July 10, 2006 / Vol. 4, No. 7 / CHINESE OPTICS LETTERS 397
Effects of cross-correlated noises on the intensity
fluctuation of the single-mode laser system
Bing Wang (
ÏÏÏ
)
1,2
, Shuwen Dai (
)
1
, and Shuping Ge (
)
2
1
Department of Physics and Mathematics, Anhui University of Science and Technology, Huainan 232001
2
Departmant of Physics, Yunnan University, Kunming 650091
Received December 7, 2005
A single-mode laser model with cross-correlated additive and multiplicative noise terms is considered, and
the effects of correlation between noises on the relaxation time and the intensity correlation function are
studied. Using the projection operator method and taking into account the effects of the memory kernels
of the intensity correlation function, the analytic expressions for the relaxation time and the correlation
function are derived. Based on numerical computations, it is found that the self-correlation time and the
cross-correlation time have the same effects on the single-mode laser system.
OCIS codes: 140.3570, 140.0140, 140.3430.
It is known that the single-mode dye laser model
with additive white noise and multiplicative white noise
is often used as a prototype to investigate the laser
fluctuations
[1−5]
. Using this method, the previous stud-
ies have revealed that the consideration of additive and
multiplicative noises simultaneously is of importance to
deeply understand statistical properties of the single-
mode laser system
[6]
. In most of these existing theo-
retical studies, the multiplicative noise (pump noise) and
the additive noise (quantum noise) are both modeled as
Gaussian white noise and are treated as uncorrelated.
Recently, the effects of correlation between additive
and multiplicative noises on the statistical fluctuation
of the single-mode laser model have attracted close
attention
[7−10]
.Xieet al.
[11]
considered the correlation
between the two noises and studied the effects of cor-
relation intensity λ on the relaxation time T
c
and the
intensity correlation function C. But that study only
considered that the self-correlation time τ
0
and the cross-
correlation time τ of the two noises are both zero. In this
paper, we consider that τ
0
and τ are not zero and inves-
tigate the effects of τ
0
and τ on the single-mode laser
system.
The complex field-amplitude E of the cubic model of
a single-mode laser system can be described by Langevin
equation (LE)
[3]
,
dE
dt
= a
0
E − A |E|
2
E +˜p(t)E +˜q(t), (1)
where a
0
and A are real and respectively stand for the net
gain and the self-saturation coefficients, ˜p(t) is the pump
noise and ˜q(t) is the quantum noise. Performing the po-
lar coordinate transform E = xe
iϕ
, Eq. (1) can be trans-
formed into two coupling LEs for the field-amplitude x
and phase ϕ. By decoupling them, the LE of x can be
obtained as
[12]
dx
dt
= a
0
x − Ax
3
+
D
2x
+ xp(t)+q(t). (2)
We only consider the intensity fluctuation of the laser
system. Assuming I is the laser intensity (I = x
2
), Eq.
(2) is readily written for I as
dI
dt
=(2a
0
− AI)I + D +2I
1/2
q(t)+2Ip(t). (3)
The multiplicative noise p(t) and the additive noise q(t)
are considered to be Gaussian-type noise,
q(t)q(t
) =2Dδ(t − t
), (4)
p(t)p(t
) =
Q
τ
0
exp(−|t − t
|/τ
0
) → 2Qδ(t − t
), (5)
and
p(t)q(t
) = q(t)p(t
) =
λ
√
QD
τ
exp [−|t − t
|/τ]
→ 2λ
QDδ(t −t
), (6)
where Q and D are the multiplicative and additive noise
intensities respectively, τ
0
and τ are the self-correlation
time and cross-correlation time respectively, λ is the in-
tensity of correlation between p(t)andq(t). Applying the
Novikov theorem
[13]
and the Fox’s approach
[14]
,theap-
proximate Fokker-Planck equation corresponding to Eq.
(3) reads
∂P(I,t)
∂t
= L
FP
P (I,t), (7)
with
L
FP
= −
∂
∂I
F (I)+
∂
2
∂I
2
B(I). (8)
The drift coefficient F (I) and the diffusion coefficient
B(I)aregivenby
F (I)=2(a
0
+
Q
1+2a
0
τ
0
− AI)I +
3λ
√
DQ
1+2a
0
τ
I
1/2
+2D,
(9)
B(I)=2
Q
1+2a
0
τ
0
I
2
+4
λ
√
DQ
1+2a
0
τ
I
3/2
+2DI. (10)
1671-7694/2006/070397-04 http://www.col.org.cn
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