Tunneling delays in frustrated total internal reflection

We demonstrate a direct relationship between tunneling delays in a one-dimensional potential barrier and the delays observed in frustrated total internal reflection. Expressions relating the group delays in these two cases have been formulated in the past [A. M. Steinberg and R. Y. Chiao, Phys. Rev. A 49, 3283 (1994)], but only for certain limits of the normalized particle energy $E/V_0$. Here, we derive a single expression relating these two group delays that is valid for arbitrary particle energy by decomposing the group delay into dwell-time and self-interference components. This decomposition also explains why the predicted delay differs in the limiting cases observed by Steinberg and Chiao.

Figure 1

Schematic diagram for one-dimensional barrier tunneling. A particle of energy $E$ approaches a barrier of potential $V_0$. $\psi$ is the wave function in each region.

Figure 2

Schematic diagram for two-dimensional barrier tunneling.

Figure 3

Group delay, dwell time, and self-interference delay as a function of normalized particle energy $E/V_0$ for a simple one-dimensional barrier. The delays are normalized to the equal time ${\tau }_0=L/v_0$, with $v_0=\hslash \gamma /m$ and $\gamma =\sqrt{2m{V}_0}/\hslash$. This equal time is the delay a particle of energy $V_0$ would experience when propagating through a distance $L$ in free space. The barrier height in this plot is defined by $\gamma L=3\pi$.

Figure 4

TIR at the interface between an equilateral prism and a dielectric slab. The Goos-Hänchen shift $\Delta y$ causes a change in the glass propagation length, which has ramifications for an experimental measurement of tunneling delay in the double-prism geometry.