mode competition in wave-chaotic microlasers
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Mode Competition in Mode Competition in Wave-Chaotic Wave-Chaotic MicrolasersMicrolasers
Hakan THakan TüürecireciPhysics Department, Yale UniversityPhysics Department, Yale University
Recent Results and Open QuestionsRecent Results and Open Questions
TheoryTheoryHarald G. Schwefel - YaleHarald G. Schwefel - Yale
A. Douglas Stone - YaleA. Douglas Stone - Yale
Philippe Jacquod - GenevaPhilippe Jacquod - Geneva
Evgenii Narimanov - Evgenii Narimanov - PrincetonPrinceton
ExperimentsExperimentsNathan B. Rex - Yale Nathan B. Rex - Yale
Grace Chern - YaleGrace Chern - Yale
Richard K. Chang – YaleRichard K. Chang – Yale
Joseph Zyss – ENS CachanJoseph Zyss – ENS Cachan
&&
Michael Kneissl, Noble Michael Kneissl, Noble Johnson - PARCJohnson - PARC
• Conventional Resonators: Fabry-Perot
• Dielectric Micro-resonators:
Trapping light by Trapping light by TIRTIR
Trapping Light : Optical Trapping Light : Optical μ-μ-ResonatorsResonators
total total internal internal reflectionreflection
n
no=1 t
Whispers in Whispers in μμDisks and Disks and μμSpheresSpheres
Microdisks, Slusher et al.
Lasing Droplets, Chang et al.
Very High-Q whispering gallery modesVery High-Q whispering gallery modes Small but finite lifetime due to tunnelingSmall but finite lifetime due to tunneling But:But: Isotropic EmissionIsotropic Emission Low output powerLow output power
Wielding the Light - Breaking the Wielding the Light - Breaking the SymmetrySymmetry
Spiral LasersSpiral Lasers
Local deformations ~ Local deformations ~ λλ Short- λ limit not applicableShort- λ limit not applicable No intuitive picture of emissionNo intuitive picture of emission
G. Chern, HE Tureci, et al. G. Chern, HE Tureci, et al. ””Unidirectional lasing from InGaN multiple-quantum-well spiral-shaped micropillars”,”,to be published in Applied Physics Lettersto be published in Applied Physics Letters
Smooth deformations Characteristic emission anisotropy High-Q modes still exist Theoretical Description: rays Connection to classical and Wave chaos Qualitative understanding > Shape engineering
AAsymmetric symmetric RResonant esonant CCavitiesavities
Maxwell Equations
The Helmholtz Eqn for The Helmholtz Eqn for DielectricsDielectrics
Continuity Conditions:
convenient family of deformationsconvenient family of deformations::
I(1
70
°)
kR
Im[k
R]
Re[kR]
Scattering vs. EmissionScattering vs. Emission
I(1
70
°)
Re[kR]
Im[kR]
Quasi-bound modes only exist at discrete, complex kQuasi-bound modes only exist at discrete, complex k
AAsymmetric symmetric RResonant esonant CCavitiesavities
Generically non-separable Generically non-separable NO `GOOD’ MODE INDICES NO `GOOD’ MODE INDICES
Numerical solution possible, but not poweful aloneNumerical solution possible, but not poweful alone
Small parameter : Small parameter : (kR)(kR)-1 -1 10 10-1-1 – 10 – 10-5-5
Ray-optics equivalent to Ray-optics equivalent to billiardbilliard problemproblem with refractive escape with refractive escape
KAM transition to chaos KAM transition to chaos
CLASSIFY MODES using PHASE SPACE STRUCTURESCLASSIFY MODES using PHASE SPACE STRUCTURES
OverviewOverview
1. Ray-Wave Connection1. Ray-Wave Connection Multi-dimensional WKB, Billiards, SOSMulti-dimensional WKB, Billiards, SOS
2. 2. Scattering quantization for dielectric resonatorsScattering quantization for dielectric resonators A numerical approach to resonatorsA numerical approach to resonators
3. Low index lasers3. Low index lasers Ray models, dynamical eclipsingRay models, dynamical eclipsing
4. High-index lasers - The Gaussian-Optical Theory4. High-index lasers - The Gaussian-Optical Theory Modes of stable ray orbitsModes of stable ray orbits
5. Non-linear laser theory for ARCs5. Non-linear laser theory for ARCs Mode selection in dielectric lasersMode selection in dielectric lasers
Multi-dimensional WKB (EBK)Multi-dimensional WKB (EBK)
• The EBK ansatz:
• Quantum Billiard Problem:
N=2, Integrable ray dynamics N, Chaotic ray dynamics
Quantization Integrable systemsQuantization Integrable systems
• Quantum Billiard: 2 irreducible loops 2 quantization conditions
• Dielectric billiard:
b(
)
Non-integrable ray dynamics (Einstein,1917) Quantum Chaos
Quantization of non-integrable systemsQuantization of non-integrable systems
Mixed dynamics: Local asymptotics possible
Globally chaotic dynamics: statistical description of spectra
Periodic Orbit Theory
Gutzwiller trace formula
Random matrix theory
Berry-Robnik conjecture
PoincarPoincaréé SSurface-urface-oof-f-SSectionection
Boundary deformations: SOS coordinates:
Billiard map:Billiard map:
The Numerical MethodThe Numerical Method
Internal Scattering Eigenvalue Problem:
Regularity at origin:
Numerical ImplementationNumerical Implementation
• Non-unitary S-matrix • Quantization condition • Complex k-values determined by a two-dimensional root search
• Interpolate to obtain and use to construct the quantized wavefunction
A numerical interpolation scheme:
Classical phase space structures
• Follow over an interval
NO root search!
Quasi-bound states and Classical Quasi-bound states and Classical Phase Space StructuresPhase Space Structures
Low-index LasersLow-index Lasers
Low index (polymers,glass,liquid droplets) n<1.5
Ray Models account for:Ray Models account for: Emission DirectionalityEmission Directionality LifetimesLifetimes
NNööckel & Stone, 1997 ckel & Stone, 1997
Polymer microdisk LasersPolymer microdisk Lasers
n=1.49
Quadrupoles: Ellipses:
Low index (polymers,glass,liquid droplets) n<1.5
Semiconductor LasersSemiconductor Lasers
High index (semiconductors) n=2.5 – 3.5
λ=5.2μm , n=3.3
Bell Labs QC ARC:
Bowtie LasersBowtie Lasers
““High Power Directional Emission from lasers with chaotic ResonatorsHigh Power Directional Emission from lasers with chaotic Resonators””
C.Gmachl,F.Capasso,EE Narimanov,JU Noeckel,AD Stone, A ChoC.Gmachl,F.Capasso,EE Narimanov,JU Noeckel,AD Stone, A Cho
Science Science 280280 1556 (1998) 1556 (1998)
• High directionalityHigh directionality• 1000 x Power wrt 1000 x Power wrt =0=0
=0.0=0.0
=0.14=0.14
=0.16=0.16
kR
Dielectric Gaussian OpticsDielectric Gaussian Optics
HE Tureci, HG Schwefel, AD Stone, and EE HE Tureci, HG Schwefel, AD Stone, and EE Narimanov Narimanov ””Gaussian optical approach to stable periodic Gaussian optical approach to stable periodic orbitorbit resonances of partially chaotic dielectric resonances of partially chaotic dielectric cavitiescavities”,”, Optics Express, Optics Express, 1010, 752-776 (2002), 752-776 (2002)
The Parabolic The Parabolic Equation Equation ApproximationApproximation
Single-valuedness:
Gaussian QuantizationGaussian Quantization
• Quantization Condition:
• Transverse Excited States:
Fresnel Transmission AmplitudeFresnel Transmission Amplitude
• Dielectric Resonator Quantization conditions:
• Comparison to numerical calculations:
““Exact”Exact” Gaussian Q.Gaussian Q.
Semiclassical theory of LasingSemiclassical theory of Lasing
Uniformly distributed, homogeneously broadenedUniformly distributed, homogeneously broadeneddistribution of two-level atomsdistribution of two-level atoms
Maxwell-Bloch equationsMaxwell-Bloch equationsHaken(1963), Sargent, Scully & Lamb(1964)Haken(1963), Sargent, Scully & Lamb(1964)
Rich spatio-temporal dynamicsRich spatio-temporal dynamics Classification of the solutions:Classification of the solutions: Time scales of the problem -Time scales of the problem - Adiabatic eliminationAdiabatic elimination Most semiconductor ARCs : Class BMost semiconductor ARCs : Class B
Reduction of MB equationsReduction of MB equations
The single-mode instabilitiesThe single-mode instabilities
Complete LComplete L22 basis basis
Modal treatment of MBE:Modal treatment of MBE:
Single-mode solutions:Single-mode solutions:
Solutions classified by:Solutions classified by:
1.1. Fixed pointsFixed points2.2. Limit cycles – steady state lasing solutions Limit cycles – steady state lasing solutions 3.3. attractorsattractors
Single-mode LasingSingle-mode Lasing
Gain clamping D ->D_sGain clamping D ->D_s
Pump rate=Loss rate -> Pump rate=Loss rate -> Steady StateSteady State
Multi-mode laser equationsMulti-mode laser equations
Eliminate polarization:Eliminate polarization:
Look for Look for StationaryStationary photon numberphoton number solutions: solutions:
Ansatz:Ansatz:
A Model for mode competitionA Model for mode competition
Mode competition - “Spatial Hole burning”
Positivity constraint : Positivity constraint : Multiple solutions possible!Multiple solutions possible!
““Diagonal Lasing”:Diagonal Lasing”:
How to choose the solutions?How to choose the solutions?
(Haken&Sauermann,19(Haken&Sauermann,1963)63)
““Off-Diagonal” LasingOff-Diagonal” Lasing
• Beating terms down by Beating terms down by • Quasi-multiplets Quasi-multiplets mode-lockmode-lock to a common lasing freq. to a common lasing freq.
Steady-state equations:Steady-state equations:
Lasing in Circular cylindersLasing in Circular cylinders
Introduce linear absorption:Introduce linear absorption:
Output Power Dependence: Output Power Dependence: εε=0=0
Internal Photon #Internal Photon # Output Photon #Output Photon #
Output strongly suppressedOutput strongly suppressed
Cylinder Laser-ResultsCylinder Laser-Results
Non-linear thresholdsNon-linear thresholds Output power OptimizationOutput power Optimization
Output Power Dependence: Output Power Dependence: εε=0.16=0.16
Flood PumpingFlood Pumping Spatially non-uniform PumpSpatially non-uniform Pump
Pump Pump diameter=0.6diameter=0.6
Output Power DependenceOutput Power Dependence
Compare Photon Numbers of different deformationsCompare Photon Numbers of different deformations
EllipseEllipse
QuadQuad
Output Power DependenceOutput Power Dependence
Pump Pump diameter=0.6diameter=0.6
Spatially selective PumpingSpatially selective Pumping
Model: A globally Chaotic LaserModel: A globally Chaotic Laser
RMT:RMT:
RMT-Power dependenceRMT-Power dependence
Ellipse lifetime distributions + RMT overlaps (+absorption)Ellipse lifetime distributions + RMT overlaps (+absorption)
RMT model-Photon number RMT model-Photon number distributionsdistributions
Output power increases because modes become leakierOutput power increases because modes become leakier
Ellipse lifetime distributions + RMT overlaps (+absorption)Ellipse lifetime distributions + RMT overlaps (+absorption)
How to treat Degenerate Lasing?How to treat Degenerate Lasing?
Existence of quasi-degenerate modes: Existence of quasi-degenerate modes:
Conclusion & OutlookConclusion & Outlook
Classical phase space dynamics good in predicting emission properties of dielectric resonators
Local asymptotic approximations are powerful but have to be supplemented by numerical calculations
Tunneling processes yet to be incorporated into semiclassical quantization
A non-linear theory of dielectric resonators: mode-selection, spatial hole-burning, mode-pulling/pushing cooperative mode-locking, fully non-linear modes, and… more chaos!!!
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