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.