Abstract

Very good approximations to many optical fields are provided by free fields, i.e., fields that do not contain any evanescent waves. New conservation laws are derived for partially coherent fields of this kind, propagating into a half-space. It is found that the cross-spectral density, integrated over either the average or the difference of spatial coordinates in any cross section of the field, is invariant on propagation. It is also shown that, as a consequence of these conservation laws, both the spectrum and the so-called antispectrum of the field, integrated over cross-sectional planes, are conserved. The new conservation laws are illustrated by application to Gaussian Schell-model beams.

© 1993 Optical Society of America

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  1. An excellent collection of papers dealing with scalar wave fields was recently published in K. E. Oughtstun, ed., Selected Papers on Scalar Wave Diffraction, Vol. MS51 of Milestone Series (SPIE, Bellingham, Wash., 1992).
  2. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), Secs. 5.1 and 6.8.
  3. H. S. Green, E. Wolf, “A scalar representation of electromagnetic fields,” Proc. Phys. Soc. A 66, 1129–1137 (1953).
    [Crossref]
  4. P. M. Morse, K. U. Ingard, Theoretical Acoustics (Princeton U. Press, Princeton, N. J., 1968), Sec. 6.2.
  5. A. Rubinowicz, Die Beugungswelle in der Kirchhoffschen Theorie der Beugung (Pan`stwowe Wydawnictwo Naukowe, Warsaw, 1957), Chap. 1, Sec. 5.
  6. G. C. Sherman, “Diffracted wave fields expressible by plane-wave expansions containing only homogeneous waves,” Phys. Rev. Lett. 9, 761–764 (1968);“Diffracted wave fields expressible by plane-wave expansions containing only homogeneous waves,” J. Opt. Soc. Am. 59, 697–711 (1968).
    [Crossref]
  7. P. Roman, E. Wolf, “Correlation theory of stationary electromagnetic fields. Part II—conservation laws,” Nuovo Cimento 17, 477–490 (1960).
    [Crossref]
  8. E. Lalor, “Conditions for the validity of the angular spectrum of plane waves,” J. Opt. Soc. Am. 58, 1235–1237 (1968).
    [Crossref]
  9. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Sec. 3.7.
  10. 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);“New theory of partial coherence in the space-frequency domain. Part II: Steady state fields and higher-order correlations,” J. Opt. Soc. Am. A 3, 76–85 (1986).
    [Crossref]
  11. The conservation laws imply that the zero-frequency components of the ambiguity function and of the Wigner distribution function are conserved. This result is in agreement with Eqs. (25) and (28) of K. H. Brenner, J. Ojeda-Castañeda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta 31, 213–223 (1984).
    [Crossref]
  12. If the domain in which the Helmholtz equation is solved is a slab occupying the domain 0 ≤ z ≤ Z, rather than the half-space z≥ 0, the angular spectrum representation for the free field has the formUF(r;ω)=∫u⊥2≤1a(u⊥;ω)exp(iku+⋅r)d2u⊥+∫u⊥2≤1b(u⊥;ω)exp(iku−⋅r)d2u⊥,where u+= (u⊥,uz) and u−= (u⊥,−uz). For such fields, the integrals ∫WF(R,Δ,z;ω)d2R and ∫WF(R,Δ,z;ω)d2Δ may readily be shown to depend on z.
  13. The quantity given by Eqs. (3.11) can be shown to be conserved on propagation, even when the field is not free, i.e., even when it contains evanescent waves.
  14. R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1978), Chap. 3.
  15. The first part of this theorem, i.e., the conservation of the autocorrelation, was noted previously by H. G. Booker, J. A. Ratcliffe, D. H. Shinn, “Diffraction from an irregular screen with applications to ionosphere problems,” Phil. Trans. R. Soc. London Ser. A 242, 579–607 (1950).See also G. C. Sherman, Ref. 6, Theorem V.Some related results for a statistically homogeneous field are given in T. S. McKechnie, “Propagation of the spectral correlation function in a homogeneous medium,” J. Opt. Soc. Am. A 8, 339–345 (1991).
    [Crossref]
  16. The spectrum conservation law [Eq. (3.13)] was previously given by A. W. Lohman, “Three-dimensional properties of wave-fields,” Optik 51, 105–117 (1978).S. M. Rytov, Yu. A. Kravstov, V. I. Tatarskii, Principles of Statistical Radiophysics 4 (Springer-Verlag, New York, 1989), p. 101, give the result within the accuracy of the paraxial approximation.
  17. If the secondary source is not finite but the field across it falls off sufficiently rapidly with distance, e.g., when the field distribution is Gaussian, the subsequent analysis can be shown to remain valid.
  18. This point has been discussed, within the context of scattering, by E. Wolf, M. Nieto-Vesperinas, “Analyticity of the angular spectrum of scattered fields and some of its consequences,” J. Opt. Soc. Am. A 2, 886–890 (1985).
    [Crossref]
  19. See, for example, K. Miyamoto, E. Wolf, “Generalization of the Maggi–Rubinowicz theory of the boundary–diffraction wave–Part I,” J. Opt. Soc. Am. 52, 615–625 (1962), App.
    [Crossref]
  20. In cases of special symmetry, there may be some udirections for which this formula is not valid. An example of such a situation is the z axis for the diffraction by a circular aperture of a spherical wave converging to a point on the axis. This case is examined in Ref. 21. More generally, asymptotic approximations to the angular spectrum, valid for all directions, are discussed in Refs. 22 and 23.
  21. G. C. Sherman, W. C. Chew, “Aperture and far-field distributions expressed by the Debye integral representation of focused fields,” J. Opt. Soc. Am. 72, 1076–1083 (1982).
    [Crossref]
  22. G. C. Sherman, J. J. Stamnes, A. J. Devaney, É. Lalor, “Contribution of the inhomogeneous waves in angular-spectrum representations,” Opt. Commun. 8, 271–274 (1973).
    [Crossref]
  23. G. C. Sherman, J. J. Stamnes, É. Lalor, “Asymptotic approximations to angular spectrum representations,” J. Math. Phys. 17, 760–776 (1976).
    [Crossref]
  24. 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]
  25. J. T. Foley, M. S. Zubairy, “The directionality of Gaussian Schell-model beams,” Opt. Commun. 26, 297–300 (1978).
    [Crossref]
  26. A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
    [Crossref]
  27. P. De Santis, F. Gori, G. Guattari, C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
    [Crossref]
  28. 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]
  29. A. T. Friberg, R. J. Sudol, “The spatial coherence properties of Gaussian Schell-model beams,” Opt. Acta 30, 1075–1097(1983).
    [Crossref]
  30. L. Mandel, E. Wolf, “Spectral coherence and the concept of cross-spectral purity,” J. Opt. Soc. Am. 66, 529–535 (1976).
    [Crossref]
  31. The equations for the energy density and the energy flux vector are easy to derive from the results found in Refs. 3–5.The formula for the energy flux vector is also given in V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Va., 1971), p. 367.
  32. The beam condition is also derived in L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, to be published), Sec. 5.7.

1985 (1)

1984 (2)

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]

The conservation laws imply that the zero-frequency components of the ambiguity function and of the Wigner distribution function are conserved. This result is in agreement with Eqs. (25) and (28) of K. H. Brenner, J. Ojeda-Castañeda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta 31, 213–223 (1984).
[Crossref]

1983 (1)

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

1982 (3)

1979 (1)

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

1978 (3)

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]

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

The spectrum conservation law [Eq. (3.13)] was previously given by A. W. Lohman, “Three-dimensional properties of wave-fields,” Optik 51, 105–117 (1978).S. M. Rytov, Yu. A. Kravstov, V. I. Tatarskii, Principles of Statistical Radiophysics 4 (Springer-Verlag, New York, 1989), p. 101, give the result within the accuracy of the paraxial approximation.

1976 (2)

G. C. Sherman, J. J. Stamnes, É. Lalor, “Asymptotic approximations to angular spectrum representations,” J. Math. Phys. 17, 760–776 (1976).
[Crossref]

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

1973 (1)

G. C. Sherman, J. J. Stamnes, A. J. Devaney, É. Lalor, “Contribution of the inhomogeneous waves in angular-spectrum representations,” Opt. Commun. 8, 271–274 (1973).
[Crossref]

1968 (2)

G. C. Sherman, “Diffracted wave fields expressible by plane-wave expansions containing only homogeneous waves,” Phys. Rev. Lett. 9, 761–764 (1968);“Diffracted wave fields expressible by plane-wave expansions containing only homogeneous waves,” J. Opt. Soc. Am. 59, 697–711 (1968).
[Crossref]

E. Lalor, “Conditions for the validity of the angular spectrum of plane waves,” J. Opt. Soc. Am. 58, 1235–1237 (1968).
[Crossref]

1962 (1)

1960 (1)

P. Roman, E. Wolf, “Correlation theory of stationary electromagnetic fields. Part II—conservation laws,” Nuovo Cimento 17, 477–490 (1960).
[Crossref]

1953 (1)

H. S. Green, E. Wolf, “A scalar representation of electromagnetic fields,” Proc. Phys. Soc. A 66, 1129–1137 (1953).
[Crossref]

1950 (1)

The first part of this theorem, i.e., the conservation of the autocorrelation, was noted previously by H. G. Booker, J. A. Ratcliffe, D. H. Shinn, “Diffraction from an irregular screen with applications to ionosphere problems,” Phil. Trans. R. Soc. London Ser. A 242, 579–607 (1950).See also G. C. Sherman, Ref. 6, Theorem V.Some related results for a statistically homogeneous field are given in T. S. McKechnie, “Propagation of the spectral correlation function in a homogeneous medium,” J. Opt. Soc. Am. A 8, 339–345 (1991).
[Crossref]

Booker, H. G.

The first part of this theorem, i.e., the conservation of the autocorrelation, was noted previously by H. G. Booker, J. A. Ratcliffe, D. H. Shinn, “Diffraction from an irregular screen with applications to ionosphere problems,” Phil. Trans. R. Soc. London Ser. A 242, 579–607 (1950).See also G. C. Sherman, Ref. 6, Theorem V.Some related results for a statistically homogeneous field are given in T. S. McKechnie, “Propagation of the spectral correlation function in a homogeneous medium,” J. Opt. Soc. Am. A 8, 339–345 (1991).
[Crossref]

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1978), Chap. 3.

Brenner, K. H.

The conservation laws imply that the zero-frequency components of the ambiguity function and of the Wigner distribution function are conserved. This result is in agreement with Eqs. (25) and (28) of K. H. Brenner, J. Ojeda-Castañeda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta 31, 213–223 (1984).
[Crossref]

Chew, W. C.

Collett, E.

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]

De Santis, P.

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

Devaney, A. J.

G. C. Sherman, J. J. Stamnes, A. J. Devaney, É. Lalor, “Contribution of the inhomogeneous waves in angular-spectrum representations,” Opt. Commun. 8, 271–274 (1973).
[Crossref]

Foley, J. T.

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, 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]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Sec. 3.7.

Gori, F.

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

Green, H. S.

H. S. Green, E. Wolf, “A scalar representation of electromagnetic fields,” Proc. Phys. Soc. A 66, 1129–1137 (1953).
[Crossref]

Guattari, G.

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

Ingard, K. U.

P. M. Morse, K. U. Ingard, Theoretical Acoustics (Princeton U. Press, Princeton, N. J., 1968), Sec. 6.2.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), Secs. 5.1 and 6.8.

Lalor, E.

Lalor, É.

G. C. Sherman, J. J. Stamnes, É. Lalor, “Asymptotic approximations to angular spectrum representations,” J. Math. Phys. 17, 760–776 (1976).
[Crossref]

G. C. Sherman, J. J. Stamnes, A. J. Devaney, É. Lalor, “Contribution of the inhomogeneous waves in angular-spectrum representations,” Opt. Commun. 8, 271–274 (1973).
[Crossref]

Lohman, A. W.

The spectrum conservation law [Eq. (3.13)] was previously given by A. W. Lohman, “Three-dimensional properties of wave-fields,” Optik 51, 105–117 (1978).S. M. Rytov, Yu. A. Kravstov, V. I. Tatarskii, Principles of Statistical Radiophysics 4 (Springer-Verlag, New York, 1989), p. 101, give the result within the accuracy of the paraxial approximation.

Mandel, L.

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

The beam condition is also derived in L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, to be published), Sec. 5.7.

Miyamoto, K.

Morse, P. M.

P. M. Morse, K. U. Ingard, Theoretical Acoustics (Princeton U. Press, Princeton, N. J., 1968), Sec. 6.2.

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]

Nieto-Vesperinas, M.

Ojeda-Castañeda, J.

The conservation laws imply that the zero-frequency components of the ambiguity function and of the Wigner distribution function are conserved. This result is in agreement with Eqs. (25) and (28) of K. H. Brenner, J. Ojeda-Castañeda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta 31, 213–223 (1984).
[Crossref]

Palma, C.

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

Ratcliffe, J. A.

The first part of this theorem, i.e., the conservation of the autocorrelation, was noted previously by H. G. Booker, J. A. Ratcliffe, D. H. Shinn, “Diffraction from an irregular screen with applications to ionosphere problems,” Phil. Trans. R. Soc. London Ser. A 242, 579–607 (1950).See also G. C. Sherman, Ref. 6, Theorem V.Some related results for a statistically homogeneous field are given in T. S. McKechnie, “Propagation of the spectral correlation function in a homogeneous medium,” J. Opt. Soc. Am. A 8, 339–345 (1991).
[Crossref]

Roman, P.

P. Roman, E. Wolf, “Correlation theory of stationary electromagnetic fields. Part II—conservation laws,” Nuovo Cimento 17, 477–490 (1960).
[Crossref]

Rubinowicz, A.

A. Rubinowicz, Die Beugungswelle in der Kirchhoffschen Theorie der Beugung (Pan`stwowe Wydawnictwo Naukowe, Warsaw, 1957), Chap. 1, Sec. 5.

Sherman, G. C.

G. C. Sherman, W. C. Chew, “Aperture and far-field distributions expressed by the Debye integral representation of focused fields,” J. Opt. Soc. Am. 72, 1076–1083 (1982).
[Crossref]

G. C. Sherman, J. J. Stamnes, É. Lalor, “Asymptotic approximations to angular spectrum representations,” J. Math. Phys. 17, 760–776 (1976).
[Crossref]

G. C. Sherman, J. J. Stamnes, A. J. Devaney, É. Lalor, “Contribution of the inhomogeneous waves in angular-spectrum representations,” Opt. Commun. 8, 271–274 (1973).
[Crossref]

G. C. Sherman, “Diffracted wave fields expressible by plane-wave expansions containing only homogeneous waves,” Phys. Rev. Lett. 9, 761–764 (1968);“Diffracted wave fields expressible by plane-wave expansions containing only homogeneous waves,” J. Opt. Soc. Am. 59, 697–711 (1968).
[Crossref]

Shinn, D. H.

The first part of this theorem, i.e., the conservation of the autocorrelation, was noted previously by H. G. Booker, J. A. Ratcliffe, D. H. Shinn, “Diffraction from an irregular screen with applications to ionosphere problems,” Phil. Trans. R. Soc. London Ser. A 242, 579–607 (1950).See also G. C. Sherman, Ref. 6, Theorem V.Some related results for a statistically homogeneous field are given in T. S. McKechnie, “Propagation of the spectral correlation function in a homogeneous medium,” J. Opt. Soc. Am. A 8, 339–345 (1991).
[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]

Stamnes, J. J.

G. C. Sherman, J. J. Stamnes, É. Lalor, “Asymptotic approximations to angular spectrum representations,” J. Math. Phys. 17, 760–776 (1976).
[Crossref]

G. C. Sherman, J. J. Stamnes, A. J. Devaney, É. Lalor, “Contribution of the inhomogeneous waves in angular-spectrum representations,” Opt. Commun. 8, 271–274 (1973).
[Crossref]

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. 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]

Tatarskii, V. I.

The equations for the energy density and the energy flux vector are easy to derive from the results found in Refs. 3–5.The formula for the energy flux vector is also given in V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Va., 1971), p. 367.

Wolf, E.

This point has been discussed, within the context of scattering, by E. Wolf, M. Nieto-Vesperinas, “Analyticity of the angular spectrum of scattered fields and some of its consequences,” J. Opt. Soc. Am. A 2, 886–890 (1985).
[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);“New theory of partial coherence in the space-frequency domain. Part II: Steady state fields and higher-order correlations,” J. Opt. Soc. Am. A 3, 76–85 (1986).
[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]

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

See, for example, K. Miyamoto, E. Wolf, “Generalization of the Maggi–Rubinowicz theory of the boundary–diffraction wave–Part I,” J. Opt. Soc. Am. 52, 615–625 (1962), App.
[Crossref]

P. Roman, E. Wolf, “Correlation theory of stationary electromagnetic fields. Part II—conservation laws,” Nuovo Cimento 17, 477–490 (1960).
[Crossref]

H. S. Green, E. Wolf, “A scalar representation of electromagnetic fields,” Proc. Phys. Soc. A 66, 1129–1137 (1953).
[Crossref]

The beam condition is also derived in L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, to be published), Sec. 5.7.

Zubairy, M. S.

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

J. Math. Phys. (1)

G. C. Sherman, J. J. Stamnes, É. Lalor, “Asymptotic approximations to angular spectrum representations,” J. Math. Phys. 17, 760–776 (1976).
[Crossref]

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (1)

Nuovo Cimento (1)

P. Roman, E. Wolf, “Correlation theory of stationary electromagnetic fields. Part II—conservation laws,” Nuovo Cimento 17, 477–490 (1960).
[Crossref]

Opt. Acta (2)

The conservation laws imply that the zero-frequency components of the ambiguity function and of the Wigner distribution function are conserved. This result is in agreement with Eqs. (25) and (28) of K. H. Brenner, J. Ojeda-Castañeda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta 31, 213–223 (1984).
[Crossref]

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

Opt. Commun. (5)

G. C. Sherman, J. J. Stamnes, A. J. Devaney, É. Lalor, “Contribution of the inhomogeneous waves in angular-spectrum representations,” Opt. Commun. 8, 271–274 (1973).
[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]

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

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

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

Optik (1)

The spectrum conservation law [Eq. (3.13)] was previously given by A. W. Lohman, “Three-dimensional properties of wave-fields,” Optik 51, 105–117 (1978).S. M. Rytov, Yu. A. Kravstov, V. I. Tatarskii, Principles of Statistical Radiophysics 4 (Springer-Verlag, New York, 1989), p. 101, give the result within the accuracy of the paraxial approximation.

Phil. Trans. R. Soc. London Ser. A (1)

The first part of this theorem, i.e., the conservation of the autocorrelation, was noted previously by H. G. Booker, J. A. Ratcliffe, D. H. Shinn, “Diffraction from an irregular screen with applications to ionosphere problems,” Phil. Trans. R. Soc. London Ser. A 242, 579–607 (1950).See also G. C. Sherman, Ref. 6, Theorem V.Some related results for a statistically homogeneous field are given in T. S. McKechnie, “Propagation of the spectral correlation function in a homogeneous medium,” J. Opt. Soc. Am. A 8, 339–345 (1991).
[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]

Phys. Rev. Lett. (1)

G. C. Sherman, “Diffracted wave fields expressible by plane-wave expansions containing only homogeneous waves,” Phys. Rev. Lett. 9, 761–764 (1968);“Diffracted wave fields expressible by plane-wave expansions containing only homogeneous waves,” J. Opt. Soc. Am. 59, 697–711 (1968).
[Crossref]

Proc. Phys. Soc. A (1)

H. S. Green, E. Wolf, “A scalar representation of electromagnetic fields,” Proc. Phys. Soc. A 66, 1129–1137 (1953).
[Crossref]

Other (12)

P. M. Morse, K. U. Ingard, Theoretical Acoustics (Princeton U. Press, Princeton, N. J., 1968), Sec. 6.2.

A. Rubinowicz, Die Beugungswelle in der Kirchhoffschen Theorie der Beugung (Pan`stwowe Wydawnictwo Naukowe, Warsaw, 1957), Chap. 1, Sec. 5.

An excellent collection of papers dealing with scalar wave fields was recently published in K. E. Oughtstun, ed., Selected Papers on Scalar Wave Diffraction, Vol. MS51 of Milestone Series (SPIE, Bellingham, Wash., 1992).

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), Secs. 5.1 and 6.8.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Sec. 3.7.

If the secondary source is not finite but the field across it falls off sufficiently rapidly with distance, e.g., when the field distribution is Gaussian, the subsequent analysis can be shown to remain valid.

If the domain in which the Helmholtz equation is solved is a slab occupying the domain 0 ≤ z ≤ Z, rather than the half-space z≥ 0, the angular spectrum representation for the free field has the formUF(r;ω)=∫u⊥2≤1a(u⊥;ω)exp(iku+⋅r)d2u⊥+∫u⊥2≤1b(u⊥;ω)exp(iku−⋅r)d2u⊥,where u+= (u⊥,uz) and u−= (u⊥,−uz). For such fields, the integrals ∫WF(R,Δ,z;ω)d2R and ∫WF(R,Δ,z;ω)d2Δ may readily be shown to depend on z.

The quantity given by Eqs. (3.11) can be shown to be conserved on propagation, even when the field is not free, i.e., even when it contains evanescent waves.

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1978), Chap. 3.

In cases of special symmetry, there may be some udirections for which this formula is not valid. An example of such a situation is the z axis for the diffraction by a circular aperture of a spherical wave converging to a point on the axis. This case is examined in Ref. 21. More generally, asymptotic approximations to the angular spectrum, valid for all directions, are discussed in Refs. 22 and 23.

The equations for the energy density and the energy flux vector are easy to derive from the results found in Refs. 3–5.The formula for the energy flux vector is also given in V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Va., 1971), p. 367.

The beam condition is also derived in L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, to be published), Sec. 5.7.

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Figures (5)

Fig. 1
Fig. 1

Illustrating the notation for the angular spectrum representation [Eq. (2.3)] when uz is real.

Fig. 2
Fig. 2

Illustrating the conservation laws, Eqs. (3.7a), (3.7b), (3.13), and (3.15). The planes z = z1 and z = z2 are two arbitrary planes parallel to the boundary plane z = 0, in the half-space z ≥ 0.

Fig. 3
Fig. 3

Changes in (a) the spectrum and (b) the degree of coherence of a Gaussian Schell-model beam on propagation, calculated from Eqs. (5.13) and (5.15), respectively. These curves pertain to the source plane z = 0 and to the planes z = 105λ and z = 2 ×105λ. The parameters specifying the source were taken to be A = 1, σI (0) = 200λ, and σμ(0) = 100λ.

Fig. 4
Fig. 4

Illustrating that the functions (a) I+(Δ, z; ω) and (b) I(R, z; ω) are independent of the propagation distance z. These curves were calculated from Eqs. (5.16a) and Eq. (5.16b) for a Gaussian Schell-model beam. The parameters specifying the source are the same as in Fig. 3.

Fig. 5
Fig. 5

Illustrating the notation relating to the radiant intensity J(θ; ω) in Eq. (A1). r denotes the radial distance from the origin O to the point of observation P, and θ is the angle that the unit vector u makes with the z axis.

Equations (72)

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( 2 + k 2 ) U ( r ; ω ) = 0 ,
k = ω / c
U ( r ; ω ) = a ( u ; ω ) exp ( i k u r ) d 2 u ,
u z = 1 u 2 for u 2 1 ,
u z = i u 2 1 for u 2 > 1 .
u 2 1
U F ( r ; ω ) = u 2 1 a ( u ; ω ) exp ( i k u r ) d 2 u ,
W F ( r 1 , r 2 ; ω ) = U F * ( r 1 ; ω ) U F ( r 2 ; ω ) ,
W F ( r 1 , r 2 ; ω ) = D 1 , D 2 A ( u 1 , u 2 ; ω ) × exp [ i k ( u 2 r 2 u 1 r 1 ) ] d 2 u 1 d 2 u 2 .
u 1 2 1 , u 2 2 1 ,
A ( u 1 , u 2 ; ω ) = a * ( u 1 ; ω ) a ( u 2 ; ω ) .
W F ( ρ 1 , ρ 2 , z ; ω ) = D 1 , D 2 A ( u 1 , u 2 ; ω ) × exp [ i k ( u 2 ρ 2 u 1 ρ 1 ) ] × exp [ i k ( u 2 z u 1 z ) z ] d 2 u 1 d 2 u 2 .
R = ρ 2 + ρ 1 2 ,
Δ = ρ 2 ρ 1 .
W F ( R , Δ , z ; ω ) = D 1 , D 2 A ( u 1 , u 2 ; ω ) × exp [ i k ( u 2 u 1 ) R ] × exp [ i k ( u 2 + u 1 ) Δ / 2 ] × exp [ i k ( u 2 z u 1 z ) z ] d 2 u 1 d 2 u 2 ,
W F ( R , Δ , z ; ω ) W F ( R Δ / 2 , R + Δ / 2 , z ; ω ) .
W F ( R , Δ , z ; ω ) d 2 R = ( 2 π k ) 2 u 2 1 A ( u , u ; ω ) × exp ( i k u Δ ) d 2 u ,
W F ( R , Δ , z ; ω ) d 2 Δ = ( 4 π k ) 2 u 2 1 A ( u , u ; ω ) × exp ( i 2 k u R ) d 2 u ,
1 ( 2 π ) 2 exp ( i κ η ) d 2 η = δ 2 ( κ )
W F ( R , Δ , z 1 ; ω ) d 2 R = W F ( R , Δ , z 2 ; ω ) d 2 R ,
W F ( R , Δ , z 1 ; ω ) d 2 Δ = W F ( R , Δ , z 2 ; ω ) d 2 Δ .
W F ( R , Δ , z ; ω ) d 2 R = I + ( Δ , z ; ω ) ,
W F ( R , Δ , z ; ω ) d 2 Δ = I ( R , z ; ω ) ,
I + ( Δ , z 1 ; ω ) = I + ( Δ , z 2 ; ω ) ,
I ( R , z 1 ; ω ) = I ( R , z 2 ; ω ) .
W F * ( R , Δ , z ; ω ) = W F ( R , Δ , z ; ω ) .
I + ( Δ , z ; ω ) d 2 Δ = I ( R , z ; ω ) d 2 R = W F ( R , Δ , z ; ω ) d 2 R d 2 Δ
I + ( Δ , z ; ω ) U F * ( R Δ / 2 , z ; ω ) U F ( R + Δ / 2 , z ; ω ) d 2 R ,
I ( R , z ; ω ) U F * ( R Δ / 2 , z ; ω ) U F ( R + Δ / 2 , z ; ω ) d 2 Δ ,
S F ( ρ , z 1 ; ω ) d 2 ρ = S F ( ρ , z 2 ; ω ) d 2 ρ ,
S F ( ρ , z ; ω ) U F * ( ρ , z ; ω ) U F ( ρ , z ; ω )
= W F ( ρ , 0 , z ; ω ) .
S ¯ F ( ρ , z 1 ; ω ) d 2 ρ = S ¯ F ( ρ , z 2 ; ω ) d 2 ρ ,
S ¯ F ( ρ , z ; ω ) U F * ( ρ , z ; ω ) U F ( ρ , z ; ω )
= W F ( 0 , 2 ρ , z ; ω )
U ( ) ( r u ; ω ) = 2 π i k u z a ( u ; ω ) exp ( i k r ) r .
W ( ) ( r u 1 , r u 2 ; ω ) = ( 2 π k r ) 2 u 1 z u 2 z A ( u 1 , u 2 ; ω ) .
I + ( Δ , z , ω ) = r 2 u 2 1 W ( ) ( r u , r u z , r u , r u z ; ω ) 1 u 2 × exp ( i k u Δ ) d 2 u ,
I ( R , z , ω ) = 4 r 2 u 2 1 W ( ) ( r u , r u z , r u , r u z ; ω ) 1 u 2 × exp ( i 2 k u R ) d 2 u .
I + ( Δ , z , ω ) = r 2 u 2 1 S ( ) ( r u , r u z ; ω ) 1 u 2 × exp ( i k u Δ ) d 2 u ,
I ( R , z , ω ) = 4 r 2 u 2 1 S ¯ ( ) ( r u , r u z ; ω ) 1 u 2 × exp ( i 2 k u R ) d 2 u ,
S ( ) ( r u , r u z ; ω ) = W ( ) ( r u , r u z , r u , r u z ; ω ) ,
S ¯ ( ) ( r u , r u z ; ω ) = W ( ) ( r u , r u z , r u , r u z ; ω ) ,
S F ( ρ , z ; ω ) d 2 ρ = r 2 u 2 1 S ( ) ( r u , r u z ; ω ) 1 u 2 d 2 u ,
S ¯ F ( ρ , z ; ω ) d 2 ρ = r 2 u 2 1 S ¯ ( ) ( r u , r u z ; ω ) 1 u 2 d 2 u .
W ( ρ 1 , ρ 2 , z = 0 ; ω ) = [ S ( ρ 1 , z = 0 ; ω ) S ( ρ 2 , z = 0 ; ω ) ] 1 / 2 × μ ( ρ 2 ρ 1 , z = 0 ; ω ) ,
S ( ρ , z = 0 ; ω ) = A exp [ ρ 2 / 2 σ I 2 ( 0 ) ] ,
μ ( Δ , z = 0 ; ω ) = exp [ Δ 2 / 2 σ μ 2 ( 0 ) ] .
1 σ μ 2 ( 0 ) + 1 4 σ I 2 ( 0 ) 2 π 2 λ 2 ,
W ( ρ 1 , ρ 2 , z ; ω ) = A σ I 2 ( 0 ) σ I 2 ( z ) exp [ ( ρ 2 2 + ρ 1 2 ) / 4 σ I 2 ( z ) ] × exp [ ( ρ 2 ρ 1 ) 2 / 2 σ μ 2 ( z ) ] exp [ i k ( ρ 2 2 ρ 1 2 ) / 2 L ( z ) ] ,
σ I 2 ( z ) = σ I 2 ( 0 ) β 2 ( z ) ,
σ μ 2 ( z ) = σ μ 2 ( 0 ) β 2 ( z ) ,
L ( z ) = z β 2 ( z ) β 2 ( z ) 1 .
β 2 ( z ) 1 + z 2 k 2 γ 2 ( 0 ) σ I 2 ( 0 ) ,
1 γ 2 ( 0 ) 1 σ μ 2 ( 0 ) + 1 4 σ I 2 ( 0 ) .
W ( R , Δ , z ; ω ) = A σ I 2 ( 0 ) σ I 2 ( z ) exp [ R 2 / 2 σ I 2 ( z ) ] × exp [ Δ 2 / 2 γ 2 ( z ) ] × exp [ i k R Δ / L ( z ) ] ,
γ 2 ( z ) = γ 2 ( 0 ) β 2 ( z ) .
S ( ρ , z ; ω ) = A σ I 2 ( 0 ) σ I 2 ( z ) exp [ ρ 2 / 2 σ I 2 ( z ) ] .
μ ( ρ 1 , ρ 2 , z ; ω ) = W ( ρ 1 , ρ 2 , z ; ω ) [ W ( ρ 1 , ρ 1 , z ; ω ) ] 1 / 2 W ( ρ 1 , ρ 2 , z ; ω ) 1 / 2 ,
μ ( R Δ / 2 , R + Δ / 2 , z ; ω ) = exp [ Δ 2 / 2 σ μ 2 ( z ) ] exp [ i k R Δ / L ( z ) ] .
I + ( Δ , z ; ω ) = 2 π A σ I 2 ( 0 ) exp [ Δ 2 / 2 γ 2 ( 0 ) ] ,
I ( R , z ; ω ) = 2 π A γ 2 ( 0 ) exp [ R 2 / 2 σ I 2 ( 0 ) ] ,
W F ( R , Δ , z 1 ; ω ) d 2 R = W F ( R , Δ , z 2 ; ω ) d 2 R , W F ( R , Δ , z 1 ; ω ) d 2 Δ = W F ( R , Δ , z 2 ; ω ) d 2 Δ , S F ( ρ , z 1 ; ω ) d 2 ρ = S F ( ρ , z 2 ; ω ) d 2 ρ , S ¯ F ( ρ , z 1 ; ω ) d 2 ρ = S ¯ F ( ρ , z 2 ; ω ) d 2 ρ .
W F ( R , Δ , z ; ω ) U F * ( R Δ / 2 , z ; ω ) U F ( R + Δ / 2 , z ; ω ) , S F ( ρ , z ; ω ) U F * ( ρ , z ; ω ) U F ( ρ , z ; ω ) , S ¯ F ( ρ , z ; ω ) U F * ( ρ , z ; ω ) U F ( ρ , z ; ω ) .
E ( r ; ω ) = α [ k 2 U * ( r ; ω ) U ( r ; ω ) + U * ( r ; ω ) U ( r ; ω ) ] , F ( r ; ω ) = i ω α [ U * ( r ; ω ) U ( r ; ω ) U ( r ; ω ) U * ( r ; ω ) ] ,
J ( θ ; ω ) r 2 S ( ) ( r , θ ; ω ) ,
J ( θ ; ω ) 0 unless sin 2 θ 1 .
J ( θ ; ω ) = A k 2 γ 2 ( 0 ) σ I 2 ( 0 ) cos 2 θ exp { [ k 2 γ 2 ( 0 ) sin 2 θ ] / 2 } .
exp { [ k 2 γ 2 ( 0 ) sin 2 θ ] / 2 } 0 unless sin 2 θ 1 .
1 γ 2 ( 0 ) 2 π 2 λ 2 .
1 σ μ 2 ( 0 ) + 1 4 σ I 2 ( 0 ) 2 π 2 λ 2 .
UF(r;ω)=u21a(u;ω)exp(iku+r)d2u+u21b(u;ω)exp(ikur)d2u,

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