320
4 Speckle-Optical Methods and Devices for Studying …
Moment of the second order is determined as
(
p 1 ,
p 2 , τ ) =
V (
p 1 , t)V
∗
(
p 2 , t + τ )
(4.3)
Then (4.1) can be rewritten in the following way:
C 1 (
p 1 ,
p 2 , τ ) =
|(
p 1 ,
p 2 , τ )|
2
(
p 1 ,
p 1 , 0)(
p 2 ,
p 2 , 0)
= |γ (
p 1 ,
p 2 , τ )|
2 ,
(4.4)
where γ (
p 1 ,
p 2 , τ) is the normalized second moment of the complex amplitude of
the field.
Using proximal approximation and the Huygens–Fresnel’s principle [98], the field
amplitude in observation surface can be expressed through the amplitude in object
surface:
V (
p, t) =
k
2πi z
exp(ikz)
V (
ρ, t)exp
ik
2z
(
p − −
ρ)
2
d
2
ρ
(4.5)
where k is the optical wave number; z is the distance between the observation surface
and the object surface.
Then it is possible to assume that within the limits of the Fresnel’s diffraction
field
(
p 1 ,
p 2 , τ ) =
k
2π z
2 ¨
(
ρ 1 ,
ρ 2 , τ )
exp
ik
2z
(
p 1 − −
ρ 1 )
2 − (
p 2 − −
ρ 2 )
2
d
2
ρ 1 d
2
ρ 2
(4.6)
We consider that dispersion from different points is not correlated. Let us investigate the center of dispersion within the disk situated in the point with coordinate
ρ 1
at time t. If the same diffuser is situated in
ρ 1 at time t + τ, then the moment of the
second order of the scattered field complex amplitude in the object plane is given as
in the work [99]:
(
ρ 1 ,
ρ 2 , τ ) = V 1 (
ρ 1 , t)V
∗
1 (
ρ 2 , t + τ )δ(
ρ 2 − −
ρ 1 ),
(4.7)
where V 1 is the amplitude of the illuminating field in the object plane; δ is the Dirac
delta function and ensemble averaging taken on all possible spatial distributions of
point-scattering centers within the disk.
The disk is rotating around point
ρ = 0 at a frequency ω. In this case
ρ
1 = =
ρ 1 cos(ωτ ) +
A
ρ 1 sin(ωτ ),
(4.8)
where
A is the π /2-turn operator.
4 Speckle-Optical Methods and Devices for Studying …
Moment of the second order is determined as
(
p 1 ,
p 2 , τ ) =
V (
p 1 , t)V
∗
(
p 2 , t + τ )
(4.3)
Then (4.1) can be rewritten in the following way:
C 1 (
p 1 ,
p 2 , τ ) =
|(
p 1 ,
p 2 , τ )|
2
(
p 1 ,
p 1 , 0)(
p 2 ,
p 2 , 0)
= |γ (
p 1 ,
p 2 , τ )|
2 ,
(4.4)
where γ (
p 1 ,
p 2 , τ) is the normalized second moment of the complex amplitude of
the field.
Using proximal approximation and the Huygens–Fresnel’s principle [98], the field
amplitude in observation surface can be expressed through the amplitude in object
surface:
V (
p, t) =
k
2πi z
exp(ikz)
V (
ρ, t)exp
ik
2z
(
p − −
ρ)
2
d
2
ρ
(4.5)
where k is the optical wave number; z is the distance between the observation surface
and the object surface.
Then it is possible to assume that within the limits of the Fresnel’s diffraction
field
(
p 1 ,
p 2 , τ ) =
k
2π z
2 ¨
(
ρ 1 ,
ρ 2 , τ )
exp
ik
2z
(
p 1 − −
ρ 1 )
2 − (
p 2 − −
ρ 2 )
2
d
2
ρ 1 d
2
ρ 2
(4.6)
We consider that dispersion from different points is not correlated. Let us investigate the center of dispersion within the disk situated in the point with coordinate
ρ 1
at time t. If the same diffuser is situated in
ρ 1 at time t + τ, then the moment of the
second order of the scattered field complex amplitude in the object plane is given as
in the work [99]:
(
ρ 1 ,
ρ 2 , τ ) = V 1 (
ρ 1 , t)V
∗
1 (
ρ 2 , t + τ )δ(
ρ 2 − −
ρ 1 ),
(4.7)
where V 1 is the amplitude of the illuminating field in the object plane; δ is the Dirac
delta function and ensemble averaging taken on all possible spatial distributions of
point-scattering centers within the disk.
The disk is rotating around point
ρ = 0 at a frequency ω. In this case
ρ
1 = =
ρ 1 cos(ωτ ) +
A
ρ 1 sin(ωτ ),
(4.8)
where
A is the π /2-turn operator.