In this paper we investigate the inverse determination of a spacewise dependent index of refraction of a dielectric obstacle. Such a dielectric specimen could be an optical anti-reflection coating structure as is used in various optical instruments. The mathematical model is based on solving an inverse coefficient identification problem for the one-dimensional Helmholtz equation. The numerical method and solution are first validated in terms of accuracy and stability for a benchmark test example, after which the technique is applied to a case study concerning inverting real experimentally measured reflectance data supplied by Oxford Advanced Surfaces Ltd. A better fit to the data is obtained when a continuous index of refraction is sought, rather than a piecewise constant function, as in previous studies.