Abstract

The Wigner distribution function of optical signals and systems has been introduced. The concept of such functions is not restricted to deterministic signals, but can be applied to partially coherent light as well. Although derived from Fourier optics, the description of signals and systems by means of Wigner distribution functions can be interpreted directly in terms of geometrical optics: (i) for quadratic-phase signals (and, if complex rays are allowed to appear, for Gaussian signals, too), it leads immediately to the curvature matrix of the signal; (ii) for Luneburg’s first-order system, it directly yields the ray transformation matrix of the system; (iii) for the propagation of quadratic-phase signals through first-order systems, it results in the well-known bilinear transformation of the signal’s curvature matrix. The zeroth-, first-, and second-order moments of the Wigner distribution function have been interpreted in terms of the energy, the center of gravity, and the effective width of the signal, respectively. The propagation of these moments through first-order systems has been derived. Since a Gaussian signal is completely described by its three lowest-order moments, the propagation of such a signal through first-order systems is known as well.

© 1980 Optical Society of America

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  1. H. J. Butterweck, "General theory of linear, coherent optical data-processing systems," J. Opt. Soc. Am. 67, 60–70 (1977).
  2. R. K. Luneburg, Mathematical Theory of Optics (University of California, Berkeley and Los Angeles, 1966).
  3. E. Wigner, "On the quantum correction for thermodynamic equilibrium," Phys. Rev. 40, 749–759 (1932).
  4. H. Mori, I. Oppenheim, and J. Ross, "Some topics in quantum statistics: The Wigner function and transport theory," in Studies in Statistical Mechanics, edited by J. de Boer and G. E. Uhlenbeck (North-Holland, Amsterdam, 1962), Vol. 1, 213–298.
  5. N. G. de Bruijn, "A theory of generalized functions, with applications to Wigner distribution and Weyl correspondence," Nieuw Archief voor Wiskunde 21 (3), 205–280 (1973).
  6. M. J. Bastiaans, "The Wigner distribution function applied to optical signals and systems," Opt. Commun. 25, 26–30 (1978).
  7. G. A. Deschamps, "Ray techniques in electromagnetics," Proc. IEEE 60, 1022–1035 (1972).
  8. A. Walther, "Radiometry and coherence," J. Opt. Soc. Am. 58, 1256–1259 (1968).
  9. 9 E. Wolf, "Coherence and radiometry," J. Opt. Soc. Am. 68, 6–17 (1978).
  10. A. Papoulis, "Amnbiguity function in Fourier optics," J. Opt. Soc. Am. 64, 779–788 (1974).
  11. A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1977).
  12. M. J. Bastiaans, "Transport equations for the Wigner distribution function," Opt. Acta (to be published).
  13. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).
  14. M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1975).
  15. A. Papoulis, The Fourier Integral and Its Applications (McGraw-Hill, New York, 1962).
  16. N. G. de Bruijn, "Uncertainty principles in Fourier analysis," in Inequalities, edited by O. Shisha (Academic, New York, 1967), pp. 57–71.

1978 (2)

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

9 E. Wolf, "Coherence and radiometry," J. Opt. Soc. Am. 68, 6–17 (1978).

1977 (1)

1974 (1)

1973 (1)

N. G. de Bruijn, "A theory of generalized functions, with applications to Wigner distribution and Weyl correspondence," Nieuw Archief voor Wiskunde 21 (3), 205–280 (1973).

1968 (1)

1932 (1)

E. Wigner, "On the quantum correction for thermodynamic equilibrium," Phys. Rev. 40, 749–759 (1932).

Bastiaans, M. J.

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

M. J. Bastiaans, "Transport equations for the Wigner distribution function," Opt. Acta (to be published).

Born, M.

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

Butterweck, H. J.

de Bruijn, N. G.

N. G. de Bruijn, "A theory of generalized functions, with applications to Wigner distribution and Weyl correspondence," Nieuw Archief voor Wiskunde 21 (3), 205–280 (1973).

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

Deschamps, G. A.

G. A. Deschamps, "Ray techniques in electromagnetics," Proc. IEEE 60, 1022–1035 (1972).

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (University of California, Berkeley and Los Angeles, 1966).

Mori, H.

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

Oppenheim, I.

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

Papoulis, A.

A. Papoulis, "Amnbiguity function in Fourier optics," J. Opt. Soc. Am. 64, 779–788 (1974).

A. Papoulis, The Fourier Integral and Its Applications (McGraw-Hill, New York, 1962).

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

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

Ross, J.

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

Walther, A.

Wigner, E.

E. Wigner, "On the quantum correction for thermodynamic equilibrium," Phys. Rev. 40, 749–759 (1932).

Wolf, E.

9 E. Wolf, "Coherence and radiometry," J. Opt. Soc. Am. 68, 6–17 (1978).

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

J. Opt. Soc. Am. (4)

Nieuw Archief voor Wiskunde (1)

N. G. de Bruijn, "A theory of generalized functions, with applications to Wigner distribution and Weyl correspondence," Nieuw Archief voor Wiskunde 21 (3), 205–280 (1973).

Opt. Commun. (1)

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

Phys. Rev. (1)

E. Wigner, "On the quantum correction for thermodynamic equilibrium," Phys. Rev. 40, 749–759 (1932).

Other (9)

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

R. K. Luneburg, Mathematical Theory of Optics (University of California, Berkeley and Los Angeles, 1966).

G. A. Deschamps, "Ray techniques in electromagnetics," Proc. IEEE 60, 1022–1035 (1972).

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

M. J. Bastiaans, "Transport equations for the Wigner distribution function," Opt. Acta (to be published).

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

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

A. Papoulis, The Fourier Integral and Its Applications (McGraw-Hill, New York, 1962).

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

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