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Advances in Optics and Photonics

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  • Editor: Bahaa E. A. Saleh
  • Vol. 3, Iss. 4 — Dec. 31, 2011

Wigner functions in optics: describing beams as ray bundles and pulses as particle ensembles

Miguel A. Alonso  »View Author Affiliations


Advances in Optics and Photonics, Vol. 3, Issue 4, pp. 272-365 (2011)
http://dx.doi.org/10.1364/AOP.3.000272


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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.

© 2011 OSA

OCIS Codes
(030.5620) Coherence and statistical optics : Radiative transfer
(070.2590) Fourier optics and signal processing : ABCD transforms
(080.2730) Geometric optics : Matrix methods in paraxial optics
(080.5084) Geometric optics : Phase space methods of analysis
(070.7345) Fourier optics and signal processing : Wave propagation
(070.7425) Fourier optics and signal processing : Quasi-probability distribution functions

ToC Category:
Fourier Optics and Signal Processing

History
Original Manuscript: June 3, 2011
Revised Manuscript: October 4, 2011
Manuscript Accepted: October 5, 2011
Published: November 21, 2011

Virtual Issues
(2011) Advances in Optics and Photonics

Citation
Miguel A. Alonso, "Wigner functions in optics: describing beams as ray bundles and pulses as particle ensembles," Adv. Opt. Photon. 3, 272-365 (2011)
http://www.opticsinfobase.org/aop/abstract.cfm?URI=aop-3-4-272


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References

  1. N. L. Balazs and B. K. Jennings, "Wigner’s function and other distribution functions in mock phase spaces," Phys. Rep. 104, 347‒391 (1984). [CrossRef]
  2. E. P. Wigner, "On the quantum correction for thermodynamic equilibrium," Phys. Rev. 40, 749‒759 (1932). [CrossRef]
  3. J. Ville, "Thèorie et applications de la notion de signal analytique," Cables Transm. 2A, 6174 (1948).
  4. L. S. Dolin, "Beam description of weakly inhomogeneous wave fields," Izv. Vyssh. Uchebn. Zaved. Radiofiz. 7, 559‒563 (1964).
  5. A. Walther, "Radiometry and coherence," J. Opt. Soc. Am. 58, 1256‒1259 (1968). [CrossRef]
  6. R. W. Boyd, Radiometry and the Detection of Optical Radiation, Wiley, 1983, pp. 13‒27.
  7. P. Moon and D. E. Spencer, The Photic Field, MIT Press, 1981.
  8. L. Cohen, Time–Frequency Analysis, Prentice Hall, 1995.
  9. W. Mecklenbräuker and F. Hlawatsch, The Wigner Distribution: Theory and Applications in Signal Processing, Elsevier, 1997.
  10. H. W. Lee, "Theory and application of the quantum phase-space distribution functions," Phys. Rep. 259, 147‒211 (1995). [CrossRef]
  11. G. S. Agarwal and E. Wolf, "Calculus for functions of noncommuting operators and general phase-space methods in quantum mechanics. I. Mapping theorems and ordering of functions of noncommuting operators," Phys. Rev. D 2, 2161‒2186 (1970). [CrossRef]
  12. G. S. Agarwal and E. Wolf, "Calculus for functions of noncommuting operators and general phase-space methods in quantum mechanics. II. Quantum mechanics in phase space," Phys. Rev. D 2, 2187‒2205 (1970). [CrossRef]
  13. G. S. Agarwal and E. Wolf, "Calculus for functions of noncommuting operators and general phase-space methods in quantum mechanics. III. A generalized Wick theorem and multitime mapping," Phys. Rev. D 2, 2206‒2225 (1970). [CrossRef]
  14. M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, "Distribution functions in physics: Fundamentals," 106 121‒167 (1984).
  15. M. E. Testorf, B. M. Hennelly, and J. Ojeda-Castañeda, ed., Phase-Space Optics: Fundamentals and Applications, McGraw-Hill, 2009.
  16. A. Torre, Linear Ray and Wave Optics in Phase Space: Bridging Ray and Wave Optics via the Wigner Phase-Space Picture, Elsevier, 2005.
  17. D. Dragoman, E. Wolf, ed., "The Wigner distribution function in optics and optoelectronics," Progress in Optics XXXVII, Elsevier, 1997, pp. 1‒56.
  18. J. T. Sheridan, W. T. Rhodes, and B. M. Hennelly, "Wigner Optics," Proc. SPIE 5827, 627‒638 (2005).
  19. A. Papoulis, The Fourier Integral and its Applications, McGraw-Hill, 1962.
  20. G. Folland and A. Sitaram, "The uncertainty principle: a mathematical survey," J. Fourier Anal. Appl. 3, 207‒238 (1997). [CrossRef]
  21. E. U. Condon, "Immersion of the Fourier transform in a continuous group of functional transformations," Proc. Natl. Acad. Sci. 23, 158‒163 (1937). [CrossRef]
  22. V. Namias, "The fractional order Fourier transform and its application to quantum mechanics," IMA J. Appl. Math. 25, 241‒265 (1980). [CrossRef]
  23. A. W. Lohmann, D. Mendlovic, and Z. Zalevsky, E. Wolf, ed., "Fractional transformations in optics," Progress in Optics XXXVIII, 1998, pp. 263‒242.
  24. H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing, John Wiley & Sons, 2001.
  25. A. Erdélyi, Asymptotic Expansions, Dover, 1956.
  26. D. Mendlovic, Y. Bitran, R. G. Dorsch, C. Ferreira, J. Garcia, and H. M. Ozaktaz, "Anamorphic fractional Fourier transform: optical implementation and applications," Appl. Opt. 34, 7451‒7456 (1995). [CrossRef] [PubMed]
  27. S. A. Collins, "Lens-system diffraction integral written in terms of matrix optics," J. Opt. Soc. Am. 60, 1168‒1177 (1970). [CrossRef]
  28. M. Moshinsky and C. Quesne, "Linear canonical transformations and their unitary representations," J. Math. Phys. 12, (8), 1772‒1783 (1971). [CrossRef]
  29. K. B. Wolf, Integral Transforms in Science and Engineering, Plenum Press, 1979, Ch. 9, 10.
  30. T. Alieva and M. J. Bastiaans, "Properties of the linear canonical integral transformation," J. Opt. Soc. Am. A 24, 3658‒3665 (2007). [CrossRef]
  31. D. Mustard, "The fractional Fourier transform and the Wigner distribution," J. Aust. Math. Soc. B-Appl. Math. 38, 209‒219 (1996) Published earlier as Applied Mathematics Preprint AM89/6 School of Mathematics, UNSW, Sydney, Australia (1989). [CrossRef]
  32. A. W. Lohmann, "Image rotation, Wigner rotation, and the fractional Fourier transform," J. Opt. Soc. Am. A 10, 2181‒2186 (1993). [CrossRef]
  33. A. W. Lohmann and B. H. Soffer, "Relationships between the Radon–Wigner and fractional Fourier transforms," J. Opt. Soc. Am. A 11, 1798‒1801 (1994). [CrossRef]
  34. See Chapter 4 by G. Saavedra and W. Furlan in Ref. [15], pp. 107–164
  35. J. Bertrand and P. Bertrand, "A tomographic approach to Wigner’s function," Found. Phys. 17, 397‒405 (1987). [CrossRef]
  36. D. F. McAlister, M. Beck, L. Clarke, A. Mayer, and M. G. Raymer, "Optical phase retrieval by phase space tomography and fractional-order Fourier transforms," Opt. Lett. 20, 1181‒1183 (1995). [CrossRef] [PubMed]
  37. M. Beck, M. G. Raymer, I. A. Walmsley, and V. Wong, "Chronocyclic tomography for measuring the amplitude and phase structure of optical pulses," Opt. Lett. 18, 2041‒2043 (1993). [CrossRef] [PubMed]
  38. D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, "Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: application to squeezed states and the vacuum," Phys. Rev. Lett. 70, 1244‒1247 (1993). [CrossRef] [PubMed]
  39. U. Leonhardt, Measuring the Quantum State of Light, Cambridge U. Press, 1997.
  40. C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics, Vol. 1, Wiley, 1977, pp. 214‒227.
  41. J. E. Moyal, "Quantum mechanics as a statistical theory," Math. Proc. Camb. Phil. Soc. 45, 99‒124 (1949). [CrossRef]
  42. R. G. Littlejohn and R. Winston, "Generalized radiance and measurement," J. Opt. Soc. Am. A 12, 2736‒2743 (1995). [CrossRef]
  43. D. Dragoman, "Wigner-distribution-function representation of the coupling coefficient," Appl. Opt. 34, 6758‒6763 (1995). [CrossRef] [PubMed]
  44. A. Wax and J. E. Thomas, "Optical heterodyne imaging and Wigner phase space distributions," Opt. Lett. 21, 1427‒1429 (1996). [CrossRef] [PubMed]
  45. M. A. Alonso, "Measurement of Helmholtz wave fields," J. Opt. Soc. Am. A 17, 1256‒1264 (2000). [CrossRef]
  46. M. V. Berry, "Semi-classical mechanics in phase space: a study of Wigners function," Philos. Trans. R. Soc. Lond. 287, 237‒271 (1977). [CrossRef]
  47. M. V. Berry and N. L. Balazs, "Evolution of semiclassical quantum states in phase space," J. Phys. A Math. Phys. 12, 625‒642 (1979). [CrossRef]
  48. M. V. Berry, "Quantum scars of classical closed orbits in phase space," Proc. R. Soc. Lond. Ser. A 423, 219‒231 (1989). [CrossRef]
  49. M. A. Alonso and G. W. Forbes, "Phase-space distributions for high-frequency fields," J. Opt. Soc. Am. A 17, 2288‒2300 (2000). [CrossRef]
  50. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, Cambridge University Press, 1995, Sec. 4.7.
  51. M. J. Bastiaans, "Uncertainty principle for partially coherent light," J. Opt. Soc. Am. 73, 251‒255 (1983). [CrossRef]
  52. A. Starikov, "Effective number of degrees of freedom of partially coherent sources," J. Opt. Soc. Am. 72, 1538‒1544 (1982). [CrossRef]
  53. See Ref. [50], p. 261
  54. L. Cohen, "Time–frequency distributions—a review," Proc. IEEE 77, 941‒981 (1989). [CrossRef]
  55. J. H. Eberly and K. Wódkiewicz, "The time-dependent physical spectrum of light," J. Opt. Soc. Am. 67, 1252‒1261 (1977). [CrossRef]
  56. K. Husimi, "Some formal properties of the density matrix," Proc. Phys. Math. Soc. Jpn. 22, 264‒314 (1940).
  57. Y. Kano, "A new phase-space distribution function in the statistical theory of the electromagnetic field," J. Math. Phys. 6, 1913‒1915 (1965). [CrossRef]
  58. C. L. Mehta and E. C. G. Sudarshan, "Relation between Quantum and Semiclassical Description of Optical Coherence," Phys. Rev. 138, B274‒B280 (1965). [CrossRef]
  59. R. J. Glauber, C. Dewitt, A. Blandin, and C. Cohen-Tannoudji, ed., "Optical coherence and photon statistics," Quantum Optics and Electronics, Gordon and Breach, 1965, p. 65.
  60. R. J. Glauber, "Coherent and incoherent states of the radiation field," Phys. Rev. 131, 2766‒2788 (1963). [CrossRef]
  61. J. R. Klauder, "Continuous representation theory. I. Postulates of continuous representation theory," J. Math. Phys. 4, 1055‒1058 (1963). [CrossRef]
  62. E. C. G. Sudarshan, "Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams," Phys. Rev. Lett. 10, 277‒279 (1963). [CrossRef]
  63. J. G. Kirkwood, "Quantum statistics of almost classical ensembles," Phys. Rev. 44, 31‒37 (1933). [CrossRef]
  64. A. W. Rihaczek, "Signal energy distribution in time and frequency," IEEE Trans. Info. Theory 14, 369‒374 (1968). [CrossRef]
  65. H. Margenau and R. N. Hill, "Correlation between measurements in quantum theory," Prog. Theoret. Phys. 26, 722‒738 (1961). [CrossRef]
  66. P. M. Woodward, Probability and Information Theory with Applications to Radar, Pergamon, 1953.
  67. A. Papoulis, "Ambiguity function in Fourier optics," J. Opt. Soc. Am. 64, 779‒788 (1974). [CrossRef]
  68. L.-P. Guigay, Ambiguity Function in Optical Imaging, Chapter 2 of Ref. [15], pp. 45–62.
  69. L. Cohen, "Generalized phase-space distributions," J. Math. Phys. 7, 781‒786 (1966). [CrossRef]
  70. M. J. Bastiaans, Wigner Distribution in Optics, Chapter 1 of Ref. [15], pp. 1–44.
  71. R. K. Luneburg, Mathematical Theory of Optics, University of California Press, 1966, pp. 246‒257.
  72. See Ref. [71], pp. 103–110
  73. M. Born and E. Wolf, Principles of Optics, 7th Ed., Cambridge University Press, 1999, pp. 142‒144.
  74. H. A. Buchdahl, Hamiltonian Optics, Dover, 1993, pp. 7‒12.
  75. See Ref. [6], pp. 13–25
  76. See Ref. [6], pp. 75–80
  77. A. Walther, "Lenses, wave optics and eikonal functions," J. Opt. Soc. Am. 59, 1325‒1333 (1969). [CrossRef]
  78. A. Walther, The Ray and Wave Theory of Lenses, Cambridge University Press, 1995, pp. 169‒187.
  79. M. A. Alonso and G. W. Forbes, "Semigeometrical estimation of Green’s functions and wave propagators in optics," J. Opt. Soc. Am. A 14, 1076‒1086 (1997). [CrossRef]
  80. J. H. Van Vleck, "The correspondence principle in the statistical interpretation of quantum mechanics," Proc. Natl. Acad. Sci. USA 14, 178‒188 (1928). [CrossRef]
  81. M. C. Gutzwiller, "Periodic orbits and classical quantization conditions," J. Math. Phys. 12, 343‒358 (1971). [CrossRef]
  82. D. Mendlovic and H. M. Ozaktas, "Fractional Fourier transforms and their optical implementation: I," J. Opt. Soc. Am. A 10, 1875‒1881 (1993). [CrossRef]
  83. M. J. Bastiaans, "Wigner distribution function applied to optical signals and systems," Opt. Commun. 25, 26‒30 (1978). [CrossRef]
  84. M. J. Bastiaans, "Transport equations for the Wigner distribution function," J. Mod. Opt. 26, 1265‒1272 (1979).
  85. M. J. Bastiaans, "The Wigner distribution function and Hamilton’s characteristics of a geometric–optical system," Opt. Commun. 30, 321‒326 (1979). [CrossRef]
  86. M. J. Bastiaans, "Wigner distribution function and its application to first-order optics," J. Opt. Soc. Am. 69, 1710‒1716 (1979). [CrossRef]
  87. J. W. Goodman, Introduction to Fourier Optics, McGraw-Hill, 1988, pp. 101‒136.
  88. K.-H. Brenner, A. W. Lohmann, and J. Ojeda-Castañeda, "The ambiguity function as a polar display of the OTF," Opt. Commun. 44, 323‒326 (1983). [CrossRef]
  89. S. B. Oh and G. Barbastathis, "Axial imaging necessitates loss of lateral shift invariance: proof with the Wigner analysis," Appl. Opt. 48, 5881‒5888 (2009). [CrossRef] [PubMed]
  90. 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). [CrossRef]
  91. J. E. Durnin, J. J. Miceli, and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1499‒1501 (1987). [CrossRef] [PubMed]
  92. 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). [CrossRef]
  93. M. V. Berry and N. L. Balazs, "Nonspreading wave packets," Am. J. Phys. 47, 264‒267 (1979). [CrossRef]
  94. M. V. Berry and F. J. Wright, "Phase-space projection identities for diffraction catastrophes," J. Phys. A. Math. Gen. 13, 149‒160 (1980). [CrossRef]
  95. R.-P. Chen, H.-P. Zheng, and C.-Q. Dai, "Wigner distribution function of an Airy beam," J. Opt. Soc. Am. A 28, 1307‒1311 (2011). [CrossRef]
  96. G. Siviloglou and D. Christodoulides, "Accelerating finite energy Airy beams," Opt. Lett. 32, 979‒981 (2007). [CrossRef] [PubMed]
  97. G. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, "Observation of accelerating Airy beams," Phys. Rev. Lett. 99, 213901-1-213901-4 (2007). [CrossRef]
  98. E. R. Dowski and W. T. Cathey, "Extended depth of field through wave-front coding," Appl. Opt. 34, 1859‒1866 (1995). [CrossRef] [PubMed]
  99. W. T. Welford, "Use of annular apertures to increase focal depth," J. Opt. Soc. Am. 50, 749 (1960). [CrossRef]
  100. C. J. R. Sheppard, D. K. Hamilton, and I. J. Cox, "Optical microscopy with extended depth of field," Proc. R. Soc. Lond. A 387, 171‒186 (1983). [CrossRef]
  101. J. Ojeda-Castañeda, L. R. Berriel-Valdos, and E. Montes, "Bessel annular apodizers: imaging characteristics," Appl. Opt. 26, 2770‒2772 (1987). [CrossRef] [PubMed]
  102. J. Ojeda-Castañeda, L. R. Berriel-Valdos, and E. Montes, "Ambiguity function as a design tool for high focal depth," Appl. Opt. 27, 790‒795 (1988). [CrossRef] [PubMed]
  103. Q. Yang, L. Liu, J. Sun, Y. Zhu, and W. Lu, "Analysis of optical systems with extended depth of field using the Wigner distribution function," Appl. Opt. 45, 8586‒8595 (2006). [CrossRef] [PubMed]
  104. M. Testorf and J. Ojeda-Castañeda, "Fractional Talbot effect: analysis in phase space," J. Opt. Soc. Am. A 13, 119‒125 (1996). [CrossRef]
  105. M. Testorf, "Designing Talbot array illuminators with phase-space optics," J. Opt. Soc. Am. A 23, 187‒192 (2006). [CrossRef]
  106. M. E. Testorf, Self-imaging in Phase Space, Chapter 9 of Ref. [15], pp. 279–307.
  107. J. Lancis, E. E. Sicre, A. Pons, and G. Saavedra, "Achromatic white-light self-imaging phenomenon: an approach using the Wigner distribution function," J. Mod. Opt. 42, 425‒434 (1995). [CrossRef]
  108. A. Stern and B. Javidi, "Sampling in the light of Wigner distribution," J. Opt. Soc. Am. A 21, 360‒366 (2004)  errata, 21, 1602–1612 (2004). [CrossRef]
  109. B. M. Hennelly, J. J. Healy, and J. T. Sheridan, Sampling and Phase Space, Chapter 10 of Ref. [15], pp. 309–336.
  110. F. Gori, "Fresnel transform and sampling theorem," Opt. Commun. 39, 293‒297 (1981). [CrossRef]
  111. A. Stern and B. Javidi, "Improved-resolution digital holography using the generalized sampling theorem for locally band-limited fields," J. Opt. Soc. Am. A 23, 1227‒1235 (2006). [CrossRef]
  112. A. Stern and B. Javidi, "Space-bandwidth conditions for efficient phase-shifting digital holographic microscopy," J. Opt. Soc. Am. A 25, 736‒741 (2008). [CrossRef]
  113. K. B. Wolf and A. L. Rivera, "Holographic information in the Wigner function," Opt. Commun. 144, 36‒42 (1997). [CrossRef]
  114. A. W. Lohmann, M. E. Testorf, and J. Ojeda-Castañeda, H. J. Caulfield, ed., "Holography and the Wigner function," The Art and Science of Holography: A Tribute to Emmett Leith and Yuri Denisyuk, SPIE Press, 2004, pp. 127‒144.
  115. A. W. Lohmann, M. E. Testorf, J. Ojeda-Castañeda, and A. W. Lohmann, ed., "The space–bandwidth product, applied to spatial filtering and to holography," Selected Papers on Phase-Space Optics, SPIE Press, 2006, pp. 11‒32.
  116. M. Testorf and A. W. Lohmann, "Holography in phase space," Appl. Opt. 47, A70‒A77 (2008). [CrossRef] [PubMed]
  117. S. B. Oh and G. Barbastathis, "Wigner distribution function of volume holograms," Opt. Lett. 34, 2584‒2586 (2009). [CrossRef] [PubMed]
  118. M. Testorf, "Analysis of the moiré effect by use of the Wigner distribution function," J. Opt. Soc. Am. A 17, 2536‒2542 (2000). [CrossRef]
  119. N. Morelle, M. E. Testorf, N. Thirion, and M. Saillard, "Electromagnetic probing for target detection: rejection of surface clutter based on the Wigner distribution," J. Opt. Soc. Am. A 26, 1178‒1186 (2009). [CrossRef]
  120. L. E. Vicent and M. A. Alonso, "Generalized radiometry as a tool for the propagation of partially coherent fields," Opt. Commun. 207, 101‒112 (2002). [CrossRef]
  121. E. Wolf, "Radiometric model for propagation of coherence," Opt. Lett. 19, 2024‒2026 (1994). [CrossRef] [PubMed]
  122. A. T. Friberg and S. Yu. Popov, "Radiometric description of intensity and coherence in generalized holographic axicon images," Appl. Opt. 35, 3039‒3046 (1996). [CrossRef] [PubMed]
  123. K. Duan and B. Lü, "Wigner-distribution-function matrix and its application to partially coherent vectorial nonparaxial beams," J. Opt. Soc. Am. B 22, 1585‒1593 (2005). [CrossRef]
  124. R. Castañeda and J. Garcia-Sucerquia, "Electromagnetic spatial coherence wavelets," J. Opt. Soc. Am. A 23, 81‒90 (2006). [CrossRef]
  125. A. Luis, "Spatial-angular Mueller matrices," Opt. Commun. 263, 141‒146 (2006). [CrossRef]
  126. A. Luis, "Ray picture of polarization and coherence in a Young interferometer," J. Opt. Soc. Am. A 23, 2855‒2860 (2006). [CrossRef]
  127. R. Castañeda, J. Carrasquilla, and J. Herrera, "Radiometric analysis of diffraction of quasi-homogeneous optical fields," Opt. Commun. 273, 8‒20 (2007). [CrossRef]
  128. R. Castañeda and J. Carrasquilla, "Spatial coherence wavelets and phase-space representation of diffraction," Appl. Opt. 47, E76‒E87 (2008). [CrossRef] [PubMed]
  129. M. A. Alonso, "Diffraction of paraxial partially coherent fields by planar obstacles in the Wigner representation," J. Opt. Soc. Am. A 26, 1588‒1597 (2009). [CrossRef]
  130. 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). [CrossRef]
  131. H. T. Yura, L. Thrane, and P. E. Andersen, "Closed-form solution for the Wigner phase-space distribution function for diffuse reflection and small-angle scattering in a random medium," J. Opt. Soc. Am. A 17, 2464‒2474 (2000). [CrossRef]
  132. C.-C. Cheng and M. G. Raymer, "Propagation of transverse optical coherence in random multiple-scattering media," Phys. Rev. A 62, 023811 (2000). [CrossRef]
  133. A. C. Fannjiang, "White-noise and geometrical optics limits of Wigner–Moyal equation for wave beams in turbulent media," Commun. Math. Phys. 254, 289‒322 (2005). [CrossRef]
  134. D. Dragoman, "Wigner distribution function in nonlinear optics," Appl. Opt. 35, 4142‒4146 (1996). [CrossRef] [PubMed]
  135. D. Dragoman, J. P. Meunier, and M. Dragoman, "Beam-propagation method based on the Wigner transform: a new formulation," Opt. Lett. 22, 1050‒1052 (1997). [CrossRef] [PubMed]
  136. B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov, "Statistical theory for incoherent light propagation in nonlinear media," Phys. Rev. E 65, 035602R (2002). [CrossRef]
  137. M. Lisak, L. Helczynski, and D. Anderson, "Relation between different formalisms describing partially incoherent wave propagation in nonlinear optical media," Opt. Commun 220, 321‒323 (2003). [CrossRef]
  138. 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). [CrossRef] [PubMed]
  139. S. Abe and J. T. Sheridan, "Wigner optics in the metaxial regime," Optik 114, 139‒141. [CrossRef]
  140. Yu. A. Kravtsov and L. A. Apresyan, E. Wolf, ed., "Radiative transfer: new aspects of the old theory," Progress in Optics, Vol. XXXVI, North Holland, 1996, pp. 179‒244.
  141. L. A. Apresyan and Yu. A. Kravtsov, Radiation Transfer: Statistical and Wave Aspects, Gordon and Breach, 1996.
  142. A. T. Friberg, ed., Selected Papers on Coherence and Radiometry, Milestone Series, Vol. MS69, SPIE Optical Engineering Press, 1993.
  143. E. Wolf, "Coherence and radiometry," J. Opt. Soc. Am. 68, 6‒17 (1978). [CrossRef]
  144. A. T. Friberg, "Effects of coherence in radiometry," Opt. Eng. 21, 927‒936 (1982).
  145. G. S. Agarwal, J. T. Foley, and E. Wolf, "The radiance and phase-space representations of the cross-spectral density operator," Opt. Commun. 62, 67‒72 (1987). [CrossRef]
  146. A. Walther, "Radiometry and coherence," J. Opt. Soc. Am. 63, 1622‒1623 (1973). [CrossRef]
  147. A. T. Friberg, "On the generalized radiance associated with radiation from a quasihomogeneous planar source," Opt. Acta 28, 261‒277 (1981). [CrossRef]
  148. J. T. Foley and E. Wolf, "Radiometry as a short-wavelength limit of statistical wave theory with globally incoherent sources," Opt. Commun. 55, 236‒241 (1985). [CrossRef]
  149. J. Azaña, "Time–frequency (Wigner) analysis of linear and nonlinear pulse propagation in optical fibers," EURASIP J. Appl. Signal Process. 2005, 1554‒1565 (2005). [CrossRef]
  150. P. Loughlin and L. Cohen, "A Wigner approximation method for wave propagation," J. Acoust. Soc. Am. 118, 1268‒1271 (2005). [CrossRef]
  151. J. Ojeda-Castañeda, J. Lancis, C. M. Gómez-Sarabia, V. Torres-Company, and P. Andrés, "Ambiguity function analysis of pulse train propagation: applications to temporal Lau filtering," J. Opt. Soc. Am. A 24, 2268‒2273 (2007). [CrossRef]
  152. P. Loughlin and L. Cohen, "Approximate wave function from approximate non-representable Wigner distributions," J. Mod. Opt. 55, 3379‒3387 (2008). [CrossRef]
  153. L. Cohen, P. Loughlin, and G. Okopal, "Exact and approximate moments of a propagating pulse," J. Mod. Opt. 55, 3349‒3358 (2008). [CrossRef]
  154. C. Dorrer and I. A. Walmsley, Phase space in ultrafast optics, Chapter 11 of Ref. [15], pp. 337–383.
  155. I. A. Walmsley and C. Dorrer, "Characterization of ultrashort electromagnetic pulses," Adv. Opt. Phot. 1, 308‒437 (2009). [CrossRef]
  156. B. H. Kolner, "Space–time duality and the theory of temporal imaging," IEEE J. Quantum Electron. 30, 1951‒1963 (1994). [CrossRef]
  157. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed., Wiley, 2007, p. 188.
  158. H. Lajunen, J. Tervo, J. Turunen, P. Vahimaa, and F. Wyrowski, "Spectral coherence properties of temporally modulated stationary light sources," Opt. Express 11, 1894‒1899 (2003). [CrossRef] [PubMed]
  159. B. J. Davis, "Observable coherence theory for statistically periodic fields," Phys. Rev. A 76, 043843 (2007). [CrossRef]
  160. K. Vogel and H. Risken, "Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase," Phys. Rev. A 40, 2847‒2849 (1989). [CrossRef] [PubMed]
  161. C. C. Gerry and P. L. Knight, Introductory Quantum Optics, Cambridge University Press, 2005, pp. 56‒71.
  162. W. P. Schleich, Quantum Optics in Phase Space, Wiley-VCH, 2001.
  163. C. Dorrer and I. Kang, "Complete temporal characterization of short optical pulses by simplified chronocyclic tomography," Opt. Lett. 28, 1481‒1483 (2003). [CrossRef] [PubMed]
  164. K. A. Nugent, "Wave field determination using three-dimensional intensity information," Phys. Rev. Lett. 68, 2261‒2264 (1992). [CrossRef] [PubMed]
  165. 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). [CrossRef]
  166. C. Q. Tran, A. P. Mancuso, B. B. Dhal, K. A. Nugent, A. G. Peele, Z. Cai, and D. Paterson, "phase space reconstruction of focused x-ray fields," J. Opt. Soc. Am. A 23, 1779‒1786 (2006). [CrossRef]
  167. J. Tu and S. Tamura, "Wave field determination using tomography of the ambiguity function," Phys. Rev. E 55, 1946‒1949 (1997). [CrossRef]
  168. 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). [CrossRef]
  169. 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). [CrossRef] [PubMed]
  170. R. Horstmeyer, S. B. Oh, O. Gupta, and R. Raskar, "Partially coherent ambiguity functions for depth-variant point spread function design,"  presentation during PIERS, Marrakesh, March, 2011
  171. G. Hazak, "Comment on ‘Wave field determination using three-dimensional intensity information’," Phys. Rev. Lett. 69, 2874‒2874 (1992). [CrossRef] [PubMed]
  172. F. Gori, M. Santarsiero, and G. Guattari, "Coherence and the spatial distribution of intensity," J. Opt. Soc. Am. A 10, 673‒679 (1993). [CrossRef]
  173. M. G. Raymer, M. Beck, and D. F. McAlister, "Complex wave-field reconstruction using phase-space tomography," Phys. Rev. Lett. 72, 1137‒1140 (1994). [CrossRef] [PubMed]
  174. A. Cámara, T. Alieva, J. A. Rodrigo, and M. L. Calvo, "Phase space tomography reconstruction of the Wigner distribution for optical beams separable in Cartesian coordinates," J. Opt. Soc. Am. A 26, 1301‒1306 (2009). [CrossRef]
  175. P. Rojas, R. Blaser, Y. M. Sua, and K. F. Lee, "Optical phase-space-time-frequency tomography," Opt. Express 19, 7480‒7490 (2011). [CrossRef] [PubMed]
  176. J. C. Petruccelli and M. A. Alonso, "Phase space distributions tailored for dispersive media," J. Opt. Soc. Am. A 27, 1194‒1201 (2010). [CrossRef]
  177. J. C. Petruccelli, Generalized Wigner Functions, Ph.D. Thesis, University of Rochester (Rochester, NY, 2010)
  178. 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). [CrossRef]
  179. M. A. Alonso, "Radiometry and wide-angle wave fields. I. Coherent fields in two dimensions," J. Opt. Soc. Am. A 18, 902‒909 (2001). [CrossRef]
  180. C. J. R. Sheppard and K. G. Larkin, "Wigner function for nonparaxial wave fields," J. Opt. Soc. Am. A 18, 2486‒2490 (2001). [CrossRef]
  181. C. J. R. Sheppard and K. G. Larkin, "Wigner function for highly convergent three-dimensional wave fields," Opt. Lett. 26, 968‒970 (2001). [CrossRef] [PubMed]
  182. M. A. Alonso, "Radiometry and wide-angle wave fields. II. Coherent fields in three dimensions," J. Opt. Soc. Am. A 18, 910‒918 (2001). [CrossRef]
  183. M. A. Alonso, "Radiometry and wide-angle wave fields. III. Partial coherence," J. Opt. Soc. Am. A. 18, 2502‒2511 (2001). [CrossRef]
  184. M. A. Alonso, "Exact description of free electromagnetic wave fields in terms of rays," Opt. Express 11, 3128‒3135 (2003). [CrossRef] [PubMed]
  185. M. A. Alonso, "Wigner functions for nonparaxial, arbitrarily polarized electromagnetic wave fields in free-space," J. Opt. Soc. Am. A. 21, 2233‒2243 (2004). [CrossRef]
  186. 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). [CrossRef]
  187. J. C. Petruccelli and M. A. Alonso, "Ray-based propagation of the cross-spectral density," J. Opt. Soc. Am. A 25, 1395‒1405 (2008). [CrossRef]
  188. J. C. Petruccelli and M. A. Alonso, "Propagation of partially coherent fields through planar dielectric boundaries using angle-impact Wigner functions I. Two dimensions," J. Opt. Soc. Am. A 24, 2590‒2603 (2007). [CrossRef]
  189. J. C. Petruccelli and M. A. Alonso, "Propagation of nonparaxial partially coherent fields across interfaces using generalized radiometry," J. Opt. Soc. Am. A 26, 2012‒2022 (2009). [CrossRef]
  190. 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). [CrossRef]
  191. S. Cho and M. A. Alonso, "Ambiguity function and phase-space tomography for nonparaxial fields," J. Opt. Soc. Am. A 28, 897‒902 (2011). [CrossRef]
  192. G. I. Ovchinnikov and V. I. Tatarskii, "On the problem of the relationship between coherence theory and the radiation-transfer equation," Radiophys. Quantum Electron. 15, 1087‒1089 (1972). [CrossRef]
  193. H. M. Pedersen, J. H. Eberly, L. Mandel, and E. Wolf, ed., "Exact geometrical description of free space radiative energy transfer for scalar wavefields," Coherence and Quantum Optics VI, Plenum, 1990, pp. 883‒887.
  194. H. M. Pedersen, "Exact theory of free-space radiative energy transfer," J. Opt. Soc. Am. A 8, 176‒185 (1991)  errata, 8, 1518 (1991). [CrossRef]
  195. H. M. Pedersen, "Geometric theory of fields radiated from three-dimensional, quasi-homogeneous sources," J. Opt. Soc. Am. A 9, 1626‒1632 (1992). [CrossRef]
  196. R. G. Littlejohn and R. Winston, "Corrections to classical radiometry," J. Opt. Soc. Am. A 10, 2024‒2037 (1993). [CrossRef]
  197. S. Cho, J. C. Petruccelli, and M. A. Alonso, "Wigner functions for paraxial and nonparaxial fields," J. Mod. Opt. 56, 1843‒1852 (2009). [CrossRef]
  198. M. A. Alonso, T. Setälä, and A. T. Friberg, "Optimal pulses for arbitrary dispersive media," J. Eur. Opt. Soc. R.P. 6, 1100 (2011).
  199. 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). [CrossRef]
  200. 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). [CrossRef]
  201. A. Papandreou-Suppappola, R. L. Murray, B.-G. Iem, and G. F. Boudreaux-Bartels, "Group delay shift covariant quadratic time–frequency representations," IEEE Trans. Signal Process. 49, 2549‒2564 (2001). [CrossRef]
  202. J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780‒1782 (2006). [CrossRef] [PubMed]
  203. R. W. Robinett, "Quantum wave packet revivals," Phys. Rep. 392, 1‒119 (2004). [CrossRef]

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