MAXIMUM OF THE FACTOR OF ENCIRCLED ENERGY

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−x2)p−1 or a polynomial Tn(x) of degree n−1 in terms of (1−x2).The corresponding amplitudes in the diffraction pattern are Г(W) and Гn(W); the maximum factor of encircled energy E(Wm) and En(Wm). The following converge...