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A new methodology for realizing magnification of holographic images
A new methodology for realizing magnification of holographic images S. A N A N D A RAO, S.V. PAPPU The article describes a new method for obtaining a holographic image of desired magnification, consistent with the stipulated criteria for its resolution and aberrations.
Several methods are available for the achievement ot image magnification from a hologram 1"9 and of these the most widely used involves the use of light, during reconstruction, of a wavelength different from the wavelength of light employed s,6. The drawbacks of the various methods are well known. In view of the realization of the potential for the application of holographic techniques in mensuration and mapping, a statement has been made by Kurtz e t a P ° as follows: "A study should be made of the possibility of achieving image magnification by some means other than simple change of wavelength ratio." It goes without saying that one desires to produce with such means a magnified image which is free from aberrations. Of course it is impossible to get rid of aberrations completely and hence one should look for a magnified image with its aberrations within some stipulated tolerance limits. We have made such a study envisaged by Kurtz e t a l a n d the purpose of this paper is to demonstrate firstly that a modified lensless Fourier transform (MLLFT).hologram recording technique could be successfully exploited for obtaining holographic image magnification, without taking recourse to either the technique of employing a wavelength ratio or the methods of scaling the hologram, or using external optical systems and secondly, we hope to show that such a magnified image could have its aberrations within tolerance limits.
Theoretical background In order to set the problem in the proper theoretical perspective, recourse shall be taken to the widely used analysis of holographic image magnification by Meier n, who derived the following formula for lateral magnification of a holographic image: Mv(lat)
= m(1 + m 2 Z o / l a Z c
- Zo/Zr)-'
(1)
where rn = scaring factor for the hologram, tt = wavelength ratio between reconstructing and reference waves, and Zo, z , , z c = z components of the position coordinates of the object, reference and reconstruction sources. The origin of the coordinate system is located at the centre of the hologram which then lies in the X - Y plane. The authors are at the Department of Electrical Communication Engineering, the Indian Institute of Science, Bangalore-560 012, India. Received 8 August 1979.
Meier concluded that 'all aberrations can be made to disappear simultaneously for one image at a time by using parallel reference and illuminating beams, by giving the latter an off-axis position, proportional to that of the reference beam, and by scaling up the hologram according to wavelength ratio. The resulting lateral magnification for the real image is M = m =/a, but the required enlargement of the hologram is highly undesirable because it involves a conventional imaging set-up.' It is evident from this conclusion that keeping the curvatures of the reference and the reconstruction beams the same ensures the generation of an aberration-free image. Its magnification however, will only be unity in the context of our problem requirements viz., # = m = 1. In order to realise magnification under these requirements, it appears necessary to use a reconstruction beam of curvature different from that of the reference beam. In the past such a method has been used but it suffers from the drawback of yielding an aberrated image ~'4. Hence a different technique has been adopted, which is described in the following pages. For this purpose note that under the conditions of/.t = m = 1 and z c = oo eqn. (1), reduces to, Mv
(lat) = Zr/A
(2)
where A = z r - z o. Thus it is evident from eqn. (2) that the achievement of desired magnification for a holographic image implies the manipulation of the parameter A and this is the crux of the new method.
Aberrations of a holographic image It is noted that the expressions of Meier for various aberration coefficients are considerably simplified under the conditions of/a = m = 1, z c = 0% z r = finite and x r = Y r = 0 (See Table 1). The third requirement means that the reference source is located on the z axis. Under this situation the distortion coefficient becomes'zero, which is a desirable feature to have, especially in the context of mensuration and mapping. By multiplying the simplified expressions for aberration coefficients given in Table 1 with appropriate d n factor (where d is the semi-aperture of the hologram) and making use of the fact that z r - z o = A, the expressions for aberrations become
0030-3992/80/010035-04 $02.00 © 1980 IPC Business Press OPTICS AND LASER TECHNOLOGY. FEBRUARY 1980
35
Table 1.
aberrations in conventional optical systems may be used for our purpose here. These values are + 0.95X +- 0.6X and -+ 0.35X for spherical aberration, coma and astigmatism respectively) ~
Simplified MeWs aberration coefficients
Aberration
Expression u n d e r t h e requirement of/~ = m
Expression under the requirement of/~ = m
coefficient
= 1 and zc = ~°
= l z c = ~ and xr = 0
Spherical (S)
3 ZoZr
Coma (Cx)
(1/z °
_
1/zA
x-9-° [ 1/z~ - ( 1 / z o
Zo
3 (1/z ° ZoZr
x°
ZoZr
_
1/zr ) Recording and reconstruction geometries
( 2 / z o - 1/z r)
- 1/zr) 2 ]
_xr
zr
[ 1/z~
- (1/zo
- 1/zr) 2]
Astigmatism
x o2/ z o 3 -
(Xo/Zo - xr/z~)~ (1/z o
Distortion (Dx)
- 1/zr)
(3X~Xr) (z~zr)
(3x°X2r ) (ZoZ2r)
Since keeping the value of A as small as possible is an important step to achieve the desired magnification, it is clear that a recording geometry must be chosen wherein the object and the reference sources are located such that the value of z o differs from that of z r by as small a value as possible. Such a recording geometry could be realised by employing a modified lensless Fourier transform (MLLFT) configuration. The superior resolution capabilities of LLFT holograms are well known 14'is. As small values for A are to be realized, the actual recording set-up needs to be assembled with care so as to mix the object beam efficiently with the reference beam in the plane of the holographic plate. The detailed schematic of the actual set-up used is shown in Fig. 1. The reconstruction geometry involves the use of a plane wavefront illuminating the hologram at a suitable angle, which is dictated by the fact that X c = Y c va 0. Since no focusing restriction is involved in the case of an LLFT hologram is, the image of desired magnification could be obtained by locating the observation plane suitably from the hologram during reconstruction.
(Ax) -
Furthermore, it is necessary to answer the following questions: what should be the recording and reconstruction geometries? what should be the adjustments and/or tradeoffs between the three crucial parameters of magnification, aberrations and resolution limit?
0
M a g n i f i c a t i o n , aberrations and resolving l i m i t
-3Ad 4
(3)
Spherical aberration 8(ZoZ,.) 2 (zr +
Coma
A) d a Xo
2(ZoZ,) 2
-x~ d 2 Astigmatism
2zo~ z~
The equation for the resolving limit in the case of Fourier transform hologram is derived to be XZo
R -
(5)
Conclusions drawn from eqns. (3) to (5) are that: spherical aberration can be made zero by choosing z r = z o but this is an unrealistic condition in view of the requirement that the image magnification should be finite and greater than unity; coma and astigmatism could be eliminated by putting xo = 0 which means the generation of an in-line hologram and is lint of interest in this particular context. Hence the only alternative left is to choose Xo and d judiciously, so as to keep these aberrations within tolerance limits, having noted that the distortion is eliminated. The question then is: what values should be accepted as tolerance limits for the various aberrations of a holographic image?
(6)
4d
(4)
The resolution in such holograms is aperture limited and the film resolution has no role whatsoever to play 16. Quantitative expressions for the aberrations and the resolving limit in terms of the variables Xo and d are obtained for a typical set of values chosen for zr, z o a n d M and they are shown in Table-2. The particular choice of values for z r, z o a n d M has been made keeping in view certain applications of a magnified holographic image. For example, small values for zr, Zo and large values for M would be of interest in microscopy, whereas large values of zr, Zo and small values for M could be of interest in mensuration and mapping. As stated earlier,
w
>+
, >-
¢
Tolerance limits for aberrations of a holographic
image L
Champagne and Massey 12 chose arbitrarily OAk as the tolerance limit and found it to be a stringent requirement. Latta ~a chose the value of 1X without specifying any reason for selecting it. The existing situation is that there are no logically arrived-at values for the tolerance limits of aberrations for a holographic image. In view of this situation it is felt that the stipulated values for the tolerance limits of
36
Fig. 1 Schematic of the set-up used for obtaining a modified lensless Fourier transform hologram. LA = 0.5 mW HeNe laser; Sh = Shutter; BS t = Beam splitter; BS 2 = Beam splitter used as reflector and attenuator; PHi, PH 2 = Pin hole assembly; L = lens;0 = Object;W = Window for reference beam; H = Holographic plate (thick line representing emulsion side); M1, M~ = Plane mirror; CL = Collimating lens and A = Attenuator.
OPTICS AND
LASER TECHNOLOGY.
FEBRUARY
1980
Table 2 Quantitative expressions for aberrations and resolving limit for different values of zr,
Case No.
1
zr
Zo
A
(cm)
(cm)
(cm)
-1
-0.9
0.1
M
zi = M z o
(cm) IOX
9
S
from eqn. (3) (cm) +4.6x10 -2 04
C
from eqn. (4) (cm) -6.8x10 - t X0 d 3
2
-1
-0.99
0.01
-10
-9.0
1.0
A
and M
from eqn. (5)
(cm) + 6.2x10 -]
R
from eqn. (6)
(cm) 0.15x104/d
Xo2 d 2
100X
99
+3.8x10-3 d 4
-5.2x 10 -1 Xo d3
+ 5.1x10 -1 x2 d 2
0.16 x l 0 - 4 / d
1OX
90
+4.6x10 -s o4
-67.9x10 -s
+61.7x10 -s
1.43x10-4/d
Xo d3
3
zo
x~d 2
-10
-9.9
0.1
100X
990
+3.0x 10-604
-5.2x10 -4 x o d3
+ 5.1x10 -4 x ~d 2
1.57x10-4/d
-50
-45.0
5.0
10X
450
+3.7x10-7 04
-5.4x10 -6
+ 4.9x 10-6
7.12x10-4/d
Xo d3 -50
-49.5
0.5
100X
4950
+3.0x 10-8 d 4
Tolerance limits with +6.0x10 -s
-4.1x10 -6 x 0d 3 0 X = 6328 A _+3.8x10-s
X2od2 + 4.1x10 -6 x~ d 2
7.84x10-4/d
+2.2x 10-s
for any chosen set of values for zr, z o and M it is necessary to choose judiciously Xo and d in order to keep the aberrations within tolerance limits and to attain the required resolving limit. An important point to note here is that the values for A will be large in the cases where z,~ Zo are large and M is small. The validity of the use of LLFT type configuration may become questionable under such a situation. To overcome this problem, we propose that the system may be initially designed for a higher magnification. Then exploitation of the nonfocusing property of a LLFT hologram can catch the image of desired magnification by suitably locating the observation plane from the hologram. Thus the design considerations and the procedure for obtaining a holographic image of desired magnification under the various conditions stipulated can now be summarized. Choose the parameters zr, Zo so as to keep A small and to achieve the specific value for M. Choose the values for Xo and d so as to keep the aberrations within the tolerance limits and to achieve simultaneously the required resolving limit. Record the hologram using the above mentioned chosen parameters in the set-up of Fig. 1 and observe the reconstruction from such a hologram. Experimental results
The modified LLFT holograms have been prepared according to the design prescriptions mentioned above and making use of the set-up shown in Fig. 1. The source is a 0.5 mW Spectra Physics Model 155 He-Ne laser and Agfa 8E75 plates were used for recording the holograms. The following special precautions, in addition to the usual ones, were taken: long exposure and short development times were arranged so that the emulsion shrinkage might be negligible ]7 and the nonemulsion side was made to face the reconstructing parallel beam of light ~8.
OPTICS A N D LASER T E C H N O L O G Y . F E B R U A R Y 1980
Fig. 2
Image of an ant obtained at M = 4 0 X b y plane wave
illuminations of a M L L F T hologram constructed using the set-up of Fig. 1 and the analysis presented in the t e x t .
37
Table 3 Object and reference coordinates, values for magnification, aberrations and resolving limit, for the experimental cases Object coordinates
Reference coordinates
Spherical aberration
Coma
Astigmatism
(cm)
(cm)
(cm)
(cm)
(cm)
Xo
Yo
Zo
xr
Yr
zr
d
A
Numerical Resolving aperture limit = d/z o (micron)
M
(cm) (cm) 0.2
0.2
-9.9
0
0
-10
0.5 0.1
100X
+2.4x10 -s
-1.3x10 -s
+0.5x10 -s
0.05
3.2
1
1
-49.5
0
0
-50
1
0.5
100X
+3.0x10 -a
-0.4x10 -s
+0.4x10 -s
0.02
7.9
1
1
-49.5
0
0
-50
2
0.5
100X
+48.0x10 -s
-3.3x10 -s
+1.6x10 -s
0.04
3.9
Tolerance limits with
o X = 6328 A
+6.0x10 -s
+3.8x10 -s
As an example, a hologram of an ant, which was kept at a distance z o = -49.5 cm, was recorded using the MLLFT scheme. The values for xo as well as for d were chosen to keep the aberrations within tolerance limits and to achieve the required resolution. The values for the various parameters tried in our experiments are assembled in Table 3. The reconstructed image of the ant obtained at a magnification of 40X is shown in Fig. 2. Though the chosen value for M is 100X, only the image obtained at 40X is displayed because of the low power of the laser used.
S u m m a r y and conclusions The problem of holographic image magnification is examined in order to answer the question: is it possible at all to realize a magnification of desired value for the holographic image under the conditions of:/a = m = 1 ; z c = oo and zr = finite; abberations to be within tolerance limits, and required resolution? It is demonstrated by analysis backed up by experimental results that the answer to the above question is yes. Thus it is shown that techniques (eg modified lensless Fourier transform hologram recording) other than the use of wavelength ratio between the reconstruction and reference beams could be devised for the achievement of desired image magnification from a hologram.
of Education, Government of India, for pursuing his studies at the Institute leading to a Ph D degree.
References 1 2
3 4 5 6 7 8 9 10
11 12 13 14 15
Acknowledgement One of the authors, S.A. Rao acknowledges the support from the Quality Improvement Programme (QIP), Ministry
38
+2.2x10 -s
16 17 18
Collier, RJ., Butckhttdt, C.B. Lin. L.H. 'Optical Holography' (Academic Press, 1971) Stroke, G.W., Falconer, D.G., 'Theory and experimental foundations of wavefront reconstruction imaging' Symposium on optical and electrooptical information processing technology, Eds. J.T. Tippet, L.C. Clapp, D, Berkowitz., C.J. Koester, (MIT Press, Massachusetts 1964) Stroke, G.W., Falconer, D.G., Phys. Lett 13 (1964) 306 Leith, E.N., Upatnieks, J., JOSA 55 (1965) 569 El-Sum, H.M.A., 'Reconstructed wavefront microscopy' Ph D thesis (Stanford University 1952) El-Sum, H.M.A., Blez, A.V.,Phys Rev 99 (1955) 624 Vanligten, R.F., Ostetberg, H., Nature 211 (1966) 282 Ellis,G.W., Science 154 (1966) 1195 Van Listen, R.F., Opt Las Tech l (1969) 71 Kurtz, M.K., Ballmubrlmlanilm, N., Miklutil, E.M., Stevenson, W.H., 'Study of potential applications of holographic techniques to mapping' tech report US A r m y Engineer Topographic Laboratories Research Institute ET-CR-71-17 (October 1971) 178 Meier,R.W.,JOSA, 55 (1965) 989 Champagne, E.B., Massey, N.G.,Appl Opt 8 (1969) 1879 Latta, J.N., 'Computer based analysis of holography' Ph D thesis University Microfllmg (Michigan 1971) Stroke, G.W., an Introduction to coherent optics and holography (Academic Press, 1969) 120-27 De Veils, J.B., Reynolds, G.O., Theory and applications of holography (Addison-Wesley, Mass, 1967) 110 Di~-nond,F.I.JOSA 57 (1967) 503 Kellie,T.F. Stevenson, W.S., Opt Eng 12 (1973) 47 Welford, W.T., Aberrations of the Symmetrical Optical System (Academic Press, 1974) 207,198
OPTICS AND LASER T E C H N O L O G Y . FEBRUARY 1980
Several methods are available for the achievement ot image magnification from a hologram 1"9 and of these the most widely used involves the use of light, during reconstruction, of a wavelength different from the wavelength of light employed s,6. The drawbacks of the various methods are well known. In view of the realization of the potential for the application of holographic techniques in mensuration and mapping, a statement has been made by Kurtz e t a P ° as follows: "A study should be made of the possibility of achieving image magnification by some means other than simple change of wavelength ratio." It goes without saying that one desires to produce with such means a magnified image which is free from aberrations. Of course it is impossible to get rid of aberrations completely and hence one should look for a magnified image with its aberrations within some stipulated tolerance limits. We have made such a study envisaged by Kurtz e t a l a n d the purpose of this paper is to demonstrate firstly that a modified lensless Fourier transform (MLLFT).hologram recording technique could be successfully exploited for obtaining holographic image magnification, without taking recourse to either the technique of employing a wavelength ratio or the methods of scaling the hologram, or using external optical systems and secondly, we hope to show that such a magnified image could have its aberrations within tolerance limits.
Theoretical background In order to set the problem in the proper theoretical perspective, recourse shall be taken to the widely used analysis of holographic image magnification by Meier n, who derived the following formula for lateral magnification of a holographic image: Mv(lat)
= m(1 + m 2 Z o / l a Z c
- Zo/Zr)-'
(1)
where rn = scaring factor for the hologram, tt = wavelength ratio between reconstructing and reference waves, and Zo, z , , z c = z components of the position coordinates of the object, reference and reconstruction sources. The origin of the coordinate system is located at the centre of the hologram which then lies in the X - Y plane. The authors are at the Department of Electrical Communication Engineering, the Indian Institute of Science, Bangalore-560 012, India. Received 8 August 1979.
Meier concluded that 'all aberrations can be made to disappear simultaneously for one image at a time by using parallel reference and illuminating beams, by giving the latter an off-axis position, proportional to that of the reference beam, and by scaling up the hologram according to wavelength ratio. The resulting lateral magnification for the real image is M = m =/a, but the required enlargement of the hologram is highly undesirable because it involves a conventional imaging set-up.' It is evident from this conclusion that keeping the curvatures of the reference and the reconstruction beams the same ensures the generation of an aberration-free image. Its magnification however, will only be unity in the context of our problem requirements viz., # = m = 1. In order to realise magnification under these requirements, it appears necessary to use a reconstruction beam of curvature different from that of the reference beam. In the past such a method has been used but it suffers from the drawback of yielding an aberrated image ~'4. Hence a different technique has been adopted, which is described in the following pages. For this purpose note that under the conditions of/.t = m = 1 and z c = oo eqn. (1), reduces to, Mv
(lat) = Zr/A
(2)
where A = z r - z o. Thus it is evident from eqn. (2) that the achievement of desired magnification for a holographic image implies the manipulation of the parameter A and this is the crux of the new method.
Aberrations of a holographic image It is noted that the expressions of Meier for various aberration coefficients are considerably simplified under the conditions of/a = m = 1, z c = 0% z r = finite and x r = Y r = 0 (See Table 1). The third requirement means that the reference source is located on the z axis. Under this situation the distortion coefficient becomes'zero, which is a desirable feature to have, especially in the context of mensuration and mapping. By multiplying the simplified expressions for aberration coefficients given in Table 1 with appropriate d n factor (where d is the semi-aperture of the hologram) and making use of the fact that z r - z o = A, the expressions for aberrations become
0030-3992/80/010035-04 $02.00 © 1980 IPC Business Press OPTICS AND LASER TECHNOLOGY. FEBRUARY 1980
35
Table 1.
aberrations in conventional optical systems may be used for our purpose here. These values are + 0.95X +- 0.6X and -+ 0.35X for spherical aberration, coma and astigmatism respectively) ~
Simplified MeWs aberration coefficients
Aberration
Expression u n d e r t h e requirement of/~ = m
Expression under the requirement of/~ = m
coefficient
= 1 and zc = ~°
= l z c = ~ and xr = 0
Spherical (S)
3 ZoZr
Coma (Cx)
(1/z °
_
1/zA
x-9-° [ 1/z~ - ( 1 / z o
Zo
3 (1/z ° ZoZr
x°
ZoZr
_
1/zr ) Recording and reconstruction geometries
( 2 / z o - 1/z r)
- 1/zr) 2 ]
_xr
zr
[ 1/z~
- (1/zo
- 1/zr) 2]
Astigmatism
x o2/ z o 3 -
(Xo/Zo - xr/z~)~ (1/z o
Distortion (Dx)
- 1/zr)
(3X~Xr) (z~zr)
(3x°X2r ) (ZoZ2r)
Since keeping the value of A as small as possible is an important step to achieve the desired magnification, it is clear that a recording geometry must be chosen wherein the object and the reference sources are located such that the value of z o differs from that of z r by as small a value as possible. Such a recording geometry could be realised by employing a modified lensless Fourier transform (MLLFT) configuration. The superior resolution capabilities of LLFT holograms are well known 14'is. As small values for A are to be realized, the actual recording set-up needs to be assembled with care so as to mix the object beam efficiently with the reference beam in the plane of the holographic plate. The detailed schematic of the actual set-up used is shown in Fig. 1. The reconstruction geometry involves the use of a plane wavefront illuminating the hologram at a suitable angle, which is dictated by the fact that X c = Y c va 0. Since no focusing restriction is involved in the case of an LLFT hologram is, the image of desired magnification could be obtained by locating the observation plane suitably from the hologram during reconstruction.
(Ax) -
Furthermore, it is necessary to answer the following questions: what should be the recording and reconstruction geometries? what should be the adjustments and/or tradeoffs between the three crucial parameters of magnification, aberrations and resolution limit?
0
M a g n i f i c a t i o n , aberrations and resolving l i m i t
-3Ad 4
(3)
Spherical aberration 8(ZoZ,.) 2 (zr +
Coma
A) d a Xo
2(ZoZ,) 2
-x~ d 2 Astigmatism
2zo~ z~
The equation for the resolving limit in the case of Fourier transform hologram is derived to be XZo
R -
(5)
Conclusions drawn from eqns. (3) to (5) are that: spherical aberration can be made zero by choosing z r = z o but this is an unrealistic condition in view of the requirement that the image magnification should be finite and greater than unity; coma and astigmatism could be eliminated by putting xo = 0 which means the generation of an in-line hologram and is lint of interest in this particular context. Hence the only alternative left is to choose Xo and d judiciously, so as to keep these aberrations within tolerance limits, having noted that the distortion is eliminated. The question then is: what values should be accepted as tolerance limits for the various aberrations of a holographic image?
(6)
4d
(4)
The resolution in such holograms is aperture limited and the film resolution has no role whatsoever to play 16. Quantitative expressions for the aberrations and the resolving limit in terms of the variables Xo and d are obtained for a typical set of values chosen for zr, z o a n d M and they are shown in Table-2. The particular choice of values for z r, z o a n d M has been made keeping in view certain applications of a magnified holographic image. For example, small values for zr, Zo and large values for M would be of interest in microscopy, whereas large values of zr, Zo and small values for M could be of interest in mensuration and mapping. As stated earlier,
w
>+
, >-
¢
Tolerance limits for aberrations of a holographic
image L
Champagne and Massey 12 chose arbitrarily OAk as the tolerance limit and found it to be a stringent requirement. Latta ~a chose the value of 1X without specifying any reason for selecting it. The existing situation is that there are no logically arrived-at values for the tolerance limits of aberrations for a holographic image. In view of this situation it is felt that the stipulated values for the tolerance limits of
36
Fig. 1 Schematic of the set-up used for obtaining a modified lensless Fourier transform hologram. LA = 0.5 mW HeNe laser; Sh = Shutter; BS t = Beam splitter; BS 2 = Beam splitter used as reflector and attenuator; PHi, PH 2 = Pin hole assembly; L = lens;0 = Object;W = Window for reference beam; H = Holographic plate (thick line representing emulsion side); M1, M~ = Plane mirror; CL = Collimating lens and A = Attenuator.
OPTICS AND
LASER TECHNOLOGY.
FEBRUARY
1980
Table 2 Quantitative expressions for aberrations and resolving limit for different values of zr,
Case No.
1
zr
Zo
A
(cm)
(cm)
(cm)
-1
-0.9
0.1
M
zi = M z o
(cm) IOX
9
S
from eqn. (3) (cm) +4.6x10 -2 04
C
from eqn. (4) (cm) -6.8x10 - t X0 d 3
2
-1
-0.99
0.01
-10
-9.0
1.0
A
and M
from eqn. (5)
(cm) + 6.2x10 -]
R
from eqn. (6)
(cm) 0.15x104/d
Xo2 d 2
100X
99
+3.8x10-3 d 4
-5.2x 10 -1 Xo d3
+ 5.1x10 -1 x2 d 2
0.16 x l 0 - 4 / d
1OX
90
+4.6x10 -s o4
-67.9x10 -s
+61.7x10 -s
1.43x10-4/d
Xo d3
3
zo
x~d 2
-10
-9.9
0.1
100X
990
+3.0x 10-604
-5.2x10 -4 x o d3
+ 5.1x10 -4 x ~d 2
1.57x10-4/d
-50
-45.0
5.0
10X
450
+3.7x10-7 04
-5.4x10 -6
+ 4.9x 10-6
7.12x10-4/d
Xo d3 -50
-49.5
0.5
100X
4950
+3.0x 10-8 d 4
Tolerance limits with +6.0x10 -s
-4.1x10 -6 x 0d 3 0 X = 6328 A _+3.8x10-s
X2od2 + 4.1x10 -6 x~ d 2
7.84x10-4/d
+2.2x 10-s
for any chosen set of values for zr, z o and M it is necessary to choose judiciously Xo and d in order to keep the aberrations within tolerance limits and to attain the required resolving limit. An important point to note here is that the values for A will be large in the cases where z,~ Zo are large and M is small. The validity of the use of LLFT type configuration may become questionable under such a situation. To overcome this problem, we propose that the system may be initially designed for a higher magnification. Then exploitation of the nonfocusing property of a LLFT hologram can catch the image of desired magnification by suitably locating the observation plane from the hologram. Thus the design considerations and the procedure for obtaining a holographic image of desired magnification under the various conditions stipulated can now be summarized. Choose the parameters zr, Zo so as to keep A small and to achieve the specific value for M. Choose the values for Xo and d so as to keep the aberrations within the tolerance limits and to achieve simultaneously the required resolving limit. Record the hologram using the above mentioned chosen parameters in the set-up of Fig. 1 and observe the reconstruction from such a hologram. Experimental results
The modified LLFT holograms have been prepared according to the design prescriptions mentioned above and making use of the set-up shown in Fig. 1. The source is a 0.5 mW Spectra Physics Model 155 He-Ne laser and Agfa 8E75 plates were used for recording the holograms. The following special precautions, in addition to the usual ones, were taken: long exposure and short development times were arranged so that the emulsion shrinkage might be negligible ]7 and the nonemulsion side was made to face the reconstructing parallel beam of light ~8.
OPTICS A N D LASER T E C H N O L O G Y . F E B R U A R Y 1980
Fig. 2
Image of an ant obtained at M = 4 0 X b y plane wave
illuminations of a M L L F T hologram constructed using the set-up of Fig. 1 and the analysis presented in the t e x t .
37
Table 3 Object and reference coordinates, values for magnification, aberrations and resolving limit, for the experimental cases Object coordinates
Reference coordinates
Spherical aberration
Coma
Astigmatism
(cm)
(cm)
(cm)
(cm)
(cm)
Xo
Yo
Zo
xr
Yr
zr
d
A
Numerical Resolving aperture limit = d/z o (micron)
M
(cm) (cm) 0.2
0.2
-9.9
0
0
-10
0.5 0.1
100X
+2.4x10 -s
-1.3x10 -s
+0.5x10 -s
0.05
3.2
1
1
-49.5
0
0
-50
1
0.5
100X
+3.0x10 -a
-0.4x10 -s
+0.4x10 -s
0.02
7.9
1
1
-49.5
0
0
-50
2
0.5
100X
+48.0x10 -s
-3.3x10 -s
+1.6x10 -s
0.04
3.9
Tolerance limits with
o X = 6328 A
+6.0x10 -s
+3.8x10 -s
As an example, a hologram of an ant, which was kept at a distance z o = -49.5 cm, was recorded using the MLLFT scheme. The values for xo as well as for d were chosen to keep the aberrations within tolerance limits and to achieve the required resolution. The values for the various parameters tried in our experiments are assembled in Table 3. The reconstructed image of the ant obtained at a magnification of 40X is shown in Fig. 2. Though the chosen value for M is 100X, only the image obtained at 40X is displayed because of the low power of the laser used.
S u m m a r y and conclusions The problem of holographic image magnification is examined in order to answer the question: is it possible at all to realize a magnification of desired value for the holographic image under the conditions of:/a = m = 1 ; z c = oo and zr = finite; abberations to be within tolerance limits, and required resolution? It is demonstrated by analysis backed up by experimental results that the answer to the above question is yes. Thus it is shown that techniques (eg modified lensless Fourier transform hologram recording) other than the use of wavelength ratio between the reconstruction and reference beams could be devised for the achievement of desired image magnification from a hologram.
of Education, Government of India, for pursuing his studies at the Institute leading to a Ph D degree.
References 1 2
3 4 5 6 7 8 9 10
11 12 13 14 15
Acknowledgement One of the authors, S.A. Rao acknowledges the support from the Quality Improvement Programme (QIP), Ministry
38
+2.2x10 -s
16 17 18
Collier, RJ., Butckhttdt, C.B. Lin. L.H. 'Optical Holography' (Academic Press, 1971) Stroke, G.W., Falconer, D.G., 'Theory and experimental foundations of wavefront reconstruction imaging' Symposium on optical and electrooptical information processing technology, Eds. J.T. Tippet, L.C. Clapp, D, Berkowitz., C.J. Koester, (MIT Press, Massachusetts 1964) Stroke, G.W., Falconer, D.G., Phys. Lett 13 (1964) 306 Leith, E.N., Upatnieks, J., JOSA 55 (1965) 569 El-Sum, H.M.A., 'Reconstructed wavefront microscopy' Ph D thesis (Stanford University 1952) El-Sum, H.M.A., Blez, A.V.,Phys Rev 99 (1955) 624 Vanligten, R.F., Ostetberg, H., Nature 211 (1966) 282 Ellis,G.W., Science 154 (1966) 1195 Van Listen, R.F., Opt Las Tech l (1969) 71 Kurtz, M.K., Ballmubrlmlanilm, N., Miklutil, E.M., Stevenson, W.H., 'Study of potential applications of holographic techniques to mapping' tech report US A r m y Engineer Topographic Laboratories Research Institute ET-CR-71-17 (October 1971) 178 Meier,R.W.,JOSA, 55 (1965) 989 Champagne, E.B., Massey, N.G.,Appl Opt 8 (1969) 1879 Latta, J.N., 'Computer based analysis of holography' Ph D thesis University Microfllmg (Michigan 1971) Stroke, G.W., an Introduction to coherent optics and holography (Academic Press, 1969) 120-27 De Veils, J.B., Reynolds, G.O., Theory and applications of holography (Addison-Wesley, Mass, 1967) 110 Di~-nond,F.I.JOSA 57 (1967) 503 Kellie,T.F. Stevenson, W.S., Opt Eng 12 (1973) 47 Welford, W.T., Aberrations of the Symmetrical Optical System (Academic Press, 1974) 207,198
OPTICS AND LASER T E C H N O L O G Y . FEBRUARY 1980