1. 

2. Random Lasers Gregor Hackenbroich, Carlos Viviescas, F. H.

3. Random Lasers wavelength (nm) Intensity (a.u.) 376 384 392 400 pumping Disordered ZnO- clusters and films 1  m pumping H. Cao et al., 1999: feedback by chaotic scattering of light

4. Theoretical Ideas Lethokov: 1967 –photon diffusion + linear gain John, Wiersma & Lagendijk: 1995-- –photon & atomic diffusion –non-linear coupling Beenakker & coworkers: 1998-- –random scattering approach –focus on linear regime below laser threshold –rate equations for strongly confined chaotic resonator Soukoulis & coworkers: 2000-- –numerical simulation of time-dependent Maxwell-equations coupled to active medium

5. issues to be discussed here: - quantization for bad chaotic /disordered resonators - statistics for # of laser peaks - increase of laser linewidth for bad resonators - statistics for # of emitted photons

6. Why modifications to laser theory? spectrally overlapping modes have simple spatial form approximation: eigenmodes of closed resonator modes have complex spatial distribution

7. field quantization for open resonators PRL 89, 083902 (2002), PRA 67, 013805 (2003) J. Opt. B 6, 211 (2004) Integrals of mode functions along the openings system bath drop antiresonant terms!

8. Dynamics: Heisenberg picture damping noise standard laser: open laser:

9. master equation Schrödinger picture, low temperature: generalization of single-mode master equation to many overlapping modes G. Hackenbroich et al., PRA 013805 (2003): 

10. open laser pumping emission field modes polarization of p-th atom inversion of p-th atom non-linear coupled Heisenberg equations field-atom coupling(spatially random for chaotic modes) modes coupled not only via atoms but also through damping

11. increase of laser linewidth regime of a single laser oscillation laser linewidth Petermann factor wavelength Intensity PRL 89, 083902 (2002):

12. multimode lasing and mode competition modes of passive system overlap, fast atomic medium loss linear gain saturation mode competition for atomic gain wavelength Intensity

13. mean number of lasing modes pumping strength neglecting mode competition including mode competition linear gain = universal exponent: 1/3 non-linear mode competition relevant no-comp

14. summary - theory of (weakly confined) random lasers - generalization of standard laser theory - characteristics: increase of laser linewidth universal increase of # of lasing modes thanks to: D. Savin, H. Cao mode coupling through damping increased intensity fluctuations in output (not discussed here)

15. Random matrix-model ensemble of non-Hermitian random matrices internal coupling chaotic dynamics M: # channels T: channel trans- mission coefficient statistics of eigenfunctions: Fyodorov & Sommers: 1997 eigenvalues of { frequency- separation     