Transcript Document

Mode Competition in
Wave-Chaotic Microlasers
Recent Results and Open Questions
Hakan Türeci
Physics Department, Yale University
Theory
Experiments
Harald G. Schwefel - Yale
A. Douglas Stone - Yale
Philippe Jacquod - Geneva
Evgenii Narimanov - Princeton
Nathan B. Rex - Yale
Grace Chern - Yale
Richard K. Chang – Yale
Joseph Zyss – ENS Cachan
&
Michael Kneissl, Noble Johnson PARC
Trapping Light : Optical μ-Resonators
• Conventional
Resonators:
Fabry-Perot
no=1
• Dielectric
Micro-resonators:
n

t

total
internal
reflection
Trapping light by
TIR
Whispers in μDisks and μSpheres
Lasing Droplets, Chang et al.
Microdisks, Slusher et al.
 Very High-Q whispering gallery modes
 Small but finite lifetime due to tunneling
But:
 Isotropic Emission
 Low output power
Wielding the Light - Breaking the Symmetry
Asymmetric Resonant Cavities
 Smooth deformations
 Characteristic emission anisotropy
 High-Q modes still exist
 Theoretical Description: rays
 Connection to classical and Wave chaos
 Qualitative understanding
> Shape engineering
Spiral Lasers
 Local deformations ~ λ
 Short- λ limit not applicable
 No intuitive picture of emission
G. Chern, HE Tureci, et al. ”Unidirectional lasing from
InGaN multiple-quantum-well spiral-shaped micropillars”,
to be published in Applied Physics Letters
The Helmholtz Eqn for Dielectrics
 Maxwell Equations
convenient family of deformations:
Continuity Conditions:
Im[kR]
I(170°)
I(170°)
Scattering vs. Emission
kR
Re[kR]
Re[kR]
Quasi-bound modes only exist at discrete, complex k
Im[kR]
Asymmetric Resonant Cavities
 Generically non-separable  NO `GOOD’ MODE INDICES
 Numerical solution possible, but not poweful alone
 Small parameter : (kR)-1  10-1 – 10-5
 Ray-optics equivalent to billiard problem with refractive escape
 KAM transition to chaos
 CLASSIFY MODES using PHASE SPACE STRUCTURES
Overview
1. Ray-Wave Connection
Multi-dimensional WKB, Billiards, SOS
2. Scattering quantization for dielectric resonators
A numerical approach to resonators
3. Low index lasers
Ray models, dynamical eclipsing
4. High-index lasers - The Gaussian-Optical Theory
Modes of stable ray orbits
5. Non-linear laser theory for ARCs
Mode selection in dielectric lasers
Multi-dimensional WKB (EBK)
• Quantum Billiard Problem:
• The EBK ansatz:
N=2, Integrable ray dynamics
N, Chaotic ray dynamics
Quantization Integrable systems
• Quantum Billiard:
2 irreducible loops  2 quantization conditions
b()
• Dielectric billiard:
Quantization of non-integrable systems
Non-integrable ray dynamics (Einstein,1917)
Quantum Chaos
 Mixed dynamics: Local asymptotics possible
 Berry-Robnik conjecture
 Globally chaotic dynamics: statistical description of spectra
 Periodic Orbit Theory
 Gutzwiller trace formula
 Random matrix theory
Poincaré Surface-of-Section
 Boundary deformations:
 SOS coordinates:
Billiard map:
The Numerical Method
Regularity at origin:
Internal Scattering
Eigenvalue Problem:
Numerical Implementation
• Non-unitary S-matrix
• Quantization condition
• Complex k-values determined by
a two-dimensional root search
A numerical interpolation scheme:
• Follow
over an interval
 Classical phase space structures
• Interpolate
to obtain
quantized wavefunction
and use
to construct the
NO root search!
Quasi-bound states and Classical
Phase Space Structures
Low-index Lasers
Low index (polymers,glass,liquid droplets)
n<1.5
 Ray Models account for:
 Emission Directionality
 Lifetimes
Nöckel & Stone, 1997
Polymer microdisk Lasers
Quadrupoles:
Ellipses:
n=1.49
Semiconductor Lasers
Low index (polymers,glass,liquid droplets)
n<1.5
High index (semiconductors)
n=2.5 – 3.5
Bowtie Lasers
Bell Labs QC ARC:
=0.0
=0.14
=0.16
λ=5.2μm , n=3.3
• High directionality
• 1000 x Power wrt =0
“High Power Directional Emission from lasers with chaotic Resonators”
C.Gmachl,F.Capasso,EE Narimanov,JU Noeckel,AD Stone, A Cho
Science 280 1556 (1998)
Dielectric Gaussian Optics
HE Tureci, HG Schwefel, AD Stone, and EE Narimanov
”Gaussian optical approach to stable periodic orbit
resonances of partially chaotic dielectric cavities”,
Optics Express, 10, 752-776 (2002)
kR
The Parabolic Equation Approximation
Single-valuedness:
Gaussian Quantization
• Quantization Condition:
• Transverse Excited States:
• Dielectric Resonator Quantization conditions:
Fresnel Transmission Amplitude
• Comparison to numerical calculations:
“Exact”
Gaussian Q.
Semiclassical theory of Lasing
Uniformly distributed, homogeneously broadened
distribution of two-level atoms
Maxwell-Bloch equations
Haken(1963), Sargent, Scully & Lamb(1964)
Reduction of MB equations
 Rich spatio-temporal dynamics
 Classification of the solutions:
Time scales of the problem  Adiabatic elimination
 Most semiconductor ARCs : Class B
The single-mode instabilities
Modal treatment of MBE:
Single-mode solutions:
Complete L2 basis
Solutions classified by:
1. Fixed points
2. Limit cycles – steady state lasing solutions
3. attractors
Single-mode Lasing
Gain clamping D ->D_s
Pump rate=Loss rate -> Steady State
Multi-mode laser equations
Eliminate polarization:
Look for Stationary photon number solutions:
Ansatz:
A Model for mode competition
(Haken&Sauermann,1963)
“Diagonal Lasing”:
Mode competition - “Spatial Hole burning”
 Positivity constraint :
 Multiple solutions possible!
How to choose the solutions?
“Off-Diagonal” Lasing
• Beating terms down by
• Quasi-multiplets
mode-lock to a common lasing freq.
Steady-state equations:
Lasing in Circular cylinders
Introduce linear absorption:
Output Power Dependence: ε=0
Internal Photon #
Output Photon #
Output strongly suppressed
Cylinder Laser-Results
Non-linear thresholds
Output power Optimization
Output Power Dependence: ε=0.16
Flood Pumping
Spatially non-uniform Pump
Pump diameter=0.6
Output Power Dependence
Compare Photon Numbers of different deformations
Quad
Ellipse
Output Power Dependence
Spatially selective Pumping
Pump diameter=0.6
Model: A globally Chaotic Laser
RMT:
RMT-Power dependence
Ellipse lifetime distributions + RMT overlaps (+absorption)
RMT model-Photon number distributions
Ellipse lifetime distributions + RMT overlaps (+absorption)
Output power increases because modes become leakier
How to treat Degenerate Lasing?
Existence of quasi-degenerate modes:
Conclusion & 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!!!