354
4 Speckle-Optical Methods and Devices for Studying …
Fig. 4.22 Scheme for circular speckle-interferograms of longitudinal shear control. Reprinted from
[2] with permission
Equation (4.82) decomposes into three components, two of which describe intensity of waves diffracted on speckles registered during the first and the second exposures, respectively, and the third one describes their interference. As I 1 ( x) and I 2 ( x)
are the random functions, then the resulting distribution I p ( x) in observation plane
will be stochastic as well. For elimination of this noise and separation of interference
fringes, we average (4.82) on the ensemble of specklograms. We analyze the latter
compound in (4.82), which describes formation of interference fringes:
I
X
= 2Re
¨
I 1 ( x)
I 2
x
K
x,
X
K
∗
x
,
X
d
2
xd
2
x
(4.83)
The shape of the speckle-interferogram is determined by the correlation between
speckle-structures I 1 ( x) and I 2 ( x) and properties of the optical system. In fact, the
expressions describe almost every speckle-interferogram. In the considered case,
the object experiences longitudinal shift between exposures, and two speckle-fields
I 1 ( x) and I 2 ( x) are alike and radially shifted. The common expression of correlation function
I 1 ( x)I 2
x
is presented in the work [118]. But radial “recession” of
speckles is most important for forming a speckle-interferogram of longitudinal shift
and clarifying the main conditions of their observation. Neglecting decorrelation of
speckles and their final size, correlation function of intensity can be written as
I 1 ( x)I 2
x
= I
2
δ
x − α
x
(4.84)
where α is the coefficient describing radial expansion of the speckle-image, the value
of which is close to one in real conditions and is determined by longitudinal shift of
the object; I is the average value of illumination of photomaterial during exposure.
For calculation of (4.83), we suppose that the optical system is a thin lens with
apodizing Gaussian screen. It should be mentioned that the results of calculations
4 Speckle-Optical Methods and Devices for Studying …
Fig. 4.22 Scheme for circular speckle-interferograms of longitudinal shear control. Reprinted from
[2] with permission
Equation (4.82) decomposes into three components, two of which describe intensity of waves diffracted on speckles registered during the first and the second exposures, respectively, and the third one describes their interference. As I 1 ( x) and I 2 ( x)
are the random functions, then the resulting distribution I p ( x) in observation plane
will be stochastic as well. For elimination of this noise and separation of interference
fringes, we average (4.82) on the ensemble of specklograms. We analyze the latter
compound in (4.82), which describes formation of interference fringes:
I
X
= 2Re
¨
I 1 ( x)
I 2
x
K
x,
X
K
∗
x
,
X
d
2
xd
2
x
(4.83)
The shape of the speckle-interferogram is determined by the correlation between
speckle-structures I 1 ( x) and I 2 ( x) and properties of the optical system. In fact, the
expressions describe almost every speckle-interferogram. In the considered case,
the object experiences longitudinal shift between exposures, and two speckle-fields
I 1 ( x) and I 2 ( x) are alike and radially shifted. The common expression of correlation function
I 1 ( x)I 2
x
is presented in the work [118]. But radial “recession” of
speckles is most important for forming a speckle-interferogram of longitudinal shift
and clarifying the main conditions of their observation. Neglecting decorrelation of
speckles and their final size, correlation function of intensity can be written as
I 1 ( x)I 2
x
= I
2
δ
x − α
x
(4.84)
where α is the coefficient describing radial expansion of the speckle-image, the value
of which is close to one in real conditions and is determined by longitudinal shift of
the object; I is the average value of illumination of photomaterial during exposure.
For calculation of (4.83), we suppose that the optical system is a thin lens with
apodizing Gaussian screen. It should be mentioned that the results of calculations