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.