Finite-pupil admissibility constraints on the Gaussian Stiles-Crawford function

The Stiles-Crawford effect of the first kind (SCE-I) describes the directional sensitivity of retinal photoreceptors and is commonly represented by a Gaussian function with a directionality parameter. Existing waveguide, layered-scattering, optical coupling and apodization models explain its physical basis but do not explicitly consider constraints imposed by finite-pupil normalization. Here, we examine the Gaussian representation using the finite pupil as the integration domain. The finite-pupil response is derived analytically using the error function, and an admissibility criterion is introduced by requiring the normalized response to retain a prescribed fraction of the ideal infinite-pupil response. This yields a quantitative relationship among pupil radius, Gaussian directionality and tolerance. The criterion is independent of biophysical mechanisms and therefore broadly applicable. For realistic pupil radii, reported directionality values retain approximately 90-99% of the ideal response. Finite-pupil normalization thus provides a model-independent geometric consistency condition that constrains Gaussian directionality while complementing existing physical theories.