Skew ray tracing and error analysis of optical elements with ellipsoid boundary surfaces

Utilizing a ray tracing approach based upon Snell's optical laws and a 4×4 homogeneous coordinate transformation matrix, this study conducts a systematic analysis of the errors induced in the paths of skew rays as they pass through an optical element with an ellipsoid boundary surface. The error analysis performed in this study considers two principal sources of light path error, namely (1) translational and rotational errors of the boundary surfaces, and (2) differential changes induced in the position of the incident ray on the boundary surface and the unit directional vector of the refracted/reflected ray as a result of changes in the position and unit directional vector of the light source. The validity of the proposed methodology is demonstrated by analyzing an optical element with an arbitrary ellipsoid boundary surface.