Measurement of superluminal optical tunneling times in doublebarrier photonic band gaps.
ABSTRACT Tunneling of optical pulses at 1.5 microm wavelength through doublebarrier periodic fiber Bragg gratings is experimentally investigated in this paper. Tunneling time measurements as a function of the barrier distance show that, far from resonances of the structure, the transit time is paradoxically shortimplying superluminal propagationand almost independent of the barrier distance. This result is in agreement with theoretical predictions based on phasetime analysis and provides, in the optical context, an experimental evidence of the analogous phenomenon in quantum mechanics of nonresonant superluminal tunneling of particles across two successive potential barriers.

Article: Investigation on the superluminality of evanescent modes via quantum Lorentz transformation
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ABSTRACT: Applying the fact that guided photons inside a waveguide can be treated as massive particles, one can study the superluminality of evanescent modes via showing that a massive particle can propagate over a spacelike interval, which corresponds to quantum tunneling effects. For this purpose, we treat the particle as a quantum reference frame, while attach an inertia observer to a classical reference frame, and then quantize the formulae for the Lorentz transformation between the quantum and classical reference frames, from which we obtain the conclusion that, owing to the Heisenberg's uncertainty relation, the particle can propagate over a spacelike interval.09/2007;  SourceAvailable from: Guillaume Petite[Show abstract] [Hide abstract]
ABSTRACT: We analyze the optical (or microwave) tunneling properties of electromagnetic waves passing through thin films presenting a specific index profile that provides a cutoff frequency when the films are used below this frequency. We show that, contrary to the usual case of a square index profile, where tunneling is accompanied by a strong attenuation of the wave due to reflection, such films present the possibility of reflectionless tunneling, where the incoming intensity is totally transmitted.Optics Letters 05/2006; 31(8):11279. · 3.39 Impact Factor  SourceAvailable from: Erasmo Recami[Show abstract] [Hide abstract]
ABSTRACT: In recent times it has been paid attention to the fact that (linear) wave equations admit of "solitonlike" solutions, known as Localized Waves or Nondiffracting Waves, which propagate without distortion in one direction. Such Localized Solutions (existing also for KG and Dirac equations) are a priori suitable, more than Gaussian's, for describing elementary particle motion. In this paper we show that, mutatis mutandis, Localized Solutions exist even for the ordinary Schroedinger equation, within standard Quantum Mechanics; and we obtain both approximate and exact solutions, setting forth particular examples for them. In the ideal case such solutions bear infinite energy, as well as plane or spherical waves: we also demonstrate, therefore, how to obtain finiteenergy solutions. At last, we briefly consider solutions for a particle moving in the presence of a potential. Some physical comments are added.Journal of Mathematical Physics 01/2012; 53:052102 [19 pages]. · 1.30 Impact Factor
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arXiv:physics/0201013v1 [physics.optics] 9 Jan 2002
Measurement of Superluminal optical tunneling times in
doublebarrier photonic bandgaps
S. Longhi and P. Laporta
Istituto Nazionale di Fisica per la Materia, Dipartimento di Fisica, Politecnico di Milano, Piazza
L. da Vinci 32, I20133 Milan, Italy
M. Belmonte
CorningOptical Technologies Italia S.p.A., V.le Sarca 222, 20126 Milan, Italy
E. Recami
Facolt` a di Ingegneria, Universit` a statale di Bergamo, Dalmine (BG), Italy;
INFN–Sezione di Milano, Milan, Italy; and
C.C.S., UNICAMP, Campinas, S.P., Brasil.
Abstract
Tunneling of optical pulses at 1.5 µm wavelength through doublebarrier
periodic fiber Bragg gratings is experimentally investigated. Tunneling time
measurements as a function of barrier distance show that, far from the res
onances of the structure, the transit time is paradoxically short, implying
Superluminal propagation, and almost independent of the distance between
the barriers. These results are in agreement with theoretical predictions based
on phase time analysis and also provide an experimental evidence, in the op
tical context, of the analogous phenomenon expected in Quantum Mechanics
for nonresonant Superluminal tunneling of particles across two successive po
tential barriers. [Attention is called, in particular, to our last Figure].
PACS: 42.50.Wm, 03.65.Xp, 42.70.Qs, 03.50.De, 03.65.w, 73.40.Gk
Typeset using REVTEX
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I. INTRODUCTION
Tunneling of a particle through a potential barrier is one of the most intriguing phenomena
in quantum mechanics, that continues to attract a great attention both theoretically and experi
mentally. The reason thereof stems from the fact that some questions related to the dynamics of
tunneling, such as the definition and measure of tunneling transit times [1–3], have not get a gen
eral acceptance yet. In addition, tunneling hides some amazing timedomain phenomena, the most
notable one being Superluminal propagation (for related experiments see, e.g., [4,5] and references
therein). In case of opaque barriers, it is known that the tunneling time becomes independent of
the barrier width (Hartman effect [6]; see also [2]) and can become so short to imply apparent
Superluminal motion, which has been the subject of a lively debate in recent years [4,5,7,8]. Since
tunneling time measurements for electrons are usually difficult to achieve and of uncertain inter
pretation, experimental validation of the Hartman effect and direct measurements of Superluminal
tunneling times have been successfully obtained for the closelyrelated problem of tunneling of pho
tons through photonic barriers in a series of famous experiments performed at either microwave
[9,10] and optical wavelengths [11–13]. In particular, in Refs. [11,12] onedimensional photonic
bandgaps were used as photonic barriers, realizing the optical analogue of electron Bragg scat
tering in the Kr¨ onigPenney model of solidstate physics. A different but related issue is that of
particle tunneling through a doublebarrier (DB) potential structure, such as electron tunneling
in semiconductor superlattice structures. In this case, the resonant behavior of tunneling escape
versus energy [14] is a clear manifestation of the wave nature of electrons and is of major impor
tance for ultrahighspeed resonanttunneling devices [15]. Exploiting the analogy between electron
and photon tunneling [16,17], resonant tunneling phenomena have been also studied and observed
in connection with microwave propagation in undersized waveguides and in periodic layered struc
tures [8,9,18], and general relations have been derived between the traversal time at resonances and
the lifetime of the resonant states [18]. Besides of realizing resonance tunneling, it has been theo
retically recognized that DB structures are also of interest to study offresonance tunneling times
[8,9,18,19]. In this case, for opaque barriers it turns out that the transit time to traverse the DB
structure is independent not only of the barrier width, but even of the length of the intermediate
(classically allowed) region that separates the barriers. So far, measurements of Superluminal tun
neling in DB structures have been performed by G. Nimtz and coworkers in a series of microwave
transmission experiments [9]; however, no experimental study on offresonant tunneling times in
DB photonic structures at optical wavelengths has been reported yet. In recent works [20,21], some
of the present authors have shown that fiber Bragg gratings (FBGs) can provide versatile tools for
the study of tunneling phenomena. Besides for their potential relevance in applications to optical
communications, the use of FBGs as photonic barriers is very appealing from an experimental
viewpoint because the tunneling times in FBG structures fall in the tens of picoseconds time scale,
which can be easily and precisely detected by standard optoelectronic means.
In this work we report on measurement of tunneling delay times in DB photonic structures at the
1.5 µm wavelength of optical communications. Our results represent an extension at optical wave
lenghts of similar experimental achievements previously reported at microwaves [9] and provide a
clear experimental evidence that, for opaque barriers, the traversal time is independent of barrier
distance (generalized Hartman effect). The paper is organized as follows. In Sec.II, the basic model
of tunneling in a DB rectangular FBG is reviewed and the quantummechanical analogy of electron
tunneling is outlined. In Sec.III the experimental measurements of tunneling times are presented.
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Finally, in Sec.IV the main conclusions are outlined.
II. OPTICAL TUNNELING IN A DB FBG: BASIC EQUATIONS AND
QUANTUMMECHANICAL ANALOGY
We consider tunneling of optical pulses through a DB photonic structure achieved in a
monomode optical fiber by writing onto it two periodic Bragg gratings, each of width L0, sep
arated by a distance L, which realize a weak modulation of the refractive index n along the fiber
axis z according to n(z) = n0[1 + 2V (z)cos(2πz/Λ)], where n0is the average refractive index of
the structure, Λ is the Bragg modulation period, and V (z) is profiled to simulate a symmetric
rectangular DB structure, i.e. V (z) = V0constant for 0 < z < L0and for L + L0< z < L + 2L0,
and V (z) = 0 otherwise (see Fig.1). For such a structure, Bragg scattering of counterpropagating
waves at a frequency ω close to the Bragg resonance ωB≡ c0π/(n0Λ) (c0is the speed of light in
vacuum) occurs in the grating regions, whereas multiple wave interference between the two barriers
leads to FabryPerot resonances in the transmission spectrum. The tunneling problem in the DB
FBG structure bears a close connection to that of nonrelativistic electrons through a symmet
ric rectangular DB potential, which has been widely investigated in literature (see, for instance,
[14,22]). The analogy is summarized in Table I, where the basic equations and the expressions
for barrier transmission and group delay are given in the two cases [17,23]. In the electromag
netic case, a monochromatic field E(z,t) at an optical frequency ω close to the Bragg frequency
ωBpropagating inside the fiber can be written as a superposition of counterpropagating waves,
E(z,t) = u(z)exp(−iωt + ikBz) + v(z)exp(−iωt − ikBz) + c.c., where kB = π/Λ is the Bragg
wavenumber; for a small index modulation (V0 ≪ 1), the envelopes u,v of counterpropagating
waves satisfy the following coupledmode equations [24]:
du
dz= iδu + ikBV (z)u
dv
dz= −iδv − ikBV (z)v
(1a)
(1b)
where δ ≡ k − kB= n0(ω − ωB)/c0is the detuning parameter between wavenumber k = n0ω/c0
of counterpropagating waves and Bragg wavenumber kB. The wave envelopes u and v are os
cillatory (propagative) in the region L0 < z < L0+ L, whereas they are exponential (evanes
cent) inside the gratings when δ < kBV0. The spectral transmission of the structure, given by
t(ω) = [u(L)/u(0)]v(L)=0, can be analytically determined by standard transfer matrix methods
[24]. As an estimate of the tunneling time for a wavepacket crossing the structure, we use the
group delay (or phase time) as calculated by the method of stationary phase, which is given by
[25] τ = Im{∂ln(t)/∂ω}. A typical behavior of power transmission T = t2and group delay τ
versus frequency detuning ν = (ω − ωB)/(2π), computed for one of the DB structures used in the
experiments, is shown in Fig.2. Notice that, far from the sharp FabryPerot resonances, the group
delay is shorter than that for free propagation from input to output planes, implying Superluminal
propagation. At the center of the bandgap (δ = 0), simple analytical expressions for the power
transmission and group delay can be derived and read:
T =
1
cosh2(2kBV0L0)
(2)
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τ = τ1+ τ2
(3)
where:
τ1=
n0
c0kBV0tanh(2kBV0L0) =
τ2=n0L
c0
√1 − T
√Tn0L
c0
n0
c0kBV0
(4a)
1
cosh(2kBV0L0)=
(4b)
Equations (3) and (4) clearly show that two distinct contributions are involved in the expression
for the group delay. The former term, τ1, is independent of the barrier separation, and coincides
with the tunneling time of a single barrier of width 2L0. For an opaque barrier (L0V0kB ≫ 1),
τ1becomes independent of barrier width and saturates to the value τ1∼ n0/(c0kBV0) (Hartman
effect). Conversely, the latter contribution, τ2, is always shorter than the freepropagation time
over a length L and goes to zero for an opaque barrier, implying that the tunneling time be
comes independent of barrier distance (generalized Hartman effect). Similar results are obtained
for offresonance tunneling of a nonrelativistic electron through a rectangular DB potential V (z)
assuming that the incident wavepacket has a belowbarrier mean energy E half the barrier width
V0; the corresponding expressions for barrier transmission and group delay in this case are given
in Table I (for details, see, e.g., [22]). The tunneling through a DB FBG structure can hence be
used as an experimental verifiable model for the quantummechanical case.
III. TUNNELING TIME MEASUREMENTS
We performed a series of tunneling time measurements through DB FBG structures operating
at around 1.5 µm in order to assess the independence of the peak pulse transit times with barrier
distance L. The FBGs used for the experiments were manufactured by using standard writing
techniques, with an exposure time to UV laser beam and phase mask length such as to realize a
grating with sharp falloff edges of length L0≃ 8.5 mm and with a refractive index modulation
V0≃ 0.9×10−4. For such a refractive index modulation, a minimum power transmission T ≃ 0.8%
at antiresonance is achieved for a DB structure, which is low enough to get the opaque barrier
limit but yet large enough to perform time delay measurements at reasonable power levels. The
period of the phase mask was chosen to achieve Bragg resonance at around 1550 nm wavelength.
Five different DB structures were realized with grating separation L of 18, 27, 35, 42 and 47 mm.
For such structures, both transmission spectra and group delays were measured using a phase shift
technique [26] with a spectral resolution of ≃ 2 pm; an example of measured transmission spectrum
and group delay versus frequency for the 42mm separation DB FBG is shown in Fig.2. Notice
that, according to the theoretical curve shown in the same figure, far from the FabryPerot reso
nances the group delay is Superluminal, with expected time advancement of the order of 240–250
ps. Notice also that the sharp FabryPerot resonances are not fully resolved in the experimental
curves due to bandwidth limitations (∼ 2 pm) of the measurement apparatus.
Direct timedomain measurements of tunneling delay times were performed in transmission ex
periments using probing optical pulse with ≃1.3 ns duration, corresponding to a spectral pulse
bandwidth which is less than the frequency separation of FabryPerot resonances for all the five DB
structures. Since both transmission and group delay are slowly varying functions of frequency far
from FabryPerot resonances (see Fig.2), weak pulse distortions are thus expected for offresonance
4
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pulse transmission. The experimental setup for delay time measurements is shown in Fig.3. A
pulse train, at a repetition frequency fm= 300 MHz, was generated by external modulation of a
singlefrequency continuouswave tunable laser diode (Santec mod. ECL200/210), equipped with
both a coarse and a fine (thermal) tuning control of frequency emission with a resolution of ∼ 100
MHz. The fibercoupled ∼ 10 mW output power emitted by the laser diode was amplified using a
highpower erbiumdoped fiber amplifier (IPG Mod. EAD2PM; EDFA1 in Fig.3), and then sent
to a LiNbO3based MachZehnder modulator, sinusoidally driven at a frequency fm= 300 MHz by
a lownoise radiofrequency (RF) synthetizer. The bias point of the modulator and the RF modu
lation power level were chosen to generate a train with a pulse duration (FWHM) of ≃ 1.3 ns; the
measured average output power of the pulse train available for the transmission experiments was
∼ 130 mW. The pulse train was sent to the DB FBG through a threeport optical circulator, that
enables both transmitted and reflected signals to be simultaneously detected. The signal transmit
ted through the DB FBG was sent to a lownoise erbiumdoped fiber amplifier (OptoCom Mod.
OI LNPA; EDFA2 in Fig.3) with a low saturation power (≃ 30 µW at 1550 nm) that maintains
the average power level of the output optical signal at a constant level (≃ 18 mW). In this way, the
power levels transmitted through the DB FBG, for the laser emission tuned either at FabryPerot
resonances or antiresonances of the structure or outside the stopband, were comparable. The
transmitted pulse train was detected in the time domain by a fast sampling oscilloscope (Agilent
Mod. 86100A), with a low jitter noise and an impulsive response of ≃ 15 ps; a portion of the sinu
soidal RF signal that drives the MachZehnder modulator was used as an external trigger for the
oscilloscope, thus providing precise synchronism among successive pulses. Offresonance tunneling
was achieved by a careful tuning control of the laser spectrum which was detected by monitoring
the reflected signal, available at port three of the optical circulator, using both an optical spec
trum analyzer (Anritsu Mod. MS9710B) with a resolution of 0.07 nm, and a planeplane scanning
FabryPerot interferometer (Burleigh Model RC1101R) with a freespectral range of ≃50 GHz and
a measured finesse of ∼ 180, which permits to resolve the FabryPerot resonances of the DB FBG
structures. The reflectivity spectrum of the DB FBG was first measured by the FabryPerot inter
ferometer and recorded on a digital oscilloscope by disconnecting the laser diode from the input
port of EDFA1 and sending to the DB FBG the broadband amplifiedspontaneous emission signal
of the optical amplifier. This trace is then used as a reference to tune the pulse spectrum at the
center of the offresonance plateau between the two central FabryPerot resonances of the recorded
DB FBG structure.
Figure 4 shows a typical trace [curve (1)], averaged over 64 acquisitions, of the tunneled optical
pulses under offresonance tuning condition, as measured on the sampling oscilloscope, for the 42
mm separation DB FBG structure, and compared to the corresponding trace [curve (2)] recorded
when the laser was detuned apart by ∼ 200 GHz, i.e. far away from the stopband of the DB FBG
structure. A comparison of the two traces clearly shows that tunneled pulses are almost undistorted
with a peak pulse advancement of ≃ 248 ps; repeated measurements showed that the measured
pulse peak advancement is accurate within ≃ ± 15 ps, the main uncertainty in the measure being
determined by the achievement of the optimal tuning condition. We checked that propagation
through EDFA2 does not introduce any appreciable pulse distortion nor any measurable time de
lay dependence on the amplification level. Time delay measurements were repeated for the five
DB FBG structures, and the experimental results are summarized in Fig.5 and compared with the
theoretical predictions of tunneling time as given by Eqs.(3) and (4). The dashed line in the figure
shows the theoretical transit time, from input (z = 0) to output (z = 2L0+L) planes, versus barrier
5
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separation L for pulses tuned far away from the bandgap of FBG; in this case, the transit time is
given merely by the time spent by a pulse traveling along the fiber for a distance L + 2L0with a
velocity c0/n0. The solid line is, in turn, the expected transit time for offresonance tunneling of
pulses, according to the phase time analysis [see Eqs.(3) and (4)]], which shows that the transit time
does not substantially increase as the barrier separation is increased (generalized Hartman effect).
The points in the figure are obtained by subtracting to the dashed curve the measured pulse peak
advancements for the five DB FBGs, thus providing an experimental estimate of the tunneling
transit time. Notice that, within the experimental errors, the agreement between measured and
predicted transit times is rather satisfactory. For each of the five DB FBG structures, the measured
transit times are Superluminal; it is remarkable that, for the longest barrier separation used, the
transit time leads to a Superluminal velocity of about 5c0, the largest one measured in tunneling
experiments at optical wavelengths [27]. These paradoxically small transit times do not represent,
however, a genuine violation of Einstein causality [4,11,12], and may be qualitatively explained
as the result of two simultaneous effects that are a signature of the wave nature of the tunneling
processes [28]. On the one hand, following Refs. [2,11,12], peak pulse advancement occurs at each
of the two barriers as a result of a reshaping phenomenon in which the trailing edge of the pulse is
preferentially attenuated than the leading one; on the other hand, the independence of tunneling
time on the barrier distance can be explained, following Ref. [19], as an effective “acceleration” of
the forward traveling waves in the intermediate classicallyallowed region that arises in consequence
of the destructive interference between the two barriers.
IV. CONCLUSIONS
In this paper offresonant tunneling of optical pulses has been experimentally investigated in
fiber Bragg photonic barriers. Tunneling time measurements have been shown to be in good agree
ment with theoretical predictions based on phase time analysis and have unambiguously confirmed
that, for opaque barriers, the tunneling time is independent not only of the barrier width, as pre
viously shown in Ref. [12], but even of the barrier separation. Our results extend to the optical
region previous experimental achievements performed at microwaves [9] and may be of interest in
the field of optical tunneling and related issue of Superluminal propagation.
ACKNOWLEDGMENTS
The authors acknowledge M. Marano for his help in the experimental measurements, and F.
Fontana and G. Salesi for many useful discussions.
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REFERENCES
[1] E.H. Hauge and J.A. Støvneng, Rev. Mod. Phys. 61, 917 (1989).
[2] V.S. Olkhovsky and E. Recami, Phys. Rep. 214, 339 (1992); V.S.Olkhovsky, E.Recami,
F.Raciti and A.K.Zaichenko, J. de Physique–I (France) 5, 1351 (1995); V.S.Olkhovsky,
E.Recami and J.Jakiel, “Unified time analysis of photon and nonrelativistic particle tun
nelling”, Lanl Archives # quantph/0102007.
[3] R. Landauer and Th. Martin, Rev. Mod. Phys. 66, 217 (1994).
[4] R.Y. Chiao and A.M. Steinberg, Prog. Opt. 37, 345 (1997).
[5] G. Nimtz and G.W. Heitmann, Progress in Quant. Electron. 21, 81 (1997).
[6] T.E. Hartman, J. Appl. Phys. 33, 3427 (1962).
[7] E. Recami, F. Fontana and R. Garavaglia, Int. J. Mod. Phys. A 15, 2793 (2000), and refs.
therein. For an extended review about Superluminal motions within special relativity, see:
E.Recami, Rivista N. Cim. 9 (6), pp.1–178 (1986); Found. Phys. 31, 1119 (2001).
[8] A.P. Barbero, H.E. HernandezFigueroa and E. Recami, Phys. Rev. E 62, 8628 (2000).
[9] A. Enders and G. Nimtz, Phys. Rev. B 47, 9605 (1993), and refs. therein; G. Nimtz, A.
Enders and H. Spieker, J. Phys.–I (France) 4, 565 (1994); H.M.Brodowsky, W.Heitmann and
G.Nimtz, Phys. Lett. A 222, 125 (1996). For an earlier experiment on microwave tunneling
see also: G. Nimtz and A. Enders, J. Phys. I (France) 2, 1693 (1992).
[10] A.Ranfagni, P.Fabeni, G.P.Pazzi and D.Mugnai, Phys. Rev. E 48, 1453 (1993); D. Mugnai,
A. Ranfagni and L. Ronchi, Phys. Lett. A 247, 281 (1998).
[11] A.M. Steinberg, P.G. Kwiat and R.Y. Chiao, Phys. Rev. Lett. 71, 708 (1993).
[12] Ch. Spielmann, R. Szip¨ ocs, A. Stingl and F. Krausz, Phys. Rev. Lett. 73, 2308 (1994).
[13] P. Balcou and L. Dutriaux, Phys. Rev. Lett. 78, 851 (1997).
[14] See, for example, E. Merzbacher, Quantum Mechanics (Wiley, New York, 1970), Chaps. 6
and 7.
[15] F. Capasso, K. Mohammed and A.Y. Cho, IEEE J. Quant. Electron. QE22, 1853 (1986).
[16] R.Y. Chiao, P.G. Kwiat and A.M. Steinberg, Physica B 175, 257 (1991); Th. Martin and R.
Landauer, Phys. Rev. A 45, 2611 (1992); J. Jakiel, V.S. Olkhovsky and E. Recami, Phys.
Lett. A 248, 156 (1998).
[17] For early works on the analogy between electron and photon tunneling, see: T. Tsai and G.
Thomas, Am. J. Phys. 44, 636 (1976); J.J. Hupert, Am. J. Phys. 44, 636 (1977) and references
therein.
[18] E. Cuevas, V. Gasparian, M. Ortu˜ no and J. Ruiz, Zeit. Phys. B 100, 595 (1996).
[19] V.S. Olkhovsky, E. Recami and G. Salesi, “Tunneling through two successive barriers and the
Hartman (Superluminal) effect”, Lanl Archives # quantph/0002022.
[20] S. Longhi, Phys. Rev. E 64, 037601 (2001).
[21] S. Longhi, M. Marano, P. Laporta, and M. Belmonte, Phys. Rev. E 64, 055602(R) (2001).
[22] J. Headings, J. Atm. Terr. Phys. 25, 519 (1963); P. Thanikasalam, R. Venkatasubramanian,
and M. Cahay, IEEE J. Quant. Electron. 29, 2451 (1993); H. Yammamoto, K. Miyamoto, and
T. Hayashi, Phys. Stat. Sol. B 209, 305 (1998).
[23] The analogy between tunneling of electrons and photons in superlattice structures and closed
form solutions for the tunneling times have been recently reported, in the general case, by
P. Pereyra [P. Pereyra, Phys. Rev. Lett. 84, 1772 (2000)]. The derivation of photon tunnel
ing times presented here follows a different and simpler approach, based on coupledmode
equations for Bragg scattering, which is suited for FBG structures.
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[24] See, for instance, T. Ergodan, J. Lightwave Technol. 15, 1277 (1997).
[25] Among others, the phase time has been shown to be consistent with tunneling time measure
ments in previous experiments on optical tunneling through photonic bandgaps [11,12].
[26] S. Ryu, Y. Horiuchi and K. Mochizuky, J. Lightwave Technol. 7, 1177 (1989).
[27] We mention that Superluminal group velocities as large as ∼ 10c0were observed by G. Nimtz
and coworkers, but in microwave transmission experiments.
[28] We remark that the front velocity of a discontinuous stepwise signal that propagates through
the DB FBG structure is always equal to c0, so there is no violation of Einstein’s causality in
our experimental findings; this circumstance follows technically from the analytic properties
of spectral transmission t(ω) and has been discussed in Ref. [21]; for general discussions on
this point see also: G. Nimtz, Eur. Phys. J. B 7, 523 (1999); M. Mojahedi, E. Schamiloglu, F.
Hegeler, and K.J. Malloy, Phys. Rev. E 62, 5758 (2000) and references therein.
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TABLES
TABLE I. Analogies between tunneling of optical waves and electrons in a symmetric rectan
gular DB potential.
Photons
Equations
du/dz =
dv/dz = −iδv − ikBV (z)u
DB Transmissiona(offresonance)
T = t2= 1/cosh2(2kBV0L0)
Phase timea(offresonance)
τ = Im
?∂ln(t)
∂ω
= τ1+ τ2
τ1= [n0/(c0kBV0)]tanh(2kBV0L0)
τ2= (n0L/c0)/cosh(2kBV0L0)
Electrons
iδu + ikBV (z)v
d2ψ
dz2+2m
¯ h2[E − V (z)]ψ = 0
T = t2= 1/cosh2(2χL0)
?
τ = ¯ h Im
?∂ln(t)
∂E
?
= τ1+ τ2
τ1= [2/(χvg)]tanh(2χL0)
τ2= (L/vg)/cosh(2χL0)
aFor electrons, calculations are made assuming a meanenergy of incident wavepacket equal to half of the barrier height, i.e.
E = V0/2, and assuming offresonance tunneling, i.e. χL is an integer multiple of π, where χ ≡√mV0/¯ h is the wavenumber
of oscillatory wavefunction between the two barriers. vg ≡ ¯ hχ/m is the groupvelocity of free wavepacket.
9
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Figure Captions.
Fig.1. Schematic of tunneling through a rectangular DB photonic structure.
Fig.2. Spectral power transmission (left) and group delay (right) for a DB FBG structure for
L0= 8.5 mm, L = 42 mm, V0= 0.9 × 10−4, n0= 1.452, and ωB= 1.261 × 1015rad/s. Upper and
lower figures refer to measured and predicted spectral curves, respectively.
Fig.3.
waveguide modulator; OC: optical circulator; EDFA1 and EDFA2: erbiumdoped fiber amplifiers;
RF: radiofrequency synthetizer.
Schematic of the experimental setup. LD: tunable laser diode; MZM: MachZehnder
Fig.4. Pulse traces recorded on the sampling oscilloscope corresponding to the transmitted pulse
for offresonance tunneling (curve 1) and reference pulse propagating outside the stopband of the
structure (curve 2) for the 42mm separation DB FBG.
Fig.5.
DB FBG structure. The solid line is the theoretical prediction based on group delay calculations
(Table I); dots are the experimental points as obtained by time delay measurements; the dashed
curve is the transit time from input (z = 0) to output (z = L + 2L0) planes for a pulse tuned far
away from the stopband of the FBGs.
Offresonance tunneling time versus barrier separation L for a rectangular symmetric
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S. Longhi et al., "Measurement of superluminal ..."
Fig.1
z
z
=
=2
L+ L0
L0
L
L
0
Incident
Pulse
Transmitted
Pulse
( )
z
( )
zV
()
[
1
]
Λ+=
znn
π
2 cos2
0
z
Page 12
Group Delay [ps]
Power Transmission
Frequency [GHz]
S. Longhi et al., "Measurement of superluminal ..."
Fig.2
250 25
0.0
0.5
1.0
250 25
0
600
1200
1800
25025
0.0
0.5
1.0
25025
0
1000
2000
3000
Page 13
~
DB FBG
MZM
RF
EDFA2
EDFA1
LD
to Sampling
Oscilloscope
to Spectrum
Analyzer
60/40
to FabryPerot
Interferometer
S. Longhi et al., "Measurement of superluminal ..."
Fig.3
OC
1
2
3
Page 14
0.51.52.53.5
0.0
0.5
1.0
Time [ns]
Normalized Intensity
(1)
248 ps
(2)
S. Longhi et al., “Measurement of superluminal…”
Fig.4
Page 15
1020304050
Barrier separation [mm]
0
100
200
300
Tunneling Time [ps]
S. Longhi et al., "Measurement of superluminal..."
Fig. 5
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 Available from Erasmo Recami · May 21, 2014
 Available from Erasmo Recami · May 21, 2014
 Available from ArXiv