Electromagnetic barrier penetration in a dispersive medium: Tunneling times and dispersion relations

We consider a scattering process inside a dispersive and absorptive medium, and derive an analytical expression for the scattering-time parameter that characterizes the duration of this process. We consider a microwave wave packet that scatters from a dielectric inhomogeneity inside an otherwise homogeneous waveguide. The waveguide is filled with a dielectric medium with a finite inhomogeneity region in the direction of the wave-packet propagation. The permittivity of the entire medium is described by a complex function of the frequency. Hence both absorption and dispersion affect the propagation of the wave packet. We define the scattering-time parameter to be the difference in the “average arrival time” between the free and actual propagation. We show that this time difference (in the limit of a narrow-band wave packet) is given by a linear combination of the real and imaginary parts of the “complex tunneling time.” In the special case where there is no dissipation or attenuation outside the scatterer, we recover the well-known result that the scattering-time parameter is identical to the real part of the complex tunneling time. The contribution of the imaginary part of the complex tunneling time to the scattering-time parameter can, therefore, be attributed to absorption and attenuation outside the scatterer. We further show that the real and imaginary-time parameters satisfy dispersion relations. Causality in this process is briefly discussed.