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

2. Conventional Resonators: Fabry-Perot Dielectric Micro-resonators: Trapping light by TIR TIR Trapping Light : Optical μ-Resonators  total internal reflection n n o =1  t

3. Whispers in μDisks and μSpheres Microdisks, Slusher et al. Lasing Droplets, Chang et al.  Very High-Q whispering gallery modes  Small but finite lifetime due to tunneling But: But:  Isotropic Emission  Low output power

4. Wielding the Light - Breaking the Symmetry Spiral Lasers  Local deformations ~ λ  Short- λ limit not applicable  No 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 Letters   Smooth deformations   Characteristic emission anisotropy   High-Q modes still exist   Theoretical Description: rays   Connection to classical and Wave chaos   Qualitative understanding > Shape engineering Asymmetric Resonant Cavities

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

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

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 billiard problem with refractive escape  KAM transition to chaos  CLASSIFY MODES using PHASE SPACE STRUCTURES  CLASSIFY MODES using PHASE SPACE STRUCTURES

8. Overview 1. Ray-Wave Connection Multi-dimensional WKB, Billiards, SOS Multi-dimensional WKB, Billiards, SOS 2. Scattering quantization for dielectric resonators 2. Scattering quantization for dielectric resonators A numerical approach to resonators A numerical approach to resonators 3. Low index lasers Ray models, dynamical eclipsing Ray models, dynamical eclipsing 4. High-index lasers - The Gaussian-Optical Theory Modes of stable ray orbits Modes of stable ray orbits 5. Non-linear laser theory for ARCs Mode selection in dielectric lasers Mode selection in dielectric lasers

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

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

11. Non-integrable ray dynamics (Einstein,1917) Quantum Chaos Quantization 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

12. Poincaré Surface-of-Section  Boundary deformations:  SOS coordinates: Billiard map:

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

16. Numerical 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!

17. Quasi-bound states and Classical Phase Space Structures

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

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

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

21. λ=5.2μm, n=3.3 Bell Labs QC ARC: Bowtie Lasers “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) High directionality High directionality 1000 x Power wrt  =0 1000 x Power wrt  =0  =0.0  =0.14  =0.16

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”, resonances of partially chaotic dielectric  cavities”, Optics Express, 10, 752-776 (2002) Optics Express, 10, 752-776 (2002)

23. The Parabolic Equation Approximation Single-valuedness:

24. Gaussian Quantization Quantization Condition: Transverse Excited States: Fresnel Transmission Amplitude Dielectric Resonator Quantization conditions: 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.  Rich spatio-temporal dynamics  Classification of the solutions: Time scales of the problem - Time scales of the problem -  Adiabatic elimination  Most semiconductor ARCs : Class B Reduction of MB equations

27. The single-mode instabilities Complete L 2 basis Modal treatment of MBE: Single-mode solutions: Solutions classified by: 1.Fixed points 2.Limit cycles – steady state lasing solutions 3.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 Stationary photon number solutions: Look for Stationary photon number solutions: Ansatz:

30. A Model for mode competition Mode competition - “Spatial Hole burning”  Positivity constraint :  Multiple solutions possible! “Diagonal Lasing”: How to choose the solutions? (Haken&Sauermann,1963)

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

32. Lasing in Circular cylinders Introduce linear absorption:

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

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

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

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

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

38. Model: A globally Chaotic Laser RMT:

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

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

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!!!