Transmission and reflection coefficients from the tunneling equation with absorption in the complex z plane

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.

Analytic calculation of the nonzero fast wave reflection coefficient from the tunneling equation with absorption

“…Also, solutions Fk and f 1 'k have been shown to be analytic everywhere. 6 Then, following Ref. 6, we can change the path of integration of Ijk, defined in Eq.…”

“…6 Then, following Ref. 6, we can change the path of integration of Ijk, defined in Eq. (9), to the semicircles Co:…”

“…In Ref. 6, solutions fk and Opk have been extended to the complex z-plane, and it was shown that on these new contours C,, both fl and l& 1 cc exp ( r iz) and are thus exponentially small for both cases of -(y + 1) > 0, and that f2 and iP2 c exp (iz) and are thus exponentially large. It was then concluded that 111=I12=I21=0, which means that T 1 =T 2 =exp(-s7), and R 1 =0, independent of absorption.…”

“…It was then concluded that 111=I12=I21=0, which means that T 1 =T 2 =exp(-s7), and R 1 =0, independent of absorption. 6 Our approach here to calculate I22 is to expand the integrand F 2 hT 2 in an asymptotic series on C,. Since I z | co on the contour, all terms in this asymptotic series may be kept.…”

“…(I) and (2) into the complex z-plane, some of these integrals were Waves propagating in weakly inhomogeneous media shown to be zero identically. 6 Thus, fast wave transmission through a back-to-back resonance-cutoff region, in general, official entsa al To each othe promote experience effects of transmission, reflection, and mode conversion with or without absorption. These phenomena sides and the fast wave reflection coefficient (which is idenconverslon ~ ~ ~ ~ ~ ~ ~ ~~~icl zero fro theou side which encounters resonance can be modeled by different kinds of high order ordinary typically zero) from the side which encounters the resonance these is before the cutoff, were shown to be independent of absorpa fourth order equation of the form (e.g., Refs.…”