Third-order derivative matrix of a skew ray with respect to the source ray vector at a flat boundary

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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 R¯m on an image plane. This finding inspires us to determine the third-order derivative matrix of R¯m with respect to the vector X¯0 of the source ray, i.e., ∂R¯ 3m/∂X¯ 30, under reflection/refraction at a flat boundary. Finite difference methods using the second-order derivative matrix, ∂R¯ 2m/∂X¯ 20, require multiple rays to compute ∂R¯ 3m/∂X¯ 30 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.

Original languageEnglish
Pages (from-to)1435-1441
Number of pages7
JournalJournal of the Optical Society of America A: Optics and Image Science, and Vision
Issue number9
Publication statusPublished - 2019 Sep

All Science Journal Classification (ASJC) codes

  • Electronic, Optical and Magnetic Materials
  • Atomic and Molecular Physics, and Optics
  • Computer Vision and Pattern Recognition


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