International Journal of Applied & Experimental Mathematics
108 (1-7 pp.)
A brief summary of the physical context to this paper is provided, and the deviation angle undergone by an incident ray after k internal reflections inside a transparent unit sphere is formulated. For radially inhomogeneous spheres (in particular) this angle is related to a ray-path integral; an improper integral for which there are relatively few known exact analytical forms, even for simple refractive index profiles n(r). Thus for a linear profile the integral is a combination of incomplete elliptic integrals of the first and third kinds (though not all are as complicated as this). The ray-path integral is evaluated for ten different n(r) profiles, many of which have not been provided elsewhere. In the appendix a mirage theorem is proved for horizontally stratified media. This illustrates the more general principle in geometrical optics, namely that a ray path is always concave towards regions of higher refractive index.
Original Publication Citation
Adam, J., & Pohrivchak, M. (2016). Evaluation of ray-path integrals in geometrical optics. International Journal of Applied & Experimental Mathematics, 1(2), Article 108. https://doi.org/10.15344/ijaem/2016/108
Adam, John A. and Pohrivchak, Michael, "Evaluation of Ray-Path Integrals in Geometrical Optics" (2016). Mathematics & Statistics Faculty Publications. 188.