344
4 Speckle-Optical Methods and Devices for Studying …
K
x,
X , d, p
= exp
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
i
k
2
⎛
⎜
⎝
| x|
2
d
+
X
2
p
⎞
⎟
⎠
2
−
x
d
−
X
p
2
4k
2 q
2
− i
2
k
ε
⎫
⎪ ⎪ ⎬
⎪ ⎪ ⎭
,
(4.68)
where k =
2π
λ
, ε =
1
d
+
1
p
−
1
f
; f is the focal distance O b ; q is the effective radius.
Using (4.67) and (4.68), we define normalized correlation function of the
amplitude and its connection to the correlated function of intensities fluctuations.
γ I = |γ ν |
2
.
(4.69)
And after some algebraic transformations, we obtain
γ I = 4α
2
α
2
+ 1
2 + β
2
α
2
σ − 1
2 −1
× exp
⎧
⎪ ⎨
⎪ ⎩
−
k
2 q
2
2
α
2
+ 1
α 2 + 1
2
X 1
p
− α
X 2
p
2
α 2 + 1
2 + β 2
α 2 σ − 1
2
⎫
⎪ ⎬
⎪ ⎭
,
(4.70)
where α = d 2 /d 1 ; σ = ε 2 /ε 1 ; β =
kq
2 ε 1
2
.
Equation (4.70) depends on a number of parameters that is why we should consider
particular cases to illustrate it. If α = 1, σ = 1, then the field is examined in some plane
behind the objective. Analysis of the exponential factor leads to the known result that
the fields stay correlated for the points, the distance between ich is determined by
the observation aperture (in particular, it corresponds to the resolution limit for the
image plane) and these fields are stationary. If the correlation function alters under
small defocusing (d 2 ≈ d 1 ) in the same plane p 2 ≈ p 1 in the same point
X 1 =
X 2 ,
then we come to the situation, which was considered in the work [84]. It is worth
mentioning that in common case, the maximum of the correlation function will be
observed for points connected by the correlation.
d 1
p 1
X 1 =
d 2
p 2
X 2
(4.71)
When the correlation function is determined for points, which are situated in the
image planes P 1 and P 2, then (4.71) means that maximal correlation is observed
for points of the images, which correspond to the same point of the object. From
a physical point of view, interpretation of this result is simple: Identical intensity
distributions are observed where phase correlations between elementary waves are
saved. Such interpretation is actually analogous to the considerations [66] conducted
for the scheme where there is no optical system. It follows from (4.70) that if corresponding points
d 1
p 1
X 1 =
d 2
p 2
X 2 are in the image plane, then the correlation function
alters due to a simple law:
4 Speckle-Optical Methods and Devices for Studying …
K
x,
X , d, p
= exp
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
i
k
2
⎛
⎜
⎝
| x|
2
d
+
X
2
p
⎞
⎟
⎠
2
−
x
d
−
X
p
2
4k
2 q
2
− i
2
k
ε
⎫
⎪ ⎪ ⎬
⎪ ⎪ ⎭
,
(4.68)
where k =
2π
λ
, ε =
1
d
+
1
p
−
1
f
; f is the focal distance O b ; q is the effective radius.
Using (4.67) and (4.68), we define normalized correlation function of the
amplitude and its connection to the correlated function of intensities fluctuations.
γ I = |γ ν |
2
.
(4.69)
And after some algebraic transformations, we obtain
γ I = 4α
2
α
2
+ 1
2 + β
2
α
2
σ − 1
2 −1
× exp
⎧
⎪ ⎨
⎪ ⎩
−
k
2 q
2
2
α
2
+ 1
α 2 + 1
2
X 1
p
− α
X 2
p
2
α 2 + 1
2 + β 2
α 2 σ − 1
2
⎫
⎪ ⎬
⎪ ⎭
,
(4.70)
where α = d 2 /d 1 ; σ = ε 2 /ε 1 ; β =
kq
2 ε 1
2
.
Equation (4.70) depends on a number of parameters that is why we should consider
particular cases to illustrate it. If α = 1, σ = 1, then the field is examined in some plane
behind the objective. Analysis of the exponential factor leads to the known result that
the fields stay correlated for the points, the distance between ich is determined by
the observation aperture (in particular, it corresponds to the resolution limit for the
image plane) and these fields are stationary. If the correlation function alters under
small defocusing (d 2 ≈ d 1 ) in the same plane p 2 ≈ p 1 in the same point
X 1 =
X 2 ,
then we come to the situation, which was considered in the work [84]. It is worth
mentioning that in common case, the maximum of the correlation function will be
observed for points connected by the correlation.
d 1
p 1
X 1 =
d 2
p 2
X 2
(4.71)
When the correlation function is determined for points, which are situated in the
image planes P 1 and P 2, then (4.71) means that maximal correlation is observed
for points of the images, which correspond to the same point of the object. From
a physical point of view, interpretation of this result is simple: Identical intensity
distributions are observed where phase correlations between elementary waves are
saved. Such interpretation is actually analogous to the considerations [66] conducted
for the scheme where there is no optical system. It follows from (4.70) that if corresponding points
d 1
p 1
X 1 =
d 2
p 2
X 2 are in the image plane, then the correlation function
alters due to a simple law: