The Influence of Spherical Aberration on the Response Function of an Optical System

The integral expressing the frequency response of an optical system in different focal planes in the presence of primary and secondary spherical aberration has been evaluated for a large number of cases employing a Deuce computer, programmed for the numerical integration of double integrals by a method described by Hopkins. Detailed results are given for values of the secondary spherical (wave front) aberration w 60 = -4λ, -6λ, -9λ, -12λ, each with three different amounts of primary aberration and each in five different focal planes; and curves are given showing the frequency response that can be obtained when the performance is optimized for low frequencies according to Hopkins' criterion, where a low frequency is one for which the response is greater than or equal to 0.8. For these low frequencies it is confirmed that an under-corrected marginal ray gives the best image quality. For frequencies greater than that for which the response is approximately 0.6, an over-corrected margin is to be preferred. It is found that, in agreement with the prediction of geometrical optics, the values of the response, D ( s, ψ ), plotted as functions of the product sw 60 (or sw 40 , for the case w 60 = 0), where s is the reduced spatial frequency, lie closely on a single curve, when sw 60 ⩽ 1.2 (or sw 40 ⩽ 0.3, for w 60 = 0).