Second-order derivatives of a ray with respect to the variables of its source ray in optical systems containing spherical boundary surfaces

The second-order derivative matrix of a scalar function with respect to a variable vector is called a Hessian matrix, which is a square matrix. Our research group previously presented a method for determination of the first-order derivatives (i.e., the Jacobian matrix) of a skew ray with respect to the variable vector of an optical system. This paper extends our previous methodology to determine the second-order derivatives (i.e., the Hessian matrix) of a skew ray with respect to the variable vector of its source ray when this ray is reflected/refracted by spherical boundary surfaces. The traditional finite-difference methods using ray-tracing data to compute the Hessian matrix suffer from various cumulative rounding and truncation errors. The proposed method uses differential geometry, giving it an inherently greater accuracy. The proposed Hessian matrix methodology has potential use in optimization methods where the merit function is defined as ray aberrations. It also can be used to investigate the shape of the wavefront for a ray traveling through an optical system.