Abstract

It is proved that prescribed distributions of the cross-spectral density across a plane can be produced by starting from a primary source that is either spatially incoherent or spatially coherent. Possible synthesis schemes are outlined, and some difficulties generally encountered in the synthesis problem are discussed.

© 1986 Optical Society of America

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References

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  1. E. Wolf, “New theory of partial coherence in the space–frequency domain. Part I: Spectra and cross-spectra of steady-state sources,” J. Opt. Soc. Am. 72, 343–351 (1982).
    [Crossref]
  2. E. Wolf, “New theory of partial coherence in the space–frequency domain. Part II: Steady-state fields and higher-order correlations,” J. Opt. Soc. Am. (to be published).
  3. R. Simon, E. C. G. Sudarshan, N. Mukunda, “Generalized rays in first-order optics: transformation properties of Gaussian Schell-model fields,” Phys. Rev. A 29, 3273–3279 (1984).
    [Crossref]
  4. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Chap. X.
  5. M. Françon, S. Mallick, “Measurement of the second order degree of coherence,” in Progress in Optics, E. Wolf, ed. (North Holland, Amsterdam, 1967), Vol. VI, pp. 71–104.
    [Crossref]
  6. T. Asakura, H. Fujii, K. Murata, “Measurement of spatial coherence using speckle patterns,” Opt. Acta 19, 273–290 (1972).
    [Crossref]
  7. Y. Ohtsuka, “Proposal for the determination of the complex degree of spatial coherence,” Opt. Lett. 1, 133–134 (1977).
    [Crossref] [PubMed]
  8. F. T. S. Yu, F. K. Hsu, T. H. Chao, “Coherence measurement of a grating-based white-light optical signal processor,” Appl. Opt. 23, 333–340 (1984).
    [Crossref] [PubMed]
  9. P. De Santis, F. Gori, G. Guattari, C. Palma, J. M. Webster, “Measurement of spatial coherence through speckle interferometry,” J. Photogr. Sci. 33, 197–199 (1985).
  10. E. Wolf, “Young’s interference fringes with narrow-band light,” Opt. Lett 8, 250–252 (1983).
    [Crossref] [PubMed]
  11. B. J. Thompson, R. Sudol, “Finite-aperture effects in the measurement of the degree of coherence,” J. Opt. Soc. Am. A 1, 598–604 (1984).
    [Crossref]
  12. A. S. Marathay, D. B. Pollock, “Young’s interference fringes with finite-sized sampling apertures,” J. Opt. Soc. Am. A 1, 1057–1059 (1984).
    [Crossref]
  13. For a discussion of the synthesis problem for coherent fields, see A. W. Lohmann, “Three-dimensional properties of wavefields,” Optik (Stuttgart) 51, 105–117 (1978); see also F. Gori, L. Ronchi, “Degrees of freedom for scatterers with circular cross section,” J. Opt. Soc. Am. 71, 250–258 (1981).
    [Crossref]
  14. E. Wolf, W. H. Carter, “Coherence and radiant intensity in scalar wave fields generated by fluctuating primary planar sources,” J. Opt. Soc. Am.953–964 (1978).
    [Crossref]
  15. P. De Santis, F. Gori, G. Guattari, C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
    [Crossref]
  16. F. Gori, “Lau effect and coherence theory,” Opt. Commun. 31, 4–8 (1979).
    [Crossref]
  17. R. Sudol, B. J. Thompson, “Lau effect, theory and experiment,” Appl. Opt. 20, 1107–1116 (1981).
    [Crossref] [PubMed]
  18. D. Courjon, J. Bulabois, “Optical processing using a peculiar extended incoherent source,” Opt. Commun. 31, 270–274 (1979).
    [Crossref]
  19. D. Courjon, J. Bulabois, “Modifications of the coherence properties of a light beam: applications in optical processing,” Proc. Soc. Photo-Opt. Instrum. Eng. 194, 129–134 (1979).
  20. J. D. Farina, L. M. Narducci, E. Collett, “Generation of highly directional bams from a globally incoherent source,” Opt. Commun. 32, 203–208 (1980).
    [Crossref]
  21. D. Courjon, J. Bulabois, W. H. Carter, “Use of a holographic filter to modify the coherence of a light field,” J. Opt. Soc. Am. 71, 469–473 (1981).
    [Crossref]
  22. J. Deschamps, D. Courjon, J. Bulabois, “Gaussian Schell-model sources: an example and some perspectives,” J. Opt. Soc. Am. 73, 256–261 (1983).
    [Crossref]
  23. A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 58, 1256–1259 (1968).
    [Crossref]
  24. E. W. Marchand, E. Wolf, “Radiometry with sources of any state of coherence,” J. Opt. Soc. Am. 64, 1219–1226 (1974).
    [Crossref]
  25. W. H. Carter, E. Wolf, “Coherence properties of Lambertian and non-Lambertian sources,” J. Opt. Soc. Am. 65, 1067–1071 (1975).
    [Crossref]
  26. B. Steinle, H. P. Baltes, “Radiant intensity and spatial coherence for finite planar sources,” J. Opt. Soc. Am. 67, 241–247 (1977).
    [Crossref]
  27. W. H. Carter, E. Wolf, “Coherence and radiometry with quasihomogeneous planar sources,” J. Opt. Soc. Am. 67, 785–796 (1977).
    [Crossref]
  28. H. P. Baltes, J. Geist, A. Walther, “Radiometry and coherence,” in Inverse Source Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1978), pp. 119–154.
    [Crossref]
  29. E. Wolf, “Coherence and radiometry,” J. Opt. Soc. Am. 68, 6–17 (1978).
    [Crossref]
  30. E. Collett, E. Wolf, “Is complete spatial coherence necessary for the generation of highly directional light beams?” Opt. Lett. 2, 27–29 (1978).
    [Crossref] [PubMed]
  31. E. Wolf, E. Collett, “Partially coherent sources which produce the same far-field intensity distribution as a laser,” Opt. Commun. 25, 293–296 (1978).
    [Crossref]
  32. W. H. Carter, “Radiant intensity from inhomogeneous sources and the concept of averaged cross-spectral density,” Opt. Commun. 26, 1–4 (1978).
    [Crossref]
  33. E. Wolf, “The radiant intensity from planar sources of any state of coherence,” J. Opt. Soc. Am. 68, 1597–1605 (1978).
    [Crossref]
  34. H. A. Ferwerda, M G. van Heel, “Determination of coherence length from directionality,” in Proceedings of the Fourth Rochester Conference on Coherence and Quantum Optics, L. Mandel, E. Wolf, eds. (Plenum, New York, 1978), pp. 443–448.
  35. B. E. A. Saleh, “Intensity distribution due to a partially coherent field and the Collett–Wolf equivalence theorem in the Fresnel zone,” Opt. Commun. 30, 135–138 (1979).
    [Crossref]
  36. F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
    [Crossref]
  37. B. E. A. Saleh, M. I. Irshid, “Collett–Wolf equivalence theorem and propagation of a pulse in a single-mode optical fiber,” Opt. Lett. 7, 342–343 (1982).
    [Crossref] [PubMed]
  38. A. T. Friberg, “Effects of coherence in radiometry,” Opt. Eng. 21, 927–936 (1982).
    [Crossref]
  39. A. T. Friberg, R. J. Sudol, “The spatial coherence properties of Gaussian Schell-model beams,” Opt. Acta 30, 1075–1098 (1983).
    [Crossref]
  40. R. Martínez-Herrero, P. M. Mejías, “Radiometric definitions for partially coherent sources,” J. Opt. Soc. Am. A 1, 556–558 (1984).
    [Crossref]
  41. W. H. Carter, “Coherence properties of some isotropic planar sources,” J. Opt. Soc. Am. A 1, 716–722 (1984).
    [Crossref]
  42. G. J. Swanson, E. N. Leith, “Analysis of the Lau effect and generalized grating imaging,” J. Opt. Soc. Am. A 2, 789–793 (1985).
    [Crossref]
  43. F. G. Tricomi, Integral Equations (Wiley Interscience, New York, 1957), Chap. III.
  44. W. Pogorzelski, Integral Equations and Their Applications (Pergamon, Oxford, 1966), Chap. VI.
  45. In order to be a possible cross-spectral density, the function W(x1x2) must satisfy certain mathematical conditions.1,2 Hence it cannot be completely arbitrary.
  46. A. C. Schell, “A technique for the determination of the radiation pattern of a partially coherent aperture,” IEEE Trans. Antennas Propag. AP-15, 187–188 (1967).
    [Crossref]
  47. L. Mandel, E. Wolf, “Spectral coherence and the concept of cross-spectral purity,” J. Opt. Soc. Am. 66, 529–535 (1976).
    [Crossref]
  48. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  49. A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1977), Chap. VII.
  50. A. Starikov, E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and of their radiation fields,” J. Opt. Soc. Am. 72, 923–928 (1982).
    [Crossref]
  51. A. Starikov, “Effective number of degrees of freedom of partially coherent sources,” J. Opt. Soc. Am. 72, 1538–1544 (1982).
    [Crossref]
  52. D. Gabor, “Les transformations de l’information en optique,” Opt. Acta 13, 299–310 (1966).
    [Crossref]
  53. G. A. Deschamps, H. S. Cabayan, “Antenna synthesis and solution of inverse problems of regularization methods,” IEEE Trans. Antennas Propag. AP-20, 268–274 (1972).
    [Crossref]
  54. M. Bertero, C. De Mol, G. A. Viano, “The stability of inverse problems,” in Inverse Scattering Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1980).
    [Crossref]

1985 (2)

P. De Santis, F. Gori, G. Guattari, C. Palma, J. M. Webster, “Measurement of spatial coherence through speckle interferometry,” J. Photogr. Sci. 33, 197–199 (1985).

G. J. Swanson, E. N. Leith, “Analysis of the Lau effect and generalized grating imaging,” J. Opt. Soc. Am. A 2, 789–793 (1985).
[Crossref]

1984 (6)

1983 (3)

E. Wolf, “Young’s interference fringes with narrow-band light,” Opt. Lett 8, 250–252 (1983).
[Crossref] [PubMed]

J. Deschamps, D. Courjon, J. Bulabois, “Gaussian Schell-model sources: an example and some perspectives,” J. Opt. Soc. Am. 73, 256–261 (1983).
[Crossref]

A. T. Friberg, R. J. Sudol, “The spatial coherence properties of Gaussian Schell-model beams,” Opt. Acta 30, 1075–1098 (1983).
[Crossref]

1982 (5)

1981 (2)

1980 (2)

F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[Crossref]

J. D. Farina, L. M. Narducci, E. Collett, “Generation of highly directional bams from a globally incoherent source,” Opt. Commun. 32, 203–208 (1980).
[Crossref]

1979 (5)

D. Courjon, J. Bulabois, “Optical processing using a peculiar extended incoherent source,” Opt. Commun. 31, 270–274 (1979).
[Crossref]

D. Courjon, J. Bulabois, “Modifications of the coherence properties of a light beam: applications in optical processing,” Proc. Soc. Photo-Opt. Instrum. Eng. 194, 129–134 (1979).

P. De Santis, F. Gori, G. Guattari, C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
[Crossref]

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

B. E. A. Saleh, “Intensity distribution due to a partially coherent field and the Collett–Wolf equivalence theorem in the Fresnel zone,” Opt. Commun. 30, 135–138 (1979).
[Crossref]

1978 (7)

E. Wolf, “Coherence and radiometry,” J. Opt. Soc. Am. 68, 6–17 (1978).
[Crossref]

E. Collett, E. Wolf, “Is complete spatial coherence necessary for the generation of highly directional light beams?” Opt. Lett. 2, 27–29 (1978).
[Crossref] [PubMed]

E. Wolf, E. Collett, “Partially coherent sources which produce the same far-field intensity distribution as a laser,” Opt. Commun. 25, 293–296 (1978).
[Crossref]

W. H. Carter, “Radiant intensity from inhomogeneous sources and the concept of averaged cross-spectral density,” Opt. Commun. 26, 1–4 (1978).
[Crossref]

E. Wolf, “The radiant intensity from planar sources of any state of coherence,” J. Opt. Soc. Am. 68, 1597–1605 (1978).
[Crossref]

For a discussion of the synthesis problem for coherent fields, see A. W. Lohmann, “Three-dimensional properties of wavefields,” Optik (Stuttgart) 51, 105–117 (1978); see also F. Gori, L. Ronchi, “Degrees of freedom for scatterers with circular cross section,” J. Opt. Soc. Am. 71, 250–258 (1981).
[Crossref]

E. Wolf, W. H. Carter, “Coherence and radiant intensity in scalar wave fields generated by fluctuating primary planar sources,” J. Opt. Soc. Am.953–964 (1978).
[Crossref]

1977 (3)

1976 (1)

1975 (1)

1974 (1)

1972 (2)

T. Asakura, H. Fujii, K. Murata, “Measurement of spatial coherence using speckle patterns,” Opt. Acta 19, 273–290 (1972).
[Crossref]

G. A. Deschamps, H. S. Cabayan, “Antenna synthesis and solution of inverse problems of regularization methods,” IEEE Trans. Antennas Propag. AP-20, 268–274 (1972).
[Crossref]

1968 (1)

1967 (1)

A. C. Schell, “A technique for the determination of the radiation pattern of a partially coherent aperture,” IEEE Trans. Antennas Propag. AP-15, 187–188 (1967).
[Crossref]

1966 (1)

D. Gabor, “Les transformations de l’information en optique,” Opt. Acta 13, 299–310 (1966).
[Crossref]

Asakura, T.

T. Asakura, H. Fujii, K. Murata, “Measurement of spatial coherence using speckle patterns,” Opt. Acta 19, 273–290 (1972).
[Crossref]

Baltes, H. P.

B. Steinle, H. P. Baltes, “Radiant intensity and spatial coherence for finite planar sources,” J. Opt. Soc. Am. 67, 241–247 (1977).
[Crossref]

H. P. Baltes, J. Geist, A. Walther, “Radiometry and coherence,” in Inverse Source Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1978), pp. 119–154.
[Crossref]

Bertero, M.

M. Bertero, C. De Mol, G. A. Viano, “The stability of inverse problems,” in Inverse Scattering Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1980).
[Crossref]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Chap. X.

Bulabois, J.

J. Deschamps, D. Courjon, J. Bulabois, “Gaussian Schell-model sources: an example and some perspectives,” J. Opt. Soc. Am. 73, 256–261 (1983).
[Crossref]

D. Courjon, J. Bulabois, W. H. Carter, “Use of a holographic filter to modify the coherence of a light field,” J. Opt. Soc. Am. 71, 469–473 (1981).
[Crossref]

D. Courjon, J. Bulabois, “Modifications of the coherence properties of a light beam: applications in optical processing,” Proc. Soc. Photo-Opt. Instrum. Eng. 194, 129–134 (1979).

D. Courjon, J. Bulabois, “Optical processing using a peculiar extended incoherent source,” Opt. Commun. 31, 270–274 (1979).
[Crossref]

Cabayan, H. S.

G. A. Deschamps, H. S. Cabayan, “Antenna synthesis and solution of inverse problems of regularization methods,” IEEE Trans. Antennas Propag. AP-20, 268–274 (1972).
[Crossref]

Carter, W. H.

Chao, T. H.

Collett, E.

J. D. Farina, L. M. Narducci, E. Collett, “Generation of highly directional bams from a globally incoherent source,” Opt. Commun. 32, 203–208 (1980).
[Crossref]

E. Collett, E. Wolf, “Is complete spatial coherence necessary for the generation of highly directional light beams?” Opt. Lett. 2, 27–29 (1978).
[Crossref] [PubMed]

E. Wolf, E. Collett, “Partially coherent sources which produce the same far-field intensity distribution as a laser,” Opt. Commun. 25, 293–296 (1978).
[Crossref]

Courjon, D.

J. Deschamps, D. Courjon, J. Bulabois, “Gaussian Schell-model sources: an example and some perspectives,” J. Opt. Soc. Am. 73, 256–261 (1983).
[Crossref]

D. Courjon, J. Bulabois, W. H. Carter, “Use of a holographic filter to modify the coherence of a light field,” J. Opt. Soc. Am. 71, 469–473 (1981).
[Crossref]

D. Courjon, J. Bulabois, “Optical processing using a peculiar extended incoherent source,” Opt. Commun. 31, 270–274 (1979).
[Crossref]

D. Courjon, J. Bulabois, “Modifications of the coherence properties of a light beam: applications in optical processing,” Proc. Soc. Photo-Opt. Instrum. Eng. 194, 129–134 (1979).

De Mol, C.

M. Bertero, C. De Mol, G. A. Viano, “The stability of inverse problems,” in Inverse Scattering Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1980).
[Crossref]

De Santis, P.

P. De Santis, F. Gori, G. Guattari, C. Palma, J. M. Webster, “Measurement of spatial coherence through speckle interferometry,” J. Photogr. Sci. 33, 197–199 (1985).

P. De Santis, F. Gori, G. Guattari, C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
[Crossref]

Deschamps, G. A.

G. A. Deschamps, H. S. Cabayan, “Antenna synthesis and solution of inverse problems of regularization methods,” IEEE Trans. Antennas Propag. AP-20, 268–274 (1972).
[Crossref]

Deschamps, J.

Farina, J. D.

J. D. Farina, L. M. Narducci, E. Collett, “Generation of highly directional bams from a globally incoherent source,” Opt. Commun. 32, 203–208 (1980).
[Crossref]

Ferwerda, H. A.

H. A. Ferwerda, M G. van Heel, “Determination of coherence length from directionality,” in Proceedings of the Fourth Rochester Conference on Coherence and Quantum Optics, L. Mandel, E. Wolf, eds. (Plenum, New York, 1978), pp. 443–448.

Françon, M.

M. Françon, S. Mallick, “Measurement of the second order degree of coherence,” in Progress in Optics, E. Wolf, ed. (North Holland, Amsterdam, 1967), Vol. VI, pp. 71–104.
[Crossref]

Friberg, A. T.

A. T. Friberg, R. J. Sudol, “The spatial coherence properties of Gaussian Schell-model beams,” Opt. Acta 30, 1075–1098 (1983).
[Crossref]

A. T. Friberg, “Effects of coherence in radiometry,” Opt. Eng. 21, 927–936 (1982).
[Crossref]

Fujii, H.

T. Asakura, H. Fujii, K. Murata, “Measurement of spatial coherence using speckle patterns,” Opt. Acta 19, 273–290 (1972).
[Crossref]

Gabor, D.

D. Gabor, “Les transformations de l’information en optique,” Opt. Acta 13, 299–310 (1966).
[Crossref]

Geist, J.

H. P. Baltes, J. Geist, A. Walther, “Radiometry and coherence,” in Inverse Source Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1978), pp. 119–154.
[Crossref]

Goodman, J. W.

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

Gori, F.

P. De Santis, F. Gori, G. Guattari, C. Palma, J. M. Webster, “Measurement of spatial coherence through speckle interferometry,” J. Photogr. Sci. 33, 197–199 (1985).

F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[Crossref]

P. De Santis, F. Gori, G. Guattari, C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
[Crossref]

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

Guattari, G.

P. De Santis, F. Gori, G. Guattari, C. Palma, J. M. Webster, “Measurement of spatial coherence through speckle interferometry,” J. Photogr. Sci. 33, 197–199 (1985).

P. De Santis, F. Gori, G. Guattari, C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
[Crossref]

Hsu, F. K.

Irshid, M. I.

Leith, E. N.

Lohmann, A. W.

For a discussion of the synthesis problem for coherent fields, see A. W. Lohmann, “Three-dimensional properties of wavefields,” Optik (Stuttgart) 51, 105–117 (1978); see also F. Gori, L. Ronchi, “Degrees of freedom for scatterers with circular cross section,” J. Opt. Soc. Am. 71, 250–258 (1981).
[Crossref]

Mallick, S.

M. Françon, S. Mallick, “Measurement of the second order degree of coherence,” in Progress in Optics, E. Wolf, ed. (North Holland, Amsterdam, 1967), Vol. VI, pp. 71–104.
[Crossref]

Mandel, L.

Marathay, A. S.

Marchand, E. W.

Martínez-Herrero, R.

Mejías, P. M.

Mukunda, N.

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Generalized rays in first-order optics: transformation properties of Gaussian Schell-model fields,” Phys. Rev. A 29, 3273–3279 (1984).
[Crossref]

Murata, K.

T. Asakura, H. Fujii, K. Murata, “Measurement of spatial coherence using speckle patterns,” Opt. Acta 19, 273–290 (1972).
[Crossref]

Narducci, L. M.

J. D. Farina, L. M. Narducci, E. Collett, “Generation of highly directional bams from a globally incoherent source,” Opt. Commun. 32, 203–208 (1980).
[Crossref]

Ohtsuka, Y.

Palma, C.

P. De Santis, F. Gori, G. Guattari, C. Palma, J. M. Webster, “Measurement of spatial coherence through speckle interferometry,” J. Photogr. Sci. 33, 197–199 (1985).

P. De Santis, F. Gori, G. Guattari, C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
[Crossref]

Papoulis, A.

A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1977), Chap. VII.

Pogorzelski, W.

W. Pogorzelski, Integral Equations and Their Applications (Pergamon, Oxford, 1966), Chap. VI.

Pollock, D. B.

Saleh, B. E. A.

B. E. A. Saleh, M. I. Irshid, “Collett–Wolf equivalence theorem and propagation of a pulse in a single-mode optical fiber,” Opt. Lett. 7, 342–343 (1982).
[Crossref] [PubMed]

B. E. A. Saleh, “Intensity distribution due to a partially coherent field and the Collett–Wolf equivalence theorem in the Fresnel zone,” Opt. Commun. 30, 135–138 (1979).
[Crossref]

Schell, A. C.

A. C. Schell, “A technique for the determination of the radiation pattern of a partially coherent aperture,” IEEE Trans. Antennas Propag. AP-15, 187–188 (1967).
[Crossref]

Simon, R.

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Generalized rays in first-order optics: transformation properties of Gaussian Schell-model fields,” Phys. Rev. A 29, 3273–3279 (1984).
[Crossref]

Starikov, A.

Steinle, B.

Sudarshan, E. C. G.

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Generalized rays in first-order optics: transformation properties of Gaussian Schell-model fields,” Phys. Rev. A 29, 3273–3279 (1984).
[Crossref]

Sudol, R.

Sudol, R. J.

A. T. Friberg, R. J. Sudol, “The spatial coherence properties of Gaussian Schell-model beams,” Opt. Acta 30, 1075–1098 (1983).
[Crossref]

Swanson, G. J.

Thompson, B. J.

Tricomi, F. G.

F. G. Tricomi, Integral Equations (Wiley Interscience, New York, 1957), Chap. III.

van Heel, M G.

H. A. Ferwerda, M G. van Heel, “Determination of coherence length from directionality,” in Proceedings of the Fourth Rochester Conference on Coherence and Quantum Optics, L. Mandel, E. Wolf, eds. (Plenum, New York, 1978), pp. 443–448.

Viano, G. A.

M. Bertero, C. De Mol, G. A. Viano, “The stability of inverse problems,” in Inverse Scattering Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1980).
[Crossref]

Walther, A.

A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 58, 1256–1259 (1968).
[Crossref]

H. P. Baltes, J. Geist, A. Walther, “Radiometry and coherence,” in Inverse Source Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1978), pp. 119–154.
[Crossref]

Webster, J. M.

P. De Santis, F. Gori, G. Guattari, C. Palma, J. M. Webster, “Measurement of spatial coherence through speckle interferometry,” J. Photogr. Sci. 33, 197–199 (1985).

Wolf, E.

E. Wolf, “Young’s interference fringes with narrow-band light,” Opt. Lett 8, 250–252 (1983).
[Crossref] [PubMed]

E. Wolf, “New theory of partial coherence in the space–frequency domain. Part I: Spectra and cross-spectra of steady-state sources,” J. Opt. Soc. Am. 72, 343–351 (1982).
[Crossref]

A. Starikov, E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and of their radiation fields,” J. Opt. Soc. Am. 72, 923–928 (1982).
[Crossref]

E. Wolf, W. H. Carter, “Coherence and radiant intensity in scalar wave fields generated by fluctuating primary planar sources,” J. Opt. Soc. Am.953–964 (1978).
[Crossref]

E. Wolf, “Coherence and radiometry,” J. Opt. Soc. Am. 68, 6–17 (1978).
[Crossref]

E. Collett, E. Wolf, “Is complete spatial coherence necessary for the generation of highly directional light beams?” Opt. Lett. 2, 27–29 (1978).
[Crossref] [PubMed]

E. Wolf, E. Collett, “Partially coherent sources which produce the same far-field intensity distribution as a laser,” Opt. Commun. 25, 293–296 (1978).
[Crossref]

E. Wolf, “The radiant intensity from planar sources of any state of coherence,” J. Opt. Soc. Am. 68, 1597–1605 (1978).
[Crossref]

W. H. Carter, E. Wolf, “Coherence and radiometry with quasihomogeneous planar sources,” J. Opt. Soc. Am. 67, 785–796 (1977).
[Crossref]

L. Mandel, E. Wolf, “Spectral coherence and the concept of cross-spectral purity,” J. Opt. Soc. Am. 66, 529–535 (1976).
[Crossref]

W. H. Carter, E. Wolf, “Coherence properties of Lambertian and non-Lambertian sources,” J. Opt. Soc. Am. 65, 1067–1071 (1975).
[Crossref]

E. W. Marchand, E. Wolf, “Radiometry with sources of any state of coherence,” J. Opt. Soc. Am. 64, 1219–1226 (1974).
[Crossref]

E. Wolf, “New theory of partial coherence in the space–frequency domain. Part II: Steady-state fields and higher-order correlations,” J. Opt. Soc. Am. (to be published).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Chap. X.

Yu, F. T. S.

Appl. Opt. (2)

IEEE Trans. Antennas Propag. (2)

A. C. Schell, “A technique for the determination of the radiation pattern of a partially coherent aperture,” IEEE Trans. Antennas Propag. AP-15, 187–188 (1967).
[Crossref]

G. A. Deschamps, H. S. Cabayan, “Antenna synthesis and solution of inverse problems of regularization methods,” IEEE Trans. Antennas Propag. AP-20, 268–274 (1972).
[Crossref]

J. Opt. Soc. Am. (14)

A. Starikov, E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and of their radiation fields,” J. Opt. Soc. Am. 72, 923–928 (1982).
[Crossref]

A. Starikov, “Effective number of degrees of freedom of partially coherent sources,” J. Opt. Soc. Am. 72, 1538–1544 (1982).
[Crossref]

L. Mandel, E. Wolf, “Spectral coherence and the concept of cross-spectral purity,” J. Opt. Soc. Am. 66, 529–535 (1976).
[Crossref]

E. Wolf, “Coherence and radiometry,” J. Opt. Soc. Am. 68, 6–17 (1978).
[Crossref]

E. Wolf, W. H. Carter, “Coherence and radiant intensity in scalar wave fields generated by fluctuating primary planar sources,” J. Opt. Soc. Am.953–964 (1978).
[Crossref]

E. Wolf, “The radiant intensity from planar sources of any state of coherence,” J. Opt. Soc. Am. 68, 1597–1605 (1978).
[Crossref]

E. Wolf, “New theory of partial coherence in the space–frequency domain. Part I: Spectra and cross-spectra of steady-state sources,” J. Opt. Soc. Am. 72, 343–351 (1982).
[Crossref]

D. Courjon, J. Bulabois, W. H. Carter, “Use of a holographic filter to modify the coherence of a light field,” J. Opt. Soc. Am. 71, 469–473 (1981).
[Crossref]

J. Deschamps, D. Courjon, J. Bulabois, “Gaussian Schell-model sources: an example and some perspectives,” J. Opt. Soc. Am. 73, 256–261 (1983).
[Crossref]

A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 58, 1256–1259 (1968).
[Crossref]

E. W. Marchand, E. Wolf, “Radiometry with sources of any state of coherence,” J. Opt. Soc. Am. 64, 1219–1226 (1974).
[Crossref]

W. H. Carter, E. Wolf, “Coherence properties of Lambertian and non-Lambertian sources,” J. Opt. Soc. Am. 65, 1067–1071 (1975).
[Crossref]

B. Steinle, H. P. Baltes, “Radiant intensity and spatial coherence for finite planar sources,” J. Opt. Soc. Am. 67, 241–247 (1977).
[Crossref]

W. H. Carter, E. Wolf, “Coherence and radiometry with quasihomogeneous planar sources,” J. Opt. Soc. Am. 67, 785–796 (1977).
[Crossref]

J. Opt. Soc. Am. A (5)

J. Photogr. Sci. (1)

P. De Santis, F. Gori, G. Guattari, C. Palma, J. M. Webster, “Measurement of spatial coherence through speckle interferometry,” J. Photogr. Sci. 33, 197–199 (1985).

Opt. Acta (3)

T. Asakura, H. Fujii, K. Murata, “Measurement of spatial coherence using speckle patterns,” Opt. Acta 19, 273–290 (1972).
[Crossref]

D. Gabor, “Les transformations de l’information en optique,” Opt. Acta 13, 299–310 (1966).
[Crossref]

A. T. Friberg, R. J. Sudol, “The spatial coherence properties of Gaussian Schell-model beams,” Opt. Acta 30, 1075–1098 (1983).
[Crossref]

Opt. Commun. (8)

P. De Santis, F. Gori, G. Guattari, C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
[Crossref]

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

D. Courjon, J. Bulabois, “Optical processing using a peculiar extended incoherent source,” Opt. Commun. 31, 270–274 (1979).
[Crossref]

E. Wolf, E. Collett, “Partially coherent sources which produce the same far-field intensity distribution as a laser,” Opt. Commun. 25, 293–296 (1978).
[Crossref]

W. H. Carter, “Radiant intensity from inhomogeneous sources and the concept of averaged cross-spectral density,” Opt. Commun. 26, 1–4 (1978).
[Crossref]

B. E. A. Saleh, “Intensity distribution due to a partially coherent field and the Collett–Wolf equivalence theorem in the Fresnel zone,” Opt. Commun. 30, 135–138 (1979).
[Crossref]

F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[Crossref]

J. D. Farina, L. M. Narducci, E. Collett, “Generation of highly directional bams from a globally incoherent source,” Opt. Commun. 32, 203–208 (1980).
[Crossref]

Opt. Eng. (1)

A. T. Friberg, “Effects of coherence in radiometry,” Opt. Eng. 21, 927–936 (1982).
[Crossref]

Opt. Lett (1)

E. Wolf, “Young’s interference fringes with narrow-band light,” Opt. Lett 8, 250–252 (1983).
[Crossref] [PubMed]

Opt. Lett. (3)

Optik (Stuttgart) (1)

For a discussion of the synthesis problem for coherent fields, see A. W. Lohmann, “Three-dimensional properties of wavefields,” Optik (Stuttgart) 51, 105–117 (1978); see also F. Gori, L. Ronchi, “Degrees of freedom for scatterers with circular cross section,” J. Opt. Soc. Am. 71, 250–258 (1981).
[Crossref]

Phys. Rev. A (1)

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Generalized rays in first-order optics: transformation properties of Gaussian Schell-model fields,” Phys. Rev. A 29, 3273–3279 (1984).
[Crossref]

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

D. Courjon, J. Bulabois, “Modifications of the coherence properties of a light beam: applications in optical processing,” Proc. Soc. Photo-Opt. Instrum. Eng. 194, 129–134 (1979).

Other (11)

H. A. Ferwerda, M G. van Heel, “Determination of coherence length from directionality,” in Proceedings of the Fourth Rochester Conference on Coherence and Quantum Optics, L. Mandel, E. Wolf, eds. (Plenum, New York, 1978), pp. 443–448.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Chap. X.

M. Françon, S. Mallick, “Measurement of the second order degree of coherence,” in Progress in Optics, E. Wolf, ed. (North Holland, Amsterdam, 1967), Vol. VI, pp. 71–104.
[Crossref]

E. Wolf, “New theory of partial coherence in the space–frequency domain. Part II: Steady-state fields and higher-order correlations,” J. Opt. Soc. Am. (to be published).

H. P. Baltes, J. Geist, A. Walther, “Radiometry and coherence,” in Inverse Source Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1978), pp. 119–154.
[Crossref]

M. Bertero, C. De Mol, G. A. Viano, “The stability of inverse problems,” in Inverse Scattering Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1980).
[Crossref]

F. G. Tricomi, Integral Equations (Wiley Interscience, New York, 1957), Chap. III.

W. Pogorzelski, Integral Equations and Their Applications (Pergamon, Oxford, 1966), Chap. VI.

In order to be a possible cross-spectral density, the function W(x1x2) must satisfy certain mathematical conditions.1,2 Hence it cannot be completely arbitrary.

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

A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1977), Chap. VII.

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Equations (26)

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V ( x ) = K ( x , y ) V 0 ( y ) d y .
W 0 ( x 1 , x 2 ) = V * 0 ( x 1 ) V 0 ( x 2 ) , W ( x 1 , x 2 ) = V * ( x 1 ) V ( x 2 ) ,
W ( x 1 , x 2 ) = K * ( x 1 , y 1 ) K ( x 2 , y 2 ) W 0 ( y 1 , y 2 ) d y 1 d y 2 .
W 0 ( y 1 , y 2 ) = A δ ( y 1 y 2 ) ,
W ( x 1 , x 2 ) = A K * ( x 1 , y ) K ( x 2 , y ) d y .
| W 1 ( x 1 , x 2 ) | 2 d x 1 d x 2 < + .
W 1 ( x 1 , x 2 ) Φ ( x 1 ) d x 1 = λ Φ ( x 2 )
W 1 ( x 1 , x 2 ) = n = 0 λ n Φ * n ( x 1 ) Φ n ( x 2 ) ,
K ( y 1 , y 2 ) = n = 0 λ n Φ n ( y 1 ) Φ * n ( y 2 ) ,
K * ( x 1 , y ) K ( x 2 , y ) d y = n = 0 λ n Φ * n ( x 1 ) Φ n ( x 2 ) ,
W ( x 1 , x 2 ) = [ I ( x 1 ) I ( x 2 ) ] 1 / 2 g ( x 1 x 2 ) ,
K ( x , y ) = P ( x ) L ( y ) exp ( 2 π i a x y ) ,
W ( x 1 , x 2 ) = A P * ( x 1 ) P ( x 2 ) | L ( y ) | 2 exp [ 2 π i a ( x 1 x 2 ) y ] d y .
W ( x 1 , x 2 ) = n = 0 λ n Φ * n ( x 1 ) Φ n ( x 2 ) .
V ( x , t ) = U ( x , ν ) exp ( 2 π i ν t ) = exp ( 2 π i ν t ) n = 0 c n Φ n ( x ) .
c * n c m = λ n δ n m ( n , m = 0 , 1 , ) ,
V * ( x 1 , t ) V ( x 1 , t ) = U * ( x 1 , ν ) U ( x 2 , ν ) = W ( x 1 , x 2 ) .
S ( x , t ) = exp ( 2 π i ν t ) n = 0 Ψ n ( x ) P [ 1 τ n ( t k = 0 n 1 τ k ) ] ,
T = n = 0 τ n < + .
S * ( x 1 , t ) S ( x 2 , t ) t = 1 T a 0 T a S * ( x 1 , t ) S ( x 2 , t ) d t ,
S * ( x 1 , t ) S ( x 2 , t ) t = 1 T a n = 0 Ψ * n ( x 1 ) Ψ n ( x 2 ) × 0 T a | P [ 1 τ n ( t k = 0 n 1 τ k ) ] | 2 d t .
S * ( x 1 , t ) S ( x 2 , t ) t = α T a n = 0 τ n Ψ * n ( x 1 ) Ψ n ( x 2 ) ,
α = 0 1 | P ( t ) | 2 d t .
Ψ n ( x ) = Φ n ( x ) ( n = 0 , 1 , )
τ n = T a α λ n ( n = 0 , 1 , ) .
W ( N ) ( x 1 , x 2 ) = n = 0 N 1 λ n Φ * n ( x 1 ) Φ n ( x 2 ) ,

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