We have studied the validity of the eikonal equation for an inhomogeneous medium. The propagation of an electromagnetic wave in a medium that is characterized by a parabolic dielectric constant variation in the transverse direction, has been studied in detail. The general path of a ray has been calculated and the irradiance distribution near the focal point has been analytically obtained.
A. K. Ghatak and K. Thyagarajan, "Ray and energy propagation in graded-index media," J. Opt. Soc. Am. 65, 169-173 (1975)