Derivatives of optical path length: from mathematical formulation to applications

The optical path length (OPL) of an optical system is a highly important parameter since it determines the phase of the light passing through the system and governs the interference and diffraction of the rays as they propagate. The Jacobian and Hessian matrices of the OPL are of fundamental importance in tuning the performance of a system. However, the OPL varies as a recursive function of the incoming ray and the boundary variable vector, and hence computing the Jacobian and Hessian matrices is extremely challenging. In an earlier study by the present group, this problem was addressed by deriving the Jacobian matrix of the OPL with respect to all of the independent system variables of a nonaxially symmetric system. In the present study, the proposed method is extended to the Hessian matrix of a nonaxially symmetric optical system. The proposed method facilitates the cross-sensitivity analysis of the OPL with respect to arbitrary system variables and provides an ideal basis for automatic optical system design applications in which the merit function is defined in terms of wavefront aberrations. An illustrative example is given. It is shown that the proposed method requires fewer iterations than that based on the Jacobian matrix and yields a more reliable and precise optimization performance.