Meshing and Discretization Error
In the Geometrical Optics interface, rays don’t interact with an exact analytical representation of the geometry, but rather they interact with the underlying finite element mesh. For example, when detecting ray-boundary interactions including reflection, refraction, and absorption, the intersection point of each ray with a surface is actually the intersection point with a boundary element on that surface.
The advantage of using a mesh representation of the geometry to detect and apply ray-boundary interactions is that the algorithm is readily extended to high-fidelity multiphysics simulation including translational motion, rotation, and structural deformation including thermal stress. In addition, this implementation allows rays to be traced through geometric entities of arbitrary shape, not just simple shapes for which a parametric representation is readily available.
To ensure that ray reflections and refractions are both detected and applied accurately, the mesh must be of sufficiently high quality. This is trivial when the surfaces are planar because even a small number of linear boundary elements can represent a planar surface to machine precision. Accurately discretizing the geometry becomes more important when the surfaces are curved, as in spherical lenses and conic mirrors, or when the surfaces may be deformed.
In the following image, the radial position over one boundary element on a coarsely meshed sphere (using quadratic elements) is compared to the exact value for an ideal sphere. The error is on the order of 10-5.
Figure 2-2: Error in the radial displacement over one boundary element of a unit sphere.
The discretization error not only applies to the position on the sphere, but also to the normal direction and Gaussian curvatures of the surface, all of which can be used in geometrical optics simulation. A relative error like the one shown above (10-5) might be sufficiently small for some simulation results, but in geometrical optics such an error might translate to tens of additional wavelengths in spot size—large enough to invalidate the results of the simulation entirely, unless adequate precautions are taken.
Similarly, in models with mesh deformation, the degrees of freedom for the displacement field must be solved for extremely accurately for the results of a coupled multiphysics model to be trusted. A good practice, as with other types of simulation, is to perform a mesh refinement study, ensuring that the results don’t change appreciably when the mesh element size is reduced further.