%0 Journal Article
%T Relativity and the Tunneling Problem in a ¡°Reduced¡± Waveguide
%A Eckehard W. Mielke
%A Miguel A. Marquina Carmona
%J International Journal of Optics
%D 2013
%I Hindawi Publishing Corporation
%R 10.1155/2013/947068
%X Wave packets are considered as solutions of the Maxwell equations in a reduced waveguide exhibiting tunneling due to a stepwise change of the index of refraction. We discuss several concepts of ¡°tunneling time¡± during the propagation of an electromagnetic pulse and analyze their compatibility with standard relativity. 1. Introduction Tunneling is often regarded as a quantum effect. Among the many recent applications is the scanning tunneling microscope exhibiting also phonon tunneling [1]. However, in optics, it was discovered already by Newton as frustrated total reflection of light, compare [2]. In physics, it is largely accepted that there is some time scale [3] associated with the duration of any tunneling process [4]. In fact, it has been directly measured, for instance, with microwaves [5¨C8]. However, there is a lack of consensus what is the exact nature of this ¡°tunneling time,¡± and a unique and simple expression [9] is still missing. Here, we will recapitulate some introductional material about its classical aspects and discuss the consequences for propagating electromagnetic waves in undersized waveguides. Our main objective is to confront them with standard relativity [10]. Recently, modern versions [11] of the Michelson-Morley experiment have provided the bound of for the isotropy of the velocity of light in vacuum, one of the most stringent experimental limits in physics. The wave operator of Jean-Baptiste le Rond d¡¯Alembert is invariant under the general Lorentz transformations where is the radius vector of an event and the Lorentz factor. It is quite remarkable that Riemann [12] proposed already in 1858 an invariant wave equation for the electromagnetic potential in an attempt to accommodate¡ªwithin his scalar electrodynamics¡ªthe 1855 experiments of Kohlrausch and Weber [13]. He estimated correctly the velocity of light in vaccum from the then known values the electromagnetic units. In 1886, Voigt [14] anticipated to some extent the invariance of the d¡¯Alembertian (1) under what is now called a Lorentz boost (2). 2. Electromagnetic Plane Waves Let us consider the electromagnetic field in a waveguide [15, 16]. To this end, we depart from the Maxwell equations of the Appendix which imply the wave equations for the electric and magnetic field. The refractive index is given in terms of the relative permittivity or dielectric constant and permeability of a medium. Let us restrict ourselves first to a plane wave solution written in the complex form where is the wave vector determining the direction of the wave propagation and the amplitudes and
%U http://www.hindawi.com/journals/ijo/2013/947068/