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 [5–8]. 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—within his scalar electrodynamics—the 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
References
[1]
I. Altfelder, A. A. Voevodin, and A. K. Roy, “Vacuum phonon tunneling,” Physical Review Letters, vol. 105, Article ID 166101, 2010.
[2]
H. M. Nussenzveig, “Light tunneling in clouds,” Applied Optics, vol. 42, no. 9, pp. 1588–1593, 2003.
[3]
E. H. Hauge and J. A. Stovneng, “Tunneling times: a critical review,” Reviews of Modern Physics, vol. 61, p. 917, 1989.
[4]
M. Razavy, Quantum Theory of Tunneling, World Scientific, Singapore, 2003.
[5]
G. Nimtz, “Superluminal signal velocity,” Annalen der Physik, vol. 7, no. 7-8, pp. 618–624, 1998.
[6]
G. Nimtz, “Evanescent modes are not necessarily Einstein causal,” European Physical Journal B, vol. 7, no. 4, pp. 523–525, 1999.
[7]
A. A. Stahlhofer and G. Nimtz, “Evanescent modes are virtual fotons,” Europhysics Letters, vol. 76,, pp. 189–195, 2006.
[8]
G. Nimtz, “Do evanescent modes violate relativistic causality?” Lecture Notes in Physics, vol. 702, pp. 506–531, 2006.
[9]
R. Landauer and T. Martin, “Barrier interaction time in tunneling,” Reviews of Modern Physics, vol. 66, no. 1, pp. 217–228, 1994.
[10]
W. Rindler, Introduction to Special Relativity, The Clarendon Press, Oxford, UK, 1982.
[11]
S. Herrmann, A. Senger, K. M？hle, M. Nagel, E. V. Kovalchuk, and A. Peters, “Rotating optical cavity experiment testing Lorentz invariance at the 10-17 level,” Physical Review D, vol. 80, no. 10, Article ID 105011, 2009.
[12]
B. Riemann, “Ein Beitrag zur Elektrodynamik,” Annalen der Physik und Chemie, vol. 131, pp. 237–243, 1867.
[13]
R. Kohlrausch and W. E. Weber, “Ueber die Elektricit？tsmenge, welche bei galvanischen Str？men durch den Querschnitt der Kette fliesst,” Annalen der Physik, vol. 99, no. 10, 1856.
[14]
W. Voigt, “Theorie des Lichtes für bewegte Medien,” Nachrichten von der K？nigl. Gesellschaft der Wissenschaften und der Georg-Augusts-Universit？t zu G？ttingen, no. 8, pp. 177–238, 1887.
[15]
S. Flügge, Rechenmethoden der Elektrodynamik, Springer, Berlin, Germany, 1986.
[16]
M. A. Heald and J. B. Marion, Classical Electromagnetic Radiation, Brooks/Cole, New York, NY, USA, 3rd edition, 1985.
[17]
T. Dereli, J. Gratus, and R. W. Tucker, “The covariant description of electromagnetically polarizable media,” Physics Letters A, vol. 361, no. 3, pp. 190–193, 2007.
[18]
G. L. J. A. Rikken and B. A. Van Tiggelen, “Measurement of the Abraham force and its predicted QED corrections in crossed electric and magnetic fields,” Physical Review Letters, vol. 107, no. 17, Article ID 170401, 2011.
[19]
F. W. Hehl, J. Lemke, and E. W. Mielke, “Two lectures on fermions and gravity,” in Geometry and Theoretical Physics (Bad Honnef, 1990), J. Debrus and A. C. Hirshfeld, Eds., pp. 56–140, Springer, Berlin, Germany, 1991.
[20]
R. Ferraro, “Testing nonlinear electrodynamics in waveguides: the effect of magnetostatic fields on the transmitted power,” Journal of Physics A, vol. 43, no. 19, Article ID 195202, 2010.
[21]
M. A. Marquina Carmona, Microondas y Tiempo de Tunelaje, Proyecto de Investigacion, UAM-I, 2010.
[22]
S. J. Orfanidis, Electromagnetic Waves and Antennas, Rutgers University, 2008.
[23]
L. Brillouin, Wave Propagation and Group Velocity, vol. 8 of Pure and Applied Physics, Academic Press, New York, NY, USA, 1960.
[24]
T. Emig, Interpretation of tunneling times of evanescent electromagnetic modes within the framework of classical electrodynamics [Diploma thesis], (Advisor: E. Mielke) Cologne, Jun 1995, (Original German title: “Deutung der Tunnelzeiten bei evaneszenten elektromagnetischen Moden im Rahmen der klassischen Elektrodynamik”.).
[25]
R. L. Smith, “The velocities of light,” American Journal of Physics, vol. 38, no. 8, 1970.
[26]
L. Zhang, L. Zhan, K. Qian et al., “Superluminal propagation at negative group velocity in optical fibers based on Brillouin lasing oscillation,” Physical Review Letters, vol. 107, no. 9, Article ID 093903, 2011.
[27]
A. Schweinsberg, N. N. Lepeshkin, M. S. Bigelow, R. W. Boyd, and S. Jarabo, “Observation of superluminal and slow light propagation in erbium-doped optical fiber,” Europhysics Letters, vol. 73, no. 2, pp. 218–224, 2006.
[28]
H. M. Nussenzveig, “Time delay in quantum scattering,” Physical Review D, vol. 6, no. 6, pp. 1534–1542, 1972.
[29]
C. A. A. de Carvalho and H. M. Nussenzveig, “Time delay,” Physics Report, vol. 364, no. 2, pp. 83–174, 2002.
[30]
G. Nimtz, “On virtual phonons, photons, and electrons,” Foundations of Physics, vol. 39, no. 12, pp. 1346–1355, 2009.
[31]
L. A. MacColl, “Note on the transmission and reflection of wave packets by potential barriers,” Physical Review, vol. 40, no. 4, pp. 621–626, 1932.
[32]
N. T. Maitra and E. J. Heller, “Barrier tunneling and reflection in the time and energy domains: the battle of the exponentials,” Physical Review Letters, vol. 78, no. 16, pp. 3035–3038, 1997.
[33]
D. Sokolovski, “Why does relativity allow quantum tunnelling to “take no time”?” Proceedings of the Royal Society A, vol. 460, no. 2042, pp. 499–506, 2004.
[34]
A. M. Steinberg, P. G. Kwiat, and R. Y. Chiao, “Measurement of the single-photon tunneling time,” Physical Review Letters, vol. 71, no. 5, pp. 708–711, 1993.
[35]
A. M. Steinberg, “Comment on “Photonic tunneling times”,” Journal de Physique I, vol. 4, pp. 1813–1816, 1994.
[36]
V. Domínguez-Rocha, C. Zagoya, and M. Martínez-Mares, “Poynting's theorem for plane waves at an interface: a scattering matrix approach,” American Journal of Physics, vol. 76, no. 7, pp. 621–629, 2008.
[37]
G. Breit and J. A. Wheeler, “Collision of two light quanta,” Physical Review, vol. 46, no. 12, pp. 1087–1091, 1934.
[38]
J. Tausch, “Mathematical and numerical techniques for open periodic waveguides,” in Progress in Computational Physics (PiCP), M. Ehrhardt, Ed., pp. 50–72, Bentham, 2010.
[39]
S. C. Miller Jr. and R. H. Good, “A WKB-type approximation to the Schr？dinger equation,” Physical Review, vol. 91, no. 1, pp. 174–179, 1953.
[40]
W. Heitmann and G. Nimtz, “On causality proofs of superluminal barrier traversal of frequency band limited wave packets,” Physics Letters A, vol. 196, no. 1-2, pp. 154–158, 1994.
[41]
H. Wilbraham, “On a certain periodic function,” The Cambridge and Dublin Mathematical Journal, vol. 3, pp. 198–201, 1848.
[42]
J. W. Gibbs, “Letter to the editor,” Nature, vol. 59, p. 606, 1899.
[43]
W. J. Thompson, “Fourier series and the Gibbs phenomenon,” American Journal of Physics, vol. 60, p. 425, 1992.
[44]
Q.-R. Zhang, “Relativity and impossibility of superluminal motion,” Chinese Physics B, vol. 21, Article ID 110301, 2012.
[45]
B. Macke and B. Ségard, “From Sommerfeld and Brillouin forerunners to optical precursors,” Physical Review A, vol. 87, Article ID 043830, 2013.
[46]
J. S. Bell and D. Weaire, “George Francis FitzGerald,” Physics World, vol. 5, no. 9, pp. 31–35, 1992.
[47]
A. Lampa, “Wie erscheint nach der Relativit？tstheorie ein bewegter Stab einem ruhenden Beobachter?” Zeitschrift für Physik, vol. 27, no. 1, pp. 138–148, 1924.
[48]
U. Kraus, “First-person visualizations of the special and general theory of relativity,” European Journal of Physics, vol. 29, no. 1, pp. 1–13, 2008.
[49]
T. Müller and D. Weiskopf, “Special-relativistic visualization,” Computing in Science and Engineering, vol. 13, no. 4, Article ID 5931490, pp. 85–93, 2011.
[50]
H. G. Winful, ““Nature of superluminal” barrier tunneling,” Physical Review Letters, vol. 90, Article ID 023901, 2003.
[51]
H. G. Winful, “Tunneling time, the Hartman effect, and superluminality: a proposed resolution of an old paradox,” Physics Reports, vol. 436, pp. 1–69, 2006.
[52]
M. Rinaldi, “Superluminal dispersion relations and the Unruh effect,” Physical Review D, vol. 77, Article ID 124029, 2008.
[53]
B. Zhang, Q.-Y. Cai, and M.-S. Zhan, “The temperature in Hawking radiation as tunneling,” Physics Letters B, vol. 671, no. 2, pp. 310–313, 2009.
[54]
R. Schützhold and W. G. Unruh, “Hawking radiation in an electromagnetic waveguide?” Physical Review Letters, vol. 95, Article ID 031301, 2005.
[55]
S. Hopper and C. R. Evans, “Metric perturbations from eccentric orbits on a Schwarzschild black hole. I. Oddparity Regge-Wheeler to Lorenz gauge transformation and two new methods to circumvent the Gibbs phenomenon,” Physical Review D, vol. 87, Article ID 064008, 2013.
[56]
A. O. Caldeira and A. J. Leggett, “Influence of dissipation on quantum tunneling in macroscopic systems,” Physical Review Letters, vol. 46, no. 4, pp. 211–214, 1981.
[57]
M. A. Sillanp？？, R. Khan, T. T. Heikkil？, and P. J. Hakonen, “Macroscopic quantum tunneling in nanoelectromagnetical systems,” Physical Review B, vol. 84, Article ID 195433, 2011.
[58]
W. Franz, “Elektroneninterferenzen im Magnetfeld,” Verhandlungen der Deutchen Physikalischen Gesellschaft, vol. 65, 1939.
[59]
W. Franz, “über zwei unorthodoxe Interferenzversuche,” Zeitschrift für Physik, vol. 184, pp. 85–91, 1965.
[60]
H. Batelaan and A. Tonomura, “The Aharonov-Bohm effects: variations on a subtle theme,” Physics Today, vol. 62, no. 9, pp. 38–43, 2009.
[61]
J. Mac Cullagh, “An essay towards a dynamical theory of cristtaline reflexion and refraction,” Transaction of the Royal Irish Academy, vol. 21, pp. 17–50, 1839.
