Universal tunneling time for all fields

Tunneling is probably the most important physical process. The observation that particles surmount a high mountain in spite of the fact that they don’t have the necessary energy can not be explained by classical physics. However, this so called tunneling became allowed by the theory of quantum mechanics. Experimental tunneling studies with different photonic barriers from microwave frequencies up to ultraviolet frequencies pointed toward a universal tunneling time [1, 2]. The observed results and calculations have shown that the tunneling time of opaque photonic barriers (for instance optical mirrors) equals approximately the reciprocal frequency of the electromagnetic wave in question. The tunneling process is described by virtual photons [3]. Virtual particles like photons or electrons are not observable. However, from the theoretical point of view, they represent necessary intermediate states between observable real states. In the case of tunneling there is a virtual particle between the incident and the transmitted particle. Tunneling modes have a purely imaginary wave number. They represent solutions of the Schrödinger equation and of the classical Helmholtz equation. The most prominent example of the occurrence of tunneling modes in optics is frustrated total internal reflection (FTIR) at double prisms. In 1949 Sommerfeld [4] pointed out that this pseudo classical optical phenomenon represents the analogy of quantum mechanical tunneling. Recent experimental and theoretical data confirmed the conjecture that the tunneling process is characterized by a universal tunneling time independent of the kind of field. Tunneling proceeds at a time of the order of magnitude of the reciprocal frequency of the wave. Optical evanescent modes and solutions of quantum mechanical tunneling have a purely imaginary wave number. This means that they do not experience a phase shift in traversing space. The delay time τ of a propagating wave packet is given by the derivative