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A Tolerancing Procedure Based On Modulation Transfer Function (MTF)

A procedure is described which calculates the variation of MTF with respect to the constructional parameters of a lens system including the effects of diffraction, spectral band, obscuration, and vignetting. The calculation is much faster than the common procedure of changing a parameter and re-evaluating, and is applicable to systems in any state of correction. Compensating parameters such as defocus and image plane tilt are included in the procedure to account for adjustments made during assembly of the lens. The basic approach is an expansion of the OTF integral into a power series up to second order in the parameter. Each of the terms in the expansion is an integral over the convolved aperture. The integration is carried out by tracing a grid of rays through the system and using the H. H. Hopkins algorithm. The system is evaluated only once; differential ray tracing being used to obtain the wavefront derivatives, and hence the MTF variation, for each parameter. The relationship between MTF and any parameter is quadratic for small variations of the parameter, but not necessarily symmetric with respect to positive and negative incremental changes in the parameter. The changes in MTF for both positive and negative perturbations of all the constructional parameters are combined in a statistical manner to arrive at a probable degradation for the manufactured and assembled system. In addition to MTF, the resulting change in distortion for each field is calculated based on the parameter changes and adjustments.