Superluminal tunneling of microwaves in smoothly varying transmission lines

Tunneling of microwaves through a smooth barrier in a transmission line is considered. In contrast to standard wave barriers, we study the case where the dielectric permittivity is positive, and the barrier is caused by the inhomogeneous dielectric profile. It is found that reflectionless, superluminal tunneling can take place for waves with a finite spectral width. The consequences of these findings are discussed, and an experimental setup testing our predictions is proposed.

Figure 1

(Color online) A schematic picture of a coaxial waveguide with a gradient wave barrier in segment 1 between the empty segments 0 and 2. A spatial distribution of the normalized impedance $W^2\left(z\right)$ plotted as a function of $z$ $\left(y^2=1/3\right)$ can also be seen.

Figure 2

(Color online) The transmittance spectra for the TEM mode tunneling through two wave barriers in the coaxial waveguide $\left({ϵ}_m=4.912,{ϵ}_{\text{min}}=3.69,m=2,y_2=1/3\right)$. $\left|T\right|$ is the transparency, $u$ is the normalized frequency, and $\phi$ is the phase of the tunneling wave. (a) The spectral window of transparency in the range $1<u<10$ and $1.02<u<1.25$. (b) The phase spectrum of the tunneling mode in the range $1<u<10$ and $1.02<u<1.25$.

Figure 3

(Color online) An interferometerlike scheme for measurement of phase shift of the tunneling mode. $\mathit{G}$ is the generator of the TEM mode, 1 is an arm containing two adjoining gradient impedance layers, 2 is an arm containing the empty coaxial, $\mathit{D}$ is a phase detecting element.