Indeterminacy in determining the refractive index distribution of a gradient medium realizing definite optical paths

Abstract

In this paper, we investigate a problem whether unique refractive index distribution of a gradient medium can be determined if definite optical paths in the medium are given previously. From the standpoint of transformations that leave the ray equation unchanged, general considerations reveal that the equation is invariant under a single-parameter scale transformation, and a type of local transformation parameterized by an arbitrary real function. A two-dimensional case involving the design of a gradient refractive index lens antenna reported by the author is re-examined by applying the obtained findings, and it is found that the indeterminacy with respect to the latter transformation disappears, especially in the case of the authors’ report.