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Abstract
Our group recently showed that the Seidel primary ray aberration coefficients of an axis-symmetrical system can be accurately determined using the third-order Taylor series expansion of a skew ray ${\bar{\text{R}}_{\text{m}}}$ on an image plane. This finding inspires us to determine the third-order derivative matrix of ${\bar{\text{R}}_{\text{m}}}$ with respect to the vector ${\bar{\text{X}}_0}$ of the source ray, i.e., ${{\partial \bar{\text{R}}_{\text{m}}^3} / {\partial \bar{\text{X}}_0^3}}$, under reflection/refraction at a flat boundary. Finite difference methods using the second-order derivative matrix, ${{\partial \bar{\text{R}}_{\text{m}}^2} / {\partial \bar{\text{X}}_0^2}}$, require multiple rays to compute ${{\partial \bar{\text{R}}_{\text{m}}^3} / {\partial \bar{\text{X}}_0^3}}$ and suffer from cumulative rounding and truncation errors. By contrast, the present method is based on differential geometry. Thus, it provides a greater inherent accuracy and requires the tracing of just one ray. The proposed method facilitates the analytical investigation of the primary aberrations of an axis-symmetrical system and can be easily extended to determine the higher-order derivative matrices required to explore higher-order ray aberration coefficients.
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