The Role of Eikonal and Matrix Methods in Contrast Transfer Calculations

Abstract

The notion that the optical contrast transfer function is a useful tool for assessing the performance of image-forming instruments has been accepted generally for some time and is now well established. This paper discusses one method of making the transition from ray-trace data to the evaluation of this important function. First, the light distribution in the point image is rigorously derived in terms of an integral over angular coordinates involving the eikonal function about a reference surface at infinity. Then, the ray-trace procedure is developed in the language of refraction and translation matrices culminating in matrix elements which are simply related to the eikonal coefficients of wave optics. Finally, the numerical evaluation of the contrast transfer function in amplitude and phase from these eikonal coefficients is presented, and the paper ends with an example showing the off-axis transfer function for line structures oriented at various azimuths. All calculations are carried out to fifth order in the eikonal coefficients, and emphasis is placed on the usefulness of this approach on relatively slow, low-capacity computing machines.