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
