fields in response to a gain medium.
Thisavoidstherequirementofusingexcessivelylongsimulationtimesrequiredto
generateathresholdcurve. Thiscomesattheexpenseoflackingagainsaturation
mechanism. Withoutgainsaturation,oursystemremainslinear,andlinearsystem
characterization techniques can be applied. We envision a similar interpolation
approach could be applied to finite-difference time-domain simulations with a
saturable gain medium. In this case, the system is no longer linear, so nonlinear
systemtechniquesmustbeapplied.
The main result of this Chapter is the verification of the laser threshold con-
dition Eq. 2.22 within the context of the finite-difference time-domain method.
However,electrodynamicssimulationswithagainmodelcanyieldresultsthatare
notdirectlyobtainablefromalgebraicmanipulations. Oneexampleisthegainsat-
uration refered to in preceding paragraph. Regions in which the optical intensity
116
is large will result in a depleted local carrier density [NB05]. This phenomenon
is known as spatial hole burning. Knowledge of the equilibrium carrier density
resultingfromitsinteractionwiththelasingmodecouldleadtoimprovedinjection
efficiencyandlowerthresholdsandoperatingtemperatures.
The other feature of active devices that has been explored using the finite-
difference time-domain method with a saturable material gain model is gain and
field dynamics on extremely short time scales. Because finite-difference time-
domainsimulationstypicallymonitorthefieldevolutionforafewtensofpicosec-
onds, analyzing laser dynamics is a natural application of this technique. Specifi-
cally,onecanobtaininformationonbandwidthlimitationsarisingfrominteraction
betweentheopticalmodeandthespatialgaindistribution.
Onefinalapplicationinwhichthisapproachwouldyieldusefulresultsisinthe
modelingofspontaneousemission. AsmentionedattheendofChapter2,amore
complete statement of the laser threshold condition would include both sponta-
neous and stimulated emission sources. Small cavities with large Q factors show
anenhancementofspontaneousemissioncouplingintotheresonantmode[BY91].
This enhanced coupling can lower thresholds and improve modulation speeds.
However, this has not been explored within the context of the finite-difference
time-domainmethod. Havinganumericaltoolthatcouldaccuratelyhandlespon-
taneous emission enhancement would represent a powerful tool as cavities with
higherQfactorsandsmallermodevolumescontinuetobedesigned.
BecausethestimulatedandspontaneousemissionratesarerelatedviaEinstein’s
AandBcoefficients,thecontributions fromstimulatedandspontaneousemission
could be distinguished within the simulation approach described in this Chapter.
The additional information required is the threshold photon density. This raises
an interesting point about the numerical approach described here: it is linear.
117
Because of the linearity of the system, the field amplitude has no meaning. The
threshold condition in Eq. 2.22 contains ratios of field amplitudes which serve as
normalizationcoefficientsandremovesanysignificanceaboutthefieldamplitude.
The only feature in which we have been interested is the field decay rate. As
soon as the field amplitude has meaning, the system becomes nonlinear. This
makes methods of efficient extraction of threshold information in the presence of
spontaneousemissionsourcesaninterestingfuturepursuit.
118
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129

Abstract (if available)

Abstract Two-dimensional photonic crystals represent a versatile technology platform for constructing photonic integrated circuits. Low-loss and small footprint waveguides and cavities can be combined to make delay lines, modulators, filters and lasers for efficient optical signal processing. However, this diverse functionality comes at the expense of higher complexity in both the fabrication and the modeling of these devices. This Thesis discusses the finite-difference time-domain numerical modeling of large quality factor photonic crystal cavities for chip-scale laser applications.

Linked assets

University of Southern California Dissertations and Theses

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Asset Metadata

Creator Mock, Adam (author)

Core Title Analysis of photonic crystal double heterostructure resonant cavities

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Electrical Engineering

Publication Date 07/21/2009

Defense Date 06/01/2009

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag cavity,finite-difference time-domain,laser,OAI-PMH Harvest,optics,photonic crystals,photonic integrated circuit,photonics,resonator,semiconductor

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor O'Brien, John D. (committee chair), Nakano, Aiichiro (committee member), Steier, William H. (committee member)

Creator Email amock77@yahoo.com,mock1ap@cmich.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m2382

Unique identifier UC1216138

Identifier etd-Mock-3011 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-563141 (legacy record id),usctheses-m2382 (legacy record id)

Legacy Identifier etd-Mock-3011.pdf

Dmrecord 563141

Document Type Dissertation

Rights Mock, Adam

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Repository Name Libraries, University of Southern California

Repository Location Los Angeles, California

Repository Email cisadmin@lib.usc.edu