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Optics Express

  • Editor: J. H. Eberly
  • Vol. 9, Iss. 11 — Nov. 19, 2001
  • pp: 561–566

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On the apparent superluminality of evanescent waves

Vera Brudny and W. Luis Mochan  »View Author Affiliations


Optics Express, Vol. 9, Issue 11, pp. 561-566 (2001)
http://dx.doi.org/10.1364/OE.9.000561


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Abstract

There have been many recent theoretical and experimental reports on the propagation of light pulses at speeds exceeding the speed of light in vacuum c within media with anomalous dispersion, either opaque or with gain. Superluminal propagation has also been reported within vacuum, in the case of inhomogeneous pulses. In this paper we show that the observations of superluminal and non-causal propagation of evanescent pulses under the conditions of frustrated internal reflection are only apparent, and that they can be simply explained employing an explicitly (sub)luminal causal theory. However, the usual one-dimensional approach to the analysis of pulse propagation has to be abandoned and the spatial extent of the incoming pulse along the directions normal to the propagation direction has to be accounted for to correctly interpret the propagation speed of these evanescent waves. We illustrate our theory with animations of the time development of a pulse built upon the Huygen’s construction.

© Optical Society of America

[Optical Society of America ]

The last decade has seen a renewed interest in the possibility of faster than light propagation. In 1993 Steinberg et al [1

A. M. Steinberg, P. G. Kwiat, and R. Y. Chiao, “Measurement of the single-photon tunneling time,” Phys. Rev. Lett. 71, 708–711 (1993) http://link.aps.org/abstract/PRL/v71/i5/p708. [CrossRef] [PubMed]

] observed that single photons tunneling across a barrier made up of a photonic band gap multilayered filter appeared earlier than photons travelling the same distance in free space. The tunneling time of classical electromagnetic pulses was later observed to become independent of the length [2

Ch. Spielmann, R. Szipöcs, A. Stingl, and F. Krausz, “Tunneling of optical pulses through photonic band gaps,” Phys. Rev. Lett. 73, 2308 (1994) http://link.aps.org/abstract/PRL/v73/p2308. [CrossRef] [PubMed]

] of very opaque barriers, in analogy to earlier theoretical predictions for electronic quantum tunneling [3

T. H. Hartman, “Tunneling of a wave packet,” J. Appl. Phys. 33, 3427 (1962). [CrossRef]

]. Other experiments have shown superluminal group velocities for light pulses crossing materials with an anomalous dispersion such as when the frequency is close to an absorption resonance [4

C. G. B. Garret and D. E. McCumber, “Propagation of a Gaussian Light Pulse through an Anomalous Dispersion Medium,” Phys. Rev. A 1, 305 (1970) http://link.aps.org/abstract/PRA/v1/i2/p305. [CrossRef]

, 5

S. Chu and S. Wong, “Linear Pulse Propagation in an Absorbing Medium,” Phys. Rev. Lett. 48, 738 (1982) http://link.aps.org/abstract/PRL/v48/i11/p738. [CrossRef]

, 6

A. Katz and R. R. Alfano, “Pulse Propagation in an Absorbing Medium,” Phys. Rev. Lett. 49, 1292 (1982) http://link.aps.org/abstract/PRL/v49/p1292. [CrossRef]

].

The theoretical analysis of the experiments above and many others has centered on the question of the possibility of communicating information or propagating energy at speeds faster than the speed of light in vacuum c, and on the consequent violation of Einstein causality. The key argument in these analyses is that an electromagnetic pulse is not a point object, but instead, has an extension along its propagation direction. Furthermore, the pulse has a shape that may vary as it propagates. This suggests a simple explanation of the apparent observation of superluminal propagation: The peak of the transmitted pulse may be fully produced by the leading tail of the incoming pulse, being causally unrelated to its original maximum [7

Rolf Landauer, “Light faster than light?” Nature 365, 692 (1993). [CrossRef]

, 8

G. Diener, “Superluminal group velocities and information transfer,” Phys. Lett. A 223, 327–331 (1996). [CrossRef]

]. The onset of a pulse can never propagate at speeds greater than c, as shown by Brillouin and Sommerfeld [9

Léon Brillouin, “Wave Propagation and Group Velocity” (Academic, New york, 1960) Ch. II , III , and IV.

], but the intensity of the pulse can be redistributed as it propagates; if the peak of the original signal is damped much more strongly than its leading front within an absorbing medium, the pulse’s maximum may appear to move at a superluminal velocity, and under certain conditions its shape may be mostly unmodified [4

C. G. B. Garret and D. E. McCumber, “Propagation of a Gaussian Light Pulse through an Anomalous Dispersion Medium,” Phys. Rev. A 1, 305 (1970) http://link.aps.org/abstract/PRA/v1/i2/p305. [CrossRef]

]. This interpretation was confirmed by the remark that the amplitude of the transmitted pulse is always lower than the amplitude that the incident pulse would have if it propagated without attenuation, and that its duration is smaller [2

Ch. Spielmann, R. Szipöcs, A. Stingl, and F. Krausz, “Tunneling of optical pulses through photonic band gaps,” Phys. Rev. Lett. 73, 2308 (1994) http://link.aps.org/abstract/PRL/v73/p2308. [CrossRef] [PubMed]

]. A careful definition of the energy velocity has lead to the conclusion that it is smaller than c even in superluminal situations [10

G. Diener, “Energy Transport in Dispersive Media and Superluminal Group Velocities,” Phys. Lett. A 235, 118 (1997). [CrossRef]

, 11

V. Romero-Rochín, R. P. Duarte-Zamorano, S. Nielsen-Hofseth, and R. G. Barrera, “Superluminal transmission of light pulses through optically opaque barriers,” Phys. Rev. E 63, 027601 (2001) http://link.aps.org/abstract/PRE/v63/e027601. [CrossRef]

]. Therefore, it has been generally agreed that there is no violation of causality and that energy and information do not propagate superluminally.

Besides transmission through opaque materials, superluminal propagation is also expected in novel passive artificial media [12

Richard W. Ziolkowski, “Superluminal Transmission of Information through an Electromagnetic Metamaterial,” Phys. Rev. E 63, 046604 (2001) http://link.aps.org/abstract/PRE/v63/e046604. [CrossRef]

] and active media with gain [13

A. N. Oraevsky, “Superluminal Waves in Amplifying Media,” Physycs-Uspekhi , 41, 1199 (1998). [CrossRef]

]. The latter has been experimentally realized [14

L. J. Wang, A. Kuzmich, and A. Dogariu, “Gain-Assisted Superluminal Light Propagation,” Nature 406, 277 (2000). [CrossRef] [PubMed]

, 15

A. Dogariu, A. Kuzmich, and L. J. Wang, “Transparent anomalous dispersion and superluminal light-pulse propagation at a negative group velocity,” Phys. Rev. A 63, 053806 (2001) http://link.aps.org/abstract/PRA/v63/e053806. [CrossRef]

] in a transparent medium with anomalous dispersion, where a negative velocity, faster than infinity, was observed: the peak of the outgoing pulse left the cell containing the medium even before the incoming peak entered it [16

A. Dogariu, Ak. Kuzmich, H. Cao, and L. J. Wang, “Superluminal Light Pulse Propagation Via Rephasing in a Transparent Anomalously Dispersive Medium,” Opt. Express 8, 344 (2001) http://www.opticsexpress.org/oearchive/source/30536.htm. [CrossRef] [PubMed]

]. A quantum mechanical discussion of the signal detection [17

A. Kuzmich, A. Dogariu, L. J. Wang, P.W. Milonni, and R. Y. Chiao, “Signal Velocity, Causality, and Quantum Noise in Superluminal Light Pulse Propagation,” Phys. Rev. Lett. 86, 3925–3929 (2001) http://link.aps.org/abstract/PRL/v86/p3925. [CrossRef] [PubMed]

] shows that even in this case the signal velocity does not exceed c. Quantum aspects of superluminal propagation [18

G. Nimtz, “Evanescent Modes are not Necessarily Einstein Causal,” Eur. Phys. J. B 7, 523 (1999). [CrossRef]

] have also been investigated for model materials [19

Yakir Aharonov, Benni Reznil, and Ady Stern, “Quantum Limitatins on Superluminal Propagation,” Phys. Rev. Lett. 81, 2190 (1998) http://link.aps.org/abstract/PRL/v81/p2190. [CrossRef]

, 20

P. W. Milonni, K. Furuya, and R. Y. Chiao, “Quantum Theory of Superluminal Pulse Propagation,” Opt. Express 8, 59 (2001) http://www.opticsexpress.org/oearchive/source/27132.htm. [CrossRef] [PubMed]

]. However, deeper understanding of the field of superluminal propagation is still required [21

Aephraim M Steinberg, “No thing goes faster than light,” Physics World 13 3 (2000) http://www.physicsweb.org/article/world/13/9/3.

, 22

Peter W. Milonni, “Causal Discussion of Superluminal Pulses,” Physics Today 54, 81 (2001) http://www.physicstoday.org/pt/vol-54/iss-2/p14b.html. [CrossRef]

].

Curiously, superluminal propagation can also occur within vacuum [23

Jacob Broe and Ole Keller, “Superluminality and spatial confinement in optical tunneling,” Opt. Commun. 194, 83 (2001). [CrossRef]

], although in this case the participating waves must be inhomogeneous. Evanescent guided microwaves whose frequency lies below the cutoff frequency of an empty small waveguide display superluminal propagation [24

A. Ranfagni, D. Mugnai, P. Fabeni, and G. P. Pazzi, “Delay-time measurements in narrowed waveguides as a test of tunneling,” Appl. Phys. Lett. 58, 774 (1991). [CrossRef]

, 25

A. Enders and G. Nimtz, “Photonic-tunneling experiments,” Phys. Rev. B 47, 9605 (1993) http://link.aps.org/abstract/PRB/v47/p9605. [CrossRef]

], in many ways analogous to that of light traversing an opaque barrier or electrons tunneling across a barrier. Although phase velocity is superluminal, Einstein causality is again preserved [26

A. Pablo, L. Barbero, Hugo E. Hernández-Figueroa, and Erasmo Recami, “Propagation Speed of Evanescent Modes,” Phys. Rev. E 62, 8628 (2000) http://link.aps.org/abstract/PRE/v62/p8628. [CrossRef]

] and pulse velocities are subluminal [27

Klass Wynne, John J. J. Carey, Justyna Zawadzka, and Dino Jaroszynski, “Tunneling of Single-Cycle Terahertz Pulses through Waveguides,” Opt. Commun. 176, 429 (2000). [CrossRef]

]. The propagation of evanescent waves across an air gap between two transparent dielectrics with index of refraction n >1 has also been predicted to be superluminal [28

S. Bosanac, “Propagation of Electromagnetic Wave Packets in Nondispersive Dielectric Media,” Phys. Rev. A 28, 577 (1983) http://link.aps.org/abstract/PRA/v28/p577. [CrossRef]

] in the frustrated-total-internal-reflection (FTIR)regime and has been verified experimentally [29

Ph. Balcou and L. Dutriaux, “Dual Optical Tunneling Times in Frustrated Total Internal Reflection,” Phys. Rev. Lett. 78, 851 (1997) http://link.aps.org/abstract/PRL/v78/p851. [CrossRef]

]. More recently, Carey et al [30

John J. Carey, Justyna Zawadzka, Dino A. Jaroszynski, and Klaas Wynne, “Noncausal Time Response in Frustrated Total Internal Reflection?” Phys. Rev. Lett. 84, 1431 (2000) http://link.aps.org/abstract/PRL/v84/p1431. [CrossRef] [PubMed]

] have performed FTIR experiments using infrared picosecond pulses with only a single oscillation, and have been able to measure the transmitted electric field directly in the time domain. Their results show that as the separation between the two prisms increases, the arrival time of the pulse is significantly advanced by more than the pulse width. Even after applying corrections due to the changes in the optical path travelled by the pulse outside of the air gap, it was found that propagation across the gap was essentially instantaneous and that part of the pulse actually travelled backwards in time. Although surprising, this noncausal propagation was seen to be in full accordance with a theoretical calculation of the time response function [30

John J. Carey, Justyna Zawadzka, Dino A. Jaroszynski, and Klaas Wynne, “Noncausal Time Response in Frustrated Total Internal Reflection?” Phys. Rev. Lett. 84, 1431 (2000) http://link.aps.org/abstract/PRL/v84/p1431. [CrossRef] [PubMed]

].

Fig. 1. Animation illustrating the non-evanescent transmission of light across an air gap from ∑1 to ∑2 separating two semi-infinite dielectrics (1.9Mb). A single incoming narrow plane wavefront, with angle of incidence θ, excites secondary waves w 1 within the air gap, which grow with speed c around successive points p 1 at the surface ∑1. As they sweep points p 2 on the other side of the air gap ∑2, these waves in turn excite further secondary waves w 2 within the second dielectric growing with speed c/n. S 1 and S 2 are perfectly opaque screens. The nominal propagation directions are indicated by red arrows. The dots indicate the centers of the secondary waves launched at equally spaced times. The figure corresponds to n=2 and θ=27°<θc .

It has frequently been argued that FTIR is the classical analogue of quantum mechanical 1D tunneling. However, there is a fundamental difference: FTIR is intrinsically at least a 2D problem. Total internal reflection requires a fairly well defined angle of incidence θ larger than the critical angle θc =sin-1(1/n). Therefore, the incident field requires equally well defined wavefronts, and thus, a non-null spatial extension along the plane normal to the propagation direction. There are uncertainty relations between propagation direction and wavefront size that cannot be violated. Thus, to fully understand propagation phenomena in the FTIR regime, account has to be taken of the spatial extention of the wavepacket, not only in the direction of propagation, as in the many examples of superluminal propagation discussed above, but also along its normal direction [31

W. Luis Mochán and Vera L. Brudny, “Comment on Noncausal Time Response in Frustrated Total Internal Reflection?” Phys. Rev. Lett. 87, 119101 (2001) http://link.aps.org/abstract/PRL/v87/e119101. [CrossRef]

].

To understand our remark above, consider a single infinitely extended plane wavefront impinging obliquely from a semi-infinite dielectric onto the front interface ∑1 with a vacuum gap that extends up to the back interface ∑2. In Figs. 1 and 2 we construct the transmitted wave employing a variation of the well known Huygen’s construction. The incoming wavefront sweeps the surface of the dielectric at a speed v ‖=c/(n sin θ) exciting secondary waves (SW) which grow in vacuum with speed c. If θ<θc , v ‖> c and each SW intersects previously launched SW’s, thus giving rise to a well defined flat envelope that propagates non-evanescently with the direction θt =sin-1(n sin θ). As each SW reaches the surface of the second dielectric, it generates a new set of SW that grow within the dielectric with speed c/n, and originate a new flat wavefront that travels in the original direction θ. If the incoming wave were an infinitely sharp pulse described by a traveling Dirac’s delta function, in the limit of an infinite aperture the transmitted field would also be a Dirac’s delta function. The blocking screens in Fig. 1 give rise to diffraction effects. For the sake of simplicity, we have purposefully disregarded multiply reflected waves in the Huygen’s construction.

Fig. 2. Animation illustrating the evanescent transmission of light across an air gap separating two semi-infinite dielectrics (2.1Mb), as in Fig. 1, but with θ=34°>θc . The vertical dashed line connects the point at which the incident wavefront intersects the front face of the air gap i with the corresponding point on the back face t. The thin black arrow connects the superluminally excited position t to its true field source s.

In Fig. 2 we consider the case of FTIR, for which θ>θc . In this case v ‖<c and therefore the SW’s within the air gap overtake the incoming wave and do not intersect one another. The transmitted electromagnetic field might reach positions on ∑2 across the gap, such as t, even before the incident wavefront reaches the corresponding positions i on the front face, thus giving the impression of superluminal and even causality violating propagation. Causality is however not violated at all, as the exciting field is not actually produced at i, but comes instead from the far-away points s and beyond, excited some time in the past, from which it could reach t traveling at speed ≤c.

Due to the generality of Huygen’s construction, our analysis leads to an apparent superluminality regardless of the polarization of the incoming light, in agreement with experiment [29

Ph. Balcou and L. Dutriaux, “Dual Optical Tunneling Times in Frustrated Total Internal Reflection,” Phys. Rev. Lett. 78, 851 (1997) http://link.aps.org/abstract/PRL/v78/p851. [CrossRef]

] but in contrast to a recent theoretical result that only p-polarized waves propagate superluminally in FTIR [23

Jacob Broe and Ole Keller, “Superluminality and spatial confinement in optical tunneling,” Opt. Commun. 194, 83 (2001). [CrossRef]

]. The mechanism for superluminality in vacuum described here is unrelated to the photon source delocalization invoked in Ref. [23

Jacob Broe and Ole Keller, “Superluminality and spatial confinement in optical tunneling,” Opt. Commun. 194, 83 (2001). [CrossRef]

].

The fact that the field at the back face of the air gap comes from previously excited positions after traveling subluminally, and not superluminally from the position immediately across can be further demonstrated by using the screens S 1 and S 2. If propagation were indeed superluminal in the direction normal to the interfaces, we would expect a transmitted field to appear across the gap as soon as the incoming wavefront leaves the screen S 1. This is not the case, as the field takes a finite time to reach ∑2 and some more time to reach the point opposite the incident wavefront and finally build up the apparently superluminal transmitted pulse. By the same token, it takes a finite time for the intensity of the transmitted field to diminish after the incoming front reaches the screen S 2.

Fig. 2 also shows that under FTIR conditions the field produced by a single wavefront spreads as it traverses the air gap in proportion to the gap’s width. Thus, its intensity decays only algebraically and not exponentially. The decay of each secondary wave and that of their superposition are only qualitatively reproduced by the figure. A monochromatic plane wave may be visualized as a series of wavefronts of alternating signs separated by half a wavelength. The regions of influence of each of them overlap those of nearby wavefronts, leading to interference and to the familiar exponential decay of evanescent wave trains [32

Vera L. Brudny and W. Luis Mochán, under preparation.

]. Smaller wavelengths yield more overlapping regions and a stronger decay, as expected.

Our qualitative results above may be verified by actually computing the field transmitted across the gap. We have constructed [32

Vera L. Brudny and W. Luis Mochán, under preparation.

] the propagator

P (x,z,t)=− cπ ∂ ∂z Θ ( ct− x2+ z2) (ct)2− x2− z2
(1)

to calculate the retarded solution of the 2D wave equation ϕ(x, z, t)=∫dt ′ dx ′ P(x-x ′, z, t-t ′)ϕ(x ′, 0, t ′) from its values ϕ(x ′, 0, t ′)at previous times on a given flat surface z=0. The Heavyside unit step function Θ(…)guarantees that our propagator is causal and (sub)luminal. Notice that, in agreement with the Huygen’s construction, the field at position (x ′, 0)at time t ′ only influences the field at positions (x, z) within a disk of radius c(t-t ′)centered at (x ′, 0)at a later time t. Applying the propagator (1)t o a single sharp incoming wavefront ϕ(x ′, 0, t ′)=ϕ 0 δ(x ′-v ‖ t ′)with v ‖<c(θ>θc )we obtain the transmitted field after first crossing the gap,

ϕ (x,z,t)= { 0 t<T (x,z; x1s) ϕ0F (x,z,t; x1s) T (x,z; x1s)<t<T (x,z; x2s) ϕ0 ( F (x,z,t; x1s)−F (x,z,t; x2s)) t>T (x,z; x2s),
(2)

where T (x,z; x2)= xs⁄ v∥+ ( x− xs)2+ z2⁄c is the time of first arrival at (x, z)of a signal excited at the edges (xs , 0), xs = x 1 s, x 2 s of the screens S 1 and S 2 by the incident wavefront, F(x, z, t; xs )=F 0(x, z, t)(1+C(x, z, t; xs )) is given by the field

F0 (x,z,t)= γz⁄π γ2 ( x− v∥t)2+ z2
(3)

that would have been excited in the absence of screens, with a correction C(x, z, t; xs ) due to diffraction [32

Vera L. Brudny and W. Luis Mochán, under preparation.

] by the screens. Here, γ≡1⁄ 1− ( v∥⁄c)2 Eq. (3) shows that the transmitted field at z is a Lorentzian of width z/γ and height ϕ 0 γ/(πz)centered on the nominal position x=v ‖ t of the incident wavefront at z=0, as if propagation were instantaneous and along the direction normal to the surface, although it was obtained from a causal and retarded propagator (1). There is no violation of the precursor theory of Brillouin and Sommerfeld [9

Léon Brillouin, “Wave Propagation and Group Velocity” (Academic, New york, 1960) Ch. II , III , and IV.

] as our system is 2D, not 1D.

In conclusion, we have shown that propagation across an air gap under FTIR conditions may seem to take place in the direction normal to the interface in a noncausal and superluminal fashion. However, a careful analysis shows that propagation is indeed causal and retarded and takes place along oblique directions. Eq. (2) suggests that an extension of the experiments reported in [30

John J. Carey, Justyna Zawadzka, Dino A. Jaroszynski, and Klaas Wynne, “Noncausal Time Response in Frustrated Total Internal Reflection?” Phys. Rev. Lett. 84, 1431 (2000) http://link.aps.org/abstract/PRL/v84/p1431. [CrossRef] [PubMed]

] adding opaque screens could settle the question on the superluminality of propagation under FTIR conditions. It has been recognized since long ago that the spatial extention of a pulse has to be taken into account to understand superluminal wave phenomena, but attention has been restricted to the length of the pulse along its propagation direction. Here we have shown that the transverse extention of the pulse should also be taken into account in order to understand the propagation of evanescent waves in vacuum. For these waves, the transmitted pulses can be derived causally from the lateral wings of the incident pulse and not only from their leading tails as in other superluminal systems. A similar discussion may clarify other experiments performed in vacuum with inhomogeneous waves [33

A. Ranfagni, P. Fabeni, G. P. Pazzi, and D. Mugnai “Anomalous pulse delay in microwave propagation: A plausible connection to the tunneling time,” Phys. Rev. E 48, 1453 (1993). http://link.aps.org/abstract/PRE/v48/p1453. [CrossRef]

, 34

D. Mugnai, A. Ranfagni, and R. Ruggeri, Phys. Rev. Lett. 84, 4830 (2000) http://link.aps.org/abstract/PRL/v84/p4830. [CrossRef] [PubMed]

].

We acknowledge partial support from UBACYT and Fundación Antorchas (VLB) and from DGAPA-UNAM under project IN110999 (WLM). VLB is a member of CON-ICET.

References and links

1.

A. M. Steinberg, P. G. Kwiat, and R. Y. Chiao, “Measurement of the single-photon tunneling time,” Phys. Rev. Lett. 71, 708–711 (1993) http://link.aps.org/abstract/PRL/v71/i5/p708. [CrossRef] [PubMed]

2.

Ch. Spielmann, R. Szipöcs, A. Stingl, and F. Krausz, “Tunneling of optical pulses through photonic band gaps,” Phys. Rev. Lett. 73, 2308 (1994) http://link.aps.org/abstract/PRL/v73/p2308. [CrossRef] [PubMed]

3.

T. H. Hartman, “Tunneling of a wave packet,” J. Appl. Phys. 33, 3427 (1962). [CrossRef]

4.

C. G. B. Garret and D. E. McCumber, “Propagation of a Gaussian Light Pulse through an Anomalous Dispersion Medium,” Phys. Rev. A 1, 305 (1970) http://link.aps.org/abstract/PRA/v1/i2/p305. [CrossRef]

5.

S. Chu and S. Wong, “Linear Pulse Propagation in an Absorbing Medium,” Phys. Rev. Lett. 48, 738 (1982) http://link.aps.org/abstract/PRL/v48/i11/p738. [CrossRef]

6.

A. Katz and R. R. Alfano, “Pulse Propagation in an Absorbing Medium,” Phys. Rev. Lett. 49, 1292 (1982) http://link.aps.org/abstract/PRL/v49/p1292. [CrossRef]

7.

Rolf Landauer, “Light faster than light?” Nature 365, 692 (1993). [CrossRef]

8.

G. Diener, “Superluminal group velocities and information transfer,” Phys. Lett. A 223, 327–331 (1996). [CrossRef]

9.

Léon Brillouin, “Wave Propagation and Group Velocity” (Academic, New york, 1960) Ch. II , III , and IV.

10.

G. Diener, “Energy Transport in Dispersive Media and Superluminal Group Velocities,” Phys. Lett. A 235, 118 (1997). [CrossRef]

11.

V. Romero-Rochín, R. P. Duarte-Zamorano, S. Nielsen-Hofseth, and R. G. Barrera, “Superluminal transmission of light pulses through optically opaque barriers,” Phys. Rev. E 63, 027601 (2001) http://link.aps.org/abstract/PRE/v63/e027601. [CrossRef]

12.

Richard W. Ziolkowski, “Superluminal Transmission of Information through an Electromagnetic Metamaterial,” Phys. Rev. E 63, 046604 (2001) http://link.aps.org/abstract/PRE/v63/e046604. [CrossRef]

13.

A. N. Oraevsky, “Superluminal Waves in Amplifying Media,” Physycs-Uspekhi , 41, 1199 (1998). [CrossRef]

14.

L. J. Wang, A. Kuzmich, and A. Dogariu, “Gain-Assisted Superluminal Light Propagation,” Nature 406, 277 (2000). [CrossRef] [PubMed]

15.

A. Dogariu, A. Kuzmich, and L. J. Wang, “Transparent anomalous dispersion and superluminal light-pulse propagation at a negative group velocity,” Phys. Rev. A 63, 053806 (2001) http://link.aps.org/abstract/PRA/v63/e053806. [CrossRef]

16.

A. Dogariu, Ak. Kuzmich, H. Cao, and L. J. Wang, “Superluminal Light Pulse Propagation Via Rephasing in a Transparent Anomalously Dispersive Medium,” Opt. Express 8, 344 (2001) http://www.opticsexpress.org/oearchive/source/30536.htm. [CrossRef] [PubMed]

17.

A. Kuzmich, A. Dogariu, L. J. Wang, P.W. Milonni, and R. Y. Chiao, “Signal Velocity, Causality, and Quantum Noise in Superluminal Light Pulse Propagation,” Phys. Rev. Lett. 86, 3925–3929 (2001) http://link.aps.org/abstract/PRL/v86/p3925. [CrossRef] [PubMed]

18.

G. Nimtz, “Evanescent Modes are not Necessarily Einstein Causal,” Eur. Phys. J. B 7, 523 (1999). [CrossRef]

19.

Yakir Aharonov, Benni Reznil, and Ady Stern, “Quantum Limitatins on Superluminal Propagation,” Phys. Rev. Lett. 81, 2190 (1998) http://link.aps.org/abstract/PRL/v81/p2190. [CrossRef]

20.

P. W. Milonni, K. Furuya, and R. Y. Chiao, “Quantum Theory of Superluminal Pulse Propagation,” Opt. Express 8, 59 (2001) http://www.opticsexpress.org/oearchive/source/27132.htm. [CrossRef] [PubMed]

21.

Aephraim M Steinberg, “No thing goes faster than light,” Physics World 13 3 (2000) http://www.physicsweb.org/article/world/13/9/3.

22.

Peter W. Milonni, “Causal Discussion of Superluminal Pulses,” Physics Today 54, 81 (2001) http://www.physicstoday.org/pt/vol-54/iss-2/p14b.html. [CrossRef]

23.

Jacob Broe and Ole Keller, “Superluminality and spatial confinement in optical tunneling,” Opt. Commun. 194, 83 (2001). [CrossRef]

24.

A. Ranfagni, D. Mugnai, P. Fabeni, and G. P. Pazzi, “Delay-time measurements in narrowed waveguides as a test of tunneling,” Appl. Phys. Lett. 58, 774 (1991). [CrossRef]

25.

A. Enders and G. Nimtz, “Photonic-tunneling experiments,” Phys. Rev. B 47, 9605 (1993) http://link.aps.org/abstract/PRB/v47/p9605. [CrossRef]

26.

A. Pablo, L. Barbero, Hugo E. Hernández-Figueroa, and Erasmo Recami, “Propagation Speed of Evanescent Modes,” Phys. Rev. E 62, 8628 (2000) http://link.aps.org/abstract/PRE/v62/p8628. [CrossRef]

27.

Klass Wynne, John J. J. Carey, Justyna Zawadzka, and Dino Jaroszynski, “Tunneling of Single-Cycle Terahertz Pulses through Waveguides,” Opt. Commun. 176, 429 (2000). [CrossRef]

28.

S. Bosanac, “Propagation of Electromagnetic Wave Packets in Nondispersive Dielectric Media,” Phys. Rev. A 28, 577 (1983) http://link.aps.org/abstract/PRA/v28/p577. [CrossRef]

29.

Ph. Balcou and L. Dutriaux, “Dual Optical Tunneling Times in Frustrated Total Internal Reflection,” Phys. Rev. Lett. 78, 851 (1997) http://link.aps.org/abstract/PRL/v78/p851. [CrossRef]

30.

John J. Carey, Justyna Zawadzka, Dino A. Jaroszynski, and Klaas Wynne, “Noncausal Time Response in Frustrated Total Internal Reflection?” Phys. Rev. Lett. 84, 1431 (2000) http://link.aps.org/abstract/PRL/v84/p1431. [CrossRef] [PubMed]

31.

W. Luis Mochán and Vera L. Brudny, “Comment on Noncausal Time Response in Frustrated Total Internal Reflection?” Phys. Rev. Lett. 87, 119101 (2001) http://link.aps.org/abstract/PRL/v87/e119101. [CrossRef]

32.

Vera L. Brudny and W. Luis Mochán, under preparation.

33.

A. Ranfagni, P. Fabeni, G. P. Pazzi, and D. Mugnai “Anomalous pulse delay in microwave propagation: A plausible connection to the tunneling time,” Phys. Rev. E 48, 1453 (1993). http://link.aps.org/abstract/PRE/v48/p1453. [CrossRef]

34.

D. Mugnai, A. Ranfagni, and R. Ruggeri, Phys. Rev. Lett. 84, 4830 (2000) http://link.aps.org/abstract/PRL/v84/p4830. [CrossRef] [PubMed]

OCIS Codes
(020.1670) Atomic and molecular physics : Coherent optical effects
(270.5530) Quantum optics : Pulse propagation and temporal solitons
(350.5500) Other areas of optics : Propagation
(350.5720) Other areas of optics : Relativity

ToC Category:
Research Papers

History
Original Manuscript: October 19, 2001
Published: November 19, 2001

Citation
Vera Brudny and W. Luis Mochan, "On the apparent superluminality of evanescent waves," Opt. Express 9, 561-566 (2001)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-9-11-561


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References

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  17. A. Kuzmich, A. Dogariu, L. J. Wang, P. W. Milonni, and R. Y. Chiao, "Signal Velocity , Causality, and Quantum Noise in Superluminal Light Pulse Propagation," Phys. Rev. Lett. 86, 3925-3929 (2001) http://link.aps.org/abstract/PRL/v86/p3925. [CrossRef] [PubMed]
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  27. Klass Wynne, John J. J. Carey, Justyna Zawadzka, and Dino Jaroszynski, "Tunneling of Single-Cycle Terahertz Pulses through Waveguides," Opt. Commun. 176, 429 (2000). [CrossRef]
  28. S. Bosanac, "Propagation of Electromagnetic Wave Packets in Nondispersive Dielectric Media," Phys. Rev. A 28, 577 (1983) http://link.aps.org/abstract/PRA/v28/p577. [CrossRef]
  29. Ph. Balcou and L. Dutriaux, "Dual Optical Tunneling Times in Frustrated Total Internal Reflection," Phys. Rev. Lett. 78, 851 (1997) http://link.aps.org/abstract/PRL/v78/p851. [CrossRef]
  30. John J. Carey, Justyna Zawadzka, Dino A. Jaroszynski, and Klaas Wynne, "Noncausal Time Response in Frustrated Total Internal Reflection?" Phys. Rev. Lett. 84, 1431 (2000) http://link.aps.org/abstract/PRL/v84/p1431. [CrossRef] [PubMed]
  31. W. Luis Mochan and Vera L. Brudny, "Comment on Noncausal Time Response in Frustrated Total Internal Reflection?" Phys. Rev. Lett. 87, 119101 (2001) http://link.aps.org/abstract/PRL/v87/e119101. [CrossRef]
  32. Vera L. Brudny and W. Luis Mochan, under preparation.
  33. A. Ranfagni, P. Fabeni, G. P. Pazzi, and D. Mugnai "Anomalous pulse delay in microwave propagation: A plausible connection to the tunneling time," Phys. Rev. E 48, 1453 (1993). http://link.aps.org/abstract/PRE/v48/p1453. [CrossRef]
  34. D. Mugnai, A. Ranfagni, and R. Ruggeri, Phys. Rev. Lett. 84, 4830 (2000) http://link.aps.org/abstract/PRL/v84/p4830. [CrossRef] [PubMed]

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