Universal Tunnelling Time in photonic Barriers

Tunnelling transit time for a frustrated total internal reflection (FTIR) in a double–prism experiment was measured using microwave radiation. We have found that the transit time is of the same order of magnitude as the corresponding transit time measured either in an undersized wave-guide (evanescent modes) or in a photonic lattice. Moreover we have established that in all such experiments the tunnelling transit time is approximately equal to the reciprocal (1/f) of the corresponding frequency of radiation. Previous photonic tunnelling transit time experiments have been carried out using electromagnetic radiation both at microwave and optical frequencies. Such experiments were stimulated by the formal analogy between the classical Helmholtz wave equation and the quantum–mechanical Schrödinger equation. The corresponding tunnelling transit time data for e.g. electrons are not yet available. In our Letter we are considering the tunnelling transit time for opaque photonic barriers [1, 2, 3, 4]. We suggest that in general the transit or delay time is approximately equal to the reciprocal frequency 1/f of the corresponding radiation and that it is independent of the type or shape of the actual barrier. The transit time or group delay time is defined as τ gr = x vgr , were x is the tunnelling distance and v gr = dω dk. This definition agrees with that introduced by Eisenbud and Wigner who put τ ϕ = dϕ dω = x dω/dk [5]. The tunnelling transit time or just tunnelling time for short, is measured as the time interval between