Exact solutions to the valley problem in inverse scattering

Abstract

Standard approximate methods involving the Abel integral equation do not allow the ionospheric electron density to be determined in the ‘‘valley’’ between two electron density peaks. Here we present analytic solutions to the Gel’fand–Levitan equation, which occurs in the exact full-wave inverse scattering theory. These exact analytic solutions exhibit multiple peaks in the electron density as a function of height and provide a solution to the valley problem.

Inverse scattering designs of dispersion-engineered single-mode planar waveguides

“…This is shown in Figure 10. Of additional interest is the fact that if the pole positions are chosen to all be located in the same neighborhood but with displacement  , thus avoiding multiplicity which would void the inverse-scattering technique 17 , then the threepole, five-pole and seven-pole cases demonstrate similar values of waveguide dispersion 2 D and dispersion curvature 4 D but significant reduction of dispersion slope as shown in Table 1. Associated with this change in dispersion slope is an expansion and exaggeration of the core, ring and trench as shown in Figure 11.…”

“…Conservation of energy, ( ) 1 rk  for all real k , dictates that the parameter space is limited. Previous authors 6 , 7 have identified and considered the allowable region bounded above by the line 2 0.5 c  and to the right by the lemniscate of Bernoulli 17 . The allowable region can be extended past these points as is illustrated in Figure 2 below which shows the dispersion 2 D , dispersion slope 3 D and dispersion curvature 4 D associated with the allowable region.…”

“…In order for a solution to exist, the reflection coefficient must obey a set of conditions 17 which are satisfied by the general form given in (9). Conservation of energy, ( ) 1 rk  for all real k , dictates that the parameter space is limited.…”

Inverse problem of reflection of electromagnetic waves from layered inhomogeneous plasma-like structures with refractive-index discontinuity

“…since the condition |r(k)| ≤ 1 holds for real k in this case To solve integral equation (24), we use the method described in [6], which yields the following result:…”

“…There exist exact analytical solutions of this equation for the reflection coefficients in the form of fractional rational functions which can be used as model approximations of the electron number density [5,6].…”