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!!!