IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 9, NO. 1, JANUARY/FEBRUARY 2003 79 Superluminal Signaling by Photonic Tunneling Günter Nimtz Abstract—Photonic tunneling has found much theoretical and applied interest recently. Superluminal photonic pulse transmis- sion and reflection have been presented at microwave and infrared frequencies. Presumably superluminal photonic and electronic de- vices can become reality soon. The author introduces new exper- imental and theoretical data on superluminal tunneling. Data of reflection by tunneling barriers have evidenced the nonlocal prop- erty of tunneling. An empirical relation was found for the photonic tunneling time independent of the system in question. The relation seems to be universal for all tunneling processes. The outstanding property of superluminal velocity can be applied to speed up pho- tonic modulation and transmission as well as to improve micro elec- tronic devices. Index Terms—Electromagnetic propagation, envelope detection, superluminal, tunneling. I. INTRODUCTION T HE MOST spectacular property of the tunneling process is its barrier traversal velocity: An imaginary tunneling time and a superluminal barrier traversal velocity have been predicted. The latter was first observed with microwave sig- nals by Enders and Nimtz [1]. They transmitted microwave sig- nals through undersized waveguides as was first suggested by Ranfagni et al. [2], [3]. Ranfagni et al., however, detected es- sentially nonevanescent subluminal propagating microwaves as they stated later in [4]. At the end of the 19th century, Rayleigh considered the group velocity to correspond to the velocity of energy or of signal transmission in a vacuum. Later, this raised difficulties in the rel- ativistic theory of dispersive media. The problem was resolved by Sommerfeld and Brillouin in the case of waves with real wave numbers [5]. However, it was not tackled for media with purely imaginary wave numbers as it is in the case of tunneling [6]. In- cidentally, an imaginary wave number may yield a superluminal velocity. Many textbooks and review articles deny the possibility of su- perluminal signal velocities; see for instance [7]–[12]. The au- thors claim that a signal velocity never can exceed the vacuum velocity of light. Allegedly, this has been proven by Einstein studying the light propagation in vacuum [13]. The classical evanescent modes represent the general wave mechanical tun- neling process [14], [15]. The outstanding property of the tun- neling process, i.e., the lack of a phase shift inside a barrier, can result in a superluminal signal velocity as discussed in [6], [7], [17]–[19]. Fig. 1 shows a measured Gaussian pulse-like digital signal, which traveled with a group velocity faster than the ve- locity of light in vacuum [6]. The signal’s information is rep- Manuscript received October 8, 2002; revised November 29, 2002. The author is with the Physics Department, University of Cologne, D-50937 Köln, Germany (e-mail: G.Nimtz@uni-koeln.de). Digital Object Identifier 10.1109/JSTQE.2002.808195 Fig. 1. Intensity versus time of a microwave pulse (2) which has tunneled a photonic barrier (i.e., an undersized wave-guide barrier of 114.2-mm length) with superluminal velocity. The tunneled signal is for comparison normalized with a pulse (1) which propagated through a normal waveguide of the same length. The tunneled signal traveled at a speed of and was measured 0.5 ns earlier than the guided one. The tunneled microwave pulse contained about 10 photons. resented by the half width which is detected only by measuring the complete signal’s envelope. Frequency band limitation and finite time duration are found for all physical signals and have been discussed and analyzed in [5], [16], [20], for instance. The impossiblity of a hypotheti- cally unlimited frequency band of signals has been explained by quantum mechanical arguments [17], [18]. Since the introduc- tion of quantum mechanics, a field does exist with a frequency only if for its energy holds. In order to avoid signal re- shaping due to the dispersion of waveguides or of any medium, the signal has to have an appropriately limited frequency spec- trum. In this paper, superluminal signal transmission is discussed in tunneling. Such electromagnetic waves with a purely imaginary wave number (often called evanescent modes) play an impor- tant role in microwave technology, in tunneling spectroscopy, in photonics, and in optoelectronics and they are useful for the simulation of quantum mechanical tunneling. Three prominent photonic tunneling barriers are presented in Fig. 2. They are based on different mechanisms but all of them are characterized by a purely imaginary wave number. The wave equation yields for the electric field in the tunneling case (1) where is the angular frequency, the time, the distance, the wave number, and the imaginary wave number. 1077-260X/03$17.00 © 2003 IEEE