4.3 Statistical Properties of Dynamic Speckle-Field Scattered …
337
β =
1
W 2 (z 1 )
+
1
W 2 (z 2 )
,
(4.51)
ρ = k
1
R 0 − z 1
−
1
R 0 − z 2
.
(4.52)
Using (4.49) and parameters of (4.50)–(4.52), the set in (4.48) autocorrelation
function of speckle intensity fluctuations can be presented as
|( x, z 1 , z 2 )|
2 =
π W
4
0 S
2
W 2 (z 1 )W 2 (z 2 )
α 2 + β 2
exp
−
βρ
2
x
2
+ y
2
2
α 2 + β 2
.
(4.53)
Parameters α, β and W(x) are presented in values z 1 , z 2 and a, using (4.38) and
(4.39) in (4.53):
α =
π (z 2 − z 1 )
z 2 z 1 − a
2
λ
z
2
1 + a 2
z
2
2 + a 2
,
(4.54)
β =
πa
z
2
1 + z
2
2 + 2a
2
λ
z
2
1 + a 2
z
2
2 + a 2
,
(4.55)
α
2
+ β
2
=
π
λ
2 (z 1 − z 2 )
2
+ 4a
2
z
2
1 + a 2
z
2
2 + a 2
,
(4.56)
W
2
(z 1 )W
2
(z 2 ) =
W
4
0
α 4
z
2
1 + a
2
z
2
2 + a
2
.
(4.57)
Substituting (4.54)–(4.57) into (4.53), we get
|( x, z 1 , z 2 )|
2 =
λ
2 a
4
S
2
(z 1 − z 2 )
2
+ 4a 2 exp
−
λa
z
2
1 + z
2
2 + 2a
2
ρ
2
2π
z
2
1 − z
2
2
+ 4a 2
x
2
+ y
2
(4.58)
It follows from this that autocorrelation function of speckle intensity fluctuations
is generally non-stationary, as (4.58) is the function of two independent variables z 1
and z 2 . Nevertheless, it becomes stationary in the middle of the far-field region, i.e.,
x = 0 (x = y = 0) being only interval function (z 1 − z 2 ) of two points along the
optical axis. Alteration of intensity is also stationary near the center in the far-field
region of diffraction as the given in (4.52) parameter ρ has a very small value in this
case.
Normalized autocorrelation function in the center
x = 0 can be obtained from
(4.58):
γ (0, z 1 , z 2 ) = |Γ (0, z 1 , z 2 )|
2
/|Γ (0, 0, 0)|
2
.
(4.59)