On the Wigner distribution moments of rotationally symmetric partially coherent light

Martin J. Bastiaans and Tatiana Alieva

After the introduction of the Wigner distribution [1] for the description of coherent and partially coherent optical fields in a phase space, it became an important tool for optical signal/image analysis and beam characterization. The Wigner distribution completely describes the complex amplitude of a coherent optical field (up to a constant phase factor), or the two-point correlation function (or cross-spectral density) of a partially coherent field. Since the Wigner distribution of a two-dimensional optical field is a function of four variables, it is difficult to analyze. Therefore, the optical field is often represented not by the Wigner distribution itself, but by its global moments. Beam characterization based on the second-order moments of the Wigner distribution, for instance, thus became the basis of an International Organization for Standardization standard. [2]

In previous papers, the special but important case of rotational symmetry has been studied extensively; we mention twisted Gaussian-Schell model light and the characterization of rotationally symmetric light in terms of second-order moments. In this paper we study the particular case of rotationally symmetric partially coherent light, and the constraints that this kind of symmetry imposes on the second- and higher-order moments. We will see that only even-order moments remain, and that the number of parameters that we need to describe these even-order moments reduces drastically compared to the general case: whereas in general we have (N+1)(N+2)(N+3)/6 different moments of order N, this number reduces to (1+N/2)^2 in the case of rotational symmetry. A way to measure the moments as intensity moments in the output planes of (generally anamorphic but separable) first-order optical systems is presented. [3]

  1. E. Wigner, "On the quantum correction for thermodynamic equilibrium," Phys. Rev., vol. 40, pp. 749-759, 1932.
  2. ISO Technical Committee/Subcommittee 172/SC9, "Lasers and laser-related equipment - test methods for laser beam parameters - beam widths, divergence angle and beam propagation factor," ISO Doc. 11146, International Organization for Standardization, Geneva, 1999.
  3. M. J. Bastiaans and T. Alieva, "Wigner distribution moments in fractional Fourier transform systems," J. Opt. Soc. Am. A, vol. 19, pp. 1763-1773, 2002.

PDF version of the poster presentation

To: Papers by Martin J. Bastiaans