Mode Competition in Wave-Chaotic Microlasers

1. Mode Competition in Wave-Chaotic Microlasers Theory Harald G. Schwefel - Yale A. Douglas Stone - Yale Philippe Jacquod - Geneva Evgenii Narimanov - Princeton Recent Results and Open Questions Hakan Türeci Physics Department, Yale University Experiments Nathan B. Rex - Yale Grace Chern - Yale Richard K. Chang – Yale Joseph Zyss – ENS Cachan & Michael Kneissl, Noble Johnson - PARC

2.  no=1 t n total internal reflection Trapping light by TIR   Trapping Light : Optical μ-Resonators • Conventional • Resonators: • Fabry-Perot • Dielectric • Micro-resonators:

3. 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

4. Spiral Lasers 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 • 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 Wielding the Light - Breaking the Symmetry

5. convenient family of deformations: Continuity Conditions: The Helmholtz Eqn for Dielectrics • Maxwell Equations

6. I(170°) I(170°) Im[kR] Im[kR] kR Re[kR] Re[kR] Quasi-bound modes only exist at discrete, complex k Scattering vs. Emission

7. 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 billiardproblem with refractive escape • KAM transition to chaos  CLASSIFY MODES using PHASE SPACE STRUCTURES

8. 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

9. N, Chaotic ray dynamics N=2, Integrable ray dynamics Multi-dimensional WKB (EBK) • Quantum Billiard Problem: • The EBK ansatz:

10. c ' • Dielectric billiard: c c f b() Quantization Integrable systems • Quantum Billiard: • 2 irreducible loops  2 quantization conditions

11. 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

12. Billiard map: PoincaréSurface-of-Section • Boundary deformations: • SOS coordinates:

15. The Numerical Method Regularity at origin: Internal Scattering Eigenvalue Problem:

16. A numerical interpolation scheme: • Follow over an interval  Classical phase space structures • Interpolate to obtain and use to construct the • quantized wavefunction Numerical Implementation • Non-unitary S-matrix • Quantization condition • Complex k-values determined by • a two-dimensional root search NO root search!

17. Quasi-bound states and Classical Phase Space Structures

18. Low index (polymers,glass,liquid droplets) n<1.5 Nöckel & Stone, 1997 Low-index Lasers • Ray Models account for: • Emission Directionality • Lifetimes

19. n=1.49 Polymer microdisk Lasers Quadrupoles: Ellipses:

20. Low index (polymers,glass,liquid droplets) n<1.5 High index (semiconductors) n=2.5 – 3.5 Semiconductor Lasers

21. Bowtie Lasers =0.0 =0.14 =0.16 Bell Labs QC ARC: • 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) λ=5.2μm , n=3.3

22. kR 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)

23. Single-valuedness: The Parabolic Equation Approximation

24. Gaussian Quantization • Quantization Condition: • Transverse Excited States: • Dielectric Resonator Quantization conditions: Fresnel Transmission Amplitude • Comparison to numerical calculations: “Exact” Gaussian Q.

25. Semiclassical theory of Lasing Uniformly distributed, homogeneously broadened distribution of two-level atoms Maxwell-Bloch equations Haken(1963), Sargent, Scully & Lamb(1964)

26. Reduction of MB equations • Rich spatio-temporal dynamics • Classification of the solutions: Time scales of the problem - • Adiabatic elimination • Most semiconductor ARCs : Class B

27. Complete L2 basis The single-mode instabilities Modal treatment of MBE: Single-mode solutions: Solutions classified by: • Fixed points • Limit cycles – steady state lasing solutions • attractors

28. Single-mode Lasing Gain clamping D ->D_s Pump rate=Loss rate -> Steady State

29. Multi-mode laser equations Eliminate polarization: Look for Stationaryphoton number solutions: Ansatz:

30. 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?

31. Beating terms down by • Quasi-multiplets mode-lock to a common lasing freq. “Off-Diagonal” Lasing Steady-state equations:

32. Lasing in Circular cylinders Introduce linear absorption:

33. Output Power Dependence: ε=0 Output Photon # Internal Photon # Output strongly suppressed

34. Cylinder Laser-Results Non-linear thresholds Output power Optimization

35. Output Power Dependence: ε=0.16 Spatially non-uniform Pump Flood Pumping Pump diameter=0.6

36. Output Power Dependence Compare Photon Numbers of different deformations Quad Ellipse

37. Output Power Dependence Spatially selective Pumping Pump diameter=0.6

38. Model: A globally Chaotic Laser RMT:

39. RMT-Power dependence Ellipse lifetime distributions + RMT overlaps (+absorption)

40. RMT model-Photon number distributions Ellipse lifetime distributions + RMT overlaps (+absorption) Output power increases because modes become leakier

41. How to treat Degenerate Lasing? Existence of quasi-degenerate modes:

42. 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!!!