Contrast Transmission Functions at Low Spatial Frequencies

Abstract

The relations existing at low spatial frequencies between the ray-theoretic ct-function t of an optical system and the more exact ct-function τ predicted by the usual scalar wave model based on Huyghens' principle are investigated by sharpening and extending a recent analysis due to K. Miyamoto [1, 2]. An approximate expression is obtained for the difference between t and τ, together with an explicit upper limit for the error of approximation in terms of the wave aberration of the system. This main result is shown, in § 3.2, to yield quick and easy proofs of most of the known results concerning the relations between the radial first and second derivatives of t and τ at the origin with each other, with aperture diffraction, and with the radius of gyration of the ray-theoretic image about the origin of coordinates. In § 4 it is shown that the above-mentioned scalar wave model has the following property: the increase, due to aberrations, in the moment of inertia of the diffraction image about any point in or near to its bright central region is equal to the increase in the moment of inertia of the ray-theoretic image about that point. The result is of some interest because of its formal equivalence to a statement of P.-M. Duffieux [3, 4] about the physical light distribution, although it does not suffice to prove that statement.