Geometric explanation of conic-section interference fringes in a Michelson interferometer

A simple geometric method based on wave-front analysis is developed to provide a concise explanation for the various interference fringes in a Michelson interferometer without the compensator plate. In view of the fact that a homocentric pencil of rays from a point light source becomes astigmatic as it passes obliquely through a glass plate, the wave-front deviation from a spherical one is obtained by calculating the astigmatic focal distance of the central ray. If the compensator plate is removed, the two central rays along the interfering paths have different astigmatic focal distances (AFDs), therefore, the optical path length difference can never be compensated with the movement of the reflected mirrors. The wave-front difference or the optical path difference is determined to the accuracy of second order by comparing the two pairs of principal curvature radii, and the trivial point-by-point calculation of the optical path length difference (OPD) is avoided. Numerical results support the theoretical analysis with great accuracy.