Elementary functions: propagation of partially coherent light

Abstract

The theory of propagation of partially coherent light is well known, but performing numerical calculations still presents a difficulty because of the dimensionality of the problem. We propose using a recently introduced method based on the use of elementary functions [Wald et al. Proc. SPIE6040, 59621G (2005)] to reduce the integrals to two dimensions. We formalize the method, describe its inherent assumptions and approximations, and introduce a sampling criterion for adequate interpolation. We present an analysis of some special cases, such as the Gaussian Schell-model beam, and briefly discuss generalized numerical propagation of two-dimensional field distributions.

Numerical modeling of spatial coherence using the elementary function method

“…Unlike the coherent-mode expansion, the elementary function method is not mathematically exact: certain approximations limit its application to relatively wellbehaved fields such as short-wavelength partially coherent excimer sources. The following summary of the theory is expanded further in [1].…”

“…Next, using the sampling criterion outlined in [1], we calculate the number of functions required to represent the source. The elementary functions are, in general, not orthogonal, and thus finding the expansion coefficients becomes more difficult than for orthogonal sets of functions.…”

“…These changes allow us to use the three conditions for expansion presented by Unser. Following the argument presented in [1], we arrive at an expression for the sampling distance Δx:…”

“…We have previously explored the elementary function method [1], originally presented by Wald et al [2], and we have demonstrated the circumstances under which this method can be used to reduce the propagation calculations of partially coherent light to two dimensions. In this paper, we apply the elementary function method to a 248 nm excimer beam and examine the effectiveness of this method in the case of a partially coherent source with a relatively high degree of spatial coherence.…”

“…The structure of this paper is as follows. In Section 2, the elementary function method is summarized and the analytical results previously presented in [1] are compared with a theoretical numerical model. To create a system-specific numerical model, the spatial coherence of a 248 nm excimer beam is measured and the results are presented in Section 3.…”