Wigner distribution function of Bessel-Gauss beams

Three equivalent expressions are presented for the Wigner distribution function (WDF) of a Bessel-Gauss (BG) beam. The first involves double summations of products of Laguerre-Gauss functions; the second is a single summation of modified Bessel functions; and the third is a compact and elegant integral representation that shows that the WDF of the BG of order m is proportional to the mth coefficient of the complex Fourier series of a 2π-periodic function. The WDF can be written as a function of three quadratic forms in phase-space, namely, the Hamiltonian, the Lagrangian, and the orbital angular momentum. Symmetries and limiting cases of the WDF-BG are also analyzed.