Abstract

Digital and optical reconstruction techniques are applied to synthetic holograms that are recorded at suboptical frequencies. The first section of the paper considers digital reconstruction that entails the application of the inverse diffraction transform to the diffraction pattern of an object illuminated with a suboptical source. Different cross sections of the object are displayed on a CRT in a sequential fashion. Image enhancement techniques are also employed in the process of digital reconstruction. The second section outlines a method for partially alleviating the longitudinal distortion that is inherent in optical reconstruction from synthetic holograms because of the difference between the recording and reconstructing wavelengths. The paper considers both the diffuse and nondiffuse illumination schemes and discusses the relative advantages and disadvantages of digital and optical reconstruction schemes.

© 1974 Optical Society of America

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References

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  1. R. Meier, J. Opt. Soc. Am. 55, 987 (1965).
    [CrossRef]
  2. T. H. Demetrakopoulos, R. Mittra, Opt. Soc. Am.1973 Spring Meeting Program, p. 15 (1973).
  3. R. Mittra, P. L. Ransom, Proc. Symposium on Modern Optics (Polytechnic Institute of Brooklyn, New York, 1967).
  4. M. M. Sondhi, J. Acoust. Soc. Am. 46, 1158 (1969).
    [CrossRef]
  5. Y. Aoki, IEEE Trans. Audio Electroacoust. AE-8, 258 (1970).
    [CrossRef]
  6. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  7. L. H. Enloe, Bell System Tech. J. 46, 1479 (1967).
  8. J. W. Cooley, J. W. Tukey, Math. Computat. 19, 297 (1965).
    [CrossRef]
  9. G. Bergland, IEEE Spectrum 6, 41 (1969).
    [CrossRef]
  10. E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley, Reading, Mass., 1963).
  11. E. Lalor, J. Math. Phys. 9, 2001 (1968).
    [CrossRef]
  12. L. B. Lesem, P. M. Hirsch, J. A. Jordan, Proc. Symposium on Modern Optics (Polytechnic Institute of Brooklyn, New York, 1967).
  13. T. S. Huang, PProc. IEEE 59, 1335 (1972).
    [CrossRef]
  14. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970).

1972 (1)

T. S. Huang, PProc. IEEE 59, 1335 (1972).
[CrossRef]

1970 (1)

Y. Aoki, IEEE Trans. Audio Electroacoust. AE-8, 258 (1970).
[CrossRef]

1969 (2)

M. M. Sondhi, J. Acoust. Soc. Am. 46, 1158 (1969).
[CrossRef]

G. Bergland, IEEE Spectrum 6, 41 (1969).
[CrossRef]

1968 (1)

E. Lalor, J. Math. Phys. 9, 2001 (1968).
[CrossRef]

1967 (1)

L. H. Enloe, Bell System Tech. J. 46, 1479 (1967).

1965 (2)

J. W. Cooley, J. W. Tukey, Math. Computat. 19, 297 (1965).
[CrossRef]

R. Meier, J. Opt. Soc. Am. 55, 987 (1965).
[CrossRef]

Aoki, Y.

Y. Aoki, IEEE Trans. Audio Electroacoust. AE-8, 258 (1970).
[CrossRef]

Bergland, G.

G. Bergland, IEEE Spectrum 6, 41 (1969).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970).

Cooley, J. W.

J. W. Cooley, J. W. Tukey, Math. Computat. 19, 297 (1965).
[CrossRef]

Demetrakopoulos, T. H.

T. H. Demetrakopoulos, R. Mittra, Opt. Soc. Am.1973 Spring Meeting Program, p. 15 (1973).

Enloe, L. H.

L. H. Enloe, Bell System Tech. J. 46, 1479 (1967).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Hirsch, P. M.

L. B. Lesem, P. M. Hirsch, J. A. Jordan, Proc. Symposium on Modern Optics (Polytechnic Institute of Brooklyn, New York, 1967).

Huang, T. S.

T. S. Huang, PProc. IEEE 59, 1335 (1972).
[CrossRef]

Jordan, J. A.

L. B. Lesem, P. M. Hirsch, J. A. Jordan, Proc. Symposium on Modern Optics (Polytechnic Institute of Brooklyn, New York, 1967).

Lalor, E.

E. Lalor, J. Math. Phys. 9, 2001 (1968).
[CrossRef]

Lesem, L. B.

L. B. Lesem, P. M. Hirsch, J. A. Jordan, Proc. Symposium on Modern Optics (Polytechnic Institute of Brooklyn, New York, 1967).

Meier, R.

Mittra, R.

T. H. Demetrakopoulos, R. Mittra, Opt. Soc. Am.1973 Spring Meeting Program, p. 15 (1973).

R. Mittra, P. L. Ransom, Proc. Symposium on Modern Optics (Polytechnic Institute of Brooklyn, New York, 1967).

O’Neill, E. L.

E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley, Reading, Mass., 1963).

Ransom, P. L.

R. Mittra, P. L. Ransom, Proc. Symposium on Modern Optics (Polytechnic Institute of Brooklyn, New York, 1967).

Sondhi, M. M.

M. M. Sondhi, J. Acoust. Soc. Am. 46, 1158 (1969).
[CrossRef]

Tukey, J. W.

J. W. Cooley, J. W. Tukey, Math. Computat. 19, 297 (1965).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970).

Bell System Tech. J. (1)

L. H. Enloe, Bell System Tech. J. 46, 1479 (1967).

IEEE Spectrum (1)

G. Bergland, IEEE Spectrum 6, 41 (1969).
[CrossRef]

IEEE Trans. Audio Electroacoust. (1)

Y. Aoki, IEEE Trans. Audio Electroacoust. AE-8, 258 (1970).
[CrossRef]

J. Acoust. Soc. Am. (1)

M. M. Sondhi, J. Acoust. Soc. Am. 46, 1158 (1969).
[CrossRef]

J. Math. Phys. (1)

E. Lalor, J. Math. Phys. 9, 2001 (1968).
[CrossRef]

J. Opt. Soc. Am. (1)

Math. Computat. (1)

J. W. Cooley, J. W. Tukey, Math. Computat. 19, 297 (1965).
[CrossRef]

PProc. IEEE (1)

T. S. Huang, PProc. IEEE 59, 1335 (1972).
[CrossRef]

Other (6)

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970).

L. B. Lesem, P. M. Hirsch, J. A. Jordan, Proc. Symposium on Modern Optics (Polytechnic Institute of Brooklyn, New York, 1967).

E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley, Reading, Mass., 1963).

T. H. Demetrakopoulos, R. Mittra, Opt. Soc. Am.1973 Spring Meeting Program, p. 15 (1973).

R. Mittra, P. L. Ransom, Proc. Symposium on Modern Optics (Polytechnic Institute of Brooklyn, New York, 1967).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

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Figures (8)

Fig. 1
Fig. 1

Geometrical arrangement for recording of diffraction pattern.

Fig. 2
Fig. 2

Recording of hologram.

Fig. 3
Fig. 3

Digitally reconstructed cross sections of object shown in Fig. 2. Nondiffuse illumination was used.

Fig. 4
Fig. 4

Digitally reconstructed cross sections of object shown in Fig. 2. Diffuse illumination was used.

Fig. 5
Fig. 5

Two-level quantization in the intensity of Fig. 3 (a), (b). Intensity of points I = 250 if I ≥ 150. (Scale 0 → 255.)

Fig. 6
Fig. 6

Two-level quantization in the intensity of Fig. 4, (a) (b). Intensity of points I = 250 if I ≥ 100. (Scale 0 → 255.)

Fig. 7
Fig. 7

Diagram of experimental setup for optical reconstruction.

Fig. 8
Fig. 8

Optically reconstructed images. In (a) the letter U is out of focus, in (b) the letter I is out of focus.

Equations (17)

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f 1 ( x 1 , y 1 , z 1 ) S o f o ( x o , y o , z o ) g ( x 1 - x o , y 1 - y o , z 1 - z o ) d S o ,
g ( x 1 - x o , y 1 - y o , z 1 - z o ) exp ( i k r 1 - r o ) i λ ( z 1 - z o ) ,
f o ( x o , y o , z o ) = n = 1 N δ ( z o - z o n ) f o n ( x o n , y o n ) ,
f o ( x o , y o , z o ) = n = 0 N δ ( z o - z o n ) i = 1 I j = 1 J f o n ( x o , y o ) δ ( x o - i Δ x ) δ ( y o - j Δ y ) = n = 0 N δ ( z o - z o n ) i = 1 I j = 1 J f o n ( i Δ x , j Δ y ) ,
f o ( x o , y o , z o ) = n = 0 N δ ( z o - z o n ) i = 1 I j = 1 J f o n ( i Δ x , j Δ y ) exp ( i 2 π R i j ) ,
f 1 ( x 1 , y 1 , z 1 ) n = 0 N 1 i λ ( z 1 - z o n ) S o n f o n ( x o n , y o n ) exp ( i k r 1 - r o n ) d S .
f 1 = f o * g ,
f 1 = n = 0 N F - 1 [ F o n G o n ] ,
f o n ( r o n ) S 1 f 1 ( r 1 ) g ˜ ( r o n , r 1 ) d S 1 ,
g ˜ ( r o n , r 1 ) = exp ( - i k r 1 - r o n ) - i λ r 1 - r o n .
f o n ( x o n , y o n , z o n ) f 1 ( r 1 ) exp ( - i k r - r o n ) - i λ z o - z o n d S 1 .
f o n ( r o n ) F - 1 { F [ f 1 ( r 1 ) ] F [ exp ( - i k r 1 - r o n ) i λ ( z 1 - z o n ) ] } .
U r = S exp ( - i k sin θ y 1 ) ,
I ( x 1 , y 1 , z 1 ) = f 1 ( x 1 , y 1 , z 1 ) + S exp ( - i k sin θ y 1
h s ( x , y ) = n = - ( N / 2 ) ( N / 2 ) - 1 m = - ( M / 2 ) ( M / 2 ) - 1 h ( n d , m d ) δ ( x - n d ) δ ( y - m d ) ,
H s ( f x , f y ) = n = - ( N / 2 ) ( N / 2 ) - 1 m = - ( M / 2 ) ( M / 2 ) - 1 H [ f x - ( n / d ) , f y - ( m / d ) ] .
Δ z = ± 2 ( z i 2 / D 1 ) λ 2 ,

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