Earlier authors have considered the apodization problem of determining the pupil function that has the maximum Strehl ratio in the class of all pupil functions that have the same Rayleigh limit of resolution, specified in advance. They did not, however, verify that the solution they offered for this problem actually has the desired, rather than a smaller, Rayleigh limit. We shall supply this verification when the specified Rayleigh limit does not exced 139% of the Rayleigh limit for the classical Airy-type objective. For larger values of the specified Rayleigh limit, the offered solution does indeed have a smaller Rayleigh limit and is accordingly not a solution to the apodization problem. We show in this case that there is no solution, but that the Strehl ratio computed for the offered solution is a least upper bound that can be approached (but not attained) arbitrarily closely by using a suitably chosen complex-valued pupil function.
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