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.

中文翻译：

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倾斜射线相对于平面边界处源射线矢量的三阶导数矩阵。

我们的小组最近表明，可以使用偏斜射线的三阶泰勒级数展开$ {\ bar {\ text {R}} _ {\ text {m来精确确定轴对称系统的Seidel主射线像差系数}}} $在图像平面上。这一发现启发我们确定相对于向量$ {\ bar {\ text {X}}的$ {\ bar {\ text {R}} _ {\ text {m}}} $$的三阶导数矩阵_0}美元的源光线，即$ {{\ partial \ bar {\ text {R}} _ {\ text {m}} ^ 3} / {\ partial \ bar {\ text {X}} _ 0 ^ 3}} $，在平坦边界的反射/折射下。使用二阶导数矩阵的有限差分方法$ {{\ partial \ bar {\ text {R}} _ {\ text {m}} ^ 2} / {\ partial \ bar {\ text {X}} _ 0 ^ 2}} $，需要多条光线来计算$ {{\ partial \ bar {\ text {R}} _ {\ text {m}} ^ 3} / {\ partial \ bar {\ text {X}} _ 0 ^ 3}} $并遭受累积的舍入和截断错误。相反，本方法基于微分几何。因此，它提供了更高的固有精度，并且仅需要跟踪一条光线。所提出的方法有助于对轴对称系统的基本像差进行分析研究，并且可以轻松地扩展为确定探索高阶射线像差系数所需的高阶导数矩阵。