MAXIMUM OF THE FACTOR OF ENCIRCLED ENERGY

Abstract

It is recognized that the most favorable distribution of radiant energy in a diffraction pattern is that which corresponds to the best concentration around the center O. This hypothesis is expressed by an extremal condition on the factor of encircled energy E(W), that is, the ratio of the energy inside a circle of radius W and centered on the diffraction pattern, to the total energy in the same.A study of the effects of spherical aberration on this factor of encircled energy has shown that aberration always tends to decrease the factor from the value obtained with an Airy pattern. However, this factor may be increased by the use of an amplitude filter at the pupil of the optical system.In treating the case of amplitude filters one may use a rigorous analysis in terms of Taylor's series in (1−x 2 ) p−1 or a polynomial T n (x) of degree n−1 in terms of (1−x 2 ).The corresponding amplitudes in the diffraction pattern are Г(W) and Г n (W); the maximum factor of encircled energy E(W m ) and E n (W m ). The following convergences are established: E n (W m ) → E(W m ), T n (x) → T(x), and Г n (W) → Г(W) as n → ∞.When the interval (0, W m ) of the diffraction pattern is made to correspond to the interval (0, 1) of the pupil by means of a suitable normalization the amplitude distributions T(x) and Г(W)—with W = W m x—are identical. Some properties are deduced from this relation; for example, the Airy pattern is the limit of Г(W) when W m → 0; on the other hand, the Gauss function [Formula: see text] is an asymptotic expression of T(x) when W m → ∞. In any case, the factor of encircled energy is connected to the marginal amplitude in the pupil by the relation E(W m ) = 1−T 2 (1).The numerical determination of E(W) given up to W = 10 and W m = 2, 3, 4, and 5 can be extended by use of an asymptotic expression of the factor of encircled energy.Finally, a curve M(W) has been obtained, which is an envelope of the curves E(W) corresponding to various values of W m . This gives the locus of the maximum factor of encircled energy and represents the limiting performance of optical systems.