8 - 1D First-Order Optical Systems: Transfer Laws for the Wigner Distribution Function

Abstract

This chapter discusses the transfer laws for the Wigner distribution function. Using the Wigner method, it is possible to calculate a class of quantum mechanical averages in the same manner as the classical phase space distribution is used to calculate classical averages. The Wigner representation is considered like an extension of quantum mechanics into the classical phase space, focusing on the operative advantages of such a representation rather than on the structure of the manifold where it is defined. It is found that as a point representation in phase space is allowed by the classical formalism, the motion can be viewed as a simple substitution of one point for another. The concept of moving point in phase space is replaced within the quantum mechanical formalism by a quasiprobability distribution, which changes with time through nonlocal transformations. It is found that for quadratic polynomial generated systems, the quantum Liouville equation turns into the classical Liouville equation. The propagation law for the Wigner distribution function is elaborated. It is found that the fractional Fourier transform bridges the gap between Fourier and Wigner optics, that is, between the optics described in terms of the complementary spaces and the spatial frequency signal representation.