Local-Frequency Description of Optical Signals and Systems

摘要

Two types of local frequency spectra are presented: the Wigner distribution function and the sliding-window spectrum; the latter function is, in fact, the Fourier transform of the signal after being multiplied by a slided window function. The Wigner distribution function can provide a link between Fourier optics and geometrical optics; many properties of the Wigner distribution function and the way in which it propagates through a linear system, can be interpreted in geometric-optical terms. The Wigner distribution function and its related signal representations can be applied to stochastic signals, leading to a signal description that is related to Walther's generalized radiance in optics. Properties of partially coherent light can thus be derived easily by means of the Wigner distribution function. The sliding-window spectrum has the advantage that a sampling theorem can be formulated for it: the sliding-window spectrum of a time signal is completely described by its values at the points of a certain time-frequency lattice, which is exactly the lattice suggested by Gabor in 1946. The sliding-window spectrum thus leads naturally to Gabor's expansion of a signal into a discrete set of properly shifted and modulated versions of an elementary signal; the latter description leads, by its discrete nature, directly to the concept of the number of degrees of freedom of a signal. Applications of Gabor's expansion to optical signals (completely coherent and partially coherent) as well as a way to generate Gabor's expansion coefficients of a time signal by optical means, are described.