An application of one-dimensional inverse-scattering theory for inhomogeneous regions

One-dimensional inverse-scattering theory is applied to the study of the reflection of electromagnetic waves from an inhomogeneous region having a refractive index
n(x) = [1-(1/k^{2})q(x)]^{1/2}
where
k = 2\pi/\lambda
, and
\lambda
is the free-space wavelength. The exact refractive index profile is obtained that will produce a reflection coefficient in which the frequency dependence is described by the Butterworth approximation.