A skew ray tracing approach for the error analysis of optical elements with paraboloidal boundary surfaces

Using the error analysis methodology developed by the current author in previous studies for optical systems comprising elements with flat boundary surfaces, this study examines the errors induced in a light ray's path as it is reflected or refracted at a paraboloidal boundary surface. In analyzing the light path, two principal sources of error are considered, namely (1) translational errors (Δxi, Δyi and Δzi) and rotational errors (ΔΓi, ΔΨi and ΔΦi), which collectively determine the deviation of the light path at each boundary surface, and (2) the differential changes induced in the incident point position and unit directional vector of the refracted/reflected ray as a result of differential changes in the position and unit directional vector of the light source. The validity of the proposed approach is verified using a generic parabaloidal boundary surface for illustration purposes. Overall, the results show that the proposed error analysis methodology provides a straightforward means of analyzing the performance of optical systems characterized by paraboloidal boundary surfaces such as headlight reflectors, optical telescope mirrors, flashlights and so forth.