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The propagation of Gaussian light through first-order optical systems is studied; in particular, the twist in the output (or input) plane is expressed in terms of the characteristics of the light in the other plane. Conditions are derived for which zero twist in the input plane corresponds to zero twist in the output plane, and vice versa. Three special cases for which zero twist is preserved, are described: (1) propagation between conjugated planes, (2) adaptation of the signal to the system, and (3) the case of symplectic Gaussian light.

A measure for the twist of Gaussian light is expressed in terms of the second-order moments of the Wigner distribution function. The propagation law for these second-order moments between the input plane and the output plane of a first- order optical system is used to express the twist in one plane in terms of moments in the other plane. Although in general the twist in one plane is determined not only by the twist in the other plane, but also by other combinations of the moments, several special cases are considered for which a direct relationship between the twists can be formulated. In particular it is shown under what conditions zero twist is preserved in a first-order optical system.© (1999) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

The paper presents a review of the Wigner distribution function--its definition and its most important properties-- and of some of its applications to optical problems, in particular to the description of partially coherent light beams and to the propagation of such light beams through first-order optical ABCD-systems. Special attention is paid to Gaussian beams and to second-order moments of general, partially coherent light.© (1999) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

The effect of truncating a light beam is expressed as a convolution of the Wigner distribution function of the light beam and the Wigner distribution function of the truncating aperture. The Wigner distribution function of a circular truncating aperture is derived and an approximate expression is presented. The approximation not only has the properties that it is exact in the spatial origin and in the spatial- frequency origin, but also that the integral over all spatial frequencies takes the exact value.© (1999) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

The complex spectrogram of a signal φ(t) is defined by ∫φ(t)g*(t-to)exp[-iwot]dt ; it is, in fact, the Fourier transform of the product of the signal and the complex conjugated and shifted version of a so-called window function g(t). From the complex spectrogram the signal can be reconstructed uniquely. It is shown that the complex spectrogram is completely determined by its values on the points (to=mT, wo=nΩ), where ΩT= 2tt and m and n take all integer values. The lattice of points (mT,nΩ) is exactly the lattice suggested by Gabor as early as 1946; it arose in connection with Gabor's suggestion to expand a signal into a discrete set of Gaussian elementary signals. Such an expansion is a special case of the general expansion ∑∑mn amng(t-mT)exp[inΩt] of a signal into a discrete set of shifted and modulated window functions. It is shown that this expansion exists. Furthermore, a set of functions is constructed, which is bi-orthonormal to the set of shifted and modulated window functions. With the help of this bi-orthonormal set of functions, the expansion coefficients amn can be determined easily.© (1980) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.