Luo A.C.J. (Ed.) Dynamical Systems: Discontinuity, Stochasticity and Time-Delay
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20 Dynamical Properties of Intensity Fluctuation 237
ˇ
2
D
F
2
1
.AD F
1
Q/
2Œ.AD F
1
Q/
2
C 4b
2
0
F
1
A
;
ˇ
3
D
b
0
q
DQ b
2
0
K
Q
F
2
1
.AD C F
1
Q/
.AD F
1
Q/
2
C 4b
2
0
F
1
A
1
;
and
ˇ
4
D
2b
0
F
5
2
1
p
A
.AD F
1
Q/
2
C 4b
2
0
F
1
A
:
2. When a
0
D 0, the probability density function is given by
P
st
.I / D N.QI C 2
p
DQI
1
2
C D/
ˇ
1
A
F
1
I C 1
ˇ
2
exp
"
ˇ
3
arctan
QI
1
2
C
p
DQ
p
DQ.1
2
/
C ˇ
4
arctan
I
1
2
s
A
F
1
!#
;
(20.13)
for 0 jj <1and
P
st
.I / D N.QI C 2
p
DQI
1
2
C D/
ˇ
1
A
F
1
I C 1
ˇ
2
exp
"
˛
1
QI
1
2
C
p
DQ
C ˇ
4
arctan
I
1
2
s
A
F
1
!#
:
(20.14)
for jjD1,where
˛
1
D
p
DQ
K
Q
F
2
1
.AD C F
1
Q/
.AD F
1
Q/
2
C 4
2
DQF
1
A
1
:
3. When a
0
<0, discriminant D4.b
2
0
DQ/D4DQ
˚
2
=Œ1 C 2K.1K=a
0
C
K/
2
1
, plus-minus of which is determined by taking values of , K, a
0
and .
(a) >0,wehave
P
st
.I / D N.QI C 2b
0
I
1
2
C D/
ˇ
1
A
F
1
I C 1
ˇ
2
0
B
@
Q
p
I C b
0
q
.b
2
0
DQ/
Q
p
I C b
0
C
q
.b
2
0
DQ/
1
C
A
˛
2
exp
"
ˇ
4
arctan
I
1
2
s
A
F
1
!#
;
(20.15)
238 P. Z h u
where
˛
2
D
b
0
2
q
b
2
0
DQ
K
Q
F
2
1
.AD C F
1
Q/
.AD F
1
Q/
2
C 4b
2
0
F
1
A
1
:
(b) D 0,wehave
P
st
.I / D N.QI C 2b
0
I
1
2
C D/
ˇ
1
A
F
1
I C 1
ˇ
2
exp
"
˛
3
QI C b
0
C ˇ
4
arctan
I
1
2
s
A
F
1
!#
; (20.16)
where
˛
3
D b
0
K
Q
F
2
1
.AD C F
1
Q/
.AD F
1
Q/
2
C 4b
2
0
F
1
A
1
:
(c) <0,wehave
P
st
.I / D N.QI C 2b
0
I
1
2
C D/
ˇ
1
A
F
1
I C 1
ˇ
2
exp
2
6
4
ˇ
3
arctan
QI
1
2
C b
0
q
DQ b
2
0
C ˇ
4
arctan
I
1
2
s
A
F
1
!
3
7
5
:
(20.17)
In (20.12)–(20.17), N is the responding normalization constant. The normalization
constant N is given by the equation
Z
1
0
P
st
.I /dI D 1:
Then, expectation values of the nth power of the laser intensity I are given by
hI
n
iD
Z
1
0
I
n
P
st
.I /dI: (20.18)
For a general laser model where a stationary state exists, the stationary correlation
function is defined by
C.s/ DhıI.t C s/ıI.t/i
st
D lim
t!1
hıI.t C s/ıI.t/i; (20.19)
where
ıI.t / D I.t/ hI.t/i:
20 Dynamical Properties of Intensity Fluctuation 239
A normalized correlation function is
C.s/ D
hıI.t C s/ıI.t/i
st
h.ıI /
2
i
st
: (20.20)
The associated relaxation time which describes the fluctuation decay of the laser
intensity variable I is defined by
T
c
D
Z
1
0
C.t/dt; (20.21)
By using the projection operator method [29], the zeroth approximation for the
relaxation time is given by
T
c
D
1
0
D
h.ıI /
2
i
st
hG.I/i
st
: (20.22)
Similarly, the first-order approximation for the relaxation time is given by
T
c
D
0
C
1
1
1
; (20.23)
where
1
D
hG.I/f
0
.I /i
st
h.ıI /
2
i
st
C
2
0
; (20.24)
and
1
D
hG.I/Œf
0
.I /
2
i
st
1
h.ıI /
2
i
st
C
3
0
1
2
0
: (20.25)
Employing (20.9), (20.10), and (20.18), we have
0
D
4k
1
h.I /
2
i
st
hI i
2
st
; (20.26)
1
D
8Œ.K C 2Q/k
1
C F
1
k
2
C
3b
0
2
k
12
h.I /
2
i
st
hI i
2
st
C
2
0
; (20.27)
1
D
1
1
.h.I /
2
i
st
hI i
2
st
/
8
ˆ
ˆ
<
ˆ
ˆ
:
16.K C 2Q/
2
k
1
C 16F
2
1
k
4
C 36b
2
0
k
0
C32.K C 2Q/F
1
k
2
C48b
0
.K C 2Q/k
12
C 48b
0
F
1
k
5
9
>
>
=
>
>
;
C
3
0
1
2
0
;
(20.28)
where
k
0
D D C 2b
0
hI
1
2
i
st
C QhI i
st
; (20.29)
240 P. Z h u
k
1
D DhI i
st
C 2b
0
hI
3
2
i
st
C QhI
2
i
st
; (20.30)
k
12
D DhI
1
2
i
st
C 2b
0
hI i
st
C QhI
3
2
i
st
; (20.31)
k
2
D D
*
I
1C
AI
F
1
2
+
st
C 2b
0
*
I
3
2
1C
AI
F
1
2
+
st
C Q
*
I
2
1C
AI
F
1
2
+
st
; (20.32)
k
4
D D
*
I
1C
AI
F
1
4
+
st
C 2b
0
*
I
3
2
1C
AI
F
1
4
+
st
C Q
*
I
2
1C
AI
F
1
4
+
st
; (20.33)
and
k
5
D D
*
I
1
2
1 C
AI
F
1
2
+
st
C 2b
0
*
I
1 C
AI
F
1
2
+
st
C Q
*
I
3
2
1 C
AI
F
1
2
+
st
:(20.34)
Here, we see that the zeroth approximation of the relaxation time T
c
D
1
0
is good agreement with that by virtue of the Stratonovich-like ansatz [41]. When
D 0 and Q D 0, the above results fall back to (2.29)–(2.31) presented in [24]. In
other words, the Stratonovich-like ansatz completely neglects the memory kernel.
Performing converse the Laplace transformation, we easily get the correlation
function of the laser intensity fluctuation
C.s/ D ˇ exp.
s/ C .1 ˇ/ exp.
C
s/; (20.35)
where
ˇ D
1
C
(20.36)
and
˙
D
0
C
1
2
˙
1
2
p
.
0
1
/
2
4
1
: (20.37)
To see the effects of cross-correlation noises on the laser intensity fluctuation, by
virtue of the expression of the associated relaxation time (20.23) and the correlation
function (20.35) and by applying the method of numerical calculation, we have
plotted the curves of C.s/ and T
c
(a
0
>0)inFigs.20.4–20.8, respectively, so that
we can discuss properties of the intensity fluctuation of a saturation laser model with
cross-correlation noises.
20.3 Discussion and Conclusion
In order to illustrate the effects of the coupling strength and the cross-correlation
time between additive and multiplicative white noises, by the means of numerical
calculations, we plot the curves of the stationery probability distribution (SPD) of
20 Dynamical Properties of Intensity Fluctuation 241
0
5
10 15 20
I
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
=-1
=-0.6
=-0.3
=0
=0.3
=0.6
=1
P
st
(I)
a
0
5
10
15
20
I
t=0
t=0.1
————————— t=0.5
————————— t=1
0.05
0.15
0.25
0.2
0.1
P
st
(I)
b
Fig. 20.1 The steady-state laser intensity distribution function P
st
.I / vs the variable I above the
threshold a
0
>0. The parameters chosen are a
0
D 5, A D 1, D D 1, Q D 0:5 and K D 30.(a)
is fixed to be 0.3 and takes different values. (b) is fixed to be 0.2 and takes different values
the saturation laser model vs. the laser intensity variable I for above the threshold
.a
0
>0/, at the threshold .a
0
D 0/, and below the threshold .a
0
<0/under the
condition of 1 C 2K.1 K=.a
0
C K// > 0 in Figs. 20.1–20.3.
When the laser is operated above the threshold .a
0
>0/, the SPD of the system
vs. the variable I is plotted in Fig. 20.1.InFig.20.1a, is fixed to be 0:3 and
takes different values. We see that the SPD exhibits one-maximum structure, the
height of the peak of P
st
.I / decreases as the coupling strength increases from 1
to 1.InFig.20.1b, is fixed to be 0:2 and takes different values, we see that for
D 0 and the smaller value of , P
st
.I / always gives a higher value as I ! 0 and
decreases monotonously as I increases; however, when is increased continuously,
P
st
.I / starts to exhibit one-maximum structure, and the height of the peak of the
SPD decreases as increases.
When the laser is operated at the threshold (a
0
D 0), the effects of the cross-
correlation time cannot occur, and the SPD of the system vs. the variable I is
plotted in Fig. 20.2.FromFig.20.2, we see that the values of the SPD decreases
as increases for a fixed I , that is, the coupling strength attenuates the SPD;
under the case of the parameters chosen, when takes larger values (>0),
P
st
.I / possesses no-maximum structure, and P
st
.I / decreases monotonously as I
242 P. Z h u
0 2
4
6
8
I
0.2
0.4
0.6
0.8
0.9
0.5
0
0.5
0.9
P
st
(I)
Fig. 20.2 The steady-state laser intensity distribution function P
st
.I / vs the variable I for different
values of the coupling strength at threshold a
0
D 0. The parameters chosen are A D 1, D D 1,
Q D 2 and K D 30
increases; however, when takes smaller values(<0), P
st
.I / possesses one-
maximum structure and the height of the speak of the SPD increases as decreases.
When ( D 0), the P
st
.I / I curve is agreeable with the result given by [13].
When the laser is operated below the threshold (a
0
<0), discriminant
D 4
b
2
0
DQ
D 4DQ
˚
2
=Œ1 C 2K.1 K=a
0
C K/
2
1
, plus–minus
of which is determined by taking values of , K, a
0
,and, and the approximate
Fokker–Plank equation [20.8] must satisfy the condition 1 C 2K.1
K
F
1
/>0.
For example, if K D 60 and a
0
D5, 0 < 0:092.InFig.20.3a, the cross-
correlation time takes D 0:05,Whenjj > 0:4545, >0;WhenjjD0:4545,
D 0;Whenjj < 0:4545; the SPD vs. the variable I for different values of the
coupling strength is plotted. P
st
.I / possesses no-maximum structure and P
st
.I /
decreases monotonously as I increases. The coupling strength enhances the SPD.
In Fig. 20.3b, is fixed to be 0.5 and takes different values, the parameters chosen
make the discriminant <0, and corresponding the SPD is plotted. P
st
.I / pos-
sesses no-maximum structure and P
st
.I / decreases monotonously as I increases.
The correlation time weakens the SPD.
When the laser starts working, it is always from below the threshold to at the
threshold, and again to above threshold. So we discuss mainly the character of the
associated time and the correlation function above the threshold and at the threshold.
The associated relaxation time T
c
gives dynamical information about the time
scale of the evolution of a spontaneous fluctuation of the system in the steady
state, which means that it reflects the evolution velocity of the system from an ar-
bitrary initial state to the steady state. The associated relaxation time distribution
diagrams of the saturation laser model vs. different parameter variables are given in
Figs. 20.4–20.6.InFig.20.4a, the associated time cures of T
c
are symmetrical
on the axes D 0.When<0, T
c
decreases as increases. On the contrary, when
>0, T
c
increases as increases. When D 0, T
c
is unchange as increases.
In Fig.20.4b, the associated time T
c
monotonously decreases as increases. For
20 Dynamical Properties of Intensity Fluctuation 243
0 0.5
1 1.5
2
I
l=0.95,D >0
l=0.4545,D =0
l=0.2,D <0
0.25
0.5
0.75
1
1.25
1.5
P
st
(I)
a
0 0.2 0.4 0.6 0.8
1
I
P
st
(I)
t=0.01
t=0.02
t=0.04
0.2
0.4
0.6
0.8
1
b
Fig. 20.3 The steady-state laser intensity distribution function P
st
.I / vs. the variable I for dif-
ferent values of the coupling strength below the threshold a
0
<0. The parameters chosen are
a
0
D5, K D 60, A D 1, D D 1,andQ D 2.(a) is fixed to be 0.05 and takes different
values. (b) is fixed to be 0.5 and takes different values
0 0.2
t
0.4 0.6 0.8
T
c
0.045
0.05
0.055
0.06
0.065
=-1
=-0.5
=0
=0.5
=1
a
Fig. 20.4 (a) The relaxation time T
c
as a function of the cross-correlated time for different
values of the coupling strength . Parameters chosen are a
0
D 5, K D 60, A D 1, D D 1 and
Q D 2.(b) The relaxation time T
c
as a function of the coupling strength for different values
of the cross-correlated time . The parameters chosen are a
0
D 5, K D 60, A D 1, D D 1
and Q D 2
244 P. Z h u
-0.75 -0.5 -0.25 0 0.25 0.5 0.75
T
c
t= 0.5
t= 0
t= 0.1
t= 0.9
0.05
0.055
0.06
0.065
b
Fig. 20.4 (continued)
0 10 20 30 40
D
T
C
l=0.9
l=0
l=−0.9
0.03
0.035
0.04
0.045
0.05
0.055
a
0 10 20 30 40
D
T
c
t=0
t=0.2
t=1.6
0.03
0.035
0.04
0.045
0.05
b
Fig. 20.5 (a) The relaxation time T
c
as a function of the quantum noise intensity D for different
values of the coupling strength . Parameters chosen are a
0
D 5, D 0:9, K D 120, A D 1,and
Q D 2.(b) The relaxation time T
c
as a function of the quantum noise intensity D for different
values of the cross-correlated time . The parameters chosen are a
0
D 5, D 0:5, K D 120,
A D 1,andQ D 2
20 Dynamical Properties of Intensity Fluctuation 245
0 20 40 60
80
100
Q
0
1
2
3
4
——————— 0.9
0————————
0.9———————
T
c
a
0 20 40 60
80
Q
0
0.5
1
1.5
2
2.5
3
T
c
———————
0
0.2————————
1.2———————
b
Fig. 20.6 (a) The relaxation time T
c
as a function of the pump noise intensity Q for different
values of the coupling strength . Parameters chosen are a
0
D 5, D 0:9, K D 120, A D 1,and
D D 1.(b) The relaxation time T
c
as a function of the pump noise intensity Q for different values
of the cross-correlated time . The parameters chosen are a
0
D 5, D 0:5, K D 120, A D 1,and
D D 1
<0, T
c
decreases as takes the larger values; for >0, the effects of are
entirely opposite.
Figure 20.5 shows the the effects of the quantum noise intensity D, the coupling
strength , and the correlation time for the associated relaxation time. T
c
–D
curves exhibit one-minimum structure. When D takes smaller values, T
c
decreases
as D increases, and the effects of attenuate the associated relaxation time. When
D takes larger values, T
c
increases as D increases, enhances the associated re-
laxation time. In contrast, the effects of are entirely opposite for the associated
relaxation time.
Figure 20.6 displays the the effects of the pump noise intensity Q, the cou-
pling strength , and the correlation time for the associated relaxation time. T
c
–Q
curves exhibit one-minimum structure. When Q takes smaller values, T
c
decreases
as Q increases and the effects of and do not almost occur. When Q takes larger
246 P. Z h u
0 0.05 0.1 0.15 0.2
s
—————————— 0.9
——————————
0
——————————
0.9
0
0.2
0.4
0.6
0.8
1
C(s)
a
0 0.02 0.04 0.06
0.08
0.1 0.12 0.14
s
0
0.2
0.4
0.6
0.8
1
C(s)
——————————
1.2
——————————
0.1
——————————
0
b
Fig. 20.7 The normalized correlation function C.s/ as a function of decay time s for a
0
D 5,
A D 1, K D 60, D D 1 and Q D 1:5.(a) is fixed to be 0.2 and takes different values.
(b) is fixed to be 0.2 and takes different values
values, T
c
increases as Q increases, attenuates the associated relaxation time,
however, the effects of are entirely opposite.
The correlated function C.s/ describes the fluctuation decay of the variable ıI
with time in the stationary state. To see that the effect of the coupling strength
for the intensity fluctuation, C.s/ as functions of the decay time s are plotted in
Fig. 20.7. It is clear that the correlation function C.s/ is an exponential function of
the variable I .InFig.20.7a, the cross-correlated time is fixed to be 0:2 and takes
different values. C.s/ increases as increases from 0:9 to 0:9.InFig.20.7b, the
correlated strength is fixed to be 0:8 and takes different values. C.s/ increases
as increases from zero to 1.
When the laser system is operated at the threshold, the associated relaxation
time and the correlation function distribution diagrams are given in Fig. 20.8.
In Fig. 20.8a, the relaxation time decreases as increases. In Fig. 20.8b, C.s/
decreases as increases from 0:9 to 0:9 for a fixed value of the decay time s.
From the foregoing, we can further understand that dynamical properties
of the saturation laser intensity, and see that the cross-correlated additive and
multiplicative noises play important roles in a laser model with a full account of the
saturation effects.