Klyatskin V.I. Lectures on Dynamics of Stochastic Systems
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386 Lectures on Dynamics of Stochastic Systems
(a)
0.4 0.8 1.2
0.1
0.2
0.3
0.4
0.5
β
0
(b)
β
0
I = 1.5
I = 2.5
0.4 0.8 1.2
0.02
0.04
0.06
0.08
0.1
β
0
I = 1.5
I = 2.5
〈l (x, l )D
−1/12
(x)〉
〈n (x, l )D
−1/6
(x)〉
Figure 13.4 (a) Average specific contour length and (b) average contour number versus para-
meter β
0
(x).
further increasing parameter β
0
(x) (β
0
(x) ≥10) statistical characteristics of intensity
approach the saturation regime, and this region of parameter β
0
(x) is called the region
of strong intensity fluctuations.
13.4.2 Strong Intensity Fluctuations
From Eq. (13.82) for probability density follows that specific average area of regions
within which I(x, R) > I is
h
s(x, I)
i
=
1
√
π
(
β(x) − 1
)
∞
Z
0
dz
z
exp
−zI −
ln z −
β(x) − 1
4
2
β(x) − 1
, (13.96)
and specific average power concentrated in these regions is given by the expression
h
e(x, I)
i
=
1
√
π
(
β(x) − 1
)
∞
Z
0
dz
z
I +
1
z
exp
−zI −
ln z −
β(x) − 1
4
2
β(x) − 1
.
(13.97)
Figure 13.5 shows functions (13.96) and (13.97) versus parameter β(x). We note
that parameter β(x) is a very slow function of β
0
(x). Indeed, limit process β
0
(x) → ∞
corresponds to β(x) = 1 and value β
0
(x) = 1 corresponds to β(x) = 1.861.
Asymptotic formulas (13.96) and (13.97) adequately describe the transition to the
region of saturated intensity fluctuations
(
β(x) → 1
)
. In this region, we have
P(I) = e
−I
,
h
s(I)
i
= e
−I
,
h
e(I)
i
=
(
I + 1
)
e
−I
. (13.98)
Caustic Structure of Wavefield in Random Media 387
(a)
1.2 1.4 1.6 1.8
0.35
0.3
0.25
0.15
0.2
β (x)
(b)
β (x)
I = 1
I = 2
〈s (x, l )〉
〈e (x, l )〉
1.2 1.4 1.6 1.8
0.5
0.6
0.7
I = 1
I = 2
Figure 13.5 (a) Average specific area and (b) average contour number in the region of strong
intensity fluctuations versus parameter β(x).
Moreover, we obtain the expression for specific average contour length
h
l(x, I)
i
= L
f
(x)
h
|p(x, R)|δ
(
I(x, R) − I
)
i
= 2L
f
(x)
√
I
h
|κ(x, R)|δ
(
I(x, R) − I
)
i
= 2L
f
(x)
√
I
h
|q(x, R)|
i
P(x;I) = L
f
(x)
q
2πσ
2
q
(x)I P(x;I), (13.99)
where the variance of amplitude level gradient in the region of saturated fluctua-
tions coincides with the variance calculated in the first approximation of Rytov’s
smooth perturbation method. Specific average contour length (13.99) is maximum at
I = 1/
√
2.
In the region of saturated intensity fluctuations, the estimator of specific average
number of contours is given by the following chain of equalities
h
n(x, I)
i
=
L
2
f
(x)
2π
h
κ(x, R, I)|p(x, R)|δ
(
I(x, R) − I
)
i
= −
L
2
f
(x)
2π
√
I
h
1
R
A(x, R)δ
(
I(x, R) − I
)
i
= −
L
2
f
(x)
π
D
κ
2
(x, R)
E
√
I
∂
∂I
√
IP(x;I)
= −
2L
2
f
(x)σ
2
q
(x)
π
√
I
∂
∂I
√
Ie
−I
=
2L
2
f
(x)σ
2
q
(x)
π
I −
1
2
e
−I
. (13.100)
Expression (13.100) is maximum at I = 3/2, and the level at which specific average
number of contours bounding region I(x, R) > I
0
is equal to specific average number
of contours corresponding to I(x, R) < I
0
is I
0
= 1/2.
We note that Eq. (13.100) fails in the narrow region around I ∼ 0. The correct
formula must vanish for I = 0 (
h
n(x, 0)
i
= 0).
388 Lectures on Dynamics of Stochastic Systems
As may be seen from Eqs. (13.99) and (13.100), average length of level lines and
average number of contours keep increasing behaviors with parameter β
0
(x) in the
region of saturated fluctuations, although the corresponding average areas and powers
remain fixed. The reason of this behavior consists in the fact that the leading (and
defining) role in this regime is played by interference of partial waves arriving from
different directions.
Behavioral pattern of level lines depends on the relationship between processes of
focusing and defocusing by separate portions of turbulent medium. Focusing by large-
scale inhomogeneities becomes apparent in random intensity relief as high peaks. In
the regime of maximal focusing (β
0
(x) ∼ 1), the narrow high peaks concentrate about
a half of total wave power. With increasing parameter β
0
(x), radiation defocusing
begins to prevail,which results in spreading the high peaks and forming highly idented
(interference) relief characterized by multiple vertices at level I ∼ 1.
In addition to parameter β
0
(x), average length of level lines and average number
of contours depend on wave parameter D(x); namely, they increase with decreasing
microscale of inhomogeneities. This follows from the fact that the large-scale relief is
distorted by fine ripples appeared due to scattering by small-scale inhomogeneities.
Thus, in this section, we attempted to qualitatively explain the cluster (caustic)
structure of the wavefield generated in turbulent medium by the plane light wave and
to quantitatively estimate the parameters of such a structure in the plane transverse to
the propagation direction. In the general case, this problem is multiparametric. How-
ever, if we limit ourselves to the problem on the plane incident wave and consider it
for a fixed wave parameter in a fixed plane, then the solution is expressed in terms of
the sole parameter, namely, the variance of intensity in the region of weak fluctuations
β
0
(x). We have analyzed two asymptotic cases corresponding to weak and saturated
intensity fluctuations. It should be noted that applicability range of these asymptotic
formulas most likely depends on intensity level I. It is expected naturally that this
applicability range will grow with decreasing the level.
As regards the analysis of the intermediate case corresponding to the developed
caustic structure (this case is the most interesting from the standpoint of applications),
it requires the knowledge of the probability density of intensity and its transverse
gradient for arbitrary distances in the medium. Such an analysis can be carried out
either by using probability density approximations for all parameters [79], or on the
basis of numerical simulations (see, e.g., [82–84]).
Problems
Problem 13.1
Find the second-order coherence function and average intensity at the axis of
parabolic waveguide in the problem formulated in terms of the dynamic equation
∂
∂x
u(x, R) =
i
2k
1
R
u(x, R) +
ik
2
h
−α
2
R
2
+ ε(x, R)
i
u(x, R)
Caustic Structure of Wavefield in Random Media 389
assuming that ε(x, R) is the homogeneous isotropic Gaussian field delta-
correlated in x and characterized by the parameters
h
ε(x, R)
i
= 0, B
ε
(x, R) = B
eff
ε
(x, R) = A(R)δ(x).
Solution
0
2
(x, R, ρ) =
e
−iαRρ tan(αx)
cos
2
(αx)
Z
dqγ
0
q,
1
cos(αx)
ρ −
q
αk
tan(αx)
× exp
i
1
cos(αx)
qR −
k
2
4
x
Z
0
dξD
cos(αξ )
cos(αx)
ρ −
1
αk
sin
[
α
(
x −ξ
)
]
cos(αx)
q
,
where
γ
0
(
q, ρ
)
=
1
(
2π
)
2
Z
dR0
2
(0, R, ρ)e
−iqR
and D(ρ) = A(0) − A(ρ).
Setting R = 0 and ρ = 0, we obtain the average intensity at waveguide axis
as a function of longitudinal coordinate x
h
I(x, 0)
i
=
1
cos
2
(αx)
Z
dqγ
0
q, −
q
αk
tan(αx)
× exp
−
k
2
4
x
Z
0
dξD
1
αk
sin
(
αξ
)
cos(αx)
q
.
Problem 13.2
Within the framework of the diffusion approximation, derive equations for ave-
rage field
h
u(x, R)
i
and second-order coherence function
0
2
(x;R, ρ) =
u
x, R +
1
2
ρ
u
∗
x, R −
1
2
ρ
starting from parabolic equation (13.1) [55, 56].
Solution
∂
∂x
−
i
2k
1
R
h
u(x, R)
i
= −
k
2
4
x
Z
0
dx
0
Z
dR
0
B
ε
(x
0
, R − R
0
)e
ix
0
2k
1
R
δ(R − R
0
)e
−
ix
0
2k
1
R
h
u(x, R)
i
,
390 Lectures on Dynamics of Stochastic Systems
∂
∂x
−
i
k
∇
R
∇
ρ
0
2
(x, R, ρ)
= −
k
2
4
x
Z
0
dx
1
Z
dR
1
h
B
ε
x
1
, R − R
1
+
ρ
2
− B
ε
x
1
, R − R
1
−
ρ
2
i
× e
i
k
x
1
∇
R
∇
ρ
δ
R − R
1
+
1
2
ρ
− δ
R − R
1
−
1
2
ρ
× e
−
i
k
x
1
∇
R
∇
ρ
0
2
(x −x
1
, R, ρ).
If we introduce now the two-dimensional spectral density of inhomogeneities
B
ε
(x, R) =
Z
dq8
(2)
ε
(x, q)e
iqR
,
8
(2)
ε
(x, q) =
1
(
2π
)
2
Z
dRB
ε
(x, R)e
−iqR
and Fourier transforms of wave field u(x, R) and coherence function 0
2
(x, R, ρ)
with respect to transverse coordinates
u(x, R) =
Z
dqeu(x, q)e
iqR
, eu(x, q) =
1
(
2π
)
2
Z
dRu(x, R)e
−iqR
,
0
2
(x, R, ρ) =
Z
dq
e
0
2
(x, q, ρ)e
iqR
,
e
0
2
(x, q, ρ) =
1
(
2π
)
2
Z
dR0
2
(x, R, ρ)e
−iqR
,
then we obtain that average field is given by the expression
h
u(x, R)
i
=
1
(
2π
)
2
Z
dq
Z
dR
0
u
0
(R
0
)
× exp
iq(R − R
0
) − i
q
2
x
2
−
k
2
2
x
Z
0
dx
0
D(x
0
, q)
,
where
D(x, q) =
x
Z
0
dξ
Z
dq
0
8
(2)
ε
(ξ, q
0
) exp
−
iξ
2k
q
02
− 2q
0
q
,
Caustic Structure of Wavefield in Random Media 391
and function
e
0
2
(x, q, ρ) satisfies the integro-differential equation
∂
∂x
+
1
k
q∇
ρ
e
0
2
(x, q, ρ) = −
k
2
4
x
Z
0
dx
1
Z
dq
1
8
(2)
ε
x
1
, q
1
×
n
cos
h
x
1
2k
q
1
(q
1
− q)
i
− cos
h
q
1
ρ −
x
1
2k
q
1
(q
1
− q)
io
e
0
2
(x, q
1
, ρ).
If distances a wave passes in medium satisfy the condition x l
k
, where l
k
is the longitudinal correlation radius of field ε(x, R), then we obtain
h
u(x, R)
i
=
1
(
2π
)
2
Z
dq
Z
dR
0
u
0
(R
0
) exp
iq(R − R
0
) − i
q
2
x
2
−
k
2
2
xD(q)
,
where
D(q) =
∞
Z
0
dξ
Z
dq
0
8
(2)
ε
(ξ, q
0
) exp
−
iξ
2k
q
02
− 2q
0
q
.
For the plane incident wave, we have u
0
(R) =1, and, consequently, average
intensity is independent of R,
h
u(x, R)
i
= e
−
1
2
k
2
xD(0)
.
The applicability range of this expression is described by the condition
k
2
2
D(0)l
k
1.
Problem 13.3
Derive operator expression for average field of plane incident wave in the gen-
eral case of the Gaussian field ε(x, R) with correlation function B
ε
(x, R) and
calculate average field in the delta-correlated approximation.
Solution Averaging Eq. (13.53), page 369, over an ensemble of realizations of
field ε(x, R), we obtain in the general case the expressionv
h
u(x, R)
i
= u
0
exp
i
2k
x
Z
0
dξ
δ
2
δv
2
(ξ)
× exp
−
k
2
8
x
Z
0
dξ
1
x
Z
0
dξ
2
B
ε
ξ
1
− ξ
2
,
ξ
2
Z
ξ
1
dη v(η)
v(x)=0
.
(13.101)
392 Lectures on Dynamics of Stochastic Systems
According to Eq. (13.6), page 356, the correlation function of the Gaussian
delta-correlated field is B
ε
(x, R) = A(R)δ(x) and, consequently, we obtain the
expression
h
u(x, R)
i
= u
0
exp
−
k
2
8
A(0)x
.
Problem 13.4
Derive operator expression for the second-order coherence function of plane
incident wave in the general case of the Gaussian field ε(x, R) with correlation
function B
ε
(x, R) and calculate the second-order coherence function in the delta-
correlated approximation.
Solution On the case of plane incident wave, the second-order coherence func-
tion is given by the operator expression
0
2
(x, R
1
− R
2
)
=
u(x, R
1
)u
∗
(x, R
2
)
= |u
0
|
2
exp
i
2k
x
Z
0
dξ
"
δ
2
δv
2
1
(ξ)
−
δ
2
δv
2
2
(ξ)
#
× exp
−
k
2
4
x
Z
0
dξ D
R
1
− R
2
+
x
Z
ξ
dη
[
v
1
(η) − v
2
(η)
]
v
i
=0
,
where D(R) = A(0) − A(R).
Changing functional variables
v
1
(x) − v
2
(x) = v(x), v
1
(x) + v
2
(x) = 2V(x)
and introducing new space variables
R
1
− R
2
= ρ, R
1
+ R
2
= 2R,
we can rewrite last expression in the form
0
2
(x, R
1
− R
2
) = |u
0
|
2
exp
i
k
x
Z
0
dξ
δ
2
δv(ξ )δV(ξ)
× exp
−
k
2
4
x
Z
0
dξ D
ρ +
x
Z
ξ
dη v(η)
v
i
=0
= |u
0
|
2
exp
−
k
2
xD
(
ρ
)
4
.
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