Determination of second-order derivatives of a skew ray with respect to the variables of its source ray in optical prism systems

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8 Citations (Scopus)

Abstract

The second-order derivative of a scalar function with respect to a variable vector is known as the Hessian matrix. We present a computational scheme based on the principles of differential geometry for determining the Hessian matrix of a skew ray as it travels through a prism system. A comparison of the proposed method and the conventional finite difference (FD) method is made at last. It is shown that the proposed method has a greater inherent accuracy than FD methods based on ray-tracing data. The proposed method not only provides a convenient means of investigating the wave front shape within complex prism systems, but it also provides a potential basis for determining the higher order derivatives of a ray by further taking higher order differentiations.

Original languageEnglish
Pages (from-to)1600-1609
Number of pages10
JournalJournal of the Optical Society of America A: Optics and Image Science, and Vision
Volume28
Issue number8
DOIs
Publication statusPublished - 2011 Aug

Fingerprint

Hessian matrices
Prisms
Finite difference method
prisms
rays
Derivatives
differential geometry
Ray tracing
wave fronts
ray tracing
travel
scalars
Geometry

All Science Journal Classification (ASJC) codes

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

Cite this

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abstract = "The second-order derivative of a scalar function with respect to a variable vector is known as the Hessian matrix. We present a computational scheme based on the principles of differential geometry for determining the Hessian matrix of a skew ray as it travels through a prism system. A comparison of the proposed method and the conventional finite difference (FD) method is made at last. It is shown that the proposed method has a greater inherent accuracy than FD methods based on ray-tracing data. The proposed method not only provides a convenient means of investigating the wave front shape within complex prism systems, but it also provides a potential basis for determining the higher order derivatives of a ray by further taking higher order differentiations.",
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AB - The second-order derivative of a scalar function with respect to a variable vector is known as the Hessian matrix. We present a computational scheme based on the principles of differential geometry for determining the Hessian matrix of a skew ray as it travels through a prism system. A comparison of the proposed method and the conventional finite difference (FD) method is made at last. It is shown that the proposed method has a greater inherent accuracy than FD methods based on ray-tracing data. The proposed method not only provides a convenient means of investigating the wave front shape within complex prism systems, but it also provides a potential basis for determining the higher order derivatives of a ray by further taking higher order differentiations.

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