Ray tracing in a finite-element domain using nodal basis functions

Abstract

A method is presented for tracing rays through a medium discretized as finite-element volumes. The ray-trajectory equations are cast into the local element coordinate frame, and the full finite-element interpolation is used to determine instantaneous index gradient for the ray-path integral equation. The finite-element methodology is also used to interpolate local surface deformations and the surface normal vector for computing the refraction angle when launching rays into the volume, and again when rays exit the medium. The procedure is applied to a finite-element model of an optic with a severe refractive-index gradient, and the results are compared to the closed-form gradient ray-path integral approach.

A Thermo-Optic Propagation Modeling Capability.

“…The primary contribution of this LDRD effort is the development of a finite-element ray-trace theory that fully exploits the rich methodologies of FE theory for interpolation and gradient calculations within an element volume defined by a collection of discrete nodes. The theory outlined in [5], enables ray-tracing directly within a finite-element volume and has broad application in the fields of optics and seismic wave propagation. The approach allows data files from the thermal and structural FE models to be imported into the optical ray-trace application and multi-spectral, multi-field analyses to be performed with no need for intermediate processing.…”

“…where r is the position vector of a point on the ray, n(r) is the refractive index of the medium at position r, and ds is a differential portion of the arc length s along the ray. Using the change of variable and , proposed in [7] and the techniques described in [5] (included in…”

“…One of the terms in equation 3is the analogue to the gradient of the indexsquared, composed in the local FE (, ,  ) domain. As explained in [5], the second term resolves the curvilinear mapping of the local FE domain to the physical coordinate space.…”

“…A sequential algorithm for tracing a given ray through a finite-element volume is also described in [5]. The basic algorithm entails the following steps:…”

“…Accurate determination of the ray incidence point and its mapping into the local (, ,  ) coordinates of the incident element is a non-trivial task that strongly impacts the accuracy of the ray trace. A gradient-descent cost-minimization approach to determining the incidence point in (, ,  ) space is proposed in Appendix B of [5].…”