Abstract

The intensity distribution in the neighborhood of the real image reconstructed from an in-line Fraunhofer hologram was investigated. Assuming a finite amount of information recorded on the hologram in terms of a limiting aperture, the image is shown to possess intensity variations that are dependent on the argument of a first-order Bessel function recorded on the hologram. The theoretical results are presented on isophote diagrams and are verified experimentally. Three measurements of the image width are presented, and the inherent errors are found. The focal tolerance of the aperture-limited image relative to the recording parameters is discussed.

© 1972 Optical Society of America

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References

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  1. J. D. Trolinger, R. A. Belz, W. M. Farmer, Appl. Opt. 8, 957 (1969).
    [CrossRef] [PubMed]
  2. R. Menzel, F. M. Shofner, Appl. Opt. 9, 2073 (1968).
    [CrossRef]
  3. W. M. Farmer, K. S. Burgess, J. D. Trolinger, AEDC-TR-70-181 (1970).
  4. J. D. Trolinger, J. E. O’Hare, AEDC-TR-70-44 (1970).
  5. J. L. Harris, J. Opt. Soc. Am. 54, 931 (1964).
    [CrossRef]
  6. R. Meier, J. Opt. Soc. Am. 55, 1693 (1965).
    [CrossRef]
  7. R. F. VanLigten, J. Opt. Soc. Am. 56, 1 (1966).
    [CrossRef]
  8. G. B. Parrent, G. O. Reynolds, SPIE J. 3, 219 (1965).
  9. A. Kozma, J. S. Zelenka, J. Opt. Soc. Am. 60, 34 (1970).
    [CrossRef]
  10. W. H. Carter, A. A. Dougal, J. Opt. Soc. Am. 56, 1754 (1966).
    [CrossRef]
  11. D. J. Stigliani, R. Mittra, R. G. Semonin, J. Opt. Soc. Am. 60, 1059 (1970).
    [CrossRef]
  12. B. J. Thompson, SPIE J. 2, 43 (1963).
  13. R. A. Belz, “An Investigation of the Real Image Reconstructed by an In-Line Fraunhofer Hologram Aperture—Limited by Film Effects” (Ph.D. Dissertation, University of Tennessee, August, 1971, available from University Microfilms).
  14. B. Carnahan, H. Luther, J. Wilkes, Applied Numerical Methods (Wiley, New York, 1969).
  15. J. W. Goodman, J. Opt. Soc. Am. 57, 493 (1967).
    [CrossRef] [PubMed]
  16. M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1965).
  17. A short depth of field of the smaller-particle images is always observed in the reconstruction irrespective of the individual hologram apertures. The intensity change of these images is much more rapid as they are moved out of focus than it is for larger particles in the same plane.
  18. R. A. Belz, Proceedings of the EOSD Conference 1971 East (1971), p. 288.

1971 (1)

R. A. Belz, Proceedings of the EOSD Conference 1971 East (1971), p. 288.

1970 (2)

1969 (1)

1968 (1)

1967 (1)

1966 (2)

1965 (2)

G. B. Parrent, G. O. Reynolds, SPIE J. 3, 219 (1965).

R. Meier, J. Opt. Soc. Am. 55, 1693 (1965).
[CrossRef]

1964 (1)

1963 (1)

B. J. Thompson, SPIE J. 2, 43 (1963).

Belz, R. A.

R. A. Belz, Proceedings of the EOSD Conference 1971 East (1971), p. 288.

J. D. Trolinger, R. A. Belz, W. M. Farmer, Appl. Opt. 8, 957 (1969).
[CrossRef] [PubMed]

R. A. Belz, “An Investigation of the Real Image Reconstructed by an In-Line Fraunhofer Hologram Aperture—Limited by Film Effects” (Ph.D. Dissertation, University of Tennessee, August, 1971, available from University Microfilms).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1965).

Burgess, K. S.

W. M. Farmer, K. S. Burgess, J. D. Trolinger, AEDC-TR-70-181 (1970).

Carnahan, B.

B. Carnahan, H. Luther, J. Wilkes, Applied Numerical Methods (Wiley, New York, 1969).

Carter, W. H.

Dougal, A. A.

Farmer, W. M.

J. D. Trolinger, R. A. Belz, W. M. Farmer, Appl. Opt. 8, 957 (1969).
[CrossRef] [PubMed]

W. M. Farmer, K. S. Burgess, J. D. Trolinger, AEDC-TR-70-181 (1970).

Goodman, J. W.

Harris, J. L.

Kozma, A.

Luther, H.

B. Carnahan, H. Luther, J. Wilkes, Applied Numerical Methods (Wiley, New York, 1969).

Meier, R.

Menzel, R.

Mittra, R.

O’Hare, J. E.

J. D. Trolinger, J. E. O’Hare, AEDC-TR-70-44 (1970).

Parrent, G. B.

G. B. Parrent, G. O. Reynolds, SPIE J. 3, 219 (1965).

Reynolds, G. O.

G. B. Parrent, G. O. Reynolds, SPIE J. 3, 219 (1965).

Semonin, R. G.

Shofner, F. M.

Stigliani, D. J.

Thompson, B. J.

B. J. Thompson, SPIE J. 2, 43 (1963).

Trolinger, J. D.

J. D. Trolinger, R. A. Belz, W. M. Farmer, Appl. Opt. 8, 957 (1969).
[CrossRef] [PubMed]

J. D. Trolinger, J. E. O’Hare, AEDC-TR-70-44 (1970).

W. M. Farmer, K. S. Burgess, J. D. Trolinger, AEDC-TR-70-181 (1970).

VanLigten, R. F.

Wilkes, J.

B. Carnahan, H. Luther, J. Wilkes, Applied Numerical Methods (Wiley, New York, 1969).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1965).

Zelenka, J. S.

Appl. Opt. (2)

J. Opt. Soc. Am. (7)

Proceedings of the EOSD Conference 1971 East (1)

R. A. Belz, Proceedings of the EOSD Conference 1971 East (1971), p. 288.

SPIE J. (2)

G. B. Parrent, G. O. Reynolds, SPIE J. 3, 219 (1965).

B. J. Thompson, SPIE J. 2, 43 (1963).

Other (6)

R. A. Belz, “An Investigation of the Real Image Reconstructed by an In-Line Fraunhofer Hologram Aperture—Limited by Film Effects” (Ph.D. Dissertation, University of Tennessee, August, 1971, available from University Microfilms).

B. Carnahan, H. Luther, J. Wilkes, Applied Numerical Methods (Wiley, New York, 1969).

M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1965).

A short depth of field of the smaller-particle images is always observed in the reconstruction irrespective of the individual hologram apertures. The intensity change of these images is much more rapid as they are moved out of focus than it is for larger particles in the same plane.

W. M. Farmer, K. S. Burgess, J. D. Trolinger, AEDC-TR-70-181 (1970).

J. D. Trolinger, J. E. O’Hare, AEDC-TR-70-44 (1970).

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Figures (7)

Fig. 1
Fig. 1

Plane wave recording and reconstruction geometry for an in-line hologram.

Fig. 2
Fig. 2

Intensity variations in the neighborhood of the real image for Ω = 3.832.

Fig. 3
Fig. 3

Intensity variations in the neighborhood of the real image for Ω = 7.016.

Fig. 4
Fig. 4

Intensity variations in the neighborhood of the real image for Ω = 13.324.

Fig. 5
Fig. 5

Experimental geometry for recording and reconstructing the in-line hologram.

Fig. 6
Fig. 6

Reconstructed images and corresponding intensity distributions from two films where z1 = 91 cm and a = 130 μm.

Fig. 7
Fig. 7

Measurements of the particle image diameter. In-focus image resulting when Ω = 19.616.

Tables (2)

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Table I Theoretical Size Measurement Errors

Tables Icon

Table II Values of Kh for Calculation of Focal Tolerance

Equations (24)

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I = 1 + ( a 2 / r 2 ) J 1 2 ( k a r / z 1 ) - 2 ( a / r ) J 1 ( k a r / z 1 ) sin ( k r 2 / 2 z 1 ) ,
I R = ( a β T λ z 2 ) 2 { [ 0 H J 1 ( k a r z 1 ) J 0 ( k w r z 2 ) cos [ k 2 ( 1 z 1 - 1 z 2 ) r 2 ] d r ] 2 + [ 0 H J 1 ( k a r z 1 ) J 0 ( k w r z 2 ) sin [ k 2 ( 1 z 1 - 1 z 2 ) r 2 ] d r ] 2 } ,
r = H ρ
I R norm = Ω 2 [ 1 - J 0 ( Ω ) ] 2 { [ 0 1 J 1 ( Ω ρ ) J 0 ( V ρ ) cos ( U 2 ρ 2 ) d ρ ] 2 + [ 0 1 J 1 ( Ω ρ ) J 0 ( V ρ ) sin ( U 2 ρ 2 ) d p ] 2 } ,
Ω = k a H / z 1 ,
U = ( k H 2 / 2 ) [ ( 1 / z 1 ) - ( 1 / z 2 ) ] ,
V = k w H / z 2 .
z 2 = z 1 + η .
U = ( λ / 2 π ) ( Ω 2 / a 2 ) [ z 1 η / ( z 1 + η ) ] ( λ / 2 π ) ( Ω 2 / a 2 ) η ,
V = ( Ω / a ) w [ z 1 / ( z 1 + η ) ] ( Ω / a ) w .
η = [ 2 π a 2 z 1 U / ( z 1 λ Ω 2 - 2 π a 2 U ) ] ( 2 π / λ ) ( a 2 / Ω 2 ) U ,
w = [ z 1 λ Ω a V / ( z 1 λ Ω 2 - 2 π a 2 U ) ] ( a / Ω ) V ,
V = [ Ω - ( π / 2 N Ω ) U ] ( w / a ) .
Percent error = Δ w / a = ( Δ V / Ω ) × 100 % ,
I R norm ( w = a ) = 1 4 [ 1 + J 0 ( Ω ) ] 2 .
Δ η = ± K h ( a 2 / λ ) ,
K h = ( 2 π 2 / Ω 2 ) ( Δ U / π ) .
I norm = [ 0 1 J 1 ( Ω ρ ) J 0 ( V ρ ) d ρ ] 2 / [ 1 - J 0 ( Ω ) 2 ] .
d I norm / d V = 0.
[ 0 1 J 1 ( Ω ρ ) J 0 ( V ρ ) d p ] [ 0 1 ρ J 1 ( Ω ρ ) J 1 ( V ρ ) d p ] = 0.
I = 0 1 ρ J 1 ( Ω ρ ) J 1 ( V ρ ) d p = 0 ,
I = 1 2 [ J 2 ( Ω ) ] 2
0 1 ρ J 1 ( Ω ρ ) J 1 ( V ρ ) d ρ = [ V J 1 ( Ω ) J 0 ( V ) - Ω J 0 ( Ω ) J 1 ( V ) ] / ( Ω 2 - V 2 )
V J 0 ( V ) / J 1 ( V ) = Ω J 0 ( Ω ) / J 1 ( Ω ) .

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