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Based on the common Hermite-Gaussian modes, a general class of orthonormal Hermite-Gaussian-type modes is introduced. Such modes can most easily be defined by means of their generating function. A propagation law for the generating function is formulated, when these modes propagate through 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.

It is shown how all global Wigner distribution moments of arbitrary order can be measured as intensity moments in the output plane of an appropriate number of fractional Fourier transform systems (generally anamorphic ones). The minimum number of (anamorphic) fractional power spectra that are needed for the determination of these moments is derived.

The main properties of n-th order power filtering in the fractional Fourier domain and its relationship to the differentiation operation are considered. The application of power filtering for solving the phase retrieval problem is proposed.

The orthonormal set of Hermite-Gaussian modes maps into another orthonormal set after propagation through a first-order optical system. A compact expression for these modes is derived. The introduction of a novel class of orthonormal modes is useful for non-interferometric phase recovery techniques and simplifies the design of mode converters and information processing systems.