Sum of Coherent Systems Decomposition by SVD

The Hopkins partially coherent imaging equation is expanded by eigenfunctions into a sum of coherent systems (SOCS). The result is a bank of linear systems whose outputs are squared, scaled and summed. This technique is useful because of the partial linearity of the resulting system approximation. The eigenfunction expansion can be accomplished by computer using the SVD algorithm. To solve this problem, the Hopkins transmission cross coe cients (TCCs) are rst obtained as a matrix, then SVD is used on the matrix. Then the system is truncated at some low order (5th or 6th) to obtain an optimal approximation to Hopkins. In e ect, the numerical implementation of this using SPLAT TCCs amounts to a direct approximation to SPLAT. 1 Linear Systems Approximation The goal in this section is to compute the value of a single intensity point centered in a nite square mask region of size Lx Lx m , as depicted in Figure 1. First, we consider the 1-D case. By periodicizing a the length Lx mask, and taking its Fourier series expansion ~ G(n), as done by Flanner [2] we obtain the following expression for the Fourier series of the resulting periodic intensity, ~ I( ): ~ I(n) = X n ~ T (n + n0; n0) ~ G(n + n0)G (n0) (1) Suppose that the transmission cross coe cient function ~ T (n; n) can be approximated by: ~ T (n0; n00) Na X k=1 k k(n 0) k(n 00) (2) for some functions k( ); k = 1 : : : Na. We call this an Nath order approximation. Then substitute equation 2 into equation 1 to yield ~ I(n) = X n Na X k=1 k k(n 0 + n) k(n 0) ~ G(n+ n0)G (n0)