Apodization for maximum Strehl criterion and specified Sparrow limit of resolution for incoherent illumination

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Abstract
In their analysis of the problem of determining the diffraction pattern, and associated pupil function, for a rotationally symmetric optical system that has a specified Sparrow limit of resolution for incoherent illumination and the maximum possible ratio of central maximum irradiance to the over-all transmittance (i.e., maximum possible Strehl criterion), Barakat and Levin encountered a critical value of the Sparrow limit, below which barrier they were unable to find a solution. We adopt a somewhat different approach to the variational problem and are able to reduce it to finding the solution of a single quartic equation that lies between the negative and smaller positive root of a certain cubic equation, followed by the solution of three simultaneous linear equations in three unknowns. The barrier occurs when the quartic and cubic equations have a common root, in which case the determinant of the system of three linear equations vanishes. There is, however, no difficulty in finding solutions below the barrier. At the barrier itself, which we find to be 70.8% of the Sparrow limit for the classical Airy-type objective, two essentially different pupil functions produce the same maximum Strehl criterion.

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