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!!!