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]

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]

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]

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]

M. A. Alonso, T. Setälä, and A. T. Friberg, "Optimal pulses for arbitrary dispersive media," J. Eur. Opt. Soc. R.P. 6, 1100 (2011).

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]

J. C. Petruccelli and M. A. Alonso, "Phase space distributions tailored for dispersive media," J. Opt. Soc. Am. A 27, 1194‒1201 (2010).

[CrossRef]

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]

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]

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]

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]

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]

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]

S. B. Oh and G. Barbastathis, "Wigner distribution function of volume holograms," Opt. Lett. 34, 2584‒2586 (2009).

[CrossRef]
[PubMed]

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]

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]

I. A. Walmsley and C. Dorrer, "Characterization of ultrashort electromagnetic pulses," Adv. Opt. Phot. 1, 308‒437 (2009).

[CrossRef]

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]

S. Cho, J. C. Petruccelli, and M. A. Alonso, "Wigner functions for paraxial and nonparaxial fields," J. Mod. Opt. 56, 1843‒1852 (2009).

[CrossRef]

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]

P. Loughlin and L. Cohen, "Approximate wave function from approximate non-representable Wigner distributions," J. Mod. Opt. 55, 3379‒3387 (2008).

[CrossRef]

L. Cohen, P. Loughlin, and G. Okopal, "Exact and approximate moments of a propagating pulse," J. Mod. Opt. 55, 3349‒3358 (2008).

[CrossRef]

M. Testorf and A. W. Lohmann, "Holography in phase space," Appl. Opt. 47, A70‒A77 (2008).

[CrossRef]
[PubMed]

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]

R. Castañeda and J. Carrasquilla, "Spatial coherence wavelets and phase-space representation of diffraction," Appl. Opt. 47, E76‒E87 (2008).

[CrossRef]
[PubMed]

R. Castañeda, J. Carrasquilla, and J. Herrera, "Radiometric analysis of diffraction of quasi-homogeneous optical fields," Opt. Commun. 273, 8‒20 (2007).

[CrossRef]

B. J. Davis, "Observable coherence theory for statistically periodic fields," Phys. Rev. A 76, 043843 (2007).

[CrossRef]

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]

G. Siviloglou and D. Christodoulides, "Accelerating finite energy Airy beams," Opt. Lett. 32, 979‒981 (2007).

[CrossRef]
[PubMed]

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]

T. Alieva and M. J. Bastiaans, "Properties of the linear canonical integral transformation," J. Opt. Soc. Am. A 24, 3658‒3665 (2007).

[CrossRef]

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]

J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780‒1782 (2006).

[CrossRef]
[PubMed]

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]

R. Castañeda and J. Garcia-Sucerquia, "Electromagnetic spatial coherence wavelets," J. Opt. Soc. Am. A 23, 81‒90 (2006).

[CrossRef]

A. Luis, "Spatial-angular Mueller matrices," Opt. Commun. 263, 141‒146 (2006).

[CrossRef]

A. Luis, "Ray picture of polarization and coherence in a Young interferometer," J. Opt. Soc. Am. A 23, 2855‒2860 (2006).

[CrossRef]

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]

M. Testorf, "Designing Talbot array illuminators with phase-space optics," J. Opt. Soc. Am. A 23, 187‒192 (2006).

[CrossRef]

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]

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]

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]

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]

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]

P. Loughlin and L. Cohen, "A Wigner approximation method for wave propagation," J. Acoust. Soc. Am. 118, 1268‒1271 (2005).

[CrossRef]

J. T. Sheridan, W. T. Rhodes, and B. M. Hennelly, "Wigner Optics," Proc. SPIE 5827, 627‒638 (2005).

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]

R. W. Robinett, "Quantum wave packet revivals," Phys. Rep. 392, 1‒119 (2004).

[CrossRef]

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]

M. A. Alonso, "Exact description of free electromagnetic wave fields in terms of rays," Opt. Express 11, 3128‒3135 (2003).

[CrossRef]
[PubMed]

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]

C. Dorrer and I. Kang, "Complete temporal characterization of short optical pulses by simplified chronocyclic tomography," Opt. Lett. 28, 1481‒1483 (2003).

[CrossRef]
[PubMed]

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]

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]

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]

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]

C. J. R. Sheppard and K. G. Larkin, "Wigner function for nonparaxial wave fields," J. Opt. Soc. Am. A 18, 2486‒2490 (2001).

[CrossRef]

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]

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]

M. A. Alonso, "Radiometry and wide-angle wave fields. III. Partial coherence," J. Opt. Soc. Am. A. 18, 2502‒2511 (2001).

[CrossRef]

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]

M. Testorf, "Analysis of the moiré effect by use of the Wigner distribution function," J. Opt. Soc. Am. A 17, 2536‒2542 (2000).

[CrossRef]

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]

C.-C. Cheng and M. G. Raymer, "Propagation of transverse optical coherence in random multiple-scattering media," Phys. Rev. A 62, 023811 (2000).

[CrossRef]

M. A. Alonso, "Measurement of Helmholtz wave fields," J. Opt. Soc. Am. A 17, 1256‒1264 (2000).

[CrossRef]

M. A. Alonso and G. W. Forbes, "Phase-space distributions for high-frequency fields," J. Opt. Soc. Am. A 17, 2288‒2300 (2000).

[CrossRef]

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]

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]

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]

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]

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]

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]

K. B. Wolf and A. L. Rivera, "Holographic information in the Wigner function," Opt. Commun. 144, 36‒42 (1997).

[CrossRef]

J. Tu and S. Tamura, "Wave field determination using tomography of the ambiguity function," Phys. Rev. E 55, 1946‒1949 (1997).

[CrossRef]

G. Folland and A. Sitaram, "The uncertainty principle: a mathematical survey," J. Fourier Anal. Appl. 3, 207‒238 (1997).

[CrossRef]

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]

A. Wax and J. E. Thomas, "Optical heterodyne imaging and Wigner phase space distributions," Opt. Lett. 21, 1427‒1429 (1996).

[CrossRef]
[PubMed]

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]

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]

M. Testorf and J. Ojeda-Castañeda, "Fractional Talbot effect: analysis in phase space," J. Opt. Soc. Am. A 13, 119‒125 (1996).

[CrossRef]

D. Dragoman, "Wigner distribution function in nonlinear optics," Appl. Opt. 35, 4142‒4146 (1996).

[CrossRef]
[PubMed]

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]

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]

R. G. Littlejohn and R. Winston, "Generalized radiance and measurement," J. Opt. Soc. Am. A 12, 2736‒2743 (1995).

[CrossRef]

D. Dragoman, "Wigner-distribution-function representation of the coupling coefficient," Appl. Opt. 34, 6758‒6763 (1995).

[CrossRef]
[PubMed]

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]

H. W. Lee, "Theory and application of the quantum phase-space distribution functions," Phys. Rep. 259, 147‒211 (1995).

[CrossRef]

E. R. Dowski and W. T. Cathey, "Extended depth of field through wave-front coding," Appl. Opt. 34, 1859‒1866 (1995).

[CrossRef]
[PubMed]

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]

E. Wolf, "Radiometric model for propagation of coherence," Opt. Lett. 19, 2024‒2026 (1994).

[CrossRef]
[PubMed]

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]

B. H. Kolner, "Space–time duality and the theory of temporal imaging," IEEE J. Quantum Electron. 30, 1951‒1963 (1994).

[CrossRef]

F. Gori, M. Santarsiero, and G. Guattari, "Coherence and the spatial distribution of intensity," J. Opt. Soc. Am. A 10, 673‒679 (1993).

[CrossRef]

A. W. Lohmann, "Image rotation, Wigner rotation, and the fractional Fourier transform," J. Opt. Soc. Am. A 10, 2181‒2186 (1993).

[CrossRef]

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]

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]

D. Mendlovic and H. M. Ozaktas, "Fractional Fourier transforms and their optical implementation: I," J. Opt. Soc. Am. A 10, 1875‒1881 (1993).

[CrossRef]

R. G. Littlejohn and R. Winston, "Corrections to classical radiometry," J. Opt. Soc. Am. A 10, 2024‒2037 (1993).

[CrossRef]

H. M. Pedersen, "Geometric theory of fields radiated from three-dimensional, quasi-homogeneous sources," J. Opt. Soc. Am. A 9, 1626‒1632 (1992).

[CrossRef]

G. Hazak, "Comment on ‘Wave field determination using three-dimensional intensity information’," Phys. Rev. Lett. 69, 2874‒2874 (1992).

[CrossRef]
[PubMed]

K. A. Nugent, "Wave field determination using three-dimensional intensity information," Phys. Rev. Lett. 68, 2261‒2264 (1992).

[CrossRef]
[PubMed]

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]

M. V. Berry, "Quantum scars of classical closed orbits in phase space," Proc. R. Soc. Lond. Ser. A 423, 219‒231 (1989).

[CrossRef]

L. Cohen, "Time–frequency distributions—a review," Proc. IEEE 77, 941‒981 (1989).

[CrossRef]

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]

J. Ojeda-Castañeda, L. R. Berriel-Valdos, and E. Montes, "Bessel annular apodizers: imaging characteristics," Appl. Opt. 26, 2770‒2772 (1987).

[CrossRef]
[PubMed]

J. E. Durnin, J. J. Miceli, and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1499‒1501 (1987).

[CrossRef]
[PubMed]

J. Bertrand and P. Bertrand, "A tomographic approach to Wigner’s function," Found. Phys. 17, 397‒405 (1987).

[CrossRef]

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]

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]

M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, "Distribution functions in physics: Fundamentals," 106 121‒167 (1984).

M. J. Bastiaans, "Uncertainty principle for partially coherent light," J. Opt. Soc. Am. 73, 251‒255 (1983).

[CrossRef]

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]

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]

A. T. Friberg, "On the generalized radiance associated with radiation from a quasihomogeneous planar source," Opt. Acta 28, 261‒277 (1981).

[CrossRef]

F. Gori, "Fresnel transform and sampling theorem," Opt. Commun. 39, 293‒297 (1981).

[CrossRef]

V. Namias, "The fractional order Fourier transform and its application to quantum mechanics," IMA J. Appl. Math. 25, 241‒265 (1980).

[CrossRef]

M. V. Berry and F. J. Wright, "Phase-space projection identities for diffraction catastrophes," J. Phys. A. Math. Gen. 13, 149‒160 (1980).

[CrossRef]

M. J. Bastiaans, "Transport equations for the Wigner distribution function," J. Mod. Opt. 26, 1265‒1272 (1979).

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. V. Berry and N. L. Balazs, "Nonspreading wave packets," Am. J. Phys. 47, 264‒267 (1979).

[CrossRef]

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]

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]

M. C. Gutzwiller, "Periodic orbits and classical quantization conditions," J. Math. Phys. 12, 343‒358 (1971).

[CrossRef]

M. Moshinsky and C. Quesne, "Linear canonical transformations and their unitary representations," J. Math. Phys. 12, (8), 1772‒1783 (1971).

[CrossRef]

S. A. Collins, "Lens-system diffraction integral written in terms of matrix optics," J. Opt. Soc. Am. 60, 1168‒1177 (1970).

[CrossRef]

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]

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]

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]

L. Cohen, "Generalized phase-space distributions," J. Math. Phys. 7, 781‒786 (1966).

[CrossRef]

Y. Kano, "A new phase-space distribution function in the statistical theory of the electromagnetic field," J. Math. Phys. 6, 1913‒1915 (1965).

[CrossRef]

C. L. Mehta and E. C. G. Sudarshan, "Relation between Quantum and Semiclassical Description of Optical Coherence," Phys. Rev. 138, B274‒B280 (1965).

[CrossRef]

L. S. Dolin, "Beam description of weakly inhomogeneous wave fields," Izv. Vyssh. Uchebn. Zaved. Radiofiz. 7, 559‒563 (1964).

R. J. Glauber, "Coherent and incoherent states of the radiation field," Phys. Rev. 131, 2766‒2788 (1963).

[CrossRef]

J. R. Klauder, "Continuous representation theory. I. Postulates of continuous representation theory," J. Math. Phys. 4, 1055‒1058 (1963).

[CrossRef]

E. C. G. Sudarshan, "Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams," Phys. Rev. Lett. 10, 277‒279 (1963).

[CrossRef]

H. Margenau and R. N. Hill, "Correlation between measurements in quantum theory," Prog. Theoret. Phys. 26, 722‒738 (1961).

[CrossRef]

J. E. Moyal, "Quantum mechanics as a statistical theory," Math. Proc. Camb. Phil. Soc. 45, 99‒124 (1949).

[CrossRef]

J. Ville, "Thèorie et applications de la notion de signal analytique," Cables Transm. 2A, 6174 (1948).

K. Husimi, "Some formal properties of the density matrix," Proc. Phys. Math. Soc. Jpn. 22, 264‒314 (1940).

E. U. Condon, "Immersion of the Fourier transform in a continuous group of functional transformations," Proc. Natl. Acad. Sci. 23, 158‒163 (1937).

[CrossRef]

J. G. Kirkwood, "Quantum statistics of almost classical ensembles," Phys. Rev. 44, 31‒37 (1933).

[CrossRef]

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

[CrossRef]

J. H. Van Vleck, "The correspondence principle in the statistical interpretation of quantum mechanics," Proc. Natl. Acad. Sci. USA 14, 178‒188 (1928).

[CrossRef]

S. Abe and J. T. Sheridan, "Wigner optics in the metaxial regime," Optik 114, 139‒141.

[CrossRef]

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]

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]

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]

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]

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]

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]

M. A. Alonso, T. Setälä, and A. T. Friberg, "Optimal pulses for arbitrary dispersive media," J. Eur. Opt. Soc. R.P. 6, 1100 (2011).

J. C. Petruccelli and M. A. Alonso, "Phase space distributions tailored for dispersive media," J. Opt. Soc. Am. A 27, 1194‒1201 (2010).

[CrossRef]

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]

S. Cho, J. C. Petruccelli, and M. A. Alonso, "Wigner functions for paraxial and nonparaxial fields," J. Mod. Opt. 56, 1843‒1852 (2009).

[CrossRef]

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]

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]

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]

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]

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]

M. A. Alonso, "Exact description of free electromagnetic wave fields in terms of rays," Opt. Express 11, 3128‒3135 (2003).

[CrossRef]
[PubMed]

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]

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]

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]

M. A. Alonso, "Radiometry and wide-angle wave fields. III. Partial coherence," J. Opt. Soc. Am. A. 18, 2502‒2511 (2001).

[CrossRef]

M. A. Alonso, "Measurement of Helmholtz wave fields," J. Opt. Soc. Am. A 17, 1256‒1264 (2000).

[CrossRef]

M. A. Alonso and G. W. Forbes, "Phase-space distributions for high-frequency fields," J. Opt. Soc. Am. A 17, 2288‒2300 (2000).

[CrossRef]

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]

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]

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]

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]

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.

L. A. Apresyan and Yu. A. Kravtsov, Radiation Transfer: Statistical and Wave Aspects, Gordon and Breach, 1996.

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]

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]

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]

M. V. Berry and N. L. Balazs, "Nonspreading wave packets," Am. J. Phys. 47, 264‒267 (1979).

[CrossRef]

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]

S. B. Oh and G. Barbastathis, "Wigner distribution function of volume holograms," Opt. Lett. 34, 2584‒2586 (2009).

[CrossRef]
[PubMed]

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]

T. Alieva and M. J. Bastiaans, "Properties of the linear canonical integral transformation," J. Opt. Soc. Am. A 24, 3658‒3665 (2007).

[CrossRef]

M. J. Bastiaans, "Uncertainty principle for partially coherent light," J. Opt. Soc. Am. 73, 251‒255 (1983).

[CrossRef]

M. J. Bastiaans, "Transport equations for the Wigner distribution function," J. Mod. Opt. 26, 1265‒1272 (1979).

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, "Wigner distribution function applied to optical signals and systems," Opt. Commun. 25, 26‒30 (1978).

[CrossRef]

M. J. Bastiaans, Wigner Distribution in Optics, Chapter 1 of Ref. [15], pp. 1–44.

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]

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]

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]

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]

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]

J. Ojeda-Castañeda, L. R. Berriel-Valdos, and E. Montes, "Bessel annular apodizers: imaging characteristics," Appl. Opt. 26, 2770‒2772 (1987).

[CrossRef]
[PubMed]

M. V. Berry, "Quantum scars of classical closed orbits in phase space," Proc. R. Soc. Lond. Ser. A 423, 219‒231 (1989).

[CrossRef]

M. V. Berry and F. J. Wright, "Phase-space projection identities for diffraction catastrophes," J. Phys. A. Math. Gen. 13, 149‒160 (1980).

[CrossRef]

M. V. Berry and N. L. Balazs, "Nonspreading wave packets," Am. J. Phys. 47, 264‒267 (1979).

[CrossRef]

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]

M. V. Berry, "Semi-classical mechanics in phase space: a study of Wigners function," Philos. Trans. R. Soc. Lond. 287, 237‒271 (1977).

[CrossRef]

J. Bertrand and P. Bertrand, "A tomographic approach to Wigner’s function," Found. Phys. 17, 397‒405 (1987).

[CrossRef]

J. Bertrand and P. Bertrand, "A tomographic approach to Wigner’s function," Found. Phys. 17, 397‒405 (1987).

[CrossRef]

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

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]

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]

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]

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

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]

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]

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

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]

R. Castañeda and J. Carrasquilla, "Spatial coherence wavelets and phase-space representation of diffraction," Appl. Opt. 47, E76‒E87 (2008).

[CrossRef]
[PubMed]

R. Castañeda, J. Carrasquilla, and J. Herrera, "Radiometric analysis of diffraction of quasi-homogeneous optical fields," Opt. Commun. 273, 8‒20 (2007).

[CrossRef]

R. Castañeda and J. Garcia-Sucerquia, "Electromagnetic spatial coherence wavelets," J. Opt. Soc. Am. A 23, 81‒90 (2006).

[CrossRef]

C.-C. Cheng and M. G. Raymer, "Propagation of transverse optical coherence in random multiple-scattering media," Phys. Rev. A 62, 023811 (2000).

[CrossRef]

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]

P. Loughlin and L. Cohen, "Approximate wave function from approximate non-representable Wigner distributions," J. Mod. Opt. 55, 3379‒3387 (2008).

[CrossRef]

L. Cohen, P. Loughlin, and G. Okopal, "Exact and approximate moments of a propagating pulse," J. Mod. Opt. 55, 3349‒3358 (2008).

[CrossRef]

P. Loughlin and L. Cohen, "A Wigner approximation method for wave propagation," J. Acoust. Soc. Am. 118, 1268‒1271 (2005).

[CrossRef]

L. Cohen, "Time–frequency distributions—a review," Proc. IEEE 77, 941‒981 (1989).

[CrossRef]

L. Cohen, "Generalized phase-space distributions," J. Math. Phys. 7, 781‒786 (1966).

[CrossRef]

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

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

E. U. Condon, "Immersion of the Fourier transform in a continuous group of functional transformations," Proc. Natl. Acad. Sci. 23, 158‒163 (1937).

[CrossRef]

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]

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]

B. J. Davis, "Observable coherence theory for statistically periodic fields," Phys. Rev. A 76, 043843 (2007).

[CrossRef]

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]

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

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]

L. S. Dolin, "Beam description of weakly inhomogeneous wave fields," Izv. Vyssh. Uchebn. Zaved. Radiofiz. 7, 559‒563 (1964).

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]

D. Dragoman, "Wigner distribution function in nonlinear optics," Appl. Opt. 35, 4142‒4146 (1996).

[CrossRef]
[PubMed]

D. Dragoman, "Wigner-distribution-function representation of the coupling coefficient," Appl. Opt. 34, 6758‒6763 (1995).

[CrossRef]
[PubMed]

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

J. E. Durnin, J. J. Miceli, and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1499‒1501 (1987).

[CrossRef]
[PubMed]

A. Erdélyi, Asymptotic Expansions, Dover, 1956.

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]

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]

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]

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]

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]

G. Folland and A. Sitaram, "The uncertainty principle: a mathematical survey," J. Fourier Anal. Appl. 3, 207‒238 (1997).

[CrossRef]

M. A. Alonso and G. W. Forbes, "Phase-space distributions for high-frequency fields," J. Opt. Soc. Am. A 17, 2288‒2300 (2000).

[CrossRef]

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]

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]

M. A. Alonso, T. Setälä, and A. T. Friberg, "Optimal pulses for arbitrary dispersive media," J. Eur. Opt. Soc. R.P. 6, 1100 (2011).

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]

A. T. Friberg, "Effects of coherence in radiometry," Opt. Eng. 21, 927‒936 (1982).

A. T. Friberg, "On the generalized radiance associated with radiation from a quasihomogeneous planar source," Opt. Acta 28, 261‒277 (1981).

[CrossRef]

C. C. Gerry and P. L. Knight, Introductory Quantum Optics, Cambridge University Press, 2005, pp. 56‒71.

R. J. Glauber, "Coherent and incoherent states of the radiation field," Phys. Rev. 131, 2766‒2788 (1963).

[CrossRef]

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.

J. W. Goodman, Introduction to Fourier Optics, McGraw-Hill, 1988, pp. 101‒136.

L.-P. Guigay, Ambiguity Function in Optical Imaging, Chapter 2 of Ref. [15], pp. 45–62.

M. C. Gutzwiller, "Periodic orbits and classical quantization conditions," J. Math. Phys. 12, 343‒358 (1971).

[CrossRef]

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]

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]

G. Hazak, "Comment on ‘Wave field determination using three-dimensional intensity information’," Phys. Rev. Lett. 69, 2874‒2874 (1992).

[CrossRef]
[PubMed]

B. M. Hennelly, J. J. Healy, and J. T. Sheridan, Sampling and Phase Space, Chapter 10 of Ref. [15], pp. 309–336.

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]

J. T. Sheridan, W. T. Rhodes, and B. M. Hennelly, "Wigner Optics," Proc. SPIE 5827, 627‒638 (2005).

B. M. Hennelly, J. J. Healy, and J. T. Sheridan, Sampling and Phase Space, Chapter 10 of Ref. [15], pp. 309–336.

R. Castañeda, J. Carrasquilla, and J. Herrera, "Radiometric analysis of diffraction of quasi-homogeneous optical fields," Opt. Commun. 273, 8‒20 (2007).

[CrossRef]

H. Margenau and R. N. Hill, "Correlation between measurements in quantum theory," Prog. Theoret. Phys. 26, 722‒738 (1961).

[CrossRef]

M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, "Distribution functions in physics: Fundamentals," 106 121‒167 (1984).

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]

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]

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

K. Husimi, "Some formal properties of the density matrix," Proc. Phys. Math. Soc. Jpn. 22, 264‒314 (1940).

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]

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]

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]

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]

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]

Y. Kano, "A new phase-space distribution function in the statistical theory of the electromagnetic field," J. Math. Phys. 6, 1913‒1915 (1965).

[CrossRef]

J. G. Kirkwood, "Quantum statistics of almost classical ensembles," Phys. Rev. 44, 31‒37 (1933).

[CrossRef]

J. R. Klauder, "Continuous representation theory. I. Postulates of continuous representation theory," J. Math. Phys. 4, 1055‒1058 (1963).

[CrossRef]

C. C. Gerry and P. L. Knight, Introductory Quantum Optics, Cambridge University Press, 2005, pp. 56‒71.

B. H. Kolner, "Space–time duality and the theory of temporal imaging," IEEE J. Quantum Electron. 30, 1951‒1963 (1994).

[CrossRef]

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.

L. A. Apresyan and Yu. A. Kravtsov, Radiation Transfer: Statistical and Wave Aspects, Gordon and Breach, 1996.

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

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

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]

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]

H. W. Lee, "Theory and application of the quantum phase-space distribution functions," Phys. Rep. 259, 147‒211 (1995).

[CrossRef]

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

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]

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]

M. Testorf and A. W. Lohmann, "Holography in phase space," Appl. Opt. 47, A70‒A77 (2008).

[CrossRef]
[PubMed]

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]

A. W. Lohmann, "Image rotation, Wigner rotation, and the fractional Fourier transform," J. Opt. Soc. Am. A 10, 2181‒2186 (1993).

[CrossRef]

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]

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

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.

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.

L. Cohen, P. Loughlin, and G. Okopal, "Exact and approximate moments of a propagating pulse," J. Mod. Opt. 55, 3349‒3358 (2008).

[CrossRef]

P. Loughlin and L. Cohen, "Approximate wave function from approximate non-representable Wigner distributions," J. Mod. Opt. 55, 3379‒3387 (2008).

[CrossRef]

P. Loughlin and L. Cohen, "A Wigner approximation method for wave propagation," J. Acoust. Soc. Am. 118, 1268‒1271 (2005).

[CrossRef]

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

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]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, Cambridge University Press, 1995, Sec. 4.7.

H. Margenau and R. N. Hill, "Correlation between measurements in quantum theory," Prog. Theoret. Phys. 26, 722‒738 (1961).

[CrossRef]

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]

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]

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

C. L. Mehta and E. C. G. Sudarshan, "Relation between Quantum and Semiclassical Description of Optical Coherence," Phys. Rev. 138, B274‒B280 (1965).

[CrossRef]

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]

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]

D. Mendlovic and H. M. Ozaktas, "Fractional Fourier transforms and their optical implementation: I," J. Opt. Soc. Am. A 10, 1875‒1881 (1993).

[CrossRef]

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

J. E. Durnin, J. J. Miceli, and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1499‒1501 (1987).

[CrossRef]
[PubMed]

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]

J. Ojeda-Castañeda, L. R. Berriel-Valdos, and E. Montes, "Bessel annular apodizers: imaging characteristics," Appl. Opt. 26, 2770‒2772 (1987).

[CrossRef]
[PubMed]

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

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]

M. Moshinsky and C. Quesne, "Linear canonical transformations and their unitary representations," J. Math. Phys. 12, (8), 1772‒1783 (1971).

[CrossRef]

J. E. Moyal, "Quantum mechanics as a statistical theory," Math. Proc. Camb. Phil. Soc. 45, 99‒124 (1949).

[CrossRef]

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]

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]

V. Namias, "The fractional order Fourier transform and its application to quantum mechanics," IMA J. Appl. Math. 25, 241‒265 (1980).

[CrossRef]

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]

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]

K. A. Nugent, "Wave field determination using three-dimensional intensity information," Phys. Rev. Lett. 68, 2261‒2264 (1992).

[CrossRef]
[PubMed]

M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, "Distribution functions in physics: Fundamentals," 106 121‒167 (1984).

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]

S. B. Oh and G. Barbastathis, "Wigner distribution function of volume holograms," Opt. Lett. 34, 2584‒2586 (2009).

[CrossRef]
[PubMed]

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]

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]

M. Testorf and J. Ojeda-Castañeda, "Fractional Talbot effect: analysis in phase space," J. Opt. Soc. Am. A 13, 119‒125 (1996).

[CrossRef]

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]

J. Ojeda-Castañeda, L. R. Berriel-Valdos, and E. Montes, "Bessel annular apodizers: imaging characteristics," Appl. Opt. 26, 2770‒2772 (1987).

[CrossRef]
[PubMed]

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]

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.

L. Cohen, P. Loughlin, and G. Okopal, "Exact and approximate moments of a propagating pulse," J. Mod. Opt. 55, 3349‒3358 (2008).

[CrossRef]

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]

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]

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]

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]

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]

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]

H. M. Pedersen, "Geometric theory of fields radiated from three-dimensional, quasi-homogeneous sources," J. Opt. Soc. Am. A 9, 1626‒1632 (1992).

[CrossRef]

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]

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.

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]

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]

J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780‒1782 (2006).

[CrossRef]
[PubMed]

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]

J. C. Petruccelli and M. A. Alonso, "Phase space distributions tailored for dispersive media," J. Opt. Soc. Am. A 27, 1194‒1201 (2010).

[CrossRef]

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]

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]

S. Cho, J. C. Petruccelli, and M. A. Alonso, "Wigner functions for paraxial and nonparaxial fields," J. Mod. Opt. 56, 1843‒1852 (2009).

[CrossRef]

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]

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]

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]

M. Moshinsky and C. Quesne, "Linear canonical transformations and their unitary representations," J. Math. Phys. 12, (8), 1772‒1783 (1971).

[CrossRef]

C.-C. Cheng and M. G. Raymer, "Propagation of transverse optical coherence in random multiple-scattering media," Phys. Rev. A 62, 023811 (2000).

[CrossRef]

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]

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]

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]

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]

J. T. Sheridan, W. T. Rhodes, and B. M. Hennelly, "Wigner Optics," Proc. SPIE 5827, 627‒638 (2005).

A. W. Rihaczek, "Signal energy distribution in time and frequency," IEEE Trans. Info. Theory 14, 369‒374 (1968).

[CrossRef]

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]

K. B. Wolf and A. L. Rivera, "Holographic information in the Wigner function," Opt. Commun. 144, 36‒42 (1997).

[CrossRef]

R. W. Robinett, "Quantum wave packet revivals," Phys. Rep. 392, 1‒119 (2004).

[CrossRef]

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]

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed., Wiley, 2007, p. 188.

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]

W. P. Schleich, Quantum Optics in Phase Space, Wiley-VCH, 2001.

J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780‒1782 (2006).

[CrossRef]
[PubMed]

M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, "Distribution functions in physics: Fundamentals," 106 121‒167 (1984).

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]

M. A. Alonso, T. Setälä, and A. T. Friberg, "Optimal pulses for arbitrary dispersive media," J. Eur. Opt. Soc. R.P. 6, 1100 (2011).

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]

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]

C. J. R. Sheppard and K. G. Larkin, "Wigner function for nonparaxial wave fields," J. Opt. Soc. Am. A 18, 2486‒2490 (2001).

[CrossRef]

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]

J. T. Sheridan, W. T. Rhodes, and B. M. Hennelly, "Wigner Optics," Proc. SPIE 5827, 627‒638 (2005).

B. M. Hennelly, J. J. Healy, and J. T. Sheridan, Sampling and Phase Space, Chapter 10 of Ref. [15], pp. 309–336.

S. Abe and J. T. Sheridan, "Wigner optics in the metaxial regime," Optik 114, 139‒141.

[CrossRef]

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]

G. Folland and A. Sitaram, "The uncertainty principle: a mathematical survey," J. Fourier Anal. Appl. 3, 207‒238 (1997).

[CrossRef]

J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780‒1782 (2006).

[CrossRef]
[PubMed]

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]

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

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]

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]

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]

C. L. Mehta and E. C. G. Sudarshan, "Relation between Quantum and Semiclassical Description of Optical Coherence," Phys. Rev. 138, B274‒B280 (1965).

[CrossRef]

E. C. G. Sudarshan, "Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams," Phys. Rev. Lett. 10, 277‒279 (1963).

[CrossRef]

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]

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed., Wiley, 2007, p. 188.

M. Testorf and A. W. Lohmann, "Holography in phase space," Appl. Opt. 47, A70‒A77 (2008).

[CrossRef]
[PubMed]

M. Testorf, "Designing Talbot array illuminators with phase-space optics," J. Opt. Soc. Am. A 23, 187‒192 (2006).

[CrossRef]

M. Testorf, "Analysis of the moiré effect by use of the Wigner distribution function," J. Opt. Soc. Am. A 17, 2536‒2542 (2000).

[CrossRef]

M. Testorf and J. Ojeda-Castañeda, "Fractional Talbot effect: analysis in phase space," J. Opt. Soc. Am. A 13, 119‒125 (1996).

[CrossRef]

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]

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.

M. E. Testorf, Self-imaging in Phase Space, Chapter 9 of Ref. [15], pp. 279–307.

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

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]

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]

J. H. Van Vleck, "The correspondence principle in the statistical interpretation of quantum mechanics," Proc. Natl. Acad. Sci. USA 14, 178‒188 (1928).

[CrossRef]

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]

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]

J. Ville, "Thèorie et applications de la notion de signal analytique," Cables Transm. 2A, 6174 (1948).

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]

I. A. Walmsley and C. Dorrer, "Characterization of ultrashort electromagnetic pulses," Adv. Opt. Phot. 1, 308‒437 (2009).

[CrossRef]

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]

C. Dorrer and I. A. Walmsley, Phase space in ultrafast optics, Chapter 11 of Ref. [15], pp. 337–383.

A. Walther, "Radiometry and coherence," J. Opt. Soc. Am. 63, 1622‒1623 (1973).

[CrossRef]

A. Walther, "Lenses, wave optics and eikonal functions," J. Opt. Soc. Am. 59, 1325‒1333 (1969).

[CrossRef]

A. Walther, "Radiometry and coherence," J. Opt. Soc. Am. 58, 1256‒1259 (1968).

[CrossRef]

A. Walther, The Ray and Wave Theory of Lenses, Cambridge University Press, 1995, pp. 169‒187.

M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, "Distribution functions in physics: Fundamentals," 106 121‒167 (1984).

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

[CrossRef]

E. Wolf, "Radiometric model for propagation of coherence," Opt. Lett. 19, 2024‒2026 (1994).

[CrossRef]
[PubMed]

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]

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]

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

[CrossRef]

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]

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]

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]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, Cambridge University Press, 1995, Sec. 4.7.

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

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]

K. B. Wolf and A. L. Rivera, "Holographic information in the Wigner function," Opt. Commun. 144, 36‒42 (1997).

[CrossRef]

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

P. M. Woodward, Probability and Information Theory with Applications to Radar, Pergamon, 1953.

M. V. Berry and F. J. Wright, "Phase-space projection identities for diffraction catastrophes," J. Phys. A. Math. Gen. 13, 149‒160 (1980).

[CrossRef]

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

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]

M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, "Distribution functions in physics: Fundamentals," 106 121‒167 (1984).

I. A. Walmsley and C. Dorrer, "Characterization of ultrashort electromagnetic pulses," Adv. Opt. Phot. 1, 308‒437 (2009).

[CrossRef]

M. V. Berry and N. L. Balazs, "Nonspreading wave packets," Am. J. Phys. 47, 264‒267 (1979).

[CrossRef]

E. R. Dowski and W. T. Cathey, "Extended depth of field through wave-front coding," Appl. Opt. 34, 1859‒1866 (1995).

[CrossRef]
[PubMed]

J. Ojeda-Castañeda, L. R. Berriel-Valdos, and E. Montes, "Bessel annular apodizers: imaging characteristics," Appl. Opt. 26, 2770‒2772 (1987).

[CrossRef]
[PubMed]

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]

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]

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]

M. Testorf and A. W. Lohmann, "Holography in phase space," Appl. Opt. 47, A70‒A77 (2008).

[CrossRef]
[PubMed]

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]

R. Castañeda and J. Carrasquilla, "Spatial coherence wavelets and phase-space representation of diffraction," Appl. Opt. 47, E76‒E87 (2008).

[CrossRef]
[PubMed]

D. Dragoman, "Wigner distribution function in nonlinear optics," Appl. Opt. 35, 4142‒4146 (1996).

[CrossRef]
[PubMed]

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]

D. Dragoman, "Wigner-distribution-function representation of the coupling coefficient," Appl. Opt. 34, 6758‒6763 (1995).

[CrossRef]
[PubMed]

J. Ville, "Thèorie et applications de la notion de signal analytique," Cables Transm. 2A, 6174 (1948).

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]

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]

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]

J. Bertrand and P. Bertrand, "A tomographic approach to Wigner’s function," Found. Phys. 17, 397‒405 (1987).

[CrossRef]

B. H. Kolner, "Space–time duality and the theory of temporal imaging," IEEE J. Quantum Electron. 30, 1951‒1963 (1994).

[CrossRef]

A. W. Rihaczek, "Signal energy distribution in time and frequency," IEEE Trans. Info. Theory 14, 369‒374 (1968).

[CrossRef]

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]

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]

V. Namias, "The fractional order Fourier transform and its application to quantum mechanics," IMA J. Appl. Math. 25, 241‒265 (1980).

[CrossRef]

L. S. Dolin, "Beam description of weakly inhomogeneous wave fields," Izv. Vyssh. Uchebn. Zaved. Radiofiz. 7, 559‒563 (1964).

P. Loughlin and L. Cohen, "A Wigner approximation method for wave propagation," J. Acoust. Soc. Am. 118, 1268‒1271 (2005).

[CrossRef]

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]

M. A. Alonso, T. Setälä, and A. T. Friberg, "Optimal pulses for arbitrary dispersive media," J. Eur. Opt. Soc. R.P. 6, 1100 (2011).

G. Folland and A. Sitaram, "The uncertainty principle: a mathematical survey," J. Fourier Anal. Appl. 3, 207‒238 (1997).

[CrossRef]

M. Moshinsky and C. Quesne, "Linear canonical transformations and their unitary representations," J. Math. Phys. 12, (8), 1772‒1783 (1971).

[CrossRef]

Y. Kano, "A new phase-space distribution function in the statistical theory of the electromagnetic field," J. Math. Phys. 6, 1913‒1915 (1965).

[CrossRef]

J. R. Klauder, "Continuous representation theory. I. Postulates of continuous representation theory," J. Math. Phys. 4, 1055‒1058 (1963).

[CrossRef]

L. Cohen, "Generalized phase-space distributions," J. Math. Phys. 7, 781‒786 (1966).

[CrossRef]

M. C. Gutzwiller, "Periodic orbits and classical quantization conditions," J. Math. Phys. 12, 343‒358 (1971).

[CrossRef]

M. J. Bastiaans, "Transport equations for the Wigner distribution function," J. Mod. Opt. 26, 1265‒1272 (1979).

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]

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]

S. Cho, J. C. Petruccelli, and M. A. Alonso, "Wigner functions for paraxial and nonparaxial fields," J. Mod. Opt. 56, 1843‒1852 (2009).

[CrossRef]

P. Loughlin and L. Cohen, "Approximate wave function from approximate non-representable Wigner distributions," J. Mod. Opt. 55, 3379‒3387 (2008).

[CrossRef]

L. Cohen, P. Loughlin, and G. Okopal, "Exact and approximate moments of a propagating pulse," J. Mod. Opt. 55, 3349‒3358 (2008).

[CrossRef]

A. Walther, "Radiometry and coherence," J. Opt. Soc. Am. 63, 1622‒1623 (1973).

[CrossRef]

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

[CrossRef]

M. J. Bastiaans, "Wigner distribution function and its application to first-order optics," J. Opt. Soc. Am. 69, 1710‒1716 (1979).

[CrossRef]

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

[CrossRef]

A. Walther, "Lenses, wave optics and eikonal functions," J. Opt. Soc. Am. 59, 1325‒1333 (1969).

[CrossRef]

W. T. Welford, "Use of annular apertures to increase focal depth," J. Opt. Soc. Am. 50, 749 (1960).

[CrossRef]

J. H. Eberly and K. Wódkiewicz, "The time-dependent physical spectrum of light," J. Opt. Soc. Am. 67, 1252‒1261 (1977).

[CrossRef]

M. J. Bastiaans, "Uncertainty principle for partially coherent light," J. Opt. Soc. Am. 73, 251‒255 (1983).

[CrossRef]

A. Starikov, "Effective number of degrees of freedom of partially coherent sources," J. Opt. Soc. Am. 72, 1538‒1544 (1982).

[CrossRef]

S. A. Collins, "Lens-system diffraction integral written in terms of matrix optics," J. Opt. Soc. Am. 60, 1168‒1177 (1970).

[CrossRef]

A. Walther, "Radiometry and coherence," J. Opt. Soc. Am. 58, 1256‒1259 (1968).

[CrossRef]

A. W. Lohmann, "Image rotation, Wigner rotation, and the fractional Fourier transform," J. Opt. Soc. Am. A 10, 2181‒2186 (1993).

[CrossRef]

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]

R. G. Littlejohn and R. Winston, "Generalized radiance and measurement," J. Opt. Soc. Am. A 12, 2736‒2743 (1995).

[CrossRef]

M. A. Alonso and G. W. Forbes, "Phase-space distributions for high-frequency fields," J. Opt. Soc. Am. A 17, 2288‒2300 (2000).

[CrossRef]

T. Alieva and M. J. Bastiaans, "Properties of the linear canonical integral transformation," J. Opt. Soc. Am. A 24, 3658‒3665 (2007).

[CrossRef]

M. A. Alonso, "Measurement of Helmholtz wave fields," J. Opt. Soc. Am. A 17, 1256‒1264 (2000).

[CrossRef]

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]

M. Testorf and J. Ojeda-Castañeda, "Fractional Talbot effect: analysis in phase space," J. Opt. Soc. Am. A 13, 119‒125 (1996).

[CrossRef]

M. Testorf, "Designing Talbot array illuminators with phase-space optics," J. Opt. Soc. Am. A 23, 187‒192 (2006).

[CrossRef]

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]

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]

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]

D. Mendlovic and H. M. Ozaktas, "Fractional Fourier transforms and their optical implementation: I," J. Opt. Soc. Am. A 10, 1875‒1881 (1993).

[CrossRef]

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]

A. Luis, "Ray picture of polarization and coherence in a Young interferometer," J. Opt. Soc. Am. A 23, 2855‒2860 (2006).

[CrossRef]

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]

M. Testorf, "Analysis of the moiré effect by use of the Wigner distribution function," J. Opt. Soc. Am. A 17, 2536‒2542 (2000).

[CrossRef]

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]

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]

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]

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]

R. Castañeda and J. Garcia-Sucerquia, "Electromagnetic spatial coherence wavelets," J. Opt. Soc. Am. A 23, 81‒90 (2006).

[CrossRef]

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]

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]

F. Gori, M. Santarsiero, and G. Guattari, "Coherence and the spatial distribution of intensity," J. Opt. Soc. Am. A 10, 673‒679 (1993).

[CrossRef]

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]

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]

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]

C. J. R. Sheppard and K. G. Larkin, "Wigner function for nonparaxial wave fields," J. Opt. Soc. Am. A 18, 2486‒2490 (2001).

[CrossRef]

J. C. Petruccelli and M. A. Alonso, "Phase space distributions tailored for dispersive media," J. Opt. Soc. Am. A 27, 1194‒1201 (2010).

[CrossRef]

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]

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]

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]

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]

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]

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]

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]

H. M. Pedersen, "Geometric theory of fields radiated from three-dimensional, quasi-homogeneous sources," J. Opt. Soc. Am. A 9, 1626‒1632 (1992).

[CrossRef]

R. G. Littlejohn and R. Winston, "Corrections to classical radiometry," J. Opt. Soc. Am. A 10, 2024‒2037 (1993).

[CrossRef]

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]

M. A. Alonso, "Radiometry and wide-angle wave fields. III. Partial coherence," J. Opt. Soc. Am. A. 18, 2502‒2511 (2001).

[CrossRef]

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]

M. V. Berry and F. J. Wright, "Phase-space projection identities for diffraction catastrophes," J. Phys. A. Math. Gen. 13, 149‒160 (1980).

[CrossRef]

J. E. Moyal, "Quantum mechanics as a statistical theory," Math. Proc. Camb. Phil. Soc. 45, 99‒124 (1949).

[CrossRef]

A. T. Friberg, "On the generalized radiance associated with radiation from a quasihomogeneous planar source," Opt. Acta 28, 261‒277 (1981).

[CrossRef]

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]

R. Castañeda, J. Carrasquilla, and J. Herrera, "Radiometric analysis of diffraction of quasi-homogeneous optical fields," Opt. Commun. 273, 8‒20 (2007).

[CrossRef]

F. Gori, "Fresnel transform and sampling theorem," Opt. Commun. 39, 293‒297 (1981).

[CrossRef]

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]

A. Luis, "Spatial-angular Mueller matrices," Opt. Commun. 263, 141‒146 (2006).

[CrossRef]

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]

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]

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]

K. B. Wolf and A. L. Rivera, "Holographic information in the Wigner function," Opt. Commun. 144, 36‒42 (1997).

[CrossRef]

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

[CrossRef]

M. J. Bastiaans, "The Wigner distribution function and Hamilton’s characteristics of a geometric–optical system," Opt. Commun. 30, 321‒326 (1979).

[CrossRef]

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]

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]

A. T. Friberg, "Effects of coherence in radiometry," Opt. Eng. 21, 927‒936 (1982).

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]

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]

M. A. Alonso, "Exact description of free electromagnetic wave fields in terms of rays," Opt. Express 11, 3128‒3135 (2003).

[CrossRef]
[PubMed]

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]

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]

C. Dorrer and I. Kang, "Complete temporal characterization of short optical pulses by simplified chronocyclic tomography," Opt. Lett. 28, 1481‒1483 (2003).

[CrossRef]
[PubMed]

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]

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]

E. Wolf, "Radiometric model for propagation of coherence," Opt. Lett. 19, 2024‒2026 (1994).

[CrossRef]
[PubMed]

S. B. Oh and G. Barbastathis, "Wigner distribution function of volume holograms," Opt. Lett. 34, 2584‒2586 (2009).

[CrossRef]
[PubMed]

G. Siviloglou and D. Christodoulides, "Accelerating finite energy Airy beams," Opt. Lett. 32, 979‒981 (2007).

[CrossRef]
[PubMed]

A. Wax and J. E. Thomas, "Optical heterodyne imaging and Wigner phase space distributions," Opt. Lett. 21, 1427‒1429 (1996).

[CrossRef]
[PubMed]

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]

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]

S. Abe and J. T. Sheridan, "Wigner optics in the metaxial regime," Optik 114, 139‒141.

[CrossRef]

M. V. Berry, "Semi-classical mechanics in phase space: a study of Wigners function," Philos. Trans. R. Soc. Lond. 287, 237‒271 (1977).

[CrossRef]

H. W. Lee, "Theory and application of the quantum phase-space distribution functions," Phys. Rep. 259, 147‒211 (1995).

[CrossRef]

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]

R. W. Robinett, "Quantum wave packet revivals," Phys. Rep. 392, 1‒119 (2004).

[CrossRef]

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

[CrossRef]

C. L. Mehta and E. C. G. Sudarshan, "Relation between Quantum and Semiclassical Description of Optical Coherence," Phys. Rev. 138, B274‒B280 (1965).

[CrossRef]

R. J. Glauber, "Coherent and incoherent states of the radiation field," Phys. Rev. 131, 2766‒2788 (1963).

[CrossRef]

J. G. Kirkwood, "Quantum statistics of almost classical ensembles," Phys. Rev. 44, 31‒37 (1933).

[CrossRef]

C.-C. Cheng and M. G. Raymer, "Propagation of transverse optical coherence in random multiple-scattering media," Phys. Rev. A 62, 023811 (2000).

[CrossRef]

B. J. Davis, "Observable coherence theory for statistically periodic fields," Phys. Rev. A 76, 043843 (2007).

[CrossRef]

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]

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]

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]

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]

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]

J. Tu and S. Tamura, "Wave field determination using tomography of the ambiguity function," Phys. Rev. E 55, 1946‒1949 (1997).

[CrossRef]

K. A. Nugent, "Wave field determination using three-dimensional intensity information," Phys. Rev. Lett. 68, 2261‒2264 (1992).

[CrossRef]
[PubMed]

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]

G. Hazak, "Comment on ‘Wave field determination using three-dimensional intensity information’," Phys. Rev. Lett. 69, 2874‒2874 (1992).

[CrossRef]
[PubMed]

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]

J. E. Durnin, J. J. Miceli, and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1499‒1501 (1987).

[CrossRef]
[PubMed]

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]

E. C. G. Sudarshan, "Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams," Phys. Rev. Lett. 10, 277‒279 (1963).

[CrossRef]

L. Cohen, "Time–frequency distributions—a review," Proc. IEEE 77, 941‒981 (1989).

[CrossRef]

E. U. Condon, "Immersion of the Fourier transform in a continuous group of functional transformations," Proc. Natl. Acad. Sci. 23, 158‒163 (1937).

[CrossRef]

J. H. Van Vleck, "The correspondence principle in the statistical interpretation of quantum mechanics," Proc. Natl. Acad. Sci. USA 14, 178‒188 (1928).

[CrossRef]

K. Husimi, "Some formal properties of the density matrix," Proc. Phys. Math. Soc. Jpn. 22, 264‒314 (1940).

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]

M. V. Berry, "Quantum scars of classical closed orbits in phase space," Proc. R. Soc. Lond. Ser. A 423, 219‒231 (1989).

[CrossRef]

J. T. Sheridan, W. T. Rhodes, and B. M. Hennelly, "Wigner Optics," Proc. SPIE 5827, 627‒638 (2005).

H. Margenau and R. N. Hill, "Correlation between measurements in quantum theory," Prog. Theoret. Phys. 26, 722‒738 (1961).

[CrossRef]

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]

J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780‒1782 (2006).

[CrossRef]
[PubMed]

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.

J. C. Petruccelli, Generalized Wigner Functions, Ph.D. Thesis, University of Rochester (Rochester, NY, 2010)

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

C. C. Gerry and P. L. Knight, Introductory Quantum Optics, Cambridge University Press, 2005, pp. 56‒71.

W. P. Schleich, Quantum Optics in Phase Space, Wiley-VCH, 2001.

C. Dorrer and I. A. Walmsley, Phase space in ultrafast optics, Chapter 11 of Ref. [15], pp. 337–383.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed., Wiley, 2007, p. 188.

P. M. Woodward, Probability and Information Theory with Applications to Radar, Pergamon, 1953.

L.-P. Guigay, Ambiguity Function in Optical Imaging, Chapter 2 of Ref. [15], pp. 45–62.

A. Walther, The Ray and Wave Theory of Lenses, Cambridge University Press, 1995, pp. 169‒187.

J. W. Goodman, Introduction to Fourier Optics, McGraw-Hill, 1988, pp. 101‒136.

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.

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.

M. E. Testorf, Self-imaging in Phase Space, Chapter 9 of Ref. [15], pp. 279–307.

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.

L. A. Apresyan and Yu. A. Kravtsov, Radiation Transfer: Statistical and Wave Aspects, Gordon and Breach, 1996.

A. T. Friberg, ed., Selected Papers on Coherence and Radiometry, Milestone Series, Vol. MS69, SPIE Optical Engineering Press, 1993.

B. M. Hennelly, J. J. Healy, and J. T. Sheridan, Sampling and Phase Space, Chapter 10 of Ref. [15], pp. 309–336.

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.

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.

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

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, Cambridge University Press, 1995, Sec. 4.7.

See Ref. [50], p. 261

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

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.

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

See Ref. [6], pp. 75–80