General method for determining the first order gradients of skew rays of optical systems with non-coplanar optical axes

Current commercial software for the analysis and design of optical systems uses finite difference (FD) approximation methodology to estimate the gradient matrix of a ray with respect to system variables. However, FD estimates are intrinsically inaccurate and are subject to gross error when the denominator is excessively small relative to the numerator. This paper avoids these problems and determines the gradient matrix of the exit ray traveling along an optical system with a non-coplanar axis. To achieve this, the gradient matrix of the rays reflected/refracted by flat or spherical boundary surfaces are first determined by directly differentiating the skew-ray tracing equations. By introducing a Jacobian matrix, which represents the partial derivatives specifying the rates of changes between boundary variables and element variables, one can obtain the gradient matrix of the exit ray of an element with respect to its independent variables. This methodology will be useful in the analysis of rays and in design of optical systems with non-coplanar axis. A right-angle prism is used as illustrative example to validate its applications.