Numerical partially coherent imaging using elementary functions

Abstract

The theory of coherence and propagation of light through imaging systems is well established. For coherent and incoherent sources, the intensity in the image plane can be predicted numerically using a straightforward convolution calculation. Image formation becomes more complicated when dealing with partially coherent light, as treating two-dimensional intensity fields (described by the four-dimensional mutual coherence function in the time domain or the cross-spectral density in the frequency domain) requires evaluating four-dimensional integrals. Thus, calculations are complex, slow to process and place demands on system memory. We present a variation of a method recently introduced [Wald et al, Proc SPIE, 59621G, 2005], in which elementary functions are used to reduce the integrals to two dimensions for light of relatively high degree of coherence. The method resembles the coherent-mode expansion, but the elementary functions are easier to find and work with than the coherent modes. We outline the method and present some numerical results. This approach has applications in modelling of photolithographic systems in which partially coherent excimer lasers operating in the Deep Ultra-Violet (DUV) regime have been used for the last decade. An accurate numerical model of such systems could prove useful in solving the classic inverse imaging problem of lithography reticle design.