The first-order derivative matrix of a function with respect to a variable vector is referred to as the Jacobian matrix in mathematics. Current commercial software packages for the analysis and design of optical systems use a finite difference (FD) approximation methodology to estimate the Jacobian matrix of the wavefront aberration with respect to all of the independent system variables in a single raytracing pass such that the change of the wavefront aberration can be determined simply by computing the product of the developed Jacobian matrix and the corresponding changes in the system variables. The proposed method provides an ideal basis for automatic optical system design applications in which the merit function is defined in terms of wavefront aberration. The validity of the proposed approach is demonstrated by means of two illustrative examples. It is shown that the proposed method requires fewer iterations than the traditional FD approach and yields a more reliable and precise optimization performance. However, the proposed method incurs an additional CPU overhead in computing the Jacobian matrix of the merit function. As a result, the CPU time required to complete the optimization process is longer than that required by the FD method.
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