Path-integral treatment of optical and microwave tunneling

A model based on a statistical method, as deduced from a path-integral treatment of the Brownian motion proposed by Feynman and Hibbs in 1965, demonstrates to be capable of interpreting the results of delay time measurements in frustrated total internal reflection experiments at an optical and microwave scale. A plausible description of the trajectories followed by the system inside the tunneling region, the air gap between the two dielectric prisms, is given.

Figure 1

(Color online) Traversal time results, as a function of the gap width between the two prisms in FTIR experiments whose geometry is sketched in the inset. For optical laser beam (a): wavelength $\lambda =3.39\mu \mathrm{m}$, $i=45.5°$, and $\rho =1.409$; for the microwave case (b): $\lambda =3.2\mathrm{cm}$, $i=60°$, and $\rho =1.49$. The curves are obtained from Eq. (21) with parameter values as, for (a): $T\left(\max \right)=23.7\mu \mathrm{m}$, $D=12.1\mu \mathrm{m}$, and ${\beta }_T=22.3°$; and for (b): $T\left(\max \right)=24\mathrm{mm}$, $D=22.6\mathrm{mm}$, and ${\beta }_T=40.5°$.

Figure 2

(Color online) Most probable paths inside the air gap in the optical case (a) and microwave case (b). The continuous curves which fit the data of Figs. 1a, 1b, respectively, are given by the function $x\left(\tilde{t}\right)$, when situated in the gap space. The dotted line in (b) resembles a typical stochastic path [3].