Propagation law for Hermite- and Laguerre-Gaussian beams in first-order optical systems

Starting from Hermite-Gaussian beams, we generate a general class of rotationally symmetric beams. These beams are Laguerre-Gaussian beams, parameterized by two parameters h and g, representing the curvature and the width of the beam, respectively. The Wigner distribution of each member of this class is readily derived from the Wigner distribution of the Hermite-Gaussian beam from which it is generated. If these Laguerre-Gaussian beams propagate through an isotropic abcd-system, they remain in their class, while the propagation of the complex beam parameter h+ig (or h-ig) satisfies the well-known abcd-law