Ray tracing based on the Wigner representation of optical fields
An optical ray is defined by its distance to the optical axis of an object, an image, or a refracting surface, and an angle giving its direction of propagation, that is, by a 2-dimensional space vector and a 2-dimensional angular frequency. Since we use spherical segments, the angular frequency is related to the angle between the ray and the normal to the spherical segment at the incident point. The Wigner representation of optical fields is a 4-dimensional space-frequency function, defined after choosing appropriate scaled space and frequency variables that are linked to "physical" space variables and angular frequencies. The effect of diffraction on the Wigner distribution of an optical field is deduced from a scalar theory of diffraction and provides a link between scaled space and frequency variables on an emitter and on a receiver at a given distance. By changing scaled variables back into physical ones, we obtain a ray tracing method which is based on a scalar theory of diffraction. An explicit example and an application to optical resonators illustrate the method.
 M. A. Alonso, "Wigner functions in optics: describing beams as ray bundles and pulses as particle ensembles", Advances in Optics and Photonics 3, 272-365 (2011).
 A. Walther, "Radiometry and Coherence", J. Opt. Soc. Am. 58, 1256-1259 (1968).
 M. J. Bastiaans, "Wigner distribution function and its application to first-order optics", J. Opt. Soc. Am. 69, 1710-1716 (1979).
 T. Cuypers, T. Haber, P. Bekaert, S. B. Oh, R. Raskar, "Reflectance Model for Diffraction", ACM Transactions on Graphics 28, Art. 106 (2009).
 T. Cuypers, R. Horstmeyer, P. Bekaert, R. Raskar, "Validity of Wigner distribution function for ray-based imaging", IEEE International Conference on Computational Photography (ICCP), Section 4 (2011).
 B. M. Mout, M. Wick, F. Bociort, H. P. Urbach, "A Wigner-based-ray-tracing method for imaging simulations", in Optical Systems Design: Computational Optics, Daniel G. Smith, Frank Wyrowski, Andreas Erdmann Eds., Proc. SPIE 9630, 96300Z-1 - 96300Z-11 (2015).
 J. W. Goodman, Introduction to Fourier optics, 3rd Ed., Englewood, Robert & Compagny (2005).
 P. Pellat-Finet, Optique de Fourier, théorie métaxiale et fractionnaire, Paris, Springer (2009).
 V. Namias, "The fractional order Fourier transform and its application to quantum mechanics", J. Inst. Maths Applics 25, 241-265 (1980).
 H. M. Ozaktas, Z. Zalzevsky, M. A. Kutay, The Fractional Fourier Transform, with Applications in Optics and Signal Processing, Chichester, John Wiley & Sons (2001).
 P. Pellat-Finet, E. Fogret, "Complex order fractional Fourier transforms and their use in diffraction theory", Opt. Comm. 258, 103-113 (2006).
 P. Pellat-Finet, E. Fogret, "A fractional Fourier Transform Theory of Optical Resonators", in Peter S. Emersone Ed. Progress in Optical Fibers, New York, Nova Science Publisher, 299-351 (2011).
 P. Pellat-Finet, P.-E. Durand, E. Fogret, "Spherical angular spectrum and the fractional order Fourier transform", Opt. Lett. 31, 3429-3431 (2006).