Apodization for maximum encircled-energy ratio and specified Rayleigh limit*

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Abstract
Earlier workers have devised theoretical and numerical techniques for determining the diffraction pattern with the property that the maximum possible fraction of the total energy in the pattern is contained within a circle of arbitrarily fixed radius (maximum encircled-energy ratio). The numerical results indicate that the Rayleigh limit of resolution associated with this diffraction pattern exceeds the arbitrarily fixed radius, but there does not appear to be any other relationship between these two quantities. Accordingly we determine in this paper the diffraction pattern, and corresponding pupil function, for which the encircled-energy ratio for an arbitrarily specified circle is a maximum, subject to the constraint that the Rayleigh limit of resolution is also arbitrarily specified. The maximizing pupil function satisfies a linear, homogeneous Fredholm integral equation, the largest eigenvalue of which is the maximum possible encircled-energy ratio for the specified circle and Rayleigh limit. We give numerical results obtained using a Rayleigh—Ritz technique to solve the integral equation in the special case when the radius of the specified circle is equal to the specified Rayleigh limit.

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