Jacobian and Hessian matrices of optical path length for computing the wavefront shape, irradiance, and caustics in optical systems

The first- and second-order derivative matrices of the ray (i.e., ∂R¯(i)/∂X¯(0) and ∂(2)R¯(i)/∂X¯02) and optical path length (i.e., ∂OPL(i)/∂X¯(0) and (2)OPL(i)/∂X¯02) were derived with respect to the variable vector X¯(0) of the source ray in an optical system by our previous papers. Using the first and second fundamental forms of the wavefront, these four matrices are used to investigate the local principal curvatures of the wavefront at each boundary surface encountered by a ray traveling through the optical system. The proposed method not only yields the data needed to compute the irradiance of the wavefront but also provides the information required to determine the caustics. Importantly, the proposed methodology is applicable to both axisymmetric and nonaxisymmetric optical systems.