Second-order derivatives of optical path length of ray with respect to variable vector of source ray

A method is proposed for determining the second-order derivatives (i.e., the Hessian matrix) of the optical path length of a ray with respect to the variable vector of the source ray in an optical system comprising both flat and spherical boundary surfaces. Several wavefront aberration problems are investigated using the Hessian matrix proposed in this study and the Jacobian (first-order derivatives) matrix presented in the literature. It is found that when using the Hessian matrix the precision of wavefront aberration is significantly improved when evaluated up to the quadratic term of the Taylor series expansion. The methodology proposed in this study not only provides the means to investigate the principal curvatures of the wavefront along a ray, but also yields the information required to determine the irradiance and caustics of both axisymmetric and nonaxisymmetric optical systems.