Abstract

In all methods of photographic optical processing presented here, the carrier frequency is derived from an intensity speckle pattern, the advantage of which is to spread out information in the Fourier plane. Furthermore, they are based on the fact that the Fourier spectrum of two laterally translated speckle patterns displays Young’s fringes, the visibility of which represents the correlation of the speckle patterns. The problems of comparing intensities of two transparencies, multiplexing optical signals, and detecting small translations of a diffuse object will be successively considered. The optical process of comparing two transparencies A and B is the following. The two signals are modulated by the same speckle pattern and recorded successively on a photographic plate which is laterally translated between the exposures. After processing, H displays Young’s fringes at infinity. If the intensity distributions of A and B are not identical, the two corresponding speckle images are not completely correlated and therefore the fringe visibility is not maximum. The difference AB can easily be obtained by filtering the minima of the fringes. For multiplexing operations, each of the signals to be stored is modulated by a speckle pattern and recorded at least twice on a photographic plate which is laterally translated between successive exposures. The amount of translation given to the plate is different for each of the signals. The spectrum of the photographic record has as many fringe systems as the signals. By filtering the maxima of a particular fringe system, the corresponding signal is reconstructed and the others are removed. Small translations of a diffuse object can be detected by illuminating the object with a speckle pattern and recording its image before and after the motion in the same way as above. The decorrelation of the two recorded speckle patterns is only due to the motion of the object which can thus be revealed by the visibility of the corresponding Young’s fringes. With such a method it is not possible to detect the direction of the translation suffered by the object. We suggest therefore an interferometer consisting of a Michelson interferometer in which both mirrors are replaced by two scattering surfaces O1 and O2 shifted longitudinally. The interferometer is illuminated by a parallel beam of laser light, and a photographic plate twice records (before and after the translation of one of the surfaces) irradiances lying in the Fourier plane of O1 and O2. After processing, H exhibits Moiré fringes. These fringes are rectilinear if the translation suffered by the object is lateral and they are circular if the translation is longitudinal. The sensitivity of the interferometer depends only on the geometrical characteristics of the recording setup.

© 1976 Optical Society of America

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References

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  1. L. I. Goldfisher, J. Opt. Soc. Am. 55, 247 (1965).
    [Crossref]
  2. A. W. Lohmann, J. Opt. Soc. Am. 55, 1030 (1965).
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    [Crossref]
  5. S. Debrus, M. Françon, and C. P. Grover, Opt. Commun. 6, 15 (1972).
    [Crossref]
  6. S. Debrus, M. Françon, and P. Koulev, Nouv. Rev. Opt. 5, 153 (1974).
    [Crossref]
  7. M. Françon, P. Koulev, and M. May, Opt. Commun. (in press).
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    [Crossref]
  9. U. Köpf, IEEE 862, 36 (1974).
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    [Crossref]
  11. M. Françon, P. Koulev, and M. May, Opt. Commun. 12, 63 (1974).
    [Crossref]
  12. B. Eliasson and F. M. Mottier, J. Opt. Soc. Am. 61, 559 (1971).
    [Crossref]
  13. E. Archbold and A. E. Ennos, Opt. Acta 19, 253 (1972).
    [Crossref]
  14. J. A. Mendez and M. L. Roblin, Opt. Commun. 11, 245 (1974).
    [Crossref]
  15. M. Françon, P. Koulev, and M. May, Opt. Commun. 13, 138 (1975).
    [Crossref]
  16. J. A. Leendertz, J. Phys. E 3, 214 (1970).
    [Crossref]
  17. Y. Dzialowski and M. May, Opt. Commun. 16, 334 (1976).
    [Crossref]

1976 (1)

Y. Dzialowski and M. May, Opt. Commun. 16, 334 (1976).
[Crossref]

1975 (1)

M. Françon, P. Koulev, and M. May, Opt. Commun. 13, 138 (1975).
[Crossref]

1974 (4)

J. A. Mendez and M. L. Roblin, Opt. Commun. 11, 245 (1974).
[Crossref]

M. Françon, P. Koulev, and M. May, Opt. Commun. 12, 63 (1974).
[Crossref]

S. Debrus, M. Françon, and P. Koulev, Nouv. Rev. Opt. 5, 153 (1974).
[Crossref]

U. Köpf, IEEE 862, 36 (1974).

1972 (3)

C. P. Grover, J. Opt. Soc. Am. 62, 1071 (1972).
[Crossref]

S. Debrus, M. Françon, and C. P. Grover, Opt. Commun. 6, 15 (1972).
[Crossref]

E. Archbold and A. E. Ennos, Opt. Acta 19, 253 (1972).
[Crossref]

1971 (3)

B. Eliasson and F. M. Mottier, J. Opt. Soc. Am. 61, 559 (1971).
[Crossref]

J. N. Butters and J. A. Leendertz, J. Phys. E 4, 277 (1971).
[Crossref]

S. Debrus, M. Françon, and C. P. Grover, Opt. Commun. 4, 172 (1971).
[Crossref]

1970 (1)

J. A. Leendertz, J. Phys. E 3, 214 (1970).
[Crossref]

1968 (1)

J. M. Burch and J. M. J. Tokarski, Opt. Acta 15, 101 (1968).

1965 (2)

Archbold, E.

E. Archbold and A. E. Ennos, Opt. Acta 19, 253 (1972).
[Crossref]

Burch, J. M.

J. M. Burch and J. M. J. Tokarski, Opt. Acta 15, 101 (1968).

Butters, J. N.

J. N. Butters and J. A. Leendertz, J. Phys. E 4, 277 (1971).
[Crossref]

Debrus, S.

S. Debrus, M. Françon, and P. Koulev, Nouv. Rev. Opt. 5, 153 (1974).
[Crossref]

S. Debrus, M. Françon, and C. P. Grover, Opt. Commun. 6, 15 (1972).
[Crossref]

S. Debrus, M. Françon, and C. P. Grover, Opt. Commun. 4, 172 (1971).
[Crossref]

Dzialowski, Y.

Y. Dzialowski and M. May, Opt. Commun. 16, 334 (1976).
[Crossref]

Eliasson, B.

Ennos, A. E.

E. Archbold and A. E. Ennos, Opt. Acta 19, 253 (1972).
[Crossref]

Françon, M.

M. Françon, P. Koulev, and M. May, Opt. Commun. 13, 138 (1975).
[Crossref]

M. Françon, P. Koulev, and M. May, Opt. Commun. 12, 63 (1974).
[Crossref]

S. Debrus, M. Françon, and P. Koulev, Nouv. Rev. Opt. 5, 153 (1974).
[Crossref]

S. Debrus, M. Françon, and C. P. Grover, Opt. Commun. 6, 15 (1972).
[Crossref]

S. Debrus, M. Françon, and C. P. Grover, Opt. Commun. 4, 172 (1971).
[Crossref]

M. Françon, P. Koulev, and M. May, Opt. Commun. (in press).

Goldfisher, L. I.

Grover, C. P.

C. P. Grover, J. Opt. Soc. Am. 62, 1071 (1972).
[Crossref]

S. Debrus, M. Françon, and C. P. Grover, Opt. Commun. 6, 15 (1972).
[Crossref]

S. Debrus, M. Françon, and C. P. Grover, Opt. Commun. 4, 172 (1971).
[Crossref]

Köpf, U.

U. Köpf, IEEE 862, 36 (1974).

Koulev, P.

M. Françon, P. Koulev, and M. May, Opt. Commun. 13, 138 (1975).
[Crossref]

M. Françon, P. Koulev, and M. May, Opt. Commun. 12, 63 (1974).
[Crossref]

S. Debrus, M. Françon, and P. Koulev, Nouv. Rev. Opt. 5, 153 (1974).
[Crossref]

M. Françon, P. Koulev, and M. May, Opt. Commun. (in press).

Leendertz, J. A.

J. N. Butters and J. A. Leendertz, J. Phys. E 4, 277 (1971).
[Crossref]

J. A. Leendertz, J. Phys. E 3, 214 (1970).
[Crossref]

Lohmann, A. W.

May, M.

Y. Dzialowski and M. May, Opt. Commun. 16, 334 (1976).
[Crossref]

M. Françon, P. Koulev, and M. May, Opt. Commun. 13, 138 (1975).
[Crossref]

M. Françon, P. Koulev, and M. May, Opt. Commun. 12, 63 (1974).
[Crossref]

M. Françon, P. Koulev, and M. May, Opt. Commun. (in press).

Mendez, J. A.

J. A. Mendez and M. L. Roblin, Opt. Commun. 11, 245 (1974).
[Crossref]

Mottier, F. M.

Roblin, M. L.

J. A. Mendez and M. L. Roblin, Opt. Commun. 11, 245 (1974).
[Crossref]

Tokarski, J. M. J.

J. M. Burch and J. M. J. Tokarski, Opt. Acta 15, 101 (1968).

IEEE (1)

U. Köpf, IEEE 862, 36 (1974).

J. Opt. Soc. Am. (4)

J. Phys. E (2)

J. A. Leendertz, J. Phys. E 3, 214 (1970).
[Crossref]

J. N. Butters and J. A. Leendertz, J. Phys. E 4, 277 (1971).
[Crossref]

Nouv. Rev. Opt. (1)

S. Debrus, M. Françon, and P. Koulev, Nouv. Rev. Opt. 5, 153 (1974).
[Crossref]

Opt. Acta (2)

J. M. Burch and J. M. J. Tokarski, Opt. Acta 15, 101 (1968).

E. Archbold and A. E. Ennos, Opt. Acta 19, 253 (1972).
[Crossref]

Opt. Commun. (6)

J. A. Mendez and M. L. Roblin, Opt. Commun. 11, 245 (1974).
[Crossref]

M. Françon, P. Koulev, and M. May, Opt. Commun. 13, 138 (1975).
[Crossref]

Y. Dzialowski and M. May, Opt. Commun. 16, 334 (1976).
[Crossref]

S. Debrus, M. Françon, and C. P. Grover, Opt. Commun. 4, 172 (1971).
[Crossref]

S. Debrus, M. Françon, and C. P. Grover, Opt. Commun. 6, 15 (1972).
[Crossref]

M. Françon, P. Koulev, and M. May, Opt. Commun. 12, 63 (1974).
[Crossref]

Other (1)

M. Françon, P. Koulev, and M. May, Opt. Commun. (in press).

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Figures (11)

FIG. 1
FIG. 1

Photographic record of the signals A and B to be compared.

FIG. 2
FIG. 2

Speckle structure of the doubly exposed photographic plate H.

FIG. 3
FIG. 3

Reconstruction of amplitude AB after filtering in Fourier plane.

FIG. 4
FIG. 4

Data storage with multiple exposures.

FIG. 5
FIG. 5

Reconstruction of one of the signals after filtering in Fourier plane.

FIG. 6
FIG. 6

Detection of lateral displacements smaller than the speckle size in image plane.

FIG. 7
FIG. 7

Detection of very small axial displacements by illuminating the object under oblique incidence with a speckle pattern.

FIG. 8
FIG. 8

Speckle interferometer: the mirrors of the Michelson interferometer are replaced by two scattering surfaces O1 and O2.

FIG. 9
FIG. 9

Amplitude Fourier spectrum of the recorded irradiance of plane P, the three orders are in line.

FIG. 10
FIG. 10

O2 is laterally shifted through x0 with respect to O1 in order to angularly separate the three orders diffracted by H.

FIG. 11
FIG. 11

Amplitude distribution reconstructed by H recorded with the configuration of Fig. 10.

Tables (1)

Tables Icon

TABLE I Scheme of exposures and translations in the three-exposure technique.

Equations (28)

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I ( ζ , η ) = A ( ζ , η ) D ( ζ , η ) + [ B ( ζ , η ) D ( ζ , η ) ] δ ( ζ - ζ 0 ) ,
t ( u , v ) = Ã ( u , v ) D ˜ ( u , v ) + [ B ˜ ( u , v ) D ˜ ( u , v ) ] exp j ( 2 π / λ ) u ζ 0 ,
t ( u , v ) = [ Ã ( u , v ) - B ˜ ( u , v ) ] D ˜ ( u , v ) + [ B ˜ ( u , v ) D ˜ ( u , v ) ] [ 1 + exp j ( 2 π / λ ) u ζ 0 ] .
U ( ζ , η ) = [ A ( ζ , η ) - B ( ζ , η ) ] D ( ζ , η ) F ˜ ( ζ , η ) ,
I ( ζ , η ) = G ( ζ , η ) O ( ζ , η ) + [ G ( ζ , η ) O ( ζ - x 0 , η ) ] δ ( ζ - ζ 0 ) ,
I ˜ ( u , v ) = G ˜ ( u , v ) Õ ( u , v ) + exp j ( 2 π / λ ) u ζ 0 × [ G ˜ ( u , v ) exp j ( 2 π / λ ) u x 0 Õ ( u , v ) ] .
I 1 ( ζ , η ) = F ( ζ , η ) O ( ζ , η ) D ˜ ( ζ , η ) 2 ,
I 2 ( ζ , η ) = F ( ζ - tan θ ) O ( ζ , η ) D ˜ ( ζ , η ) 2 δ ( ζ - ζ 0 ) .
M = 2 × 1.22 λ Ω cos θ tan θ = 2.44 λ Ω sin θ .
U ( ζ , η ) = U 1 ( ζ , η ) + U 2 ( ζ , η ) = exp ( j 4 π λ z 1 ) exp ( - j π λ z 1 ( ζ 2 + η 2 ) f 2 ) O 1 ( X , Y ) exp ( - j 2 π λ f ( ζ X + η Y ) ) d X d Y + exp ( j 4 π λ z 2 ) exp ( - j π λ z 2 ( ζ 2 + η 2 ) f 2 ) O 2 ( x , y ) exp ( - j 2 π λ f ( ζ x + η y ) ) d x d y ,
U ( ζ , η ) = exp ( j 4 π λ z 1 ) exp [ - j π λ z 1 ( ζ 2 + η 2 ) f 2 ] Õ 1 ( ζ , η ) + exp ( j 4 π λ z 2 ) exp ( - j π λ z 2 f 2 ( ζ 2 + η 2 ) ) Õ 2 ( ζ , η ) ,
I ( ζ , η ) = Õ 1 ( ζ , η ) 2 + Õ 2 ( ζ , η ) 2 + 2 Õ 1 ( ζ , η ) Õ 2 ( ζ , η ) × cos { 4 π λ ( z 2 - z 1 ) - π ( z 2 - z 1 ) λ f 2 ( ζ 2 + η 2 ) + Φ 2 ( ζ , η ) - Φ 1 ( ζ , η ) } ,
r K = f ( λ z 2 - z 1 ) 1 / 2 ( 4 ( z 2 - z 1 ) λ - 2 K + Φ 2 - Φ 1 π ) 1 / 2 ,
t ˜ ( u , v ) = I ( ζ , η ) exp [ j ( 2 π / λ ) ( u ζ + v η ) ] d ζ d η ,
t ˜ ( u , v ) = F 1 ( u , v ) + F 2 ( u , v ) + exp [ - j 4 π λ ( z 2 - z 1 ) ] { O 1 ( u , v ) O 2 * ( - u , - v ) exp [ - j π λ f 2 z 2 - z 1 ( u 2 + v 2 ) ] } + exp [ j 4 π λ ( z 2 - z 1 ) ] { O 1 * ( - u , - v ) O 2 ( u , v ) exp [ j π λ f 2 z 2 - z 1 ( u 2 + v 2 ) ] } ,
U L ( ζ , η ) = U 1 ( ζ , η ) + exp [ - j ( 2 π / λ f ) ζ x 0 ] U 2 ( ζ , η ) ,
I L ( ζ , η ) = Õ 1 ( ζ , η ) 2 + Õ 2 ( ζ , η ) 2 + Õ 1 ( ζ , η ) Õ 2 * ( ζ , η ) exp ( - j 4 π λ ( z 2 - z 1 ) ) exp ( j π λ f 2 ( z 2 - z 1 ) ( ζ 2 + η 2 ) ) exp ( j 2 π λ f ζ x 0 ) + Õ 1 * ( ζ , η ) Õ 2 ( ζ , η ) exp ( j 4 π λ ( z 2 - z 1 ) ) exp ( - j π λ f 2 ( z 2 - z 1 ) ( ζ 2 + η 2 ) ) exp ( - j 2 π λ f ζ x 0 ) .
Õ 1 Õ 2 * exp ( - j 4 π λ ( z 2 - z 1 ) + j π λ f 2 ( z 2 - z 1 ) ( ζ 2 + η 2 ) )
U ( ζ , η ) = exp ( - j 2 π λ ζ X 0 f ) U 1 ( ζ , η ) + U 2 ( ζ , η ) .
I ( ζ , η ) = Õ 1 ( ζ , η ) 2 + Õ 2 ( ζ , η ) 2 + 2 Õ 1 ( ζ , η ) Õ 2 ( ζ , η ) × cos ( 4 π λ ( z 2 - z 1 ) - π λ f 2 ( z 2 - z 1 ) ( ζ 2 + η 2 ) + 2 π λ f ζ X 0 + Φ 2 ( ζ , η ) - Φ 1 ( ζ , η ) ) .
r K = [ r K 2 + f 2 X 0 2 / ( z 2 - z 1 ) 2 ] 1 / 2 ,
M L ( ζ , η ) = 4 Õ 1 ( ζ , η ) 2 Õ 2 ( ζ , η ) 2 cos 2 ( π ζ X 0 / λ f ) .
d = λ f / X 0 .
U ( ζ , η ) = exp ( j 4 π λ ( z 1 + ) ) × exp ( - j π λ ( z 1 + ) f 2 ( ζ 2 + η 2 ) ) Õ 1 ( ζ , η ) + U 2 ( ζ , η ) .
I ( ζ , η ) = Õ 1 ( ζ , η ) 2 + Õ 2 ( ζ , η ) 2 + 2 Õ 1 ( ζ , η ) Õ 2 ( ζ , η ) × cos ( 4 π λ ( z 2 - z 1 - ) - π λ f 2 ( z 2 - z 1 - ) ( ζ 2 + η 2 ) + Φ 2 ( ζ , η ) - Φ 1 ( ζ , η ) ) .
r K = ( r K 2 - 4 f 2 z 2 - z 1 ) 1 / 2 ( 1 + z 2 - z 1 ) 1 / 2 .
M A ( ζ , η ) = 4 Õ 1 ( ζ , η ) 2 Õ 2 ( ζ , η ) 2 cos 2 ( 2 π λ - π 2 λ f 2 ( ζ 2 + η 2 ) ) .
R p = f ( 2 λ / ) 1 / 2 ( 2 / λ - p ) 1 / 2 ,