Analytic determination of the resolution-equivalent effective diameter of a pinhole collimator

To account for photon penetration, the formulas used to calculate the geometric resolution of a pinhole collimator use an equivalent diameter d(e) rather than the physical diameter of the aperture. The expression commonly used for d(e), however, was originally derived to account for penetration in sensitivity calculations. In this paper, we show that the concept of equivalent diameter is also applicable to resolution calculations, propose angular-dependent expressions for d(e) specific to resolution calculations, and discuss the limits of their applicability and how they compare to other expressions. Results show that for normal incidence Paix's expression for d(e) tends to overestimate the resolution-equivalent diameter for full-width-at-half-maximum resolution, whereas Anger's is a better approximation, but may produce underestimates for submillimeter resolution imagers, especially in the case of high-energy photons. For grazing incidence, both expressions may result in significant overestimates.