First-order derivative matrix of a ray: a simple and flexible alternative computation method

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.