Optical image assessment : a comparative study

The aberration theory of Buchdahl is extended to allow greater utilisation of it in optical image assessment and automatic design programs. Expressions are presented for the calculation of the derivatives of the seventh order aberration coefficients as well as for the determination of effects on the derivatives of OT coefficients due to pupil shifts arising from parameter changes. Where details of the theory for various constructional parameters differ attention has been confined to axial curvature derivatives. The wavefront retardation expansion has been checked for convergence and the results show general agreement with the convergence properties found by other authors for transverse aberration expansions. A series of transformations, valid over the region of convergence of the retardation expansion, is introduced to reduce the exit pupil periphery defined on a reference sphere to a circle. It is shown that, under these transformations, the form of the retardation expansion remains constant and only the coefficients need be altered. These changes are independent of the field angle but depend on the f-number of the system. A new set of assessment functions, derived from the real and imaginary parts of the optical transfer function,is introduced. It is shown that, in the geometrical limit, these approximate a set of functions defined in terms of the spot diagram distribution. Theoretical and numerical comparisons of these and some other assessment functions are presented. These show that, in general, there is agreement on the ordering of correction states when different criteria are used. However some differences in ranking do arise and these are discussed. It is found that, with a modification to allow for products of two negative quantities, a function based on the variance, V(r), of the aberration difference function provides an extremely versatile assessment quantity. The usefulness of this new quantity is shown, both theoretically and numerically, to extend far beyond the conditions under which V(r) provides a valid approximation to the modulation transfer function. Predictions of the location of optimum image planes using the various criteria are examined, and it is shown that by choosing different fractional spatial frequencies, r, at which to evaluate V(r), most of these predictions can be obtained using minimization of the variance V(r). Finally all the assessment functions introduced are used to examine two different processes of automatic optical design based on reduction of transverse aberrations. The first of these is a primitive design program, developed by the writer, which uses derivatives of the spot diagram assessment quantities introduced earlier. The second method, due to Cruickshank, reduces selected aberration coefficients to prescribed residuals determined by the designer on the basis of past experience and the requirements of the user. The results from this work indicate that the choice of assessment function is not critical in attaining an optimum region of parameter space, but the actual optimum point is dependent on the choice of assessment function. The usefulness of the modified function based on V(r) is again illustrated numerically.