No abstract

No abstract

No abstract

No abstract

While exact analytic expressions for the scattering parameters have been obtained from the tunneling equation without absorption, which describes wave propagation through a tunneling region with back-to-back resonance and cutoff layers, no exact analytic results were known with absorption. While emission from absorbing regions has traditionally been calculated from the opacity, which depends on the transmission coefficient, it is proved here that transmission is independent of absorption, so that opacity must be redefined. This has been accomplished by examining the analytic continuation of the solutions into the complex z plane, where the evaluation of certain contour integrals exactly prove that the transmission coefficient is precisely the same with and without absorption and that there is no reflection from the side which encounters the resonance first. These results are general for a broad class of localized absorption functions.