Open AccessFirst-order derivative matrix of a ray: a simple and flexible alternative computation method
Author(s) -
Psang Dain Lin
Publication year - 2019
Publication title -
optics express
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.394
H-Index - 271
ISSN - 1094-4087
DOI - 10.1364/oe.27.035402
Subject(s) - jacobian matrix and determinant , trigonometric functions , matrix (chemical analysis) , variable (mathematics) , computation , algorithm , mathematics , sine , boundary (topology) , rotation matrix , mathematical analysis , computer science , geometry , materials science , composite material
The present group previously proposed a method for determining the first-order derivative matrix (i.e., the Jacobian matrix) of a skew ray by taking all the independent variables of the optical system as the system variable vector, X¯ sys . However, many trigonometric function calls, divisions, multiplications and additions were required to determine the ray Jacobian matrix with respect to X¯ sys . Accordingly, in the present study, the angular variables in the system variable vector, X¯ sys , are replaced with their respective cosine and sine trigonometric functions. The boundary variable vector, X¯ i , is similarly redefined such that it includes no angular variables. The proposed method has three main advantages over that previously reported: 1) it is valid for any pose matrix, irrespective of the order in which the rotation and translation motions of a boundary are assigned; 2) it involves only polynomial differentiation, and is thus easily implemented in computer code; and 3) the computation speed of ∂ X¯ i / ∂ X¯ i ∂ X¯ sys ∂ X¯ sys is improved by a factor of approximately ten times.