chapter 5, Alternative Coherent-Mode Representation of a Planar Source

Excerpt

5.1 Introduction

We recall the coherent-mode representation of the cross-spectral density function (1.18) and rewrite it for a planar secondary source, omitting an explicit dependence of the considered quantities on frequency ν, as follows:

where λ_n are the eigenvalues and φ_n(x) are the eigenfunctions of the Fredholm integral equation The coherent-mode representation of the source is an essential tool in describing the processes and systems in optics. However, the practical value of this representation is essentially restricted. Really, in practice, the analytic expression for the cross-spectral density function W(x₁,x₂), as a rule, is unknown and, hence, values λ_n and functions φ_n(x) cannot be found as the solution of the Fredholm equation (5.2). Moreover, even when the cross-spectral density function is approximated by a certain analytic function, the solution of this equation in closed form may be obtained only for a very limited number of field models (see Section 1.6). Clearly, an alternative approach to the problem of the coherent-mode representation of the source, which does not involve the solution of the Fredholm equation (5.2), is desired. Here, we propose such an approach based on the replacement of an original source with an unknown cross-spectral density function by an alternative source with a cross-spectral density function that may be approximated according to the results of physical measurements.

©2006 Society of Photo-Optical Instrumentation Engineers