Abstract

The paper presents a review of the Wigner distribution function (WDF) and of some of its applications to optical problems, especially in the field of partial coherence. The WDF describes a signal in space and in spatial frequency simultaneously and can be considered the local spatial-frequency spectrum of the signal. Although derived in terms of Fourier optics, the description of an optical signal by means of its WDF closely resembles the ray concept in geometrical optics; the WDF thus presents a link between partial coherence and radiometry. Properties of the WDF and its propagation through linear optical systems are considered; again, the description of systems by WDF’s can be interpreted directly in geometric-optical terms. Some examples are included to show how the WDF can be applied to practical problems that arise in the field of partial coherence.

© 1986 Optical Society of America

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  23. M. J. Bastiaans, “Use of the Wigner distribution function in optical problems,” in 1984 European Conference on Optics, Optical Systems and Applications, Proc. Soc. Photo-Opt. Instrum. Eng.492, 251–262 (1984).
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    [CrossRef]
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    [CrossRef]
  37. M. J. Bastiaans, “The Wigner distribution function and Hamilton’s characteristics of a geometric-optical system,” Opt. Commun. 30, 321–326 (1979).
    [CrossRef]
  38. H. J. Butterweck, “General theory of linear, coherent-optical data-processing systems,” J. Opt. Soc. Am. 67, 60–70 (1977).
    [CrossRef]
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    [CrossRef]
  40. R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1966).
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    [CrossRef]
  42. M. J. Bastiaans, “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979).
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    [CrossRef]
  44. H. Bremmer, “General remarks concerning theories dealing with scattering and diffraction in random media,” Radio Sci. 8, 511–534 (1973).
    [CrossRef]
  45. J. J. McCoy, M. J. Beran, “Propagation of beamed signals through inhomogeneous media: a diffraction theory,” J. Acoust. Soc. Am. 59, 1142–1149 (1976).
    [CrossRef]
  46. I. M. Besieris, F. D. Tappert, “Stochastic wave-kinetic theory in the Liouville approximation,” J. Math. Phys. 17, 734–743 (1976).
    [CrossRef]
  47. H. Bremmer, “The Wigner distribution and transport equations in radiation problems,” J. Appl. Science Eng. A 3, 251–260 (1979).
  48. M. J. Bastiaans, “Transport equations for the Wigner distribution function,” Opt. Acta 26, 1265–1272 (1979).
    [CrossRef]
  49. M. J. Bastiaans, “Transport equations for the Wigner distribution function in an inhomogeneous and dispersive medium,” Opt. Acta 26, 1333–1344 (1979).
    [CrossRef]
  50. R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1960), Vol. 2.
  51. A. T. Friberg, “On the generalized radiance associated with radiation from a quasihomogeneous planar source,” Opt. Acta 28, 261–277 (1981).
    [CrossRef]
  52. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975).
  53. M. J. Bastiaans, “Uncertainty principle and informational entropy for partially coherent light,” J. Opt. Soc. Am. A 3, 1243–1246 (1986).
    [CrossRef]
  54. M. J. Bastiaans, “Uncertainty principle for partially coherent light,” J. Opt. Soc. Am. 73, 251–255 (1983).
    [CrossRef]
  55. M. J. Bastiaans, “New class of uncertainty relations for partially coherent light,” J. Opt. Soc. Am. A 1, 711–715 (1984).
    [CrossRef]
  56. J.-Z. Jiao, B. Wang, H. Liu, “Wigner distribution function and optical geometrical transformation,” Appl. Opt. 23, 1249–1254 (1984).
    [CrossRef] [PubMed]
  57. A. W. Lohmann, J. Ojeda-Castañeda, N. Streibl, “The influence of wave aberrations on the Wigner distribution,” Opt. Appl. 13, 465–471 (1983).
  58. S. Frankenthal, M. J. Beran, A. M. Whitman, “Caustic correction using coherence theory,” J. Acoust. Soc. Am. 71, 348–358 (1982).
    [CrossRef]
  59. A. J. E. M. Janssen, “On the locus and spread of pseudo-density functions in the time-frequency plane,” Philips J. Res. 37, 79–110 (1982).
  60. N. Marcuvitz, “Quasiparticle view of wave propagation,” Proc. IEEE 68, 1380–1395 (1980).
    [CrossRef]
  61. D. S. Bugnolo, H. Bremmer, “The Wigner distribution matrix for the electric field in a stochastic dielectric with computer simulation,” Adv. Electr. Phys. 61, 299–389 (1983).
    [CrossRef]
  62. J. Ojeda-Castañeda, E. E. Sicre, “Quasi ray-optical approach to longitudinal periodicities of free and bounded wave-fields,” Opt. Acta 32, 17–26 (1985).
    [CrossRef]
  63. W. van Etten, W. Lambo, P. Simons, “Loss in multimode fiber connections with a gap,” Appl. Opt. 24, 970–976 (1985).
    [CrossRef] [PubMed]

1986 (1)

1985 (2)

J. Ojeda-Castañeda, E. E. Sicre, “Quasi ray-optical approach to longitudinal periodicities of free and bounded wave-fields,” Opt. Acta 32, 17–26 (1985).
[CrossRef]

W. van Etten, W. Lambo, P. Simons, “Loss in multimode fiber connections with a gap,” Appl. Opt. 24, 970–976 (1985).
[CrossRef] [PubMed]

1984 (3)

1983 (4)

M. J. Bastiaans, “Lower bound in the uncertainty principle for partially coherent light,” J. Opt. Soc. Am. 73, 1320–1324 (1983).
[CrossRef]

A. W. Lohmann, J. Ojeda-Castañeda, N. Streibl, “The influence of wave aberrations on the Wigner distribution,” Opt. Appl. 13, 465–471 (1983).

M. J. Bastiaans, “Uncertainty principle for partially coherent light,” J. Opt. Soc. Am. 73, 251–255 (1983).
[CrossRef]

D. S. Bugnolo, H. Bremmer, “The Wigner distribution matrix for the electric field in a stochastic dielectric with computer simulation,” Adv. Electr. Phys. 61, 299–389 (1983).
[CrossRef]

1982 (5)

S. Frankenthal, M. J. Beran, A. M. Whitman, “Caustic correction using coherence theory,” J. Acoust. Soc. Am. 71, 348–358 (1982).
[CrossRef]

A. J. E. M. Janssen, “On the locus and spread of pseudo-density functions in the time-frequency plane,” Philips J. Res. 37, 79–110 (1982).

E. Wolf, “New theory of partial coherence in the space-frequency domain. 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]

H. M. Pedersen, “Radiometry and coherence for quasi-homogeneous scalar wavefields,” Opt. Acta 29, 877–892 (1982).
[CrossRef]

1981 (4)

A. T. Friberg, “On the generalized radiance associated with radiation from a quasihomogeneous planar source,” Opt. Acta 28, 261–277 (1981).
[CrossRef]

A. J. E. M. Janssen, “Positivity of weighted Wigner distributions,” SIAM J. Math. Anal. 12, 752–758 (1981).
[CrossRef]

A. J. E. M. Janssen, “Weighted Wigner distributions vanishing on lattices,” J. Math. Anal. Appl. 80, 156–167 (1981).
[CrossRef]

M. J. Bastiaans, “The Wigner distribution function of partially coherent light,” Opt. Acta 28, 1215–1224 (1981).
[CrossRef]

1980 (3)

T. A. C. M. Claasen, W. F. G. Mecklenbräuker, “The Wigner distribution—a tool for time-frequency signal analysis,” Part I: “Continuous-time signals”; Part II, “Discrete-time signals”; Part III, “Relations with other time-frequency signal transformations”; Philips J. Res. 35, 217–250, 276–300, 372–389 (1980).

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

N. Marcuvitz, “Quasiparticle view of wave propagation,” Proc. IEEE 68, 1380–1395 (1980).
[CrossRef]

1979 (7)

A. T. Friberg, “On the existence of a radiance function for finite planar sources of arbitrary states of coherence,” J. Opt. Soc. Am. 69, 192–198 (1979).
[CrossRef]

R. Winston, W. T. Welford, “Geometrical vector flux and some new nonimaging concentrators,” J. Opt. Soc. Am. 69, 532–536 (1979).
[CrossRef]

M. J. Bastiaans, “The Wigner distribution function and Hamilton’s characteristics of a geometric-optical system,” Opt. Commun. 30, 321–326 (1979).
[CrossRef]

H. Bremmer, “The Wigner distribution and transport equations in radiation problems,” J. Appl. Science Eng. A 3, 251–260 (1979).

M. J. Bastiaans, “Transport equations for the Wigner distribution function,” Opt. Acta 26, 1265–1272 (1979).
[CrossRef]

M. J. Bastiaans, “Transport equations for the Wigner distribution function in an inhomogeneous and dispersive medium,” Opt. Acta 26, 1333–1344 (1979).
[CrossRef]

M. J. Bastiaans, “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979).
[CrossRef]

1978 (3)

1977 (3)

1976 (3)

J. J. McCoy, M. J. Beran, “Propagation of beamed signals through inhomogeneous media: a diffraction theory,” J. Acoust. Soc. Am. 59, 1142–1149 (1976).
[CrossRef]

I. M. Besieris, F. D. Tappert, “Stochastic wave-kinetic theory in the Liouville approximation,” J. Math. Phys. 17, 734–743 (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 (2)

N. G. de Bruijn, “A theory of generalized functions, with applications to Wigner distribution and Weyl correspondence,” Nieuw Arch. Wiskunde 21, 205–280 (1973).

H. Bremmer, “General remarks concerning theories dealing with scattering and diffraction in random media,” Radio Sci. 8, 511–534 (1973).
[CrossRef]

1972 (1)

G. A. Deschamps, “Ray techniques in electromagnetics,” Proc. IEEE 60, 1022–1035 (1972).
[CrossRef]

1970 (1)

W. D. Mark, “Spectral analysis of the convolution and filtering of non-stationary stochastic processes,” J. Sound Vib. 11, 19–63 (1970).
[CrossRef]

1968 (1)

1966 (1)

L. Cohen, “Generalized phase-space distribution functions,” J. Math. Phys. 7, 781–786 (1966).
[CrossRef]

1955 (1)

E. Wolf, “A macroscopic theory of interference and diffraction of light from finite sources. II. Fields with a spectral range of arbitrary width,” Proc. R. Soc. London Ser. A 230, 246–265 (1955).
[CrossRef]

1954 (1)

E. Wolf, “A macroscopic theory of interference and diffraction of light from finite sources. I. Fields with a narrow spectral range,” Proc. R. Soc. London Ser. A 225, 96–111 (1954).
[CrossRef]

1932 (1)

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Bastiaans, M. J.

M. J. Bastiaans, “Uncertainty principle and informational entropy for partially coherent light,” J. Opt. Soc. Am. A 3, 1243–1246 (1986).
[CrossRef]

M. J. Bastiaans, “New class of uncertainty relations for partially coherent light,” J. Opt. Soc. Am. A 1, 711–715 (1984).
[CrossRef]

M. J. Bastiaans, “Uncertainty principle for partially coherent light,” J. Opt. Soc. Am. 73, 251–255 (1983).
[CrossRef]

M. J. Bastiaans, “Lower bound in the uncertainty principle for partially coherent light,” J. Opt. Soc. Am. 73, 1320–1324 (1983).
[CrossRef]

M. J. Bastiaans, “The Wigner distribution function of partially coherent light,” Opt. Acta 28, 1215–1224 (1981).
[CrossRef]

M. J. Bastiaans, “The Wigner distribution function and Hamilton’s characteristics of a geometric-optical system,” Opt. Commun. 30, 321–326 (1979).
[CrossRef]

M. J. Bastiaans, “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979).
[CrossRef]

M. J. Bastiaans, “Transport equations for the Wigner distribution function,” Opt. Acta 26, 1265–1272 (1979).
[CrossRef]

M. J. Bastiaans, “Transport equations for the Wigner distribution function in an inhomogeneous and dispersive medium,” Opt. Acta 26, 1333–1344 (1979).
[CrossRef]

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
[CrossRef]

M. J. Bastiaans, “A frequency-domain treatment of partial coherence,” Opt. Acta 24, 261–274 (1977).
[CrossRef]

M. J. Bastiaans, “The Wigner distribution function and its applications in optics,” in Optics in Four Dimensions—1980, AIP Conf. Proc. 65, M. A. Machado, L. M. Narducci, eds. (American Institute of Physics, New York, 1980), pp. 292–312.

M. J. Bastiaans, “Signal description by means of a local frequency spectrum,” in Transformations in Optical Signal Processing, W. T. Rhodes, J. R. Fienup, B. E. A. Saleh, eds., Proc. Soc. Photo-Opt. Instrum. Eng.373, 49–62 (1981).
[CrossRef]

M. J. Bastiaans, “Use of the Wigner distribution function in optical problems,” in 1984 European Conference on Optics, Optical Systems and Applications, Proc. Soc. Photo-Opt. Instrum. Eng.492, 251–262 (1984).
[CrossRef]

Beran, M. J.

S. Frankenthal, M. J. Beran, A. M. Whitman, “Caustic correction using coherence theory,” J. Acoust. Soc. Am. 71, 348–358 (1982).
[CrossRef]

J. J. McCoy, M. J. Beran, “Propagation of beamed signals through inhomogeneous media: a diffraction theory,” J. Acoust. Soc. Am. 59, 1142–1149 (1976).
[CrossRef]

Besieris, I. M.

I. M. Besieris, F. D. Tappert, “Stochastic wave-kinetic theory in the Liouville approximation,” J. Math. Phys. 17, 734–743 (1976).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975).

Bremmer, H.

D. S. Bugnolo, H. Bremmer, “The Wigner distribution matrix for the electric field in a stochastic dielectric with computer simulation,” Adv. Electr. Phys. 61, 299–389 (1983).
[CrossRef]

H. Bremmer, “The Wigner distribution and transport equations in radiation problems,” J. Appl. Science Eng. A 3, 251–260 (1979).

H. Bremmer, “General remarks concerning theories dealing with scattering and diffraction in random media,” Radio Sci. 8, 511–534 (1973).
[CrossRef]

Brenner, K.-H.

K.-H. Brenner, J. Ojeda-Castañeda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta 31, 213–223 (1984).
[CrossRef]

Bugnolo, D. S.

D. S. Bugnolo, H. Bremmer, “The Wigner distribution matrix for the electric field in a stochastic dielectric with computer simulation,” Adv. Electr. Phys. 61, 299–389 (1983).
[CrossRef]

Butterweck, H. J.

H. J. Butterweck, “General theory of linear, coherent-optical data-processing systems,” J. Opt. Soc. Am. 67, 60–70 (1977).
[CrossRef]

H. J. Butterweck, “Principles of optical data-processing,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1981), Vol. 19, pp. 211–280.
[CrossRef]

Carter, W. H.

Claasen, T. A. C. M.

T. A. C. M. Claasen, W. F. G. Mecklenbräuker, “The Wigner distribution—a tool for time-frequency signal analysis,” Part I: “Continuous-time signals”; Part II, “Discrete-time signals”; Part III, “Relations with other time-frequency signal transformations”; Philips J. Res. 35, 217–250, 276–300, 372–389 (1980).

Cohen, L.

L. Cohen, “Generalized phase-space distribution functions,” J. Math. Phys. 7, 781–786 (1966).
[CrossRef]

Courant, R.

R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1953), Vol. 1.

R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1960), Vol. 2.

de Bruijn, N. G.

N. G. de Bruijn, “A theory of generalized functions, with applications to Wigner distribution and Weyl correspondence,” Nieuw Arch. Wiskunde 21, 205–280 (1973).

N. G. de Bruijn, “Uncertainty principles in Fourier analysis,” in Inequalities, O. Shisha, ed. (Academic, New York, 1967), pp. 57–71.

Deschamps, G. A.

G. A. Deschamps, “Ray techniques in electromagnetics,” Proc. IEEE 60, 1022–1035 (1972).
[CrossRef]

Frankenthal, S.

S. Frankenthal, M. J. Beran, A. M. Whitman, “Caustic correction using coherence theory,” J. Acoust. Soc. Am. 71, 348–358 (1982).
[CrossRef]

Friberg, A. T.

A. T. Friberg, “On the generalized radiance associated with radiation from a quasihomogeneous planar source,” Opt. Acta 28, 261–277 (1981).
[CrossRef]

A. T. Friberg, “On the existence of a radiance function for finite planar sources of arbitrary states of coherence,” J. Opt. Soc. Am. 69, 192–198 (1979).
[CrossRef]

A. T. Friberg, “Phase-space methods for partially coherent wave fields,” in Optics in Four Dimensions—1980, AIP Conf. Proc. 65, M. A. Machado, L. M. Narducci, eds. (American Institute of Physics, New York, 1980), pp. 313–331.

Gamo, H.

H. Gamo, “Matrix treatment of partial coherence,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1964), Vol. 3, pp. 187–332.
[CrossRef]

Gori, F.

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

Hilbert, D.

R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1953), Vol. 1.

R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1960), Vol. 2.

Janssen, A. J. E. M.

A. J. E. M. Janssen, “On the locus and spread of pseudo-density functions in the time-frequency plane,” Philips J. Res. 37, 79–110 (1982).

A. J. E. M. Janssen, “Positivity of weighted Wigner distributions,” SIAM J. Math. Anal. 12, 752–758 (1981).
[CrossRef]

A. J. E. M. Janssen, “Weighted Wigner distributions vanishing on lattices,” J. Math. Anal. Appl. 80, 156–167 (1981).
[CrossRef]

Jiao, J.-Z.

Lambo, W.

Liu, H.

Lohmann, A. W.

A. W. Lohmann, J. Ojeda-Castañeda, N. Streibl, “The influence of wave aberrations on the Wigner distribution,” Opt. Appl. 13, 465–471 (1983).

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1966).

Mandel, L.

Marcuvitz, N.

N. Marcuvitz, “Quasiparticle view of wave propagation,” Proc. IEEE 68, 1380–1395 (1980).
[CrossRef]

Mark, W. D.

W. D. Mark, “Spectral analysis of the convolution and filtering of non-stationary stochastic processes,” J. Sound Vib. 11, 19–63 (1970).
[CrossRef]

McCoy, J. J.

J. J. McCoy, M. J. Beran, “Propagation of beamed signals through inhomogeneous media: a diffraction theory,” J. Acoust. Soc. Am. 59, 1142–1149 (1976).
[CrossRef]

Mecklenbräuker, W. F. G.

T. A. C. M. Claasen, W. F. G. Mecklenbräuker, “The Wigner distribution—a tool for time-frequency signal analysis,” Part I: “Continuous-time signals”; Part II, “Discrete-time signals”; Part III, “Relations with other time-frequency signal transformations”; Philips J. Res. 35, 217–250, 276–300, 372–389 (1980).

Mori, H.

H. Mori, I. Oppenheim, J. Ross, “Some topics in quantum statistics: the Wigner function and transport theory,” in Studies in Statistical Mechanics, J. de Boer, G. E. Uhlenbeck, eds. (North-Holland, Amsterdam, 1962), Vol. 1, pp. 213–298.

Ojeda-Castañeda, J.

J. Ojeda-Castañeda, E. E. Sicre, “Quasi ray-optical approach to longitudinal periodicities of free and bounded wave-fields,” Opt. Acta 32, 17–26 (1985).
[CrossRef]

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. W. Lohmann, J. Ojeda-Castañeda, N. Streibl, “The influence of wave aberrations on the Wigner distribution,” Opt. Appl. 13, 465–471 (1983).

Oppenheim, I.

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H. Mori, I. Oppenheim, J. Ross, “Some topics in quantum statistics: the Wigner function and transport theory,” in Studies in Statistical Mechanics, J. de Boer, G. E. Uhlenbeck, eds. (North-Holland, Amsterdam, 1962), Vol. 1, pp. 213–298.

Schell, A. C.

A. C. Schell, The Multiple Plate Antenna, Ph.D. dissertation (Massachusetts Institute of Technology, Cambridge, Mass., 1961), Sec. 7.5.

Sicre, E. E.

J. Ojeda-Castañeda, E. E. Sicre, “Quasi ray-optical approach to longitudinal periodicities of free and bounded wave-fields,” Opt. Acta 32, 17–26 (1985).
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Simons, P.

Starikov, A.

Streibl, N.

A. W. Lohmann, J. Ojeda-Castañeda, N. Streibl, “The influence of wave aberrations on the Wigner distribution,” Opt. Appl. 13, 465–471 (1983).

Sz.-Nagy, B.

F. Riesz, B. Sz.-Nagy, Functional Analysis (Ungar, New York, 1955).

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I. M. Besieris, F. D. Tappert, “Stochastic wave-kinetic theory in the Liouville approximation,” J. Math. Phys. 17, 734–743 (1976).
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van Etten, W.

Walther, A.

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E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Winston, R.

Wolf, E.

Adv. Electr. Phys. (1)

D. S. Bugnolo, H. Bremmer, “The Wigner distribution matrix for the electric field in a stochastic dielectric with computer simulation,” Adv. Electr. Phys. 61, 299–389 (1983).
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Appl. Opt. (2)

J. Acoust. Soc. Am. (2)

J. J. McCoy, M. J. Beran, “Propagation of beamed signals through inhomogeneous media: a diffraction theory,” J. Acoust. Soc. Am. 59, 1142–1149 (1976).
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S. Frankenthal, M. J. Beran, A. M. Whitman, “Caustic correction using coherence theory,” J. Acoust. Soc. Am. 71, 348–358 (1982).
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J. Appl. Science Eng. A (1)

H. Bremmer, “The Wigner distribution and transport equations in radiation problems,” J. Appl. Science Eng. A 3, 251–260 (1979).

J. Math. Anal. Appl. (1)

A. J. E. M. Janssen, “Weighted Wigner distributions vanishing on lattices,” J. Math. Anal. Appl. 80, 156–167 (1981).
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I. M. Besieris, F. D. Tappert, “Stochastic wave-kinetic theory in the Liouville approximation,” J. Math. Phys. 17, 734–743 (1976).
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M. J. Bastiaans, “Uncertainty principle for partially coherent light,” J. Opt. Soc. Am. 73, 251–255 (1983).
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M. J. Bastiaans, “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979).
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A. Walther, “Propagation of the generalized radiance through lenses,” J. Opt. Soc. Am. 68, 1606–1610 (1978).
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W. H. Carter, E. Wolf, “Coherence and radiometry with quasi-homogeneous planar sources,” J. Opt. Soc. Am. 67, 785–796 (1977).
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E. Wolf, “Coherence and radiometry,” J. Opt. Soc. Am. 68, 6–17 (1978).
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A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 58, 1256–1259 (1968).
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L. Mandel, E. Wolf, “Spectral coherence and the concept of cross-spectral purity,” J. Opt. Soc. Am. 66, 529–535 (1976).
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H. J. Butterweck, “General theory of linear, coherent-optical data-processing systems,” J. Opt. Soc. Am. 67, 60–70 (1977).
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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).
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M. J. Bastiaans, “Lower bound in the uncertainty principle for partially coherent light,” J. Opt. Soc. Am. 73, 1320–1324 (1983).
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A. T. Friberg, “On the existence of a radiance function for finite planar sources of arbitrary states of coherence,” J. Opt. Soc. Am. 69, 192–198 (1979).
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R. Winston, W. T. Welford, “Geometrical vector flux and some new nonimaging concentrators,” J. Opt. Soc. Am. 69, 532–536 (1979).
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E. Wolf, “New theory of partial coherence in the space-frequency domain. I. Spectra and cross spectra of steady state sources,” J. Opt. Soc. Am. 72, 343–351 (1982).
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M. J. Bastiaans, “The Wigner distribution function of partially coherent light,” Opt. Acta 28, 1215–1224 (1981).
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M. J. Bastiaans, “A frequency-domain treatment of partial coherence,” Opt. Acta 24, 261–274 (1977).
[CrossRef]

H. M. Pedersen, “Radiometry and coherence for quasi-homogeneous scalar wavefields,” Opt. Acta 29, 877–892 (1982).
[CrossRef]

K.-H. Brenner, J. Ojeda-Castañeda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta 31, 213–223 (1984).
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M. J. Bastiaans, “Transport equations for the Wigner distribution function,” Opt. Acta 26, 1265–1272 (1979).
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M. J. Bastiaans, “Transport equations for the Wigner distribution function in an inhomogeneous and dispersive medium,” Opt. Acta 26, 1333–1344 (1979).
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A. T. Friberg, “On the generalized radiance associated with radiation from a quasihomogeneous planar source,” Opt. Acta 28, 261–277 (1981).
[CrossRef]

J. Ojeda-Castañeda, E. E. Sicre, “Quasi ray-optical approach to longitudinal periodicities of free and bounded wave-fields,” Opt. Acta 32, 17–26 (1985).
[CrossRef]

Opt. Appl. (1)

A. W. Lohmann, J. Ojeda-Castañeda, N. Streibl, “The influence of wave aberrations on the Wigner distribution,” Opt. Appl. 13, 465–471 (1983).

Opt. Commun. (3)

M. J. Bastiaans, “The Wigner distribution function and Hamilton’s characteristics of a geometric-optical system,” Opt. Commun. 30, 321–326 (1979).
[CrossRef]

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
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F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
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Philips J. Res. (2)

T. A. C. M. Claasen, W. F. G. Mecklenbräuker, “The Wigner distribution—a tool for time-frequency signal analysis,” Part I: “Continuous-time signals”; Part II, “Discrete-time signals”; Part III, “Relations with other time-frequency signal transformations”; Philips J. Res. 35, 217–250, 276–300, 372–389 (1980).

A. J. E. M. Janssen, “On the locus and spread of pseudo-density functions in the time-frequency plane,” Philips J. Res. 37, 79–110 (1982).

Phys. Rev. (1)

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

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N. Marcuvitz, “Quasiparticle view of wave propagation,” Proc. IEEE 68, 1380–1395 (1980).
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Proc. R. Soc. London Ser. A (2)

E. Wolf, “A macroscopic theory of interference and diffraction of light from finite sources. I. Fields with a narrow spectral range,” Proc. R. Soc. London Ser. A 225, 96–111 (1954).
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E. Wolf, “A macroscopic theory of interference and diffraction of light from finite sources. II. Fields with a spectral range of arbitrary width,” Proc. R. Soc. London Ser. A 230, 246–265 (1955).
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Radio Sci. (1)

H. Bremmer, “General remarks concerning theories dealing with scattering and diffraction in random media,” Radio Sci. 8, 511–534 (1973).
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A. J. E. M. Janssen, “Positivity of weighted Wigner distributions,” SIAM J. Math. Anal. 12, 752–758 (1981).
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A. Erdélyi, ed., Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol. 2, Chap. 10.

N. G. de Bruijn, “Uncertainty principles in Fourier analysis,” in Inequalities, O. Shisha, ed. (Academic, New York, 1967), pp. 57–71.

H. J. Butterweck, “Principles of optical data-processing,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1981), Vol. 19, pp. 211–280.
[CrossRef]

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1966).

M. J. Bastiaans, “Signal description by means of a local frequency spectrum,” in Transformations in Optical Signal Processing, W. T. Rhodes, J. R. Fienup, B. E. A. Saleh, eds., Proc. Soc. Photo-Opt. Instrum. Eng.373, 49–62 (1981).
[CrossRef]

M. J. Bastiaans, “Use of the Wigner distribution function in optical problems,” in 1984 European Conference on Optics, Optical Systems and Applications, Proc. Soc. Photo-Opt. Instrum. Eng.492, 251–262 (1984).
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H. Gamo, “Matrix treatment of partial coherence,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1964), Vol. 3, pp. 187–332.
[CrossRef]

R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1953), Vol. 1.

F. Riesz, B. Sz.-Nagy, Functional Analysis (Ungar, New York, 1955).

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

H. Mori, I. Oppenheim, J. Ross, “Some topics in quantum statistics: the Wigner function and transport theory,” in Studies in Statistical Mechanics, J. de Boer, G. E. Uhlenbeck, eds. (North-Holland, Amsterdam, 1962), Vol. 1, pp. 213–298.

M. J. Bastiaans, “The Wigner distribution function and its applications in optics,” in Optics in Four Dimensions—1980, AIP Conf. Proc. 65, M. A. Machado, L. M. Narducci, eds. (American Institute of Physics, New York, 1980), pp. 292–312.

A. C. Schell, The Multiple Plate Antenna, Ph.D. dissertation (Massachusetts Institute of Technology, Cambridge, Mass., 1961), Sec. 7.5.

A. T. Friberg, “Phase-space methods for partially coherent wave fields,” in Optics in Four Dimensions—1980, AIP Conf. Proc. 65, M. A. Machado, L. M. Narducci, eds. (American Institute of Physics, New York, 1980), pp. 313–331.

R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1960), Vol. 2.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975).

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Equations (94)

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E ϕ ˜ ( x 1 , t 1 ) ϕ ˜ * ( x 2 , t 2 ) = Γ ˜ ( x 1 , x 2 , t 1 t 2 ) ,
Γ ( x 1 , x 2 , ω ) = Γ ˜ ( x 1 , x 2 , τ ) exp ( i ω τ ) d τ .
Γ ¯ ( u 1 , u 2 , ω ) = Γ ( x 1 , x 2 , ω ) exp [ i ( u 1 x 1 u 2 x 2 ) ] d x 1 d x 2 .
F ( x , u ) = Γ ( x + 1 2 x , x 1 2 x ) exp ( i u x ) d x
F ( x , u ) = 1 2 π Γ ¯ ( u + 1 2 u , u 1 2 u ) exp ( i u x ) d u .
s ¯ ( u ) = s ( x ) exp ( i u x ) d x .
Γ ( x 1 , x 2 ) = 2 σ ρ exp { π 2 ρ 2 [ σ ( x 1 + x 2 ) 2 + 1 σ ( x 1 x 2 ) 2 ] } ( ρ > 0 , 0 < σ 1 ) ;
F ( x , u ) = 2 σ exp [ σ ( 2 π ρ 2 x 2 + ρ 2 2 π u 2 ) ] ( ρ > 0 , 0 < σ 1 ) ,
f ( x , u ) = q ( x + 1 2 x ) q * ( x 1 2 x ) exp ( i u x ) d x .
1 2 π F ( x , u ) d u = Γ ( x , x )
F ( x , u ) d x = Γ ¯ ( u , u )
1 2 π F ( x , u ) d x d u = Γ ( x , x ) d x = 1 2 π Γ ¯ ( u , u ) d u .
1 2 π x 2 F ( x , u ) d x d u 1 2 π F ( x , u ) d x d u = x 2 Γ ( x , x ) d x Γ ( x , x ) d x = d x 2 ;
J z ( x ) = 1 2 π ( k 2 u 2 ) 1 / 2 k F ( x , u ) d u ,
J x ( x ) = 1 2 π u k F ( x , u ) d u ,
J ( x ) = [ J x ( x ) , J z ( x ) ] ,
1 2 π F 1 ( x , u ) F 2 ( x , u ) d x d u = Γ 1 ( x 1 , x 2 ) Γ * ( x 1 , x 2 ) d x 1 d x 2 = 1 4 π 2 Γ ¯ 1 ( u 1 , u 2 ) Γ ¯ * 2 ( u 1 , u 2 ) d u 1 d u 2 .
Γ ( x 1 , x 2 ) = 1 ρ m = 0 λ m q m ( x 1 / ρ ) q * m ( x 2 / ρ ) ;
Γ ( x 1 , x 2 ) q m ( x 2 / ρ ) d x 2 = λ m q m ( x 1 / ρ ) ( m = 0 , 1 , ) ;
q m ( ξ ) q * n ( ξ ) d ξ = { 1 m = n 0 m n ( m , n = 0 , 1 , ) .
λ m = 2 σ 1 + σ ( 1 σ 1 + σ ) m ( 0 < σ 1 , m = 0 , 1 , ) ,
exp [ π ξ 2 2 π ( ξ w ) 2 ] = 2 1 / 4 m = 0 ( m ! ) 1 / 2 ( 4 π ) m / 2 w m ψ m ( ξ ) .
F ( x , u ) = m = 0 λ m f m ( x ρ , ρ u ) ,
f m ( ξ , η ) = q m ( ξ + 1 2 ξ ) q * m ( ξ 1 2 ξ ) exp ( i η ξ ) d ξ ( m = 0 , 1 , )
1 2 π f m ( ξ , η ) f n ( ξ , η ) d ξ d η = | q m ( ξ ) q * n ( ξ ) d ξ | 2 = { 1 m = n 0 m n ( m , n = 0 , 1 , ) .
f m ( ξ , η ) = 2 ( 1 ) m exp [ ( 2 π ξ 2 + η 2 2 π ) ] L m [ 2 ( 2 π ξ 2 + η 2 2 π ) ] ( m = 0 , 1 , ) ,
1 2 π ( 2 π ρ 2 x 2 + ρ 2 2 π u 2 ) n F ( x , u ) d x d u n ! 1 2 π F ( x , u ) d x d u
1 2 π F 1 ( x , u ) F 2 ( x , u ) d x d u 0.
1 2 π F 1 ( x , u ) F 2 ( x , u ) d x d u [ 1 2 π F 1 2 ( x , u ) d x d u ] 1 / 2 [ 1 2 π F 2 2 ( x , u ) d x d u ] 1 / 2 .
1 2 π F 1 ( x , u ) F 2 ( x , u ) d x d u [ 1 2 π F 1 ( x , u ) d x d u ] [ 1 2 π F 2 ( x , u ) d x d u ] ,
1 2 π F 2 ( x , u ) d x d u [ 1 2 π F ( x , u ) d x d u ] 2 .
1 2 π F ( x , u ) d x d u = m = 0 λ m
1 2 π F 2 ( x , u ) d x d u = m = 0 λ m 2 ,
m = 0 λ m 2 ( m = 0 λ m ) 2
q 0 ( x 0 ) = h x x ( x 0 , x i ) q i ( x i ) d x i ,
q ¯ 0 ( u 0 ) = h u x ( u 0 , x i ) q i ( x i ) d x i ,
q 0 ( x 0 ) = 1 2 π h x u ( x 0 , u i ) q ¯ i ( u i ) d u i ,
q ¯ 0 ( u 0 ) = 1 2 π h u u ( u 0 , u i ) q ¯ i ( u i ) d u i .
Γ 0 ( x 1 , x 2 ) = h x x ( x 1 , ξ 1 ) Γ i ( ξ 1 , ξ 2 ) h * x x ( x 2 , ξ 2 ) d ξ 1 d ξ 2 ,
Γ ¯ 0 ( u 1 , u 2 ) = h u x ( u 1 , ξ 1 ) Γ i ( ξ 1 , ξ 2 ) h * u x ( u 2 , ξ 2 ) d ξ 1 d ξ 2 ,
Γ 0 ( x 1 , x 2 ) = 1 4 π 2 h x u ( x 1 , η 1 ) Γ ¯ i ( η 1 , η 2 ) h * x u ( x 2 , η 2 ) d η 1 d η 2 ,
Γ ¯ 0 ( u 1 , u 2 ) = 1 4 π 2 h u u ( u 1 , η 1 ) Γ ¯ i ( η 1 , η 2 ) h * u u ( u 2 , η 2 ) d η 1 d η 2 .
F 0 ( x 0 , u 0 ) = 1 2 π K ( x 0 , u 0 , x i , u i ) F i ( x i , u i ) d x i d u i ,
K ( x 0 , u 0 , x i , u i ) = h x x ( x 0 + 1 2 x 0 , x i + 1 2 x i ) × h * x x ( x 0 1 2 x 0 , x i 1 2 x i ) × exp [ i ( u 0 x 0 u i x i ) ] d x 0 d x i
K ( x 0 , u 0 , x i , u i ) = 1 2 π K 2 ( x 0 , u 0 , x , u ) K 1 ( x , u , x i , u i ) d x d u .
K ( x 0 , u 0 , x i , u i ) = 2 π δ ( x i A x 0 B u 0 ) δ ( u i C x 0 D u 0 ) ,
F 0 ( x , u ) = F i ( A x + B u , C x + D u ) .
( x i u i ) = ( A B C D ) ( x 0 u 0 ) .
i q z = ( k + 1 2 k 2 x 2 ) q ;
i Γ z = [ ( k + 1 2 k 2 x 1 2 ) ( k + 1 2 k 2 x 2 2 ) ] Γ .
u k F x + F z = 0.
F ( x , u ; z ) = F ( x u k z , u ; 0 ) ,
i Γ z = [ L ( x 1 , i x 1 ; z ) L * ( x 2 , i x 2 ; z ) ] Γ ,
F z = 2 Im [ L ( x + i 2 u , u i 2 x ; z ) ] F ,
F z = 2 Im { L ( x , u ; z ) exp [ i 2 ( x u u x ) ] } F ,
F z = 2 Im { L ( x , u ; z ) [ 1 + i 2 ( x u u x ) ] } F .
F z = 2 ( Im L ) F + Re L x F u Re L u F x ,
d x d s = Re L u , d z d s = 1 , d u d s = Re L x ;
d F d s = 2 ( Im L ) F .
i Γ z = [ ( k 2 + 2 x 1 2 ) 1 / 2 ( k 2 + 2 x 2 2 ) 1 / 2 ] Γ .
u k F x + ( k 2 u 2 ) 1 / 2 k F z = 0.
F ( x , u ; z ) = F [ x u ( k 2 u 2 ) 1 / 2 z , u ; 0 ] .
u k F x + ( k 2 u 2 ) 1 / 2 k F z + k x F u = 0 ,
d x d s = u k , d z d s = ( k 2 u 2 ) 1 / 2 k , d u d s = k x ,
d d s ( k d x d s ) = k x , d d s ( k d z d s ) = k z ,
m = 0 ( λ m n = 0 λ n ) ln ( λ m n = 0 λ n )
2 d x d u m = 0 λ m ( 2 m + 1 ) m = 0 λ m ,
exp [ m = 0 ( λ m n = 0 λ n ) ln ( λ m n = 0 λ n ) ] = 2 σ 1 σ 2 ( 1 σ 1 + σ ) 1 / 2 σ .
2 d x d u 1 σ ,
F ( x , u ) = 2 σ exp { σ [ 2 π ρ 2 ( A x + B u ) 2 + ρ 2 2 π ( C x + D u ) 2 ] } .
F ( x , u ) = 2 σ exp { σ [ ( β + α 2 β ) x 2 + 2 α β x u + 1 β u 2 ] } ( β > 0 ) ,
1 β = 2 π ρ 2 B 2 + ρ 2 2 π D 2 , α β = 2 π ρ 2 A B + ρ 2 2 π C D .
β i / β 0 = ( B α i + D ) 2 + B 2 β i 2 , α 0 β i β 0 = ( A α i + C ) ( B α i + D ) + A B β i 2 .
α 0 + i β 0 = A ( α i + i β i ) + C B ( α i + i β i ) + D .
F 0 ( x , u 0 ) = 1 2 π F i ( x , u i ) d u i m ( x + 1 2 x ) m * ( x 1 2 x ) × exp [ i ( u 0 u i ) x ] d x .
F 0 ( x , u ) = m ( x + i 2 u ) m * ( x i 2 u ) F i ( x , u ) .
F 0 ( x , u 0 ) = 1 2 π f m ( x , u 0 u i ) F i ( x , u i ) d u i ,
m ( x + 1 2 x ) m * ( x 1 2 x ) = exp [ i k = 0 2 ( 2 k + 1 ) ! γ ( 2 k + 1 ) ( x ) ( x / 2 ) 2 k + 1 ] ,
m ( x + i 2 u ) m * ( x i 2 u ) exp ( d γ d x u ) ,
f m ( x , u ) 2 π δ ( u d γ d x ) ,
F 0 ( x , u ) F i ( x , u d γ d x ) ,
F 0 ( x , u ) F i [ g x ( x , u ) , g u ( x , u ) ] ,
F ( x , u ) = π 2 x max k sin θ max × rect ( x 2 x max ) rect ( u 2 k sin θ max ) k ( k 2 u 2 ) 1 / 2 ,
1 2 π F ( x , u ) ( k 2 u 2 ) 1 / 2 k d x d u = 1.
1 2 π x max x max d x d u F [ x + u ( k 2 u 2 ) 1 / 2 z 0 , u ] ( k 2 u 2 ) 1 / 2 k = sin γ z 0 2 x max ( 1 cos γ ) sin θ max ,
u F x + υ F y + [ k 2 ( u 2 + υ 2 ) ] 1 / 2 F z + k k x F u + k k y F υ = 0 ,
( k 2 w 2 h 2 r 2 ) 1 / 2 F r + h r 2 F ϕ + w F z = 0.
w d h d z = 0 , w d w d z = 0 , w d r d z = ( k 2 w 2 h 2 r 2 ) 1 / 2 , w d ϕ d z = h r 2 ,
[ i z + L ( x 1 , i x 1 ; z ) L * ( x 2 , i x 2 ; z ) ] × Γ ( x 1 , x 2 ; z ) = 0.
1 2 π [ i z + L ( x 1 , i x 1 ; z ) L * ( x 2 , i x 2 ; z ) ] × F ( x 1 + x 2 2 , u 0 ; z ) exp [ i u 0 ( x 1 x 2 ) ] d u 0 = 0 ,
1 2 π [ i z + L ( x + 1 2 x , u 0 i 2 x ; z ) L * ( x 1 2 x , u 0 i 2 x ; z ) ] × F ( x , u 0 ; z ) exp ( i u 0 x ) d u 0 = 0.
1 2 π [ i z + L ( x + 1 2 x , u 0 i 2 x ; z ) L * ( x 1 2 x , u 0 i 2 x ; z ) ] × F ( x , u 0 ; z ) exp ( i ( u 0 u ) x ) d u 0 d x = 0 ,
1 2 π [ i z + L ( x + i 2 u , u 0 i 2 x ; z ) ] L * ( x + i 2 u , u 0 i 2 x ; z ) ] × F ( x , u 0 ; z ) exp [ i ( u 0 u ) x ] d u 0 d x = 0.
F z + 2 Im [ L ( x + i 2 u , u i 2 x ; z ) ] F = 0 ,

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