12
1 Resonance Methods for Increasing Sensitivity of Interferometry …
The main idea of the integral method is that the recording of a hologram is carried
out according to the scheme (Fig. 1.5) proposed in [69]. But unlike the methods
developed in these works, we are going to measure not the differential (local) intensity
of the reconstructed image I(R, r) but the integral intensity of the whole image:
I i (r ) =
S
I (R, r )d
2 r,
(1.16)
what is achieved due to the registration of the whole image by the photocathode
of the electron-multiplier phototube. Then according to (1.12), we have the integral
relation, which connects the intensity of the reconstructed image I I (r) and squared
absolute value of spatial coherence function |γ (R, r)|
2 :
I i (r ) =
I 0 (R, r )d
2 r = f
2
(r )I 0 (r )I b (r )
I p (R)|γ (R, r )|
2 d
2 R,
(1.17)
where I b (r) is the intensity of the reconstructing beam, I 0 (r) is the radiation intensity
in the point r of the hologram plane, and I p (r) is the radiation intensity in the point
R of the laser edge.
Consider that the recording conditions are linear. This assumption does not greatly
limit the applicability of the method for the average value determination |γ (R, r)|
2 .
Consider also that |γ (r 1 , r 2 )| = |γ (r 1 − r 2 )|, i.e., we will approximate spatial
coherence function by the homogeneous function (thus, our results will give precise
description of |γ |
2 of the sources with homogeneous spatial coherence function and
in the case of heterogeneous spatial coherence function our results will give for it
the averaged approximating form). As the function depends only on
x = =
r −
R, then
passing in the right side of (1.17) from integration on d
2 R to integration on d
2 x, we
get the following integral equation for γ (x):
I i (r )
I 0 (r )
= C
I p ( r − −
x)|γ (x)|
2 d
2 x,
(1.18)
where I i (r) is the experimentally taken dependence of the integral on the position of
the recovery point on a hologram, and I 0 (r) and I p ( r − −
x) are the light intensity in
the points of a hologram and an image, which can be experimentally detected by the
degree of the photographic-plate blackening.
The value of the constant C can be calculated, but as it can be easily seen, the
normalizing condition |γ (0)|
2
= 1 makes the value of this constant negligible during
the distribution construction γ . Equation (1.18) is the integral equation with the
difference kernel, and it can be solved by decomposition in the Fourier integral.
Then for the value |γ (x)|
2 , we have
|γ (x)|
2
= C 2
d
2 p exp(−i px)
I i (r )
I 0 (r )
exp(i pr)d
2 r
I p (r ) exp(i pr)d 2 r
,
(1.19)
1 Resonance Methods for Increasing Sensitivity of Interferometry …
The main idea of the integral method is that the recording of a hologram is carried
out according to the scheme (Fig. 1.5) proposed in [69]. But unlike the methods
developed in these works, we are going to measure not the differential (local) intensity
of the reconstructed image I(R, r) but the integral intensity of the whole image:
I i (r ) =
S
I (R, r )d
2 r,
(1.16)
what is achieved due to the registration of the whole image by the photocathode
of the electron-multiplier phototube. Then according to (1.12), we have the integral
relation, which connects the intensity of the reconstructed image I I (r) and squared
absolute value of spatial coherence function |γ (R, r)|
2 :
I i (r ) =
I 0 (R, r )d
2 r = f
2
(r )I 0 (r )I b (r )
I p (R)|γ (R, r )|
2 d
2 R,
(1.17)
where I b (r) is the intensity of the reconstructing beam, I 0 (r) is the radiation intensity
in the point r of the hologram plane, and I p (r) is the radiation intensity in the point
R of the laser edge.
Consider that the recording conditions are linear. This assumption does not greatly
limit the applicability of the method for the average value determination |γ (R, r)|
2 .
Consider also that |γ (r 1 , r 2 )| = |γ (r 1 − r 2 )|, i.e., we will approximate spatial
coherence function by the homogeneous function (thus, our results will give precise
description of |γ |
2 of the sources with homogeneous spatial coherence function and
in the case of heterogeneous spatial coherence function our results will give for it
the averaged approximating form). As the function depends only on
x = =
r −
R, then
passing in the right side of (1.17) from integration on d
2 R to integration on d
2 x, we
get the following integral equation for γ (x):
I i (r )
I 0 (r )
= C
I p ( r − −
x)|γ (x)|
2 d
2 x,
(1.18)
where I i (r) is the experimentally taken dependence of the integral on the position of
the recovery point on a hologram, and I 0 (r) and I p ( r − −
x) are the light intensity in
the points of a hologram and an image, which can be experimentally detected by the
degree of the photographic-plate blackening.
The value of the constant C can be calculated, but as it can be easily seen, the
normalizing condition |γ (0)|
2
= 1 makes the value of this constant negligible during
the distribution construction γ . Equation (1.18) is the integral equation with the
difference kernel, and it can be solved by decomposition in the Fourier integral.
Then for the value |γ (x)|
2 , we have
|γ (x)|
2
= C 2
d
2 p exp(−i px)
I i (r )
I 0 (r )
exp(i pr)d
2 r
I p (r ) exp(i pr)d 2 r
,
(1.19)