On the refractive properties of media with poles or zeros in the index of refraction

The refractive properties of inhomogeneous fibers are examined with emphasis being placed on the limiting situation where the index of refraction possesses poles or zeros. If, e.g., the index of refraction isn(xi) = xi^{m}, xi = r/a, whereais the radius of the fiber andman arbitrary constant, it is found that energy integrability is satisfied ifm > -1. Whenm leq -1energy infinities occur. The ray behavior of such media is examined in terms of geometrical optics, and corrections to geometrical optics are obtained by an asymptotic analysis of the exact solution. Form > 0, the lens is of the diverging type, and when the angle of incidence isalpha = 0, geometrical optics predicts that rays "reflect" at various angles from the origin (depending on the value ofm). Whenm < 0, rays "wrap" around the origin several times with a zero radius of curvature before they leave the lens (alpha = 0). For-1 < m < -frac{1}{2}, it is found that whenalpha= (2m + 1)/2m] picaustics occur (alpha = 0, pi/2excluded). Pictorial diagrams show the behavior of these caustics and the correction coefficients to geometrical optics are obtained.