Random Lasers

Gregor Hackenbroich, Carlos Viviescas, F. H.

Random Lasers

wavelength (nm)

Inte

nsity

(a.

u.)

Inte

nsity

(a.

u.)

Inte

nsity

(a.

u.)

376 384 392 400

pum

ping

Disordered ZnO-clusters and films

1 m

pumping

H. Cao et al., 1999:

feedback by chaotic scattering of light

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

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

Why modifications to laser theory?

• spectrally overlapping

• modes have simple spatial form • approximation: eigenmodes of

closed resonator

• modes have complex spatial distribution

field quantization for open resonators

PRL 89, 083902 (2002),PRA 67, 013805 (2003)J. Opt. B 6, 211 (2004)

†[ , ]a a †[ ( ), ( )]

( )m m

mm

b b

( ), ( ):m mV W Integrals of mode functions along the openings

system bath dropantiresonantterms!

Dynamics: Heisenberg picture

a i a a F

†WW

damping noise

standard laser:

open laser:

master equation

†

† †

,

, ,

[

[ ] [ ]

]

{ }

ia a

a a a a

Schrödinger picture, low temperature: Bk T

generalization of single-mode master equation to many overlapping modes

G. Hackenbroich et al., PRA 013805 (2003):

open laser

pumpingemission

...a

...pS

...zpS

field modes

polarization of p-th atom

inversion of p-th atom

non-linear coupled Heisenberg equations

field-atom coupling (spatially randomfor chaotic modes)

modes couplednot only via atomsbut also throughdamping

increase of laser linewidth

regime of a singlelaser oscillation

STK

2

| |1

| | |

L L R RK

L R

laser linewidth

Petermann factor

wavelength

Inte

nsity

ST

PRL 89, 083902 (2002):

multimode lasing and mode competition

modes of passive system overlap, fast atomic medium

' ''

k k kk k k kk

k kI I I I I A B

loss linear gain

saturation

mode competition for atomic gain

*

cav( ) ( )k k kdV L RA r r

* *' 'cav

cav * *' 'ca

'

v cav

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

k k k k

k k k k

kk

dV L R R RV

dV L R dV R R

Br r r r

r r r r

wavelength

Inte

nsity

kI 1kI ...

mean number of lasing modes

0.1 1 10 50

0.1

1

10

100

0.1 1 10 50

0.1

1

10

100

N

pumping strength

neglecting mode competition

including mode competition

1 linear gain = 0

• universal exponent: 1/3

• non-linear mode competition relevant

3las

1/N

ohne We1

tt/ 2N no-comp

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)

Random matrix-model

ensemble of non-Hermitian random matrices

†GOEH i WW H

internal coupling

chaoticdynamics

M: # channelsT: channel trans-

mission coefficient

statistics of eigenfunctions: ' '1 2kk kk B

Fyodorov & Sommers: 1997

( )P / 2 1

0,M 2

01/ ,

eigenvalues of :H i

{

0 4

TM

frequency-separation