Application of Apodization to Increase Two-Point Resolution by the Sparrow Criterion I Coherent Illumination

A resolution criterion due to Sparrow is studied. Unlike the well-known Rayleigh resolution criterion, the Sparrow resolution criterion is dependent upon the coherence properties of the incident light. The extreme limits of complete coherence and complete incoherence are investigated. An apodization scheme for increasing the coherent Sparrow resolution limit is formulated via the calculus of variations. The determination of the required pupil function (amplitude distribution over the exit pupil) necessitates the solution of an inhomogeneous Fredholm integral equation. Both slit and circular aperture are studied in detail.