Roadmap for geometrical optics based on Taylor series expansion of skew ray

The skew ray Rn on the image plane of an optical system possessing n boundary surfaces has the form of an n-layered deep composite function. It is hence difficult to evaluate the system performance using ray tracing alone. The present study therefore uses the Taylor series expansion to expand Rn with respect to the source ray variable vector. It is shown that the paraxial ray tracing equations, point spread function, caustic surfaces and modulation transfer function can all be explored using the first-order expansion. Furthermore, the primary and secondary ray aberrations of an axis-symmetrical system can be determined from the third- and fifth-order expansions, respectively. It is thus proposed that the Taylor series expansion of the skew ray serves as a useful basis for exploring a wide variety of problems in geometrical optics.