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Two point resolution in partially coherent Fourier holography
Volume
2, number
6
November 1970
OPTICS COMMUNICATIONS
TWO
POINT
RESOLUTION
COHERENT
FOURIER
IN
PARTIALLY
HOLOGRAPHY
R. S. SIROHI and C. S. VIKRAM Physics
Department, Indian Institute Hauz Khas, New Delhi-29, Received
3 September
of Technology, India
1970
Two point resolution in Fourier holography with partially coherent recording and illuminating waves has been theoretically investigated. The mutual coherence function of the illuminating wave at the hologram plane is assumed to be exponential.
1. INTRODUCTION The light sources for holography have been generally supposed to be completely coherent all over the illuminating beams used for recording a hologram and even for reconstructing an image from it. Since no real source is a point and even the sharpest line has a width, the complete coherence is thus a mathemathical idealisation. The effect of partial coherence on holography has thus been extensively studied by many investigators [l-11]. However, a most systematic study concerning the coherence effects on holography has been carried out by Reynolds and De Velis [6], Menzel et al. [8,9] and Murata et al. [ll]. Reynolds and De Velis [6] have treated the effect of partial coherence only in the illuminating process, while Murata et al. [ll] included the reconstruction process as well. Further in their paper Murata et al. have also investigated the effect of partial coherence on two point resolution in Fourier holography but their treatment is limited to the coherent reconstruction wave. We have extended their study by including the partially coherent wave; the mutual coherence function at the hologram plane is assumed to be exponential. The variation of the limit of resolution with the coherence parameter of the reconstructed wave has been studied for various values of coherence function of the recording wave. Following Murata et al. 1111 the intensity distribution in the hologram plane (X plane) located at a distance .z from the object plane (5 plane) is given by
(1) where I,(tl, 52) is the mutual spatial coherence function at the object plane and aGI/& derivative of the Green function satisfying the equation (V2 + k2)G1(~l,
The amplitude
= -6(x1 -
condition
transmittance
T(<) = 6(5 -a) where the reference 276
is the normal
across
the object plane is represented
by
+ D(5) , point is situated at a distance a from the object with transmittance
D(6).
OPTICS
Volume 2, number 6
November
COMMUNICATIONS
1970
In expression (l), the third and fourth terms produce the real and virtual images of the object, respectively. Restricting the attention to the third term and evaluating the normal derivatives of the Green function, this term becomes
(2) This expression is valid provided the object size is small compared with the distance z. the intensity distribution in the reconstructed By properly controlling the photographic process, image is given by
x exp {(ik/z)[a(xl
-x2) y (x1 -x2) - (~1 $1 -x2 <;)I) dxl dx2 d.61 d
where Pk(xl, ~7) is the mutual coherence function of the reconstructing constant.wh&h-depends on exposure time etc.
wave at the hologram.
(3)
A is a
2. TWO POINT RESOLUTION Assuming that the two points are placed at the distances transmittance in the object plane is given by o(51)
=6(51-b)
*b from the position
51 = 0. the amplitude
+6(51+k).
Eq. (2) then becomes k2 S(x) = $$ The resultant i(y)
=
I[
intensity distribution
rg(ba)exp(-~)+ro(-b,a)exp(+~)] .
in the reconstructed
image is given by
1roe, a) i 2ssr,&q, x2) exp (_Wz)(a - b - Y)(XI - X2))d-qdx2
+lro(-b,a) 12j’~r&q,x2) exp{W4(a+ + 2Re[ rO(b, a) l$(-b, a)] ssr&q Assuming
an exponential
Pk(xl,x2)
-X2)}dq dx2
b -y)kq
-x2) exp f(ik/z)[(a
mutual coherence
- b -y)xl
-
function for spatially
(a + b -Y) X2l)dxl dx2 stationary
process.
(4) we write
= exp[-l&l-41,
(5)
where
1, is the coherence parameter. Subsfituting eq. (5) into eq. (4) and evaluating the integrals, the normalised resultant tribution in the image plane for a hologram extending from -x0 to x0 is given by 10x0 sinhl~x~cos i(y) =
lgxi
+ Q(b, a) 1
0
Z~~~sinhZ~x~cos(k/z)~
case 10 = 0 this reduces +
rO(-b,a) Po(b,a)
dis-
(k/z)(y - a + b) x0 cash 10 x0 sin (k/z)(y - a + b) x0 ‘-
lixi
-a-b)xO
(k/z)Cy -a -b)XOcoshZ0XOsin(k/z)Oi
Z2x2 + (k2/z2)(y -a -b)2x2 0 0 0
k i(y) =[sincZ(y-a+b)xg In order
-a+ b)xO
+ (k2/z2)(y-a+b)2x2
l?O(-b,a)
In the limiting
(k/z)(y
intensity
’
+ (k2,‘z2)(y
(6)
-a+b)2xi
Z2x2 + 0 0 (k2/z2)(y-a
-a -b)xO
.
-b)2x2 0
to the well known form k sincZ(Y-a-b)xo12.
to evaluate the limit of resolution,
the Sparrow
criterion
has been employed,
2
which states
Volume
2. number
6
OPTICS
FiK. 1.
\‘:irl:rtlon of limit
November
COMMti~ICA’ITONS
of resolution
will-. I;, for
various
values
1970
of 01.
that the second derivative of Z’(J))vanishes at j’ = 0 and the calculations are made with the ICT 1909 digital computer. Fig. 1 illustrates the variation of the limit of resolution with Zb (=lOx9) for various values of (Y (= I?O(-h, ~)/rO(h, n)). It is seen that for the case of coherent recording and reconstruction, a limit of resolution of 4.16 is achieved. To make this study complete the effect of emulsion parameters should also be included.
REFERENCES [l] [2] [:
M. Lurle. J. Opt. Sot. Am. 56 (1966) 1369; 58 (1968) 614. M. Berlolotti, F. Gori and G .Guttari,
278
4 (1966)
37.
2, number
6
November 1970
OPTICS COMMUNICATIONS
TWO
POINT
RESOLUTION
COHERENT
FOURIER
IN
PARTIALLY
HOLOGRAPHY
R. S. SIROHI and C. S. VIKRAM Physics
Department, Indian Institute Hauz Khas, New Delhi-29, Received
3 September
of Technology, India
1970
Two point resolution in Fourier holography with partially coherent recording and illuminating waves has been theoretically investigated. The mutual coherence function of the illuminating wave at the hologram plane is assumed to be exponential.
1. INTRODUCTION The light sources for holography have been generally supposed to be completely coherent all over the illuminating beams used for recording a hologram and even for reconstructing an image from it. Since no real source is a point and even the sharpest line has a width, the complete coherence is thus a mathemathical idealisation. The effect of partial coherence on holography has thus been extensively studied by many investigators [l-11]. However, a most systematic study concerning the coherence effects on holography has been carried out by Reynolds and De Velis [6], Menzel et al. [8,9] and Murata et al. [ll]. Reynolds and De Velis [6] have treated the effect of partial coherence only in the illuminating process, while Murata et al. [ll] included the reconstruction process as well. Further in their paper Murata et al. have also investigated the effect of partial coherence on two point resolution in Fourier holography but their treatment is limited to the coherent reconstruction wave. We have extended their study by including the partially coherent wave; the mutual coherence function at the hologram plane is assumed to be exponential. The variation of the limit of resolution with the coherence parameter of the reconstructed wave has been studied for various values of coherence function of the recording wave. Following Murata et al. 1111 the intensity distribution in the hologram plane (X plane) located at a distance .z from the object plane (5 plane) is given by
(1) where I,(tl, 52) is the mutual spatial coherence function at the object plane and aGI/& derivative of the Green function satisfying the equation (V2 + k2)G1(~l,
The amplitude
= -6(x1 -
condition
transmittance
T(<) = 6(5 -a) where the reference 276
is the normal
across
the object plane is represented
by
+ D(5) , point is situated at a distance a from the object with transmittance
D(6).
OPTICS
Volume 2, number 6
November
COMMUNICATIONS
1970
In expression (l), the third and fourth terms produce the real and virtual images of the object, respectively. Restricting the attention to the third term and evaluating the normal derivatives of the Green function, this term becomes
(2) This expression is valid provided the object size is small compared with the distance z. the intensity distribution in the reconstructed By properly controlling the photographic process, image is given by
x exp {(ik/z)[a(xl
-x2) y (x1 -x2) - (~1 $1 -x2 <;)I) dxl dx2 d.61 d
where Pk(xl, ~7) is the mutual coherence function of the reconstructing constant.wh&h-depends on exposure time etc.
wave at the hologram.
(3)
A is a
2. TWO POINT RESOLUTION Assuming that the two points are placed at the distances transmittance in the object plane is given by o(51)
=6(51-b)
*b from the position
51 = 0. the amplitude
+6(51+k).
Eq. (2) then becomes k2 S(x) = $$ The resultant i(y)
=
I[
intensity distribution
rg(ba)exp(-~)+ro(-b,a)exp(+~)] .
in the reconstructed
image is given by
1roe, a) i 2ssr,&q, x2) exp (_Wz)(a - b - Y)(XI - X2))d-qdx2
+lro(-b,a) 12j’~r&q,x2) exp{W4(a+ + 2Re[ rO(b, a) l$(-b, a)] ssr&q Assuming
an exponential
Pk(xl,x2)
-X2)}dq dx2
b -y)kq
-x2) exp f(ik/z)[(a
mutual coherence
- b -y)xl
-
function for spatially
(a + b -Y) X2l)dxl dx2 stationary
process.
(4) we write
= exp[-l&l-41,
(5)
where
1, is the coherence parameter. Subsfituting eq. (5) into eq. (4) and evaluating the integrals, the normalised resultant tribution in the image plane for a hologram extending from -x0 to x0 is given by 10x0 sinhl~x~cos i(y) =
lgxi
+ Q(b, a) 1
0
Z~~~sinhZ~x~cos(k/z)~
case 10 = 0 this reduces +
rO(-b,a) Po(b,a)
dis-
(k/z)(y - a + b) x0 cash 10 x0 sin (k/z)(y - a + b) x0 ‘-
lixi
-a-b)xO
(k/z)Cy -a -b)XOcoshZ0XOsin(k/z)Oi
Z2x2 + (k2/z2)(y -a -b)2x2 0 0 0
k i(y) =[sincZ(y-a+b)xg In order
-a+ b)xO
+ (k2/z2)(y-a+b)2x2
l?O(-b,a)
In the limiting
(k/z)(y
intensity
’
+ (k2,‘z2)(y
(6)
-a+b)2xi
Z2x2 + 0 0 (k2/z2)(y-a
-a -b)xO
.
-b)2x2 0
to the well known form k sincZ(Y-a-b)xo12.
to evaluate the limit of resolution,
the Sparrow
criterion
has been employed,
2
which states
Volume
2. number
6
OPTICS
FiK. 1.
\‘:irl:rtlon of limit
November
COMMti~ICA’ITONS
of resolution
will-. I;, for
various
values
1970
of 01.
that the second derivative of Z’(J))vanishes at j’ = 0 and the calculations are made with the ICT 1909 digital computer. Fig. 1 illustrates the variation of the limit of resolution with Zb (=lOx9) for various values of (Y (= I?O(-h, ~)/rO(h, n)). It is seen that for the case of coherent recording and reconstruction, a limit of resolution of 4.16 is achieved. To make this study complete the effect of emulsion parameters should also be included.
REFERENCES [l] [2] [:
M. Lurle. J. Opt. Sot. Am. 56 (1966) 1369; 58 (1968) 614. M. Berlolotti, F. Gori and G .Guttari,
278
4 (1966)
37.