The Rapid Calculation of the Optical Transfer Function for On-axis Systems Using the Orthogonal Properties of Tchebycheff Polynomials

A simple and rapid numerical quadrature is developed for the evaluation of the diffraction-based optical transfer function for on-axis systems, using a Tchebycheff polynomial expansion of the pupil function. The integration of the autocorrelation integral of the pupil function is replaced by once and for all evaluations of the cross-correlation of respective polynomials. However, the expansion coefficients themselves of the Tchebycheff series are linear sums of the sampled pupil function and thus a series of coefficients can be generated that weight the pupil function at various points to give the resultant OTF. The coefficients for a tenth-order Tchebycheff expansion are included in the paper, and a set of tables of OTF values calculated with these coefficients and 64 2 64 Gaussian quadrature for a diffraction-limited system, and one with one wavelength of primary spherical aberration.