Abstract

The effect of finite-sized sampling apertures in the measurement of the complex degree of coherence of an optical field by the two-beam interference method is examined. The fringe visibility does not go to zero when the magnitude of the complex degree of coherence between the center points of the finite apertures goes to zero. This analysis explains observed errors in earlier experimental work.

© 1984 Optical Society of America

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References

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  1. E. Verdet, Intérferences en Général, Vol. 5/6 of Legons d’Opt. Phys. (L’Imprimerie Imperiale, Paris, 1869), 1, p. 106.
  2. B. J. Thompson, “Image formation with coherent light,” Proc. Soc. Photo-Opt. Instrum. Eng. 31, 31–49 (1968).
  3. A. A. Michelson, “On the application of interference methods to astronomical measurements,” Philos. Mag. 30, 1–21 (1890).
  4. A. A. Michelson, “On the application of interference methods to astronomical measures,” Am. J. Sci. 39, 579–590 (1890).
  5. A. A. Michelson, “On the application of interference methods to astronomical measurements,” Astrophys. J. 51, 257–262 (1920).
    [CrossRef]
  6. L. R. Baker, “The effect of source size on the coherence of an illuminating wave,” Proc. Phys. Soc. London Sect. B 66, 975–983 (1953).
    [CrossRef]
  7. B. J. Thompson, E. Wolf, “Two-beam interference with partially coherent light,”J. Opt. Soc. Am. 47, 895–902 (1957).
    [CrossRef]
  8. B. J. Thompson, “Illustration of the phase change in two-beam interference with partially coherent light,”J. Opt. Soc. Am. 48, 55–57 (1958).
    [CrossRef]
  9. S. T. Wu, F. T. S. Yu, “Image subtraction with encoded extended incoherent source,” Appl. Opt. 20, 4082–4088 (1981).
    [CrossRef] [PubMed]
  10. W. Rhodes, J. W. Goodman, “Interferometric technique for recording and restoring images degraded by unknown aberrations,”J. Opt. Soc. Am. 63, 647–657 (1973).
    [CrossRef]
  11. K. K. Dutta, “Sampling and restoration of images formed in partially coherent light,” Ph.D. dissertation (Stanford University, Stanford, California, 1975).
  12. E. Lau, “Interference phenomenon on double gratings,” Ann. Phys. 6, 417–420 (1948).
    [CrossRef]
  13. J. Jahns, A. W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination),” Opt. Commun. 28, 263–267 (1979).
    [CrossRef]
  14. F. Gori, “Lau effect and coherence theory,” Opt. Commun. 31, 4–8 (1979).
    [CrossRef]
  15. R. Sudol, B. J. Thompson, “An explanation of the Lau effect based on coherence theory,” Opt. Commun. 31, 105–110 (1979).
    [CrossRef]
  16. R. Sudol, B. J. Thompson, “Lau effect: theory and experiment,” Appl. Opt. 20, 1107–1116 (1981).
    [CrossRef] [PubMed]
  17. G. J. Swanson, E. N. Leith, “Lau effect and grating imaging,”J. Opt. Soc. Am. 72, 552–555 (1982).
    [CrossRef]
  18. B. J. Thompson, “A study of coherence and diffraction phenomena,” Ph.D. dissertation (University of Manchester, Manchester, England, 1959).
  19. L. Mandel, E. Wolf, “Spectral coherence and the concept of cross-spectral purity,”J. Opt. Soc. Am. 66, 529–535 (1976).
    [CrossRef]
  20. R. Sudol, “Lau effect: an interference phenomenon in partially coherent light,” Ph.D. thesis (University of Rochester, Rochester, New York, 1981).

1982 (1)

1981 (2)

1979 (3)

J. Jahns, A. W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination),” Opt. Commun. 28, 263–267 (1979).
[CrossRef]

F. Gori, “Lau effect and coherence theory,” Opt. Commun. 31, 4–8 (1979).
[CrossRef]

R. Sudol, B. J. Thompson, “An explanation of the Lau effect based on coherence theory,” Opt. Commun. 31, 105–110 (1979).
[CrossRef]

1976 (1)

1973 (1)

1968 (1)

B. J. Thompson, “Image formation with coherent light,” Proc. Soc. Photo-Opt. Instrum. Eng. 31, 31–49 (1968).

1958 (1)

1957 (1)

1953 (1)

L. R. Baker, “The effect of source size on the coherence of an illuminating wave,” Proc. Phys. Soc. London Sect. B 66, 975–983 (1953).
[CrossRef]

1948 (1)

E. Lau, “Interference phenomenon on double gratings,” Ann. Phys. 6, 417–420 (1948).
[CrossRef]

1920 (1)

A. A. Michelson, “On the application of interference methods to astronomical measurements,” Astrophys. J. 51, 257–262 (1920).
[CrossRef]

1890 (2)

A. A. Michelson, “On the application of interference methods to astronomical measurements,” Philos. Mag. 30, 1–21 (1890).

A. A. Michelson, “On the application of interference methods to astronomical measures,” Am. J. Sci. 39, 579–590 (1890).

Baker, L. R.

L. R. Baker, “The effect of source size on the coherence of an illuminating wave,” Proc. Phys. Soc. London Sect. B 66, 975–983 (1953).
[CrossRef]

Dutta, K. K.

K. K. Dutta, “Sampling and restoration of images formed in partially coherent light,” Ph.D. dissertation (Stanford University, Stanford, California, 1975).

Goodman, J. W.

Gori, F.

F. Gori, “Lau effect and coherence theory,” Opt. Commun. 31, 4–8 (1979).
[CrossRef]

Jahns, J.

J. Jahns, A. W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination),” Opt. Commun. 28, 263–267 (1979).
[CrossRef]

Lau, E.

E. Lau, “Interference phenomenon on double gratings,” Ann. Phys. 6, 417–420 (1948).
[CrossRef]

Leith, E. N.

Lohmann, A. W.

J. Jahns, A. W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination),” Opt. Commun. 28, 263–267 (1979).
[CrossRef]

Mandel, L.

Michelson, A. A.

A. A. Michelson, “On the application of interference methods to astronomical measurements,” Astrophys. J. 51, 257–262 (1920).
[CrossRef]

A. A. Michelson, “On the application of interference methods to astronomical measurements,” Philos. Mag. 30, 1–21 (1890).

A. A. Michelson, “On the application of interference methods to astronomical measures,” Am. J. Sci. 39, 579–590 (1890).

Rhodes, W.

Sudol, R.

R. Sudol, B. J. Thompson, “Lau effect: theory and experiment,” Appl. Opt. 20, 1107–1116 (1981).
[CrossRef] [PubMed]

R. Sudol, B. J. Thompson, “An explanation of the Lau effect based on coherence theory,” Opt. Commun. 31, 105–110 (1979).
[CrossRef]

R. Sudol, “Lau effect: an interference phenomenon in partially coherent light,” Ph.D. thesis (University of Rochester, Rochester, New York, 1981).

Swanson, G. J.

Thompson, B. J.

R. Sudol, B. J. Thompson, “Lau effect: theory and experiment,” Appl. Opt. 20, 1107–1116 (1981).
[CrossRef] [PubMed]

R. Sudol, B. J. Thompson, “An explanation of the Lau effect based on coherence theory,” Opt. Commun. 31, 105–110 (1979).
[CrossRef]

B. J. Thompson, “Image formation with coherent light,” Proc. Soc. Photo-Opt. Instrum. Eng. 31, 31–49 (1968).

B. J. Thompson, “Illustration of the phase change in two-beam interference with partially coherent light,”J. Opt. Soc. Am. 48, 55–57 (1958).
[CrossRef]

B. J. Thompson, E. Wolf, “Two-beam interference with partially coherent light,”J. Opt. Soc. Am. 47, 895–902 (1957).
[CrossRef]

B. J. Thompson, “A study of coherence and diffraction phenomena,” Ph.D. dissertation (University of Manchester, Manchester, England, 1959).

Verdet, E.

E. Verdet, Intérferences en Général, Vol. 5/6 of Legons d’Opt. Phys. (L’Imprimerie Imperiale, Paris, 1869), 1, p. 106.

Wolf, E.

Wu, S. T.

Yu, F. T. S.

Am. J. Sci. (1)

A. A. Michelson, “On the application of interference methods to astronomical measures,” Am. J. Sci. 39, 579–590 (1890).

Ann. Phys. (1)

E. Lau, “Interference phenomenon on double gratings,” Ann. Phys. 6, 417–420 (1948).
[CrossRef]

Appl. Opt. (2)

Astrophys. J. (1)

A. A. Michelson, “On the application of interference methods to astronomical measurements,” Astrophys. J. 51, 257–262 (1920).
[CrossRef]

J. Opt. Soc. Am. (5)

Opt. Commun. (3)

J. Jahns, A. W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination),” Opt. Commun. 28, 263–267 (1979).
[CrossRef]

F. Gori, “Lau effect and coherence theory,” Opt. Commun. 31, 4–8 (1979).
[CrossRef]

R. Sudol, B. J. Thompson, “An explanation of the Lau effect based on coherence theory,” Opt. Commun. 31, 105–110 (1979).
[CrossRef]

Philos. Mag. (1)

A. A. Michelson, “On the application of interference methods to astronomical measurements,” Philos. Mag. 30, 1–21 (1890).

Proc. Phys. Soc. London Sect. B (1)

L. R. Baker, “The effect of source size on the coherence of an illuminating wave,” Proc. Phys. Soc. London Sect. B 66, 975–983 (1953).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

B. J. Thompson, “Image formation with coherent light,” Proc. Soc. Photo-Opt. Instrum. Eng. 31, 31–49 (1968).

Other (4)

E. Verdet, Intérferences en Général, Vol. 5/6 of Legons d’Opt. Phys. (L’Imprimerie Imperiale, Paris, 1869), 1, p. 106.

K. K. Dutta, “Sampling and restoration of images formed in partially coherent light,” Ph.D. dissertation (Stanford University, Stanford, California, 1975).

R. Sudol, “Lau effect: an interference phenomenon in partially coherent light,” Ph.D. thesis (University of Rochester, Rochester, New York, 1981).

B. J. Thompson, “A study of coherence and diffraction phenomena,” Ph.D. dissertation (University of Manchester, Manchester, England, 1959).

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Figures (10)

Fig. 1
Fig. 1

Formation of two-beam interference fringes.

Fig. 2
Fig. 2

Photographic results of two-beam interference with partially coherent light (after Ref. 18.)

Fig. 3
Fig. 3

Configuration of system to be analyzed.

Fig. 4
Fig. 4

Frequency-plane analysis for the ratio a/b = 0.2. (a) Fully coherent case, (b) sinc function that actually multiplies (a), and (c) actual frequency-plane distribution.

Fig. 5
Fig. 5

Comparison of the frequency-plane distribution for various values of a with 2b and 2c fixed and 2a being varied. (a) Central peak distribution, (b) outer distribution.

Fig. 6
Fig. 6

(a) Resultant intensity distribution in the recording plane for σ = 0.2, (b) usual result that ignores the sampling effect, (c) fully coherent case.

Fig. 7
Fig. 7

Definition of parameters used in the approximate forms of the solution.

Fig. 8
Fig. 8

Correct and approximate forms for the outer portions of the spatial-frequency comparisons for σ = 0.2. (a) Correct function, (b) triangle approximation, (c) cosine approximation, (d) two-δ-function approximation.

Fig. 9
Fig. 9

Intensity distribution for the approximate forms. (a) Two-δ-function approximation, (b) cosine approximation, (c) triangular approximation.

Fig. 10
Fig. 10

(a) Approximate form of the spatial-frequency distribution that allows for a degree-of-coherence description. (b) Intensity distribution for the approximate form.

Tables (1)

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Table 1 Parameters for Approximate Forms of the Solution

Equations (18)

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I R ( x , y ) = I 1 ( x , y ) + I 2 ( x , y ) + 2 [ I 1 ( x , y ) I 2 ( x , y ) ] 1 / 2 × γ P 1 P 2 cos [ ( δ 1 - δ 2 ) + β P 1 P 2 ] ,
I 1 ( r ) = I P 1 | 2 J 1 ( k a r / f ) ( k a r / f ) | 2 ,
γ P 1 P 2 = γ P 1 P 2 exp ( i β P 1 P 2 ) ,
cos ( δ 1 - δ 2 ) = cos ( k 2 b x / f ) .
I R ( x , y ) = 2 I ( r ) { 1 + γ P 1 P 2 cos [ ( k 2 b x / f ) + β P 1 P 2 ] } .
γ ( w ) = 2 J 1 ( k c w / z ) ( k c w / z ) .
I R ( x ) = { 4 sinc 2 ( k a x / f ) cos 2 ( k b x / f ) } * Rect [ x c ] ,
I ( μ ) = - I R ( x ) exp ( - i k μ x / f ) d x ,
I ( μ ) = { Tri [ μ 2 a ] * [ 0.5 δ ( μ + 2 b ) + δ ( μ ) + 0.5 δ ( μ - 2 b ) ] } sinc ( k c μ / f ) ,
γ ( ξ 1 - ξ 2 ) = sinc [ k c ( ξ 1 - ξ 2 ) / f ] .
I ( μ ) = Tri [ μ 2 a ] + 0.5 α δ ( μ + s ) + 0.5 β δ ( μ + t ) + 0.5 α δ ( μ - s ) + 0.5 β δ ( μ - t ) .
I ( x ) = 4 a 2 sin 2 ( k a x / f ) + α cos ( k s x / f ) + β cos ( k t x / f ) .
I ( μ ) = Tri [ μ 2 a ] + { 0.5 α [ δ ( μ + s ) + δ ( μ - s ) ] } * { cos ( π μ / 2 a ) × Rect [ μ a ] } + { 0.5 β [ δ ( μ + t ) + δ ( μ - t ) ] } * { cos ( π μ / 2 a ) Rect [ μ a ] } .
I ( x ) = 4 a 2 sinc 2 ( k a x / f ) + a 2 ( sinc { 2 π a [ ( x / λ f ) - ( a / 8 ) ] } + sinc { 2 π a [ ( x / λ f ) + ( a / 8 ) ] } ) [ α cos ( k s x / f ) + β cos ( k t x / f ) ] .
I ( μ ) = Tri [ μ 2 a ] + 0.5 α [ δ ( μ + s ) + δ ( μ - s ) ] * Tri [ μ a ] + 0.5 β [ δ ( μ + t ) + δ ( μ - t ) ] * Tri [ μ a ] .
I ( x ) = a 4 a 2 sinc 2 ( k a x / f ) + a 2 sinc 2 ( k a x / 2 f ) [ α cos ( k s x / f ) + β cos ( k t x / f ) ] .
I ( μ ) = Tri [ μ 2 a ] + 0.5 α [ δ ( μ + s ) + δ ( μ - s ) ] * Tri [ μ 2 a ] + 0.5 β [ δ ( μ + t ) + δ ( μ - t ) ] * Tri [ μ 2 a ] .
I ( x ) = 4 a 2 sinc 2 ( k a x / f ) [ 1 + α cos ( k s x / f ) + β cos ( k t x / f ) ] ,

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