Probability of reflection by a random laser.

TLDR: A theory is presented (and supported by numerical simulations) for phase-coherent reflection of light by a disordered medium which either absorbs or amplifies radiation and the distribution of reflection eigenvalues is shown to be the Laguerre ensemble of random-matrix theory.

Abstract: A theory is presented (and supported by numerical simulations) for phase-coherent reflection of light by a disordered medium which either absorbs or amplifies radiation. The distribution of reflection eigenvalues is shown to be the Laguerre ensemble of random-matrix theory. The statistical fluctuations of the albedo (the ratio of reflected and incident power) are computed for arbitrary ratio of sample thickness, mean free path, and absorption or amplification length. On approaching the laser threshold all moments of the distribution of the albedo diverge. Its modal value remains finite, however, and acquires an anomalous dependence on the illuminated surface area.

Full text

VOLUME
76,
NUMBER
8
PHYSICAL
REVIEW
LEITERS
19
FEBRUARY
1996
Probability
of
Reflection
by a
Random
Laser
C.W.J.
Beenakker,
1
J.C.J.
Paasschens,
1
·
2
and
P.W.
Brouwer
1
1
Instituut-Lorentz,
University
of
Leiden,
P.O.
Box
9506,
2300
RA
Leiden,
The
Netherlands
2
Philips
Research
Laboratories,
5656
AA
Eindhoven,
The
Netherlands
(Received
14
September
1995)
A
theory
is
presented
(and
supported
by
numerical
simulations)
for
phase-coherent
reflection
of
light
by
a
disordered
medium
which
either
absorbs
or
amplifies
radiation.
The
distribution
of
reflection
eigenvalues
is
shown
to be the
Laguerre
ensemble
of
random-matrix
theory.
The
statistical
fluctuations
of
the
albedo
(the
ratio
of
reflected
and
incident
power)
are
computed
for
arbitrary
ratio
of
sample
thickness,
mean
free
path,
and
absorption
or
amplification
length.
On
approaching
the
laser
threshold
all
moments
of the
distribution
of the
albedo
diverge.
Its
modal
value
remains
finite,
however,
and
acquires
an
anomalous
dependence
on the
illuminated
surface
area.
PACS
numbers:
78.20.Ci,
05.40.+J,
42.25.Bs,
78.45.+h
Recent experiments
on
turbid laser dyes
[1—4]
have
drawn
attention
to the
remarkable
properties
of
disordered
media
which
are
optically active.
The
basic issue
is
to
understand
the
interplay
of
phase-coherent multiple
scattering
and
amplification
(or
absorption)
of
radiation.
A
quantity
which measures this interplay
is the
albedo
a,
which
is the
power reflected
by the
medium divided
by the
incident
power.
A
thick disordered
slab
which
is
optically
passive
has
a =
1.
Absorption
leads
to a < l and
amplification
to a >
1.
As the
amplification increases
the
laser threshold
is
reached,
at
which
the
äverage
albedo becomes
infinitely
large
[5].
Such
a
generator
was
referred
to by its
inventor Letokhov
äs
a
"laser
with
incoherent
feedback"
[6],
because
the
feedback
of
radiation
is
provided
by
random scattering
and not by
mirrors—äs
in a
conventional laser.
The
current
renewed interest
in
random
lasers
owes
much
to the
appreciation
that
randomness
is not the
same
äs
incoherence.
Early
theoretical work
on
this problem
was
based
on the
equation
of
radiative transfer
[7],
which
ignores phase coherence. Zyuzin
[8] and
Feng
and
Zhang
[9]
considered interference effects
on the
äverage
albedo
~ä,
averaged over different
configurations
of the
scattering centra.
Their
prediction
of a
sharpening
of
the
backscattering peak
in the
angular distribution
of the
äverage reflected intensity
has now
been observed
[3].
The
other basic interference
effect
is the
appearance
of
large,
sample-specific
fluctuations of the
albedo
around
its
äverage.
These
diverge faster than
the
äverage
on
approaching
the
laser
threshold
[10],
so
that
a is no
longer characteristic
for the
albedo
of a
given sample.
In
the
present Letter
we
show that
while
all
moments
of
the
distribution function
P(a)
of the
albedo diverge
at the
laser threshold,
its
modal value
a
max
remains
finite. The
modal value
is the
value
of a at
which
P
(a)
is
maximal,
and
hence
it is the
most
probable value measured
in a
single experiment.
The
diagrammatic
perturbation
theory
of
Refs.
[8-10]
can
give
only
the first few
moments
of
a, and
hence cannot determine
a
max
.
Here
we
develop
a
nonperturbative
random-matrix theory
for the
entire
distribution
of the
reflection matrix,
from
which
P
(a) can
be
computed directly.
We
contrast
the two
cases
of
absorption
and
amplifica-
tion.
In the
case
of
absorption, P(a)
is a
Gaussian with
a
width
δα
smaller
than
the
äverage
α by a
factor
^/~N,
where
W
—
S/A
2
»
l is the
number
of
modes associ-
ated
with
an
illuminated
area
5 and
wavelength
A. In the
case
of
amplification,
both
δα
and a
increase strongly
on
approaching
the
laser
threshold—in
a
manner
which
we
compute
precisely.
Below
threshold,
the
mean
and
modal
value
of a
coincide. Above
threshold,
the
mean
is
infinite
while
the
modal value
is
found
to be
ömax
= l +
0.8
7
7V.
(1)
Here
γ
denotes
the
amplification
per
mean
free
path,
as-
sumed
to be in the
ränge
N~
2
«:
γ
<K
1.
The
existence
of a finite
cz
max
is due to the finiteness of the
number
of
modes
N
in a
surface area
S
(ignored
in
radiative
trans-
fer
theory).
Since
a
max
scales
with
N and
hence
with
S, and the
incident
power scales
with
S, it
follows
that
the
reflected
power scales
quadratically
rather
than
lin-
early
with
the
illuminated
area.
We
suggest
the
name
"superreflection"
for
this
phenomenon.
To
measure
the
albedo
in the
unstable
regime above
the
laser threshold
we
propose
a
time-resolved
experiment,
consisting
of
il-
lumination
by a
short
intense
pulse
to
pump
the
medium
beyond
threshold,
rapidly followed
by a
weak
pulse
to
measure
the
reflected
intensity
before
spontaneous
emis-
sion
has
caused
substantial
relaxation.
Our
work
on
this
problem
was
motivated
by a
recent
paper
by
Pradhan
and
Kumar
[11]
on the
case
N = l of
a
single-mode
waveguide.
We
discovered
the
anomalous
scaling
with
area
in an
attempt
to
incorporate
the
effects
of
mode
coupling
into
their
approach.
We
consider
the
reflection
of a
monochromatic
plane
wave
(frequency
ω,
wavelength
A) by a
slab (thickness
L,
area
S)
consisting
of a
disordered
medium
(mean
free
path
/)
which
either
amplifies
or
absorbs
the
radiation.
We
denote
by σ the
amplification
per
unit
length,
a
negative
1368
0031-9007/96/76(8)/1368(4)$06.00
©
1996
The
American
Physical Society

VOLUME
76,
NUMBER
8
PHYSICAL
REVIEW
LETTERS
19FEBRUARY
1996
value
of σ
indicating
absorption.
The
parameter
γ =
σΐ
is the
amplification
(or
absorption)
per
mean
free path.
We
treat
the
case
of a
scalar
wave amplitude,
and
leave
polarization
effects
for
future
study.
A
discrete
number
N
of
scattering
channels
is
defined
by
imbedding
the
slab
in
an
optically passive waveguide
without
disorder
(see
Fig.
l,
inset).
The
number
W
is the
number
of
modes
which
can
propagate
in the
waveguide
at
frequency
ω.
The N X N
reflection
matrix
r
contains
the
amplitudes
r
mn
of
waves
reflected
into
mode
m
from
an
incident
mode
n.
(The
basis states
of r are
normalized
such
that
each
carries
unit
power.)
The
reflection eigenvalues
R
n
(n
=
l,2,...,N)
are the
eigenvalues
of the
matrix
product
rrt'.
The
matrix
r is
determined
by the
7?„'s
and
by a
unitary
matrix
U,
(2)
Note
that
r
mn
=
r
nm
because
of
time-reversal
symmetry.
From
r one can
compute
the
albedo
a of the
slab, which
is the
ratio
of the
reflected
and
incidental power:
a
=
(3)
For
a
statistical description
we
consider
an
ensemble
of
slabs
with
different
configurations
of
scatterers.
As in
earlier work
on
optically passive media
[12],
we
make
the
isotropy assumption
that
U is
uniformly
distributed
in
the
unitary
group. This assumption breaks down
if the
1.2
1.0
0.8
0.6
0.4
0.2
0.0
-0.3-0.2-0.1
0.0 0.1 0.2
7
FIG.
1.
Comparison
between
theory
and
Simulation
of the
average
albedo
Έ
(upper
curves,
squares)
and
Var
a
(lower
curves,
triangles)
for
L/1
= l .92
(dashed
curves,
open
markers)
and
L/l =
9.58
(solid
curves,
filled
markers).
Negative
y
corresponds
to
absorption,
positive
γ to
amplification.
The
curves
are the
theoretical
result
(7).
The
data
points
are a
numerical
Simulation
of a
two-dimensional
lattice
(L —
50d
and
250d,
W =
5ld,
N =
21), averaged over
100
realizations
of the
disorder.
The
inset
shows
schematically
the
System
under
consideration.
transverse dimension
W of the
slab
is
much
greater
than
its
thickness
L, but is
expected
to be
reasonable
if W S L.
As a
consequence
of
isotropy,
a
becomes statistically
independent
of the
index
n of the
incident
mode.
We
further
assume that
the
wavelength
λ is
much smaller
than
both
the
mean free path
/ and the
amplification
length
Ι/σ.
The
evolution
of the
reflection
eigenvalues with
increasing
L can
then
be
described
by a
Brownian motion
process.
To
describe
this evolution
it is
convenient
to use
the
parametrization
= l +
ε
H»,-1)
u
(o,«).
(4)
The L
dependence
of the
distribution
Ρ(μ\,
μ-2,
· ·
·.
β
Ν)
of
the
μ'
s is
governed
by the
Fokker-Planck
equation
dP_
BL
N
+
Τ-μΜ
+μ,0
X
dP
+
+
γ
(N
+
l)P
(5)
with
initial
condition
\im
L
->oP
=
ΝΥ\
ί
δ(μ
ί
+ 1). In
the
single-channel
case
(N = 1), the
term
Σ/>;
is
absent
and
Eq. (5)
reduces
to the
differential
equation studied
by
Pradhan
and
Kumar
[11,13].
The
multichannel case
is
essentially
different
due to the
coupling
of the
eigen-
values
by the
term
Σ/^
;
(/"·./
~
μ·;)"
1
·
This term induces
a
repulsion
of
closely separated eigenvalues. Equation
(5)
with
y = 0 is
known
äs
the
Dorokhov-Mello-Pereyra-
Kumar
(DMPK)
equation
[14,15],
and has
been
studied
extensively
in the
context
of
electronic
conduction
[16].
We
have
generalized
it to γ Φ 0, by
adapting
the ap-
proach
of
Ref.
[15]
to a
nonunitary
scattering matrix.
The
average
ä
=
(a) and the
variance
Var α
Ξ=
{(α
—
ä)
2
)
of the
albedo
(3) can be
computed
by first
averaging
U
over
tne
unitary
group
and
then evaluating
moments
of
the
Rk's
by
means
of Eq. (5)
[17].
In the
limit
N
—>
°°
we
obtain
the
differential
equations
ä
=
(ö-
D
2
+
2γα,
(6a)
/
—r
Var
a = 4(ä - l +
y)Var
a +
2N~
l
ä(ä
-
l)
2
.
dL
(6b)
Corrections
are
smaller
by a
factor
\γΝ
2
\
'/
2
,
which
we
assume
to be
<5Cl.
Equation
(6a)
for the
average
albedo
is an
old
result
of
radiative transfer theory
[18].
Equation
(6b)
for the
variance
is
new.
It
describes
the
sample-specific
fluctuations of the
albedo
due to
interference
of
multiply scattered waves. Integration
of
Eq.
(6)
yields
1369

VOLUME
76,
NUMBER
8
PHYSICAL REVIEW
LEITERS
19
FEBRUARY
1996
α
= l - y + (2y -
γ
2
)
172
tan/,
(7a)
Var
α =
(8JVcos
4
fr'{4y(l
-
2y)L//
+
2y(l
+ 7) -
4y
2
cos2/
+
2γ(1
-
y)cos4i
+
(2 -
y)~'(2y
-
7
2
)
1/2
[4y(l
-
r)sin2i
- (l - 4γ +
2y
2
)sin4/]}.
(7b)
We
have abbreviated
t = (2y -
y
2
)
1/2
L//
-
arcsin(l
— y).
Plots
of Eq. (7)
äs
a
function
of y are
shown
in
Fig.
l,
for
two
values
of
L/1.
(The
data points
are
numerical
simulations,
discussed
later.)
In the
case
of
absorption
(y
< 0), the
large-L
limit
y
-
(y
2
-
2y)'/
2
,
We
seek
the
probability distribution
of the
albedo
floo
=
l
-^
\2
Var
£z=o
=
2N
l - y -
(8a)
(8b)
can
be
obtained directly
from
Eq. (6) by
equating
the
right-hand side
to
zero.
The
limit
(8) is
reached when
L/l
»
(y
2
—
2y)~'/
2
.
In the
case
of
amplification
(y
>
0), Eq. (7)
holds
for L
smaller than
the
critical length
L
c
=
/(2y
-
y
2
)"
1/2
arccos(y
-
(9)
at
which
ä
and Var a
diverge.
This
is the
laser
threshold
[5,18].
For y < 0 the
large-L
limit
of the
probability
distribution
P(a)
of the
albedo
is
well
described
by a
Gaussian, with
mean
and
variance
given
by Eq.
(8).
(The
tails
are
non-Gaussian,
but
carry
negligible weight.)
The
modal
value
a
max
of the
albedo
equals
ä. For y > 0
the
large-L
limit
of P
(a)
cannot
be
reconstructed from
its
moments,
but
needs
to be
determined
directly.
We
will
see
that
while
a
diverges,
a
max
remains
finite.
The
large-L
limit
Ρ
χ
(μι,μ2,·
·
-,/ΛΝ)
of the
distrih"-
tion
of the
μ'
s
is
obtained
by
equating
to
zero
the
expres-
sion
between
square
brackets
in Eq.
(5).
The
result
is
p»
=
cf[expL-7(tf
+
Ο/*,·]
Π
h*;
-/**!'
(
10
)
l
/</
with
C a
normalization constant. Equation
(10)
holds
for
both positive
and
negative
y, but the
support
of
P
x
depends
on the
sign
of y: All μ 's
have
to be >0 for
y
> 0
(amplification)
and <
—
l
for y < 0
(absorption).
In
what follows
we
take
γ > 0. The
function
(10)
is
known
in
random-matrix
theory
äs
the
distribution
of the
Laguerre
ensemble
[19].
The
density
p (μ) =
(Χ,·
δ (μ —
μι))
of the
μ'
s
is a
series
of
Laguerre polynomials,
hence
the
name.
For
yN
2
»
l
one
has
asymptotically
ρ(μ)
=
(Ν/ιτ)
(2γ/μ
-
y
2
)
1
/
2
,
0 < μ <
2/y
.
(11)
The
square-root
singularity
at μ = 0 is cut off in the
exact
density
[20],
such
that
p =
yN
2
if μ
^
1/y/V
2
.
The
cumulative
density
is
plotted
in
Fig.
2,
together with
the
numerical
simulations (discussed
below).
P(a)
=
ιδία
- l -
£i/„*t/„W)).
\
v
k
//
(12)
The
average
{· · ·)
consists
of the
average
of U
over
the
unitary
group followed
by the
average
of the
μ,ί'β
over
the
Laguerre ensemble.
The
averages
can be
done ana-
lytically
for
N~
2
«C y
<SC
l (in the
continuum
approxi-
mation
[21], i.e.,
by
ignoring
the
discreteness
of the
eigenvalues),
and
numerically
for any N, y (by
Monte
Carlo
Integration,
i.e.,
by
randomly sampling
the La-
guerre ensemble).
The
analytical result
is an
inverse Laplace transform,
f
"
J
-/o
-
\)/yN
-
X
[l +
l
(13a)
where
/ is an
implicit
function
of the
Laplace variable
2
'
/
2
1=0.
(13b)
The
continuum approximation
(13)
is
plotted
in the
inset
of
Fig.
3
(dashed curve).
It
is
close
to the
exact numerical
1.0
S
°·
8
"a.
ο.β
7
°0.4
0.2
0.0
ι ι ι ι
MIN
l
ι
ι mm
07=1.09
Δ
7=0.66
o
7=0.22
v
7=0.072
o
7=0.036
Ίι
ι ι ι
min
ι ι ι mm ι ι ι
0.01
0.1
10
FIG.
2.
Comparison
between
theory
and
Simulation
of the
cumulative
density
of the
variables
μ
η
(related
to the
reflection
eigenvalues
by
/?„
=
!+
μ-^
1
)·
Curves
are
computed
from
the
density
(11)
of the
Laguerre
ensemble;
data
points
are
from
the
Simulation
(L =
50Qd
=
19.21,
W =
ISld,
N
=
63),
for
a
single
realization
of the
disorder.
1370

VOLUME
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PHYSICAL
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19FEBRUARY
1996
100
FIG.
3.
Comparison
between
theory
and
Simulation
of the
cumulative
probability
distribution
of the
albedo
(L —
50Qd
=
19.21,
j
=
0.07). Solid curves
are
obtained
by
numerically
averaging
over
the
Laguerre
ensemble;
data
points
are the
results
of the
Simulation,
averaged
over
100
realizations
of
the
disorder.
The
three
sets
of
data
are for W =
25d,
N
= 10
(plusses),
W =
51d,
N = 21
(triangles),
and W =
10Id,
N = 42
(diamonds).
The
inset
compares
the
continuum
approximation
(13)
for
P
(a)
(dashed)
with
the
exact
large-TV
limit
of the
Laguerre ensemble (solid).
large-N
result (solid curve).
The
modal
value
a
max
of the
albedo
is
given
by Eq.
(1).
The
distribution
P(a)
drops
off
°c
exp[—
2γΝ/(α
—
1)]
for
smaller
a and α
α~
5
/
3
for
larger
a, so
that
all
moments
diverge.
To
lest
these predictions
of
random-matrix
theory
on
a
model
System,
we
have
carried
out
numerical
simulations
of the
analogous electronic Anderson model
with
a
complex scattering potential, using
the
recursive
Green's
function
technique
[22].
The
disordered medium
is
modeled
by a
two-dimensional
square
lattice (lattice
constant
d,
length
L,
width
W). The
relative dielectric
constant
ε = ε ι + i
EI
(relative
to the
value outside
the
disordered
region)
has a
real
part
ει
which
fluctuates
from
site
to
site between
l ± δε, and a
nonfluctuating
imagi-
nary
part
ε
2
·
The
multiple
scattering
of a
scalar
wave
ψ
(wave number
k =
2π/λ)
is
described
by
discretizing
the
Helmholtz equation
(V
2
+
&
2
ε)Ψ
=
0. The
mean
free path
/
which enters
in Eq. (5) is
obtained from
the
average
albedo
ä = (l +
l/L)~
l
without
amplification
(ει
= 0). We
choose
k
2
=
1.5d~
2
,
δε = l,
leading
to
/
= 26. U. The
parameter
σ
(and
hence
y =
σΐ)
is
obtained from
the
analytical solution
of the
discretized
Helmholtz equation
in the
absence
of
disorder
(δε = 0).
The
complex longitudinal
wave
number
k„
of
transverse
mode
n
then
satisfies
the
dispersion relation
cos(k
n
d)
+
cos(mrd/W)
= 2 - γ
ie
2
),
computed
for
normal incidence. Data points
in
Figs.
1-3
are the
numerical results.
The
agreement with
the
curves
from
random-matrix theory
is
quite
remarkable,
given that
there
are
no
adjustable
parameters.
We
thank
A.
Lagendijk,
M.B.
van der
Mark,
and
D.
S.
Wiersma
for
helpful discussions. This
work
was
supported
by the
Dutch Science Foundation
NWO/FOM.
(14)
and
leads
to σ =
-2Ν~
}
Ιπ\Ση
Jt„.
The
albedo
(3) is
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1371