A Luneburg-Kline representation for wave propagation in a continuously inhomogeneous medium

In this letter, we derive a Luneburg-Kline (GK) asymptotic series represent ation for wave propagation in a one-dimensional (1-D) continuously inhomoge neous medium. We set the solution up so that the classical phase function c ommon to the Wentzel, Kramers, Brillouin, and Jeffreys (WKBJ) approximation multiplies all terms of the GK series, We develop an error criterion for t he WKBJ approximation based on the magnitude of an ignored term relative to retained terms in the governing differential equation. Finally, we note th at while the validity of the GK series solution is dependent upon a large f ree-space wavenumber for small to moderate spatial gradients in the index o f refraction, large spatial gradients can be accommodated by increasing the free-space wavenumber. Hence, there is a strong similarity of this situati on to boundary diffraction problems where rounded edges approach, but do no t achieve, absolute sharpness. Loosely speaking, in both instances a scale size in terms of the wavelength is maintained constant.