Abstract

This tutorial gives an overview of the use of the Wigner function as a tool for modeling optical field propagation. Particular emphasis is placed on the spatial propagation of stationary fields, as well as on the propagation of pulses through dispersive media. In the first case, the Wigner function gives a representation of the field that is similar to a radiance or weight distribution for all the rays in the system, since its arguments are both position and direction. In cases in which the field is paraxial and where the system is described by a simple linear relation in the ray regime, the Wigner function is constant under propagation along rays. An equivalent property holds for optical pulse propagation in dispersive media under analogous assumptions. Several properties and applications of the Wigner function in these contexts are discussed, as is its connection with other common phase-space distributions like the ambiguity function, the spectrogram, and the Husimi, P, Q, and Kirkwood–Rihaczek functions. Also discussed are modifications to the definition of the Wigner function that allow extending the property of conservation along paths to a wider range of problems, including nonparaxial field propagation and pulse propagation within general transparent dispersive media.

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J. C. Petruccelli and M. A. Alonso, "Phase space distributions tailored for dispersive media," J. Opt. Soc. Am. A 27, 1194‒1201 (2010).
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R. Horstmeyer, S. B. Oh, and R. Raskar, "Iterative aperture mask design in phase space using a rank constraint," Opt. Express 18, 22545‒22555 (2010).
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H. Gao, L. Tian, B. Zhang, and G. Barbastathis, "Iterative nonlinear beam propagation using Hamiltonian ray tracing and Wigner distribution function," Opt. Lett. 35, 4148‒4150 (2010).
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J. C. Petruccelli, N, J. Moore, and M. A. Alonso, "Two methods for modeling the propagation of the coherence and polarization properties of nonparaxial fields," Opt. Commun. 283, 4457‒4466 (2010).
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S. B. Mehta and C. J. R. Sheppard, "Using the phase-space imager to analyze partially coherent imaging systems: bright-field, phase contrast, differential interference contrast, differential phase contrast, and spiral phase contrast," J. Mod. Opt. 57, 718‒739 (2010).
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W. P. Schleich, J. P. Dahl, and S. Varró, "Wigner function for a free particle in two dimensions: a tale of interference," Opt. Commun. 283, 786‒789 (2010).
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C. Q. Tran, A. G. Peele, A. Roberts, K. A. Nugent, D. Paterson, and I. McNulty, "X-ray imaging: a generalized approach using phase-space tomography," J. Opt. Soc. Am. A 22, 1691‒1700 (2005).
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R. W. Robinett, "Quantum wave packet revivals," Phys. Rep. 392, 1‒119 (2004).
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M. A. Alonso, "Wigner functions for nonparaxial, arbitrarily polarized electromagnetic wave fields in free-space," J. Opt. Soc. Am. A. 21, 2233‒2243 (2004).
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F. Hlawatsch, A. Papandreou-Suppappola, and G. Boudreaux-Bartels, "The power classes-quadratic time–frequency representations with scale covariance and dispersive time-shift covariance," IEEE Trans. Signal Process. 47, 3067‒3083 (1999).
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K. B. Wolf, M. A. Alonso, and G. W. Forbes, "Wigner functions for Helmholtz wave fields," J. Opt. Soc. Am. A 16, 2476‒2487 (1999).
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J. Tu and S. Tamura, "Analytic relation for recovering the mutual intensity by means of intensity information," J. Opt. Soc. Am. A 15, 202‒206 (1998).
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A. Wax and J. E. Thomas, "Measurement of smoothed Wigner phase-space distributions for small-angle scattering in a turbid medium," J. Opt. Soc. Am. A 15, 1896‒1908 (1998).
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A. Papandreou-Suppappola, F. Hlawatsch, and G. Boudreaux-Bartels, "Quadratic time–frequency representations with scale covariance and generalized time-shift covariance: A unified framework for the affine, hyperbolic, and power classes," Digital Signal Processing 8, 3‒48 (1998).
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M. V. Berry and N. L. Balazs, "Evolution of semiclassical quantum states in phase space," J. Phys. A Math. Phys. 12, 625‒642 (1979).
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Alieva, T.

Alonso, M. A.

J. C. Petruccelli and M. A. Alonso, "Generalized radiometry model for the propagation of light within anisotropic and chiral media," J. Opt. Soc. Am. A 28, 791‒800 (2011).
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S. Cho and M. A. Alonso, "Ambiguity function and phase-space tomography for nonparaxial fields," J. Opt. Soc. Am. A 28, 897‒902 (2011).
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M. J. Bastiaans, Wigner Distribution in Optics, Chapter 1 of Ref. [15], pp. 1–44.

R. K. Luneburg, Mathematical Theory of Optics, University of California Press, 1966, pp. 246‒257.

See Ref. [71], pp. 103–110

M. Born and E. Wolf, Principles of Optics, 7th Ed., Cambridge University Press, 1999, pp. 142‒144.

H. A. Buchdahl, Hamiltonian Optics, Dover, 1993, pp. 7‒12.

See Ref. [6], pp. 13–25

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A. Papoulis, The Fourier Integral and its Applications, McGraw-Hill, 1962.

M. E. Testorf, B. M. Hennelly, and J. Ojeda-Castañeda, ed., Phase-Space Optics: Fundamentals and Applications, McGraw-Hill, 2009.

A. Torre, Linear Ray and Wave Optics in Phase Space: Bridging Ray and Wave Optics via the Wigner Phase-Space Picture, Elsevier, 2005.

D. Dragoman, E. Wolf, ed., "The Wigner distribution function in optics and optoelectronics," Progress in Optics XXXVII, Elsevier, 1997, pp. 1‒56.

R. W. Boyd, Radiometry and the Detection of Optical Radiation, Wiley, 1983, pp. 13‒27.

P. Moon and D. E. Spencer, The Photic Field, MIT Press, 1981.

L. Cohen, Time–Frequency Analysis, Prentice Hall, 1995.

W. Mecklenbräuker and F. Hlawatsch, The Wigner Distribution: Theory and Applications in Signal Processing, Elsevier, 1997.

U. Leonhardt, Measuring the Quantum State of Light, Cambridge U. Press, 1997.

C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics, Vol. 1, Wiley, 1977, pp. 214‒227.

See Chapter 4 by G. Saavedra and W. Furlan in Ref. [15], pp. 107–164

K. B. Wolf, Integral Transforms in Science and Engineering, Plenum Press, 1979, Ch. 9, 10.

A. W. Lohmann, D. Mendlovic, and Z. Zalevsky, E. Wolf, ed., "Fractional transformations in optics," Progress in Optics XXXVIII, 1998, pp. 263‒242.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing, John Wiley & Sons, 2001.

A. Erdélyi, Asymptotic Expansions, Dover, 1956.

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