WAVE INTERFERENCES IN RANDOM LASERS

Transcription

1 WAVE INTERFERENCES IN RANDOM LASERS Philippe Jacquod U of Arizona P. Stano and Ph. Jacquod, Nature Photonics (2013)

2 What is a laser? Light Amplification by Stimulated Emission of Radiation Three main components (i) Resonator (ii) Active medium (iii) Pump

3 What is a laser? Three main components Resonator Active medium Pump -> trap light long enough for gain -> provide gain -> population inversion in active medium

4 What is a laser? Three main components Resonator Active medium Pump -> trap light long enough for gain -> provide gain -> population inversion in active medium Conventional Lasers Fabry-Pérot resonator

5 What is a laser? Three main components Resonator Active medium Pump -> trap light long enough for gain -> provide gain -> population inversion in active medium Circular Micro-Lasers Confinement by total internal reflection Yamamoto and Slusher, Phys. Today 93

6 What is a laser? Conventional wisdom Disorder = losses

7 What is a laser? Conventional wisdom Disorder = losses = bad Lasing threshold ~ gain > loss More disorder -> higher lasing threshold

8 What is a laser? Conventional wisdom LASERS NEED TO BE ORDERED...in galaxies far, far away, they are even used to restore order...

9 WAVE INTERFERENCES IN? RANDOM LASERS? Philippe Jacquod U of Arizona P. Stano and Ph. Jacquod, Nature Photonics (2013)

10 What is a random laser? H Tureci Remark (important) At this level, only incoherent feedback Conventional ~phase Lasersmatching not Random invokedlasers Symmetric resonator No confinement, but... Path length increased multiple scattering by almost perfect diffusive, longer paths reflection gain path length > loss -> lasing Lethokov 68

11 Random Laser (i) incoherent feedback 1. Lasing dye

12 Random Laser (i) incoherent feedback 3µJ pump no scatterer 1. Lasing dye

13 Random Laser (i) incoherent feedback 1. Lasing dye 2. Scatterers (TiO2 nanoparticles)

14 Random Laser (i) incoherent feedback 2.2 µj pump with scatterer (amplitude x 10) 1. Lasing dye 2. Scatterers (TiO2 nanoparticles) 3. Pump

15 Random Laser (i) incoherent feedback 3 µj pump with scatterer (amplitude / 10) 1. Lasing dye 2. Scatterers (TiO2 nanoparticles) 3. Pump

16 Random Laser (i) incoherent feedback Signature of lasing: mode narrowing Why incoherent feedback? Lasing peak is still relatively broad. It is believed that phase matching has only a marginal influence (if any). 1. Lasing dye 2. Scatterers (TiO2 nanoparticles) 3. Pump

17 Incoherent vs. coherent feedback Commonly accepted fact random lasers have quenched scatterers (t-independent) -> no decoherence/dephasing -> theory based on the field amplitude, not intensity Coherent vs. incoherent : do interferences matter? vs.

18 Random lasers with coherent feedback From Cao, Optics and Photonics News 05 First papers: Cao et al. PRL 99; Frolov et al. PRB 99 Reviews: Cao, J. Phys. 05; Wiersma, Nat. Phys. 08

19 Random lasers with coherent feedback Smells like lasing but how did we get gain? What are the long-lived modes? Output power Narrow emission line Reviews: Cao, J. Phys. 05; Wiersma, Nat. Phys. 08 Possible answers: Anderson localized modes Jiang and Soukoulis 00 Vanneste and Sebah 01 Lucky photons on long trajs. Mujumdar et al., 04 Pre-localized modes Apalkov, Raikh, Shapiro 04 Delocalized, interacting modes Tureci, Ge, Rotter, Stone 10 Several of these scenarios Fallert et al., 09

20 Wave in random media : coherent corrections to transport!!probability to be at C:! P(C) ~ A+A =4 A! i.e. enhanced backscattering! 2! 2! A! B! C! Conductance reduction return probability! Assuming diffusion:!

21 Wave in random media : coherent corrections to transport!!break TRS - magnetic field!!!"!!probability to be at C:! A! B! C! 2! 2! P(C) ~ A+A =2 A (1+cos[!])! QM interferences disappear = positive magnetoconductance!

22 Wave in random media: weak localization / coherent backscattering Condensed matter (electrons) -> weak localization Optics (photons, EM field) -> coherent backscattering Chang et al., 94 Fiebig et al, 07

23 Anderson localization E classical particle V(x) W

24 Anderson localization E quantum particle/wave? V(x) W

25 Anderson localization Exponential localization of wavefunctions in random disordered potentials Always in 1 and 2D Above critical disorder strength W c in 3D Even for E>W! quantum particle ~ no transmission V(x) W Fig from: PJ and Shepelyansky, prl 95 Wave interference phenomenon: electronic, photonic, acoustic, atomic waves

26 Anderson localization and nonhermitian quantum mechanics Imaginary vector potential -> Energy eigenvalues are no longer real -> Time-evolution exp[-iet]=exp[-i Re(E) t] exp[im(e) t] Im(E)>0 -> amplification, eventually beating localization See also: Brouwer, Silvestrov, Beenakker PRB 97

27 The take-home message

28 Model of a complex (or not) laser Pump Cavity with active medium (2-level atoms) and with n 2 (x)=ε(x) Pump -> population inversion D(x) D+E-field -> polarization P P and E coupled via Maxwell equation

29 Maxwell-Bloch equations SEMICLASSICAL i.e.: classical EM field (1D, 2D -> scalar) coupling to a quantum active medium (2-level) Coupled electric (E), population inversion (D) and polarization (P) fields laser pumping D Haken, Light:Laser Dynamics

30 Self-consistent Ab initio Laser Theory - SALT Problem: Set of three coupled nonlinear PDE s... Maxwell-Bloch equations are usually time-evolved until steady-state is reached -> time-consuming! Tureci+Stone+friends, PRA 06, 07; Science 10

31 Self-consistent Ab initio Laser Theory - SALT Problem: Set of three coupled nonlinear PDE s... Maxwell-Bloch equations are usually time-evolved until steady-state is reached -> time-consuming! Solution: Find steady-state solution ~stationary inversion and polarization Tureci+Stone+friends, PRA 06, 07; Science 10

32 Self-consistent Ab initio Laser Theory - SALT (1) Multiperiodic expansion of E and P fields (2) Set µ=1,2,...n (to be determined for given pump) (3) Obtain self-consistent differential equation for lasing modes

33 Self-consistent Ab initio Laser Theory - SALT Pump, 0<F(x)<1, Infinite-order nonlinearity/spatial hole burning denominator!! h(x) determined by all the lasing modes!! -> self-consistency

34 SALT - enters the threshold matrix Linear equation for the expansion coefficients coefficients cast into vector form (note: greek letters for lasing modes, latin for cold-cavity states i.e. : one such equation for each lasing mode) determines if a mode lases or not - the threshold matrix : lasing mode frequency (real, tbd by solution) : cold-cavity frequency (complex)

35 Structure of the threshold matrix One line with fluctuating, exp. large entries surrounded by two blocks with small entries ~ξ Legitimate to solve the threshold equation by perturbation theory about single-pole approx.

36 Structure of the threshold matrix One line with fluctuating, exp. large entries surrounded by two blocks with small entries ~ξ Legitimate to do perturbation theory about single-pole approx. 0 th order -> lasing threshold

37 Structure of the threshold matrix One line with fluctuating, exp. large entries surrounded by two blocks with small entries ~ξ Legitimate to do perturbation theory about single-pole approx. 0 th order 1 st order -> higher order negligible if

38 Structure of the threshold matrix One line with fluctuating, exp. large entries surrounded by two blocks with small entries ~ξ Single-pole approximation rigorously justified Legitimate to do perturbation theory about single-pole approx. -> lasing modes are ~ same as cold-cavity modes in the Anderson localized regime 0 th order 1 st order Anderson localization -> f(x): algebraic in x

39 Absence of nonlinearity in the Anderson localized regime Stano and PJ, Nature Photonics 12

40 Absence of nonlinearity in the Anderson localized regime Stano and PJ, Nature Photonics 12

41 Restoring modal interactions Stano and PJ, Nature Photonics 12 1D cavity with absorption/leakage cleaner 1D cavity Tureci, Ge, Rotter, Stone, Science 10 2D diffusive cavity

42 Trademark of Anderson localization with absorption/leakage no absorption/leakage Stano and PJ, Nature Photonics 12 disordered, delocalized Tureci, Ge, Rotter, Stone, Science 10

43 Trademark of Anderson localization close to Ioffe-Regel threshold: λ=0.45 µm l=100 nm k F l = 1.3

44 Trademark of Anderson localization arxiv: D disordered waveguide Overlapping localized lasing modes In analyzing our experiment we were rather surprised about the lack of mode competition despite the fact that we pump a very small region. The only possible explanation we could think of was that it means that preferentially very localized modes are lasing. However, now it seems like your work gives a much more plausible, and surprising, explanation.

45 Conclusion The main result of the paper under review consists in the statement that if modes of a random laser are strongly Anderson localized, the nonlinear interaction between different modes become suppressed. While this statement is definitely correct, it is not at all surprising (...) Indeed, all nonlinear interactions are determined by overlap integrals (...). In Anderson localized regime these integrals are exponentially small because wavefunctions of different localized modes do not overlap this well. NO - we move until far above lasing threshold where modes spatially overlap!

46 Open questions 3D? What about time-dependent phenomena? Noise/spontaneous emission, oscillations/beatings... What about extended modes vs. modal interactions? Can we guess something useful for them from wavefunction correlations?

47 Paradigms of theoretical physics Spherical cow!linear!noninteracting!equilibrium!ordered!local Real life!nonlinear!interacting!non-equilibrium!disordered!nonlocal??? Terra incognita inspired by S. Girvin