ABCD law for partially coherent Gaussian light propagating through first-order optical systems

Abstract

It is well known that the paraxial propagation of spherical waves through Luneburg’s first-order optical systems is governed by a bilinear relationship—the so-called ABCD law—which relates the curvature of the incoming wave with the curvature of the outgoing wave. It is also well known that this ABCD law applies to completely coherent Gaussian light if we describe such light by its complex curvature. Moreover, it has been shown that the same relationship applies to partially coherent Gaussian light, at least in the one-dimensional case. In this paper we study the behavior of partially coherent Gaussian light in the two-dimensional (or, generally, n-dimensional) case, and we find the conditions under which an ABCD law can again be formulated. It is shown that in such n-dimensional cases an ABCD law holds only for partially coherent Gaussian light for which the matrix of second-order moments of the Wigner distribution function is proportional to a symplectic matrix. Furthermore, it is shown that this is the case if we are dealing with a special kind of Gaussian Schell-model light, for which the real parts of the quadratic forms that arise in the exponents of the Gaussians are described by the same real, positive definite symmetric matrix.