Abstract

Double-exposure speckle interferometry is used to measure the degree of spatial coherence from an extended quasi-monochromatic source. The method closely follows the theory of spatial coherence and allows easy interpretation of results; it can be implemented with a compact interferometer based on laser speckle patterns.

© 1992 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, UK, 1975), p. 508.
  2. H. H. Hopkins, Proc. R. Soc. London Ser. A. 208, 263 (1951); in Advanced Optical Techniques, A. C. S. Van Heel, ed. (North-Holland, Amsterdam, 1967), p. 189.
    [CrossRef]
  3. D. N. Grimes, Appl. Opt. 10, 1567 (1971).
    [CrossRef] [PubMed]
  4. B. J. Thompson, E. Wolf, J. Opt. Soc. Am. 47, 895 (1957).
    [CrossRef]
  5. T. Asakura, H. Fujii, K. Murata, Opt. Acta 19, 273 (1972).
    [CrossRef]
  6. L. G. Kazovsky, Appl. Opt. 23, 455 (1984).
    [CrossRef] [PubMed]
  7. J. C. Dainty, ed., Laser Speckle and Related Phenomena, Vol. 9 of Topics in Applied Physics (Springer-Verlag, Berlin, 1984), and references therein.
  8. J. Burch, J. M. J. Tokarsky, Opt. Acta 15, 101 (1968).
  9. R. Meynart, Appl. Opt. 23, 2235 (1984).
    [CrossRef] [PubMed]
  10. K. Hinsch, Appl. Opt. 28, 5298 (1989).
    [CrossRef] [PubMed]

1989 (1)

1984 (2)

1972 (1)

T. Asakura, H. Fujii, K. Murata, Opt. Acta 19, 273 (1972).
[CrossRef]

1971 (1)

1968 (1)

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

1957 (1)

1951 (1)

H. H. Hopkins, Proc. R. Soc. London Ser. A. 208, 263 (1951); in Advanced Optical Techniques, A. C. S. Van Heel, ed. (North-Holland, Amsterdam, 1967), p. 189.
[CrossRef]

Asakura, T.

T. Asakura, H. Fujii, K. Murata, Opt. Acta 19, 273 (1972).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, UK, 1975), p. 508.

Burch, J.

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

Fujii, H.

T. Asakura, H. Fujii, K. Murata, Opt. Acta 19, 273 (1972).
[CrossRef]

Grimes, D. N.

Hinsch, K.

Hopkins, H. H.

H. H. Hopkins, Proc. R. Soc. London Ser. A. 208, 263 (1951); in Advanced Optical Techniques, A. C. S. Van Heel, ed. (North-Holland, Amsterdam, 1967), p. 189.
[CrossRef]

Kazovsky, L. G.

Meynart, R.

Murata, K.

T. Asakura, H. Fujii, K. Murata, Opt. Acta 19, 273 (1972).
[CrossRef]

Thompson, B. J.

Tokarsky, J. M. J.

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

Wolf, E.

B. J. Thompson, E. Wolf, J. Opt. Soc. Am. 47, 895 (1957).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, UK, 1975), p. 508.

Appl. Opt. (4)

J. Opt. Soc. Am. (1)

Opt. Acta (2)

T. Asakura, H. Fujii, K. Murata, Opt. Acta 19, 273 (1972).
[CrossRef]

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

Proc. R. Soc. London Ser. A. (1)

H. H. Hopkins, Proc. R. Soc. London Ser. A. 208, 263 (1951); in Advanced Optical Techniques, A. C. S. Van Heel, ed. (North-Holland, Amsterdam, 1967), p. 189.
[CrossRef]

Other (2)

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, UK, 1975), p. 508.

J. C. Dainty, ed., Laser Speckle and Related Phenomena, Vol. 9 of Topics in Applied Physics (Springer-Verlag, Berlin, 1984), and references therein.

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Figures (4)

Fig. 1
Fig. 1

Experimental setup: a Lambertian quasi-monochromatic source of intensity γ(X, Y) at Q(X, Y) illuminates the two points P1 and P2, on a transverse plane. Secondary waves from the two points proceed to the right and finally interfere at P.

Fig. 2
Fig. 2

Simple setup used to measure the degree of spatial coherence from a quasi-monochromatic source (circular aperture of radius s). The doubly exposed speckle transparency is placed at a distance R from the source (in air). The two points P1 and P2 are two small speckle apertures that correspond in an in-plane translation d.

Fig. 3
Fig. 3

Sample pictures (left-hand side) of the Young’s fringes observed in the Fourier plane of the diffractometer shown in Fig. 2. Microdensitometer traces (right-hand side) of the interferograms show the change in contrast with increasing values of (2πp).

Fig. 4
Fig. 4

Degree of spatial coherence of a circular aperture. The solid curve is the graph of the theoretical function, and the points are experimental results obtained with the present method.

Equations (5)

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I ( σ ) = a δ ( 0 ) + b H ( σ ) [ 1 + Γ cos ( 2 π σ d ) ] ,
x = X s ,             y = Y s ,
u = ξ ( n sin α λ ) ,             v = n ( n sin α λ ) ,
Γ ( u , v ) = exp ( - i ) S γ ( x , y ) exp [ i 2 π ( x u + y v ) ] d x d y .
Γ ( p ) = exp ( - i ) [ 2 J 1 ( 2 π p ) 2 π p ] ,

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