Abstract

We derive expressions for the spectrum of the field produced by planar, secondary Gaussian Schell-model sources after propagation in free-space, homogeneous dispersive media, and graded-index fibers. Our results show, for the first time to our knowledge, the development of correlation-induced spectral changes (the Wolf effect) as a function of the propagation distance from the source plane. An important result of our study is the prediction of the enhancement of the Wolf effect for propagation in media of index of refraction larger than unity. In the case of graded-index fibers having a parabolic index profile, the source spectrum is shown to reproduce periodically at distances at which such fibers image the source.

© 1990 Optical Society of America

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  1. E. Wolf, “Non-cosmological redshifts of spectral lines,” Nature (London) 326, 363–365 (1987).
    [Crossref]
  2. E. Wolf, “Redshifts and blueshifts of spectral lines caused by source correlations,” Opt. Commun. 62, 12–16 (1987).
    [Crossref]
  3. E. Wolf, “Red shifts and blue shifts of spectral lines emitted by two correlated sources,” Phys. Rev. Lett. 58, 2646–2648 (1987).
    [Crossref] [PubMed]
  4. Z. Dacic, E. Wolf, “Changes in the spectrum of a partially coherent light beam propagating in free space,” J. Opt. Soc. Am. A 5, 1118–1126 (1988).
    [Crossref]
  5. A. Gamliel, “Spectral changes in light propagation from a class of partially coherent sources,” in Proceedings of 6th Rochester Conference on Coherence and Quantum Optics (1989), J. H. Eberly, L. Mandel, E. Wolf, eds. (Plenum, New York, 1990).
  6. E. Wolf, “Invariance of spectrum of light on propagation,” Phys. Rev. Lett. 56, 1370–1372 (1986).
    [Crossref] [PubMed]
  7. A. Gamliel, E. Wolf, “Spectral modulation by control of source correlations,” Opt. Commun. 65, 91–96 (1988).
    [Crossref]
  8. J. T. Foley, E. Wolf, “Partially coherent sources that generate the same far-field spectra as completely incoherent sources,” J. Opt. Soc. Am. A 5, 1683–1687 (1988).
    [Crossref]
  9. D. F. V. James, E. Wolf, “A spectral equivalence theorem,” Opt. Commun. 72, 1–6 (1989).
    [Crossref]
  10. J. T. Foley, “The effect of an aperture on the spectrum of partially coherent light,” Opt. Commun. 75, 347–352 (1990).
    [Crossref]
  11. G. M. Morris, D. Faklis, “Effects of source correlations on the spectrum of light,” Opt. Commun. 62, 5–11 (1987).
    [Crossref]
  12. M. F. Bocko, D. H. Douglass, R. S. Knox, “Observation of frequency shifts of spectral lines due to source correlations,” Phys. Rev. Lett. 58, 2649–2651 (1987).
    [Crossref] [PubMed]
  13. F. Gori, G. Guattari, C. Palma, G. Padovani, “Observation of optical redshifts and blueshifts produced by source correlations,” Opt. Commun. 67, 1–4 (1988).
    [Crossref]
  14. D. Faklis, G. M. Morris, “Spectral shifts produced by source correlations,” Opt. Lett. 13, 4–6 (1988).
    [Crossref] [PubMed]
  15. G. Indebetouw, “Synthesis of polychromatic light sources with arbitrary degrees of coherence: some experiments,” J. Mod. Opt. 36, 251–259 (1989).
    [Crossref]
  16. H. C. Kandpal, J. S. Vaishya, K. C. Joshi, “Wolf shift and its applications in spectroradiometry,” Opt. Commun. 73, 173–178 (1989).
    [Crossref]
  17. Statements about the variance of the spectrum are also found in L. Mandel, “Concept of cross-spectral purity in coherence theory,” J. Opt. Soc. Am. 51, 1342–1350 (1961).
    [Crossref]
  18. F. Gori, R. Grela, “Shape invariant propagation of polychromatic fields,” Opt. Commun. 49, 173–177 (1984).
    [Crossref]
  19. A. Ghatak, K. Thyagarajan, “Graded index optical waveguides: a review,” in Progress In Optics XVIII, E. Wolf, ed. (North-Holland, Amsterdam, 1980), pp. 1–126.
    [Crossref]
  20. G. P. Agrawal, A. K. Ghatak, C. L. Metha, “Propagation of a partially coherent beam through Selfoc fibers,” Opt. Commun. 12, 333–337 (1974).
    [Crossref]
  21. P. Spano, “Connection between spatial coherence and modal structure in optical fibers and semiconductor lasers,” Opt. Commun. 33, 265–270 (1980).
    [Crossref]
  22. S. Piazzola, P. Spano, “Spatial coherence in incoherently excited optical fibers,” Opt. Commun. 43, 175–179 (1982).
    [Crossref]
  23. M. Imai, Y. Ohtsuka, “The influence of mode-coupling on the degree of coherence in an optical fiber,” Opt. Commun. 45, 331–335 (1983).
    [Crossref]
  24. M. Imai, S. Satoh, Y. Ohtsuka, “Complex degree of spatial coherence in an optical fiber: theory and experiment,” J. Opt. Soc. Am. A 3, 86–93 (1986).
    [Crossref]
  25. M. Born, E. Wolf, Principles of Optics,6th ed. (Pergamon, New York, 1983), Sec. 10.3.
  26. See, for example, 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]
  27. D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972), p. 267.
  28. See also E. Wolf, “New theory of partial coherence in space-frequency domain. Part I: Spectra and cross-spectra of steady-state sources,” J. Opt. Soc. Am. 72, 343–351 (1982); “Part II: Steady-state fields and higher-order correlations,” J. Opt. Soc. Am. A 3, 76–85 (1986).
    [Crossref]
  29. 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]
  30. A. T. Friberg, J. Turunen, “Imaging of Gaussian Schell-model sources,” J. Opt. Soc. Am. A 5, 713–720 (1988).
    [Crossref]
  31. A. T. Friberg, R. J. Sudol, “The spatial coherence properties of Gaussian Schell-model beams,” Opt. Acta 30, 1075–1097 (1983).
    [Crossref]
  32. A. Gamliel, “Radiation efficiency of planar Gaussian Schell-model sources,” Opt. Commun. 60, 333–338 (1986).
    [Crossref]
  33. P. De-Santis, F. Gori, G. Guattari, C. Palma, “Anisotropic Gaussian Schell-model sources,” Opt. Acta 33, 315–326 (1986).
    [Crossref]
  34. R. Simon, E. C. G. Sudarshan, N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985).
    [Crossref] [PubMed]
  35. J. Deschamps, D. Courjon, J. Bulabois, “Gaussian Schell-model sources: an example and some perspectives,” J. Opt. Soc. Am. 73, 256–261 (1983).
    [Crossref]
  36. 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]
  37. Li Yajun, E. Wolf, “Radiation from anisotropic Gaussian Schell-model sources,” Opt. Lett. 7, 256–258 (1982).
    [Crossref]
  38. A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
    [Crossref]
  39. J. T. Foley, M. S. Zubairy, “The directionality of Gaussian Schell-model beams,” Opt. Commun. 26, 297–300 (1978).
    [Crossref]
  40. See, for example, Texas Instruments type TIL211, Bulletin No. DS-S 7412095, March1974.
  41. F. Scudieri, M. Bertolotti, R. Bartolino, “Light scattered by a liquid crystal: a new quasi-thermal source,” Appl. Opt. 13, 181–185 (1974).
    [Crossref] [PubMed]
  42. P. de Santes, F. Gori, G. Guattari, C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
    [Crossref]
  43. J. D. Farina, L. M. Narducci, E. Collett, “Generation of highly directional beams from a globally incoherent source,” Opt. Commun. 32, 203–208 (1980).
    [Crossref]
  44. A. Gamliel, “Mode analysis of spectral changes in light propagation from sources of any state of coherence,” J. Opt. Soc. Am. A 7, 1591–1597 (1990).
    [Crossref]
  45. G. P. Agrawal, A. Gamliel, “Spectrum of partially coherent light: transition from near to far zone,” Opt. Commun. 78, 1–6 (1990).
    [Crossref]
  46. G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1989), pp. 7–8.
  47. The calculations appearing in this paper were based on M. J. Adams, An Introduction to Optical Waveguides (Wiley, New York, 1981), pp. 244–245.
  48. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), Sec. 3.323.2.
  49. N. Bleistein, R. A. Handelsman, Asymptotic Expansions of Integrals (Holt, Rinehart & Winston, New York, 1975), Chap. 6.
  50. In general there may be more than one zero of ϕ′. In the case that we are considering there is a single zero and hence Eq. (B4) does not contain the usual summation over all the roots of the equation ϕ′= 0.

1990 (3)

J. T. Foley, “The effect of an aperture on the spectrum of partially coherent light,” Opt. Commun. 75, 347–352 (1990).
[Crossref]

A. Gamliel, “Mode analysis of spectral changes in light propagation from sources of any state of coherence,” J. Opt. Soc. Am. A 7, 1591–1597 (1990).
[Crossref]

G. P. Agrawal, A. Gamliel, “Spectrum of partially coherent light: transition from near to far zone,” Opt. Commun. 78, 1–6 (1990).
[Crossref]

1989 (3)

G. Indebetouw, “Synthesis of polychromatic light sources with arbitrary degrees of coherence: some experiments,” J. Mod. Opt. 36, 251–259 (1989).
[Crossref]

H. C. Kandpal, J. S. Vaishya, K. C. Joshi, “Wolf shift and its applications in spectroradiometry,” Opt. Commun. 73, 173–178 (1989).
[Crossref]

D. F. V. James, E. Wolf, “A spectral equivalence theorem,” Opt. Commun. 72, 1–6 (1989).
[Crossref]

1988 (6)

1987 (5)

E. Wolf, “Non-cosmological redshifts of spectral lines,” Nature (London) 326, 363–365 (1987).
[Crossref]

E. Wolf, “Redshifts and blueshifts of spectral lines caused by source correlations,” Opt. Commun. 62, 12–16 (1987).
[Crossref]

E. Wolf, “Red shifts and blue shifts of spectral lines emitted by two correlated sources,” Phys. Rev. Lett. 58, 2646–2648 (1987).
[Crossref] [PubMed]

G. M. Morris, D. Faklis, “Effects of source correlations on the spectrum of light,” Opt. Commun. 62, 5–11 (1987).
[Crossref]

M. F. Bocko, D. H. Douglass, R. S. Knox, “Observation of frequency shifts of spectral lines due to source correlations,” Phys. Rev. Lett. 58, 2649–2651 (1987).
[Crossref] [PubMed]

1986 (4)

E. Wolf, “Invariance of spectrum of light on propagation,” Phys. Rev. Lett. 56, 1370–1372 (1986).
[Crossref] [PubMed]

M. Imai, S. Satoh, Y. Ohtsuka, “Complex degree of spatial coherence in an optical fiber: theory and experiment,” J. Opt. Soc. Am. A 3, 86–93 (1986).
[Crossref]

A. Gamliel, “Radiation efficiency of planar Gaussian Schell-model sources,” Opt. Commun. 60, 333–338 (1986).
[Crossref]

P. De-Santis, F. Gori, G. Guattari, C. Palma, “Anisotropic Gaussian Schell-model sources,” Opt. Acta 33, 315–326 (1986).
[Crossref]

1985 (1)

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985).
[Crossref] [PubMed]

1984 (1)

F. Gori, R. Grela, “Shape invariant propagation of polychromatic fields,” Opt. Commun. 49, 173–177 (1984).
[Crossref]

1983 (3)

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

M. Imai, Y. Ohtsuka, “The influence of mode-coupling on the degree of coherence in an optical fiber,” Opt. Commun. 45, 331–335 (1983).
[Crossref]

A. T. Friberg, R. J. Sudol, “The spatial coherence properties of Gaussian Schell-model beams,” Opt. Acta 30, 1075–1097 (1983).
[Crossref]

1982 (5)

1980 (2)

P. Spano, “Connection between spatial coherence and modal structure in optical fibers and semiconductor lasers,” Opt. Commun. 33, 265–270 (1980).
[Crossref]

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

1979 (1)

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

1978 (2)

J. T. Foley, M. S. Zubairy, “The directionality of Gaussian Schell-model beams,” Opt. Commun. 26, 297–300 (1978).
[Crossref]

See, for example, 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]

1974 (2)

F. Scudieri, M. Bertolotti, R. Bartolino, “Light scattered by a liquid crystal: a new quasi-thermal source,” Appl. Opt. 13, 181–185 (1974).
[Crossref] [PubMed]

G. P. Agrawal, A. K. Ghatak, C. L. Metha, “Propagation of a partially coherent beam through Selfoc fibers,” Opt. Commun. 12, 333–337 (1974).
[Crossref]

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]

1961 (1)

Adams, M. J.

The calculations appearing in this paper were based on M. J. Adams, An Introduction to Optical Waveguides (Wiley, New York, 1981), pp. 244–245.

Agrawal, G. P.

G. P. Agrawal, A. Gamliel, “Spectrum of partially coherent light: transition from near to far zone,” Opt. Commun. 78, 1–6 (1990).
[Crossref]

G. P. Agrawal, A. K. Ghatak, C. L. Metha, “Propagation of a partially coherent beam through Selfoc fibers,” Opt. Commun. 12, 333–337 (1974).
[Crossref]

G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1989), pp. 7–8.

Bartolino, R.

Bertolotti, M.

Bleistein, N.

N. Bleistein, R. A. Handelsman, Asymptotic Expansions of Integrals (Holt, Rinehart & Winston, New York, 1975), Chap. 6.

Bocko, M. F.

M. F. Bocko, D. H. Douglass, R. S. Knox, “Observation of frequency shifts of spectral lines due to source correlations,” Phys. Rev. Lett. 58, 2649–2651 (1987).
[Crossref] [PubMed]

Born, M.

M. Born, E. Wolf, Principles of Optics,6th ed. (Pergamon, New York, 1983), Sec. 10.3.

Bulabois, J.

Collett, E.

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

See, for example, 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.

Dacic, Z.

de Santes, P.

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

De-Santis, P.

P. De-Santis, F. Gori, G. Guattari, C. Palma, “Anisotropic Gaussian Schell-model sources,” Opt. Acta 33, 315–326 (1986).
[Crossref]

Deschamps, J.

Douglass, D. H.

M. F. Bocko, D. H. Douglass, R. S. Knox, “Observation of frequency shifts of spectral lines due to source correlations,” Phys. Rev. Lett. 58, 2649–2651 (1987).
[Crossref] [PubMed]

Faklis, D.

D. Faklis, G. M. Morris, “Spectral shifts produced by source correlations,” Opt. Lett. 13, 4–6 (1988).
[Crossref] [PubMed]

G. M. Morris, D. Faklis, “Effects of source correlations on the spectrum of light,” Opt. Commun. 62, 5–11 (1987).
[Crossref]

Farina, J. D.

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

Foley, J. T.

J. T. Foley, “The effect of an aperture on the spectrum of partially coherent light,” Opt. Commun. 75, 347–352 (1990).
[Crossref]

J. T. Foley, E. Wolf, “Partially coherent sources that generate the same far-field spectra as completely incoherent sources,” J. Opt. Soc. Am. A 5, 1683–1687 (1988).
[Crossref]

J. T. Foley, M. S. Zubairy, “The directionality of Gaussian Schell-model beams,” Opt. Commun. 26, 297–300 (1978).
[Crossref]

Friberg, A. T.

A. T. Friberg, J. Turunen, “Imaging of Gaussian Schell-model sources,” J. Opt. Soc. Am. A 5, 713–720 (1988).
[Crossref]

A. T. Friberg, R. J. Sudol, “The spatial coherence properties of Gaussian Schell-model beams,” Opt. Acta 30, 1075–1097 (1983).
[Crossref]

A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
[Crossref]

Gamliel, A.

A. Gamliel, “Mode analysis of spectral changes in light propagation from sources of any state of coherence,” J. Opt. Soc. Am. A 7, 1591–1597 (1990).
[Crossref]

G. P. Agrawal, A. Gamliel, “Spectrum of partially coherent light: transition from near to far zone,” Opt. Commun. 78, 1–6 (1990).
[Crossref]

A. Gamliel, E. Wolf, “Spectral modulation by control of source correlations,” Opt. Commun. 65, 91–96 (1988).
[Crossref]

A. Gamliel, “Radiation efficiency of planar Gaussian Schell-model sources,” Opt. Commun. 60, 333–338 (1986).
[Crossref]

A. Gamliel, “Spectral changes in light propagation from a class of partially coherent sources,” in Proceedings of 6th Rochester Conference on Coherence and Quantum Optics (1989), J. H. Eberly, L. Mandel, E. Wolf, eds. (Plenum, New York, 1990).

Ghatak, A.

A. Ghatak, K. Thyagarajan, “Graded index optical waveguides: a review,” in Progress In Optics XVIII, E. Wolf, ed. (North-Holland, Amsterdam, 1980), pp. 1–126.
[Crossref]

Ghatak, A. K.

G. P. Agrawal, A. K. Ghatak, C. L. Metha, “Propagation of a partially coherent beam through Selfoc fibers,” Opt. Commun. 12, 333–337 (1974).
[Crossref]

Gori, F.

F. Gori, G. Guattari, C. Palma, G. Padovani, “Observation of optical redshifts and blueshifts produced by source correlations,” Opt. Commun. 67, 1–4 (1988).
[Crossref]

P. De-Santis, F. Gori, G. Guattari, C. Palma, “Anisotropic Gaussian Schell-model sources,” Opt. Acta 33, 315–326 (1986).
[Crossref]

F. Gori, R. Grela, “Shape invariant propagation of polychromatic fields,” Opt. Commun. 49, 173–177 (1984).
[Crossref]

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

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), Sec. 3.323.2.

Grela, R.

F. Gori, R. Grela, “Shape invariant propagation of polychromatic fields,” Opt. Commun. 49, 173–177 (1984).
[Crossref]

Guattari, G.

F. Gori, G. Guattari, C. Palma, G. Padovani, “Observation of optical redshifts and blueshifts produced by source correlations,” Opt. Commun. 67, 1–4 (1988).
[Crossref]

P. De-Santis, F. Gori, G. Guattari, C. Palma, “Anisotropic Gaussian Schell-model sources,” Opt. Acta 33, 315–326 (1986).
[Crossref]

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

Handelsman, R. A.

N. Bleistein, R. A. Handelsman, Asymptotic Expansions of Integrals (Holt, Rinehart & Winston, New York, 1975), Chap. 6.

Imai, M.

M. Imai, S. Satoh, Y. Ohtsuka, “Complex degree of spatial coherence in an optical fiber: theory and experiment,” J. Opt. Soc. Am. A 3, 86–93 (1986).
[Crossref]

M. Imai, Y. Ohtsuka, “The influence of mode-coupling on the degree of coherence in an optical fiber,” Opt. Commun. 45, 331–335 (1983).
[Crossref]

Indebetouw, G.

G. Indebetouw, “Synthesis of polychromatic light sources with arbitrary degrees of coherence: some experiments,” J. Mod. Opt. 36, 251–259 (1989).
[Crossref]

James, D. F. V.

D. F. V. James, E. Wolf, “A spectral equivalence theorem,” Opt. Commun. 72, 1–6 (1989).
[Crossref]

Joshi, K. C.

H. C. Kandpal, J. S. Vaishya, K. C. Joshi, “Wolf shift and its applications in spectroradiometry,” Opt. Commun. 73, 173–178 (1989).
[Crossref]

Kandpal, H. C.

H. C. Kandpal, J. S. Vaishya, K. C. Joshi, “Wolf shift and its applications in spectroradiometry,” Opt. Commun. 73, 173–178 (1989).
[Crossref]

Knox, R. S.

M. F. Bocko, D. H. Douglass, R. S. Knox, “Observation of frequency shifts of spectral lines due to source correlations,” Phys. Rev. Lett. 58, 2649–2651 (1987).
[Crossref] [PubMed]

Mandel, L.

Marcuse, D.

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972), p. 267.

Metha, C. L.

G. P. Agrawal, A. K. Ghatak, C. L. Metha, “Propagation of a partially coherent beam through Selfoc fibers,” Opt. Commun. 12, 333–337 (1974).
[Crossref]

Morris, G. M.

D. Faklis, G. M. Morris, “Spectral shifts produced by source correlations,” Opt. Lett. 13, 4–6 (1988).
[Crossref] [PubMed]

G. M. Morris, D. Faklis, “Effects of source correlations on the spectrum of light,” Opt. Commun. 62, 5–11 (1987).
[Crossref]

Mukunda, N.

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985).
[Crossref] [PubMed]

Narducci, L. M.

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

Ohtsuka, Y.

M. Imai, S. Satoh, Y. Ohtsuka, “Complex degree of spatial coherence in an optical fiber: theory and experiment,” J. Opt. Soc. Am. A 3, 86–93 (1986).
[Crossref]

M. Imai, Y. Ohtsuka, “The influence of mode-coupling on the degree of coherence in an optical fiber,” Opt. Commun. 45, 331–335 (1983).
[Crossref]

Padovani, G.

F. Gori, G. Guattari, C. Palma, G. Padovani, “Observation of optical redshifts and blueshifts produced by source correlations,” Opt. Commun. 67, 1–4 (1988).
[Crossref]

Palma, C.

F. Gori, G. Guattari, C. Palma, G. Padovani, “Observation of optical redshifts and blueshifts produced by source correlations,” Opt. Commun. 67, 1–4 (1988).
[Crossref]

P. De-Santis, F. Gori, G. Guattari, C. Palma, “Anisotropic Gaussian Schell-model sources,” Opt. Acta 33, 315–326 (1986).
[Crossref]

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

Piazzola, S.

S. Piazzola, P. Spano, “Spatial coherence in incoherently excited optical fibers,” Opt. Commun. 43, 175–179 (1982).
[Crossref]

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), Sec. 3.323.2.

Satoh, S.

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]

Scudieri, F.

Simon, R.

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985).
[Crossref] [PubMed]

Spano, P.

S. Piazzola, P. Spano, “Spatial coherence in incoherently excited optical fibers,” Opt. Commun. 43, 175–179 (1982).
[Crossref]

P. Spano, “Connection between spatial coherence and modal structure in optical fibers and semiconductor lasers,” Opt. Commun. 33, 265–270 (1980).
[Crossref]

Starikov, A.

Sudarshan, E. C. G.

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985).
[Crossref] [PubMed]

Sudol, R. J.

A. T. Friberg, R. J. Sudol, “The spatial coherence properties of Gaussian Schell-model beams,” Opt. Acta 30, 1075–1097 (1983).
[Crossref]

A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
[Crossref]

Thyagarajan, K.

A. Ghatak, K. Thyagarajan, “Graded index optical waveguides: a review,” in Progress In Optics XVIII, E. Wolf, ed. (North-Holland, Amsterdam, 1980), pp. 1–126.
[Crossref]

Turunen, J.

Vaishya, J. S.

H. C. Kandpal, J. S. Vaishya, K. C. Joshi, “Wolf shift and its applications in spectroradiometry,” Opt. Commun. 73, 173–178 (1989).
[Crossref]

Wolf, E.

D. F. V. James, E. Wolf, “A spectral equivalence theorem,” Opt. Commun. 72, 1–6 (1989).
[Crossref]

J. T. Foley, E. Wolf, “Partially coherent sources that generate the same far-field spectra as completely incoherent sources,” J. Opt. Soc. Am. A 5, 1683–1687 (1988).
[Crossref]

A. Gamliel, E. Wolf, “Spectral modulation by control of source correlations,” Opt. Commun. 65, 91–96 (1988).
[Crossref]

Z. Dacic, E. Wolf, “Changes in the spectrum of a partially coherent light beam propagating in free space,” J. Opt. Soc. Am. A 5, 1118–1126 (1988).
[Crossref]

E. Wolf, “Non-cosmological redshifts of spectral lines,” Nature (London) 326, 363–365 (1987).
[Crossref]

E. Wolf, “Redshifts and blueshifts of spectral lines caused by source correlations,” Opt. Commun. 62, 12–16 (1987).
[Crossref]

E. Wolf, “Red shifts and blue shifts of spectral lines emitted by two correlated sources,” Phys. Rev. Lett. 58, 2646–2648 (1987).
[Crossref] [PubMed]

E. Wolf, “Invariance of spectrum of light on propagation,” Phys. Rev. Lett. 56, 1370–1372 (1986).
[Crossref] [PubMed]

See also E. Wolf, “New theory of partial coherence in space-frequency domain. Part I: Spectra and cross-spectra of steady-state sources,” J. Opt. Soc. Am. 72, 343–351 (1982); “Part II: Steady-state fields and higher-order correlations,” J. Opt. Soc. Am. A 3, 76–85 (1986).
[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]

Li Yajun, E. Wolf, “Radiation from anisotropic Gaussian Schell-model sources,” Opt. Lett. 7, 256–258 (1982).
[Crossref]

See, for example, 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]

M. Born, E. Wolf, Principles of Optics,6th ed. (Pergamon, New York, 1983), Sec. 10.3.

Yajun, Li

Zubairy, M. S.

J. T. Foley, M. S. Zubairy, “The directionality of Gaussian Schell-model beams,” Opt. Commun. 26, 297–300 (1978).
[Crossref]

Appl. Opt. (1)

IEEE Trans. Antennas Propag. (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]

J. Mod. Opt. (1)

G. Indebetouw, “Synthesis of polychromatic light sources with arbitrary degrees of coherence: some experiments,” J. Mod. Opt. 36, 251–259 (1989).
[Crossref]

J. Opt. Soc. Am. (4)

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

Nature (London) (1)

E. Wolf, “Non-cosmological redshifts of spectral lines,” Nature (London) 326, 363–365 (1987).
[Crossref]

Opt. Acta (2)

A. T. Friberg, R. J. Sudol, “The spatial coherence properties of Gaussian Schell-model beams,” Opt. Acta 30, 1075–1097 (1983).
[Crossref]

P. De-Santis, F. Gori, G. Guattari, C. Palma, “Anisotropic Gaussian Schell-model sources,” Opt. Acta 33, 315–326 (1986).
[Crossref]

Opt. Commun. (19)

A. Gamliel, “Radiation efficiency of planar Gaussian Schell-model sources,” Opt. Commun. 60, 333–338 (1986).
[Crossref]

See, for example, 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]

G. P. Agrawal, A. K. Ghatak, C. L. Metha, “Propagation of a partially coherent beam through Selfoc fibers,” Opt. Commun. 12, 333–337 (1974).
[Crossref]

P. Spano, “Connection between spatial coherence and modal structure in optical fibers and semiconductor lasers,” Opt. Commun. 33, 265–270 (1980).
[Crossref]

S. Piazzola, P. Spano, “Spatial coherence in incoherently excited optical fibers,” Opt. Commun. 43, 175–179 (1982).
[Crossref]

M. Imai, Y. Ohtsuka, “The influence of mode-coupling on the degree of coherence in an optical fiber,” Opt. Commun. 45, 331–335 (1983).
[Crossref]

E. Wolf, “Redshifts and blueshifts of spectral lines caused by source correlations,” Opt. Commun. 62, 12–16 (1987).
[Crossref]

D. F. V. James, E. Wolf, “A spectral equivalence theorem,” Opt. Commun. 72, 1–6 (1989).
[Crossref]

J. T. Foley, “The effect of an aperture on the spectrum of partially coherent light,” Opt. Commun. 75, 347–352 (1990).
[Crossref]

G. M. Morris, D. Faklis, “Effects of source correlations on the spectrum of light,” Opt. Commun. 62, 5–11 (1987).
[Crossref]

F. Gori, R. Grela, “Shape invariant propagation of polychromatic fields,” Opt. Commun. 49, 173–177 (1984).
[Crossref]

H. C. Kandpal, J. S. Vaishya, K. C. Joshi, “Wolf shift and its applications in spectroradiometry,” Opt. Commun. 73, 173–178 (1989).
[Crossref]

A. Gamliel, E. Wolf, “Spectral modulation by control of source correlations,” Opt. Commun. 65, 91–96 (1988).
[Crossref]

F. Gori, G. Guattari, C. Palma, G. Padovani, “Observation of optical redshifts and blueshifts produced by source correlations,” Opt. Commun. 67, 1–4 (1988).
[Crossref]

G. P. Agrawal, A. Gamliel, “Spectrum of partially coherent light: transition from near to far zone,” Opt. Commun. 78, 1–6 (1990).
[Crossref]

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

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

A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
[Crossref]

J. T. Foley, M. S. Zubairy, “The directionality of Gaussian Schell-model beams,” Opt. Commun. 26, 297–300 (1978).
[Crossref]

Opt. Lett. (2)

Phys. Rev. A (1)

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985).
[Crossref] [PubMed]

Phys. Rev. Lett. (3)

E. Wolf, “Invariance of spectrum of light on propagation,” Phys. Rev. Lett. 56, 1370–1372 (1986).
[Crossref] [PubMed]

M. F. Bocko, D. H. Douglass, R. S. Knox, “Observation of frequency shifts of spectral lines due to source correlations,” Phys. Rev. Lett. 58, 2649–2651 (1987).
[Crossref] [PubMed]

E. Wolf, “Red shifts and blue shifts of spectral lines emitted by two correlated sources,” Phys. Rev. Lett. 58, 2646–2648 (1987).
[Crossref] [PubMed]

Other (10)

A. Gamliel, “Spectral changes in light propagation from a class of partially coherent sources,” in Proceedings of 6th Rochester Conference on Coherence and Quantum Optics (1989), J. H. Eberly, L. Mandel, E. Wolf, eds. (Plenum, New York, 1990).

A. Ghatak, K. Thyagarajan, “Graded index optical waveguides: a review,” in Progress In Optics XVIII, E. Wolf, ed. (North-Holland, Amsterdam, 1980), pp. 1–126.
[Crossref]

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972), p. 267.

M. Born, E. Wolf, Principles of Optics,6th ed. (Pergamon, New York, 1983), Sec. 10.3.

See, for example, Texas Instruments type TIL211, Bulletin No. DS-S 7412095, March1974.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1989), pp. 7–8.

The calculations appearing in this paper were based on M. J. Adams, An Introduction to Optical Waveguides (Wiley, New York, 1981), pp. 244–245.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), Sec. 3.323.2.

N. Bleistein, R. A. Handelsman, Asymptotic Expansions of Integrals (Holt, Rinehart & Winston, New York, 1975), Chap. 6.

In general there may be more than one zero of ϕ′. In the case that we are considering there is a single zero and hence Eq. (B4) does not contain the usual summation over all the roots of the equation ϕ′= 0.

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Figures (9)

Fig. 1
Fig. 1

Illustrating the geometry and the notation. A point in the source plane z = 0 is denoted by (ξ, η), and an observation point is denoted by (x, y, z).

Fig. 2
Fig. 2

Normalized spectral modifier Mf for propagation distance k0z = 100 in free space. The spectral modifier is shown as a function of frequency ν for k0σI = 20 and four different values of the correlation length: (a) k0σg = 1.0, (b) k0σg = 8.0, (c) k0σg = 10, and (d) k0σg = 20. The direction of the spectral shift is determined by the slope of Mf at the center frequency of the source.

Fig. 3
Fig. 3

Normalized spectral modifier Mf for three different propagation distances [(a) k0z = 100, (b) k0z = 250, and (c) k0z = 600] in free space for k0σI = 20, k0σg = 10. At ν0 = 532 THz a blue shift is obtained for k0z = 100 and a red shift for k0z = 600.

Fig. 4
Fig. 4

Normalized field spectrum for observation at an angle of 10° off axis and a propagation distance k0z = 1000. The source is characterized by k0σI = 20 and k0σg = 20. The solid curve shows the original source spectrum, and the dashed curve shows the red-shifted field spectrum.

Fig. 5
Fig. 5

Frequency shifts Δν versus propagation distance for sources characterized by the same value of k0σI = 20 and different values of k0σg: (a) k0σg = 1, (b) k0σg = 10, (c) k0σg = 20, (d) k0σg = 25.

Fig. 6
Fig. 6

Comparison of frequency shifts for propagation in nondispersive homogeneous media. The frequency shifts for a fixed angle of observation (10°) are shown for propagation in free space (a), for propagation in a homogeneous medium of an index of refraction n = 1.5 (b), and for propagation in a medium of index of refraction n = 2.0 (c). The observation angle is 10°, and the source parameters are k0σI = 20 and k0σg = 10.

Fig. 7
Fig. 7

Comparison of frequency shifts for dispersive homogeneous media. Δν is shown as a function of k0z for (a) propagation in free space, (b) propagation in pure silica, and (c) propagation in silica doped with 7.9% GeO2. The observation angle is 10°, and the source parameters are k0σI = 20 and k0σg = 10.

Fig. 8
Fig. 8

Frequency shift Δν versus the propagation distance k0z in a dispersive graded-index medium [curve (a)]. Curve (b) shows Δν when the inhomogeneous nature of the medium is ignored by setting α = 0. Curve (c) shows, for comparison, the free-space result. The observation angle is 10°, and the source parameters are k0σI = 20 and k0σg = 10.

Fig. 9
Fig. 9

Frequency shift Δν as a function of propagation distance in a graded-index fiber (solid curve). The frequency shifts are calculated for observation at a fixed distance 10/k0 from the center of the fiber and k0σI = 20 and k0σg = 10. The dashed curve shows the frequency shifts when the frequency dependence of α is ignored by setting [α(ω0)/k0 = 0.000 48].

Equations (46)

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n ˜ 2 ( x , y ; ω ) = { n 2 ( ω ) [ 1 α 2 ( ω ) ( x 2 + y 2 ) ] for x 2 + y 2 R 0 2 n 2 ( ω ) [ 1 α 2 ( ω ) R 0 2 ] for x 2 + y 2 > R 0 2
Ψ ( r ; ω ) = K ( r , ρ ; ω ) Ψ ( ρ ; ω ) d 2 ρ ,
K ( r , ρ ; ω ) = k 2 π i e i ϕ ( r ) ( α sin α z ) × exp { i k α sin α z [ cos α z 2 ( ξ 2 + η 2 ) ( x ξ + y η ) ] } ,
ϕ ( r ) = k [ z + α cot α z 2 ( x 2 + y 2 ) ] .
W ( r 1 , r 2 ; ω ) = K * ( r 1 , ρ 1 ; ω ) K ( r 2 , ρ 2 ; ω ) × W ( ρ 1 , ρ 2 ; ω ) d 2 ρ 1 d 2 ρ 2 ,
S ( r ; ω ) = K * ( r , ρ 1 ; ω ) K ( r , ρ 2 ; ω ) W ( ρ 1 , ρ 2 ; ω ) d 2 ρ 1 d 2 ρ 2 .
S ( r : ω ) = ( k α 2 π sin α z ) 2 d ξ 1 d ξ 2 d η 1 d η 2 W ( ρ 1 , ρ 2 ; ω ) × exp { i k α sin α z [ cos α z 2 ( ξ 2 2 ξ 1 2 + η 2 2 η 1 2 ) x ( ξ 2 ξ 1 ) y ( η 2 η 1 ) ] } .
W ( ρ 1 , ρ 2 ; ω ) = S ( 0 ) ( ω ) [ I ( ρ 1 ) I ( ρ 2 ) ] 1 / 2 μ ( ρ 2 ρ 1 ) .
W ( ρ 1 , ρ 2 ; ω ) = S ( 0 ) ( ω ) exp [ ( ξ 1 2 + ξ 2 2 ) + ( η 1 2 + η 2 2 ) 4 σ I 2 ( ξ 2 ξ 1 ) 2 + ( η 2 η 1 ) 2 2 σ g 2 ] ,
S ( r ; ω ) = S ( 0 ) ( ω ) M ( r ; k , α ; ω ) ,
M ( r ; k , α ; ω ) = 1 z 2 ( k σ I Δ ) 2 exp [ k 2 ( x 2 + y 2 ) / z 2 2 Δ 2 ] .
Δ = 2 α b σ I sin α z α z ,
a 2 = 1 8 σ I 2 + 1 2 σ g 2 ,
b 2 = 1 2 σ I 2 + ( k α cos α z 2 a sin α z ) 2 .
M h ( r ; k ; ω ) = lim α 0 M ( r ; k , α ; ω ) = 1 z 2 ( k σ I Δ h ) 2 exp [ k 2 ( x 2 + y 2 ) / z 2 2 Δ h 2 ] ,
Δ h = 2 a b h σ I
b h 2 = 1 2 σ I 2 + ( k 2 a z ) 2 .
S f ( r ; ω ) = S ( 0 ) ( ω ) M f ( r ; ω ) = S ( 0 ) ( ω ) M h ( r ; k 0 ; ω ) ,
S f ( ) ( r ; ω ) = S ( 0 ) ( ω ) ( k 0 σ I 2 a z ) 2 exp [ k 0 2 σ I 2 ( x 2 + y 2 ) / z 2 4 a 2 ] ,
k = n ( ω ) ω c .
n 2 ( ω ) = 1 + j = 1 3 B j ω j 2 ω j 2 ω 2 .
α ( ω ) = 1 R 0 [ 1 n 2 2 ( ω ) n 1 2 ( ω ) ] 1 / 2 .
S ( r ; ω ) = K * ( r , ρ 1 ; ω ) K ( r , ρ 2 ; ω ) W ( ρ 1 , ρ 2 ; ω ) d 2 ρ 1 d 2 ρ 2 .
S ( r ; ω ) = S ( 0 ) ( ω ) ( k α 2 π sin α z ) 2 I ( x ; ω ) I ( y ; ω ) ,
I ( x ; ω ) = d ξ 1 d ξ 2 exp { ξ 2 2 + ξ 1 2 4 σ I 2 ( ξ 2 ξ 1 ) 2 2 σ g 2 + i k α sin α z [ cos α z 2 ( ξ 2 2 ξ 1 2 ) x ( ξ 2 ξ 1 ) ] } .
γ 1 = 1 2 ( ξ 2 + ξ 1 ) ,
γ 1 = ξ 2 ξ 1 .
I ( x ; ω ) = d γ 1 exp ( γ 1 2 2 σ I 2 ) × d γ 2 exp [ γ 2 2 ( 1 8 σ I 2 + 1 2 σ g 2 ) + i k α γ 2 sin α z ( γ 1 cos α z x ) ] .
a 2 = 1 8 σ I 2 + 1 2 σ g 2 ,
exp ( p 2 x 2 ± q x ) d x = π p exp ( q 2 4 p 2 ) ,
I ( x ; ω ) = π a exp [ x 2 ( k α 2 a sin α z ) 2 ] × d γ 1 exp { γ 1 2 [ 1 2 σ I 2 + ( k α cos α z 2 a sin α z ) 2 ] } × exp [ 2 γ 1 x cos α z ( k α 2 a sin α z ) 2 ] .
b 2 = 1 2 σ I 2 + ( k α cos α z 2 α sin α z ) 2
I ( x ; ω ) = π a b exp { x 2 ( k α 2 a sin α z ) 2 [ 1 ( k α cos α z 2 a b sin α z ) 2 ] } .
[ 1 ( k α cos α z 2 a b sin α z ) 2 ] = 1 ( 2 a b sin α z ) 2 [ ( 2 a b sin α z ) 2 ( k α cos α z ) 2 ] ,
[ 1 ( k α cos α z 2 a b sin α z ) 2 ] = 1 2 b 2 σ I 2 .
S ( r ; ω ) = S ( 0 ) ( ω ) ( k α 2 a b sin α z ) 2 × exp [ ( x 2 + y 2 ) 1 2 ( k α 2 a b σ I sin α z ) 2 ] ,
S ( r ; ω ) = S ( 0 ) ( ω ) z 2 ( k σ I Δ ) 2 exp [ k 2 ( x 2 + y 2 ) / z 2 2 Δ 2 ] ,
Δ = 2 a b σ I sin α z α z .
S ( r ; ω ) = ( k α Λ 2 π ) 2 d ξ 1 d ξ 2 d η 1 d η 2 W ( ξ 1 , ξ 2 , η 1 , η 2 ; ω ) × exp { i k α Λ [ cos α z 2 ( ξ 2 2 ξ 1 2 + η 2 2 η 1 2 ) ] x ( ξ 2 ξ 1 ) y ( η 2 η 1 ) ] } ,
Λ = 1 sin α z .
I ( Λ ) f ( t ) exp [ i Λ ϕ ( t ) ] d t ,
lim Λ I ( Λ ) = exp [ i Λ ϕ ( d ) ] f ( d ) ( 2 π Λ | ϕ ( d ) | ) 1 / 2 × exp ( i π μ 4 ) + O ( Λ 3 / 2 ) .
ϕ ( ξ 2 ) = k α ( cos α z 2 ξ 2 2 x ξ 2 ) ,
f ( ξ 2 ) = W ( ξ 1 , ξ 2 , η 1 , η 2 ) ,
I ( Λ ) = ( 2 π Λ k α ) 1 / 2 exp ( i π μ 4 ) exp [ i k α cos α z x 2 ( 1 2 1 cos α z ) ] × W ( ξ 1 , x cos α z , η 1 , η 2 ) + O ( λ 3 / 2 ) .
S ( r ; ω ) = W ( x cos α z , x cos α z , y cos α z , y cos α z ; ω ) .

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