We develop a mathematical method for determining the optical path length (OPL) gradient matrix relative to all the system variables such that the effects of variable changes can be evaluated in a single pass. The approach developed avoids the requirement for multiple ray-tracing operations and is, therefore, more computationally efficient. By contrast, the effects of variable changes on the OPL of an optical system are generally evaluated by utilizing a ray-tracing approach to determine the OPL before and after the variable change and then applying a finite-difference (FD) approximation method to estimate the OPL gradient with respect to each individual variable. Utilizing a Petzval lens system for verification purposes, it is shown that the approach developed reduces the computational time by around 90% compared to that of the FD method.

This paper presents a closed-form solution to the sag of the Cartesian oval and an alternate iterative method for obtaining the sag. The emphasis is in providing a methodology for determining the sag and derivatives of a Cartesian surface for optical design, ray-tracing purposes. We verify our results by comparison of our solutions and by real ray tracing.

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