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.
© 2001 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  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  of very opaque barriers, in analogy to earlier theoretical predictions for electronic quantum tunneling . 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, 5, 6].
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, 8]. The onset of a pulse can never propagate at speeds greater than c, as shown by Brillouin and Sommerfeld , 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 . 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 . A careful definition of the energy velocity has lead to the conclusion that it is smaller than c even in superluminal situations [10, 11]. 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  and active media with gain . The latter has been experimentally realized [14, 15] 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 . A quantum mechanical discussion of the signal detection  shows that even in this case the signal velocity does not exceed c. Quantum aspects of superluminal propagation  have also been investigated for model materials [19, 20]. However, deeper understanding of the field of superluminal propagation is still required [21, 22].
Curiously, superluminal propagation can also occur within vacuum , 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, 25], 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  and pulse velocities are subluminal . 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  in the frustrated-total-internal-reflection (FTIR)regime and has been verified experimentally . More recently, Carey et al  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 .
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 .
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.
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  but in contrast to a recent theoretical result that only p-polarized waves propagate superluminally in FTIR . The mechanism for superluminality in vacuum described here is unrelated to the photon source delocalization invoked in Ref. .
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 . 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  the propagator
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,
where is the time of first arrival at (x, z)of a signal excited at the edges (xs , 0), xs =, 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
that would have been excited in the absence of screens, with a correction C(x, z, t; xs ) due to diffraction  by the screens. Here, 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  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  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, 34].
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.
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