# Radiometric consistency in source specifications for lithography

## Abstract

There is a surprising lack of clarity about the exact quantity that a lithographic source map should specify. Under the plausible interpretation that input source maps should tabulate radiance, one will find with standard imaging codes that simulated wafer plane source intensities appear to violate the brightness theorem. The apparent deviation (a cosine factor in the illumination pupil) represents one of many obliquity/inclination factors involved in propagation through the imaging system whose interpretation in the literature is often somewhat obscure, but which have become numerically significant in today's hyper-NA OPC applications. We show that the seeming brightness distortion in the illumination pupil arises because the customary direction-cosine gridding of this aperture yields non-uniform solid-angle subtense in the source pixels. Once the appropriate solid angle factor is included, each entry in the source map becomes proportional to the total |E|∧2 that the associated pixel produces on the mask. This quantitative definition of lithographic source distributions is consistent with the plane-wave spectrum approach adopted by litho simulators, in that these simulators essentially propagate |E|∧2 along the interfering diffraction orders from the mask input to the resist film. It can be shown using either the rigorous Franz formulation of vector diffraction theory, or an angular spectrum approach, that such an |E|∧2 plane-wave weighting will provide the standard inclination factor if the source elements are incoherent and the mask model is accurate. This inclination factor is usually derived from a classical Rayleigh-Sommerfeld diffraction integral, and we show that the nominally discrepant inclination factors used by the various diffraction integrals of this class can all be made to yield the same result as the Franz formula when rigorous mask simulation is employed, and further that these cosine factors have a simple geometrical interpretation. On this basis one can then obtain for the lens as a whole the standard mask-to-wafer obliquity factor used by litho simulators. This obliquity factor is shown to express the brightness invariance theorem, making the simulator's output consistent with the brightness theorem if the source map tabulates the product of radiance and pixel solid angle, as our source definition specifies. We show by experiment that dose-to-clear data can be modeled more accurately when the correct obliquity factor is used.