Aberration calculation and lens tolerancing using automatic differentiation

Automatic differentiation is a technique that allows one to easily calculate exact partial derivatives of complex, composite mathematical functions. This technique is described in a book by Rall 1 of the University of Wisconsin at Madison. Independently, S. Marshall of Xerox Corp. developed an algorithm that is effectively automatic differentiation, and extends the technique to arbitrary order partial derivatives of functions of arbitrary number of independent variables. Since aberration coefficients are merely partial derivatives of final ray coordinates with respect to initial ray coordinates, automatic differentiation can be employed to calculate any aberration coefficient desired of any lens system, including misaligned and nonsymmetric ones. Partial derivatives of these systems for use in lens tolerancing and optimization can also be directly calculated. Examples of aberration calculation and Gaussian beam tolerancing, using an implementation of ray tracing and automatic differentiation in the ADA programming language, are given. The work of Forbes 2 in the area of computation of chromatic aberration coefficients using an equivalent, highly optimized power series method, is also mentioned.