Derivation of various transfer functions of ideal or aberrated imaging systems from the three-dimensional transfer function

The three-dimensional frequency transfer function for optical imaging systems was introduced by Frieden in the 1960s. The analysis of this function and its partly back-transformed functions (two-dimensional and one-dimensional optical transfer functions) in the case of an ideal or aberrated imaging system has received relatively little attention in the literature. Regarding ideal imaging systems with an incoherently illuminated object volume, we present analytic expressions for the classical two-dimensional x-y-transfer function in a defocused plane, for the axial z-transfer function in the presence of defocusing and for the x-z-transfer function in the presence of a lateral shift δy with respect to the imaged pattern in the x-z-plane. For an aberrated imaging system we use the common expansion of the aberrated pupil function with the aid of Zernike polynomials. It is shown that the line integral appearing in Frieden's three-dimensional transfer function can be evaluated for aberrated systems using a relationship established first by Cormack between the line integral of a Zernike polynomial over a full chord of the unit disk and a Chebyshev polynomial of the second kind. Some new developments in the theory of Zernike polynomials from the last decade allow us to present explicit expressions for the line integral in the case of a weakly aberrated imaging system. We outline a similar, but more complicated, analytic scheme for the case of severely aberrated systems.