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Measurement of superluminal optical tunneling times in double-barrier photonic band gaps
S. Longhi and P. Laporta
Istituto Nazionale per la Fisica della Materia, Dipartimento di Fisica, Politecnico di Milano, Piazza Leonardo da Vinci 32, I-20133
Milan, Italy
M. Belmonte
Corning-Optical Technologies Italia S.p.A., Viale Sarca 222, 20126 Milan, Italy
E. Recami
Facolta ` di Ingegneria, Universita ` Statale di Bergamo, Dalmine, Bergamo, Italy;
INFN–Sezione di Milano, Milan, Italy;
and CCS, UNICAMP, Campinas, Sa ˜o Paulo, Brazil
?Received 23 October 2001; published 2 April 2002?
Tunneling of optical pulses at 1.5 ?m wavelength through double-barrier 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 short—implying superluminal
propagation—and almost independent of the barrier distance. This result is in agreement with theoretical
predictions based on phase-time 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.
DOI: 10.1103/PhysRevE.65.046610PACS number?s?: 42.50.?p, 03.65.Xp, 42.70.Qs
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
experimentally. 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 got a general acceptance yet. In addition, tunneling
hides some amazing time-domain phenomena, the most no-
table one being superluminal propagation ?for related experi-
ments 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
interpretation, experimental validation of the Hartman effect
and direct measurements of superluminal tunneling times
have been successfully obtained for the closely related prob-
lem of tunneling of photons through photonic barriers in a
series of famous experiments performed at either microwave
?9,10? or optical wavelengths ?11–13?. In particular, in Refs.
?11,12? one-dimensional photonic band gaps were used as
photonic barriers, realizing the optical analog of electron
Bragg scattering in the Kro ¨nig-Penney model of solid-state
physics. A different, but related, issue is that of particle tun-
neling through a double-barrier ?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 importance for
ultrahigh-speed resonant-tunneling 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 un-
dersized waveguides and in periodic layered structures
?8,9,18?, and general relations have been derived between the
traversal time at resonances and the lifetime of the resonant
states ?18?. Besides realizing resonance tunneling, it has been
theoretically recognized that DB structures are also of inter-
est to study off-resonance 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 bar-
rier width, but even of the length of the intermediate ?classi-
cally allowed? region that separates the barriers. So far, mea-
surements of superluminal tunneling in DB structures have
been performed by Nimtz and coworkers in a series of mi-
crowave transmission experiments ?9?; however, no experi-
mental study on off-resonant 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 their
potential relevance in applications to optical communica-
tions, the use of FBGs as photonic barriers is very appealing
from an experimental point of view 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 the measurement of tunneling
delay times in DB photonic structures at the 1.5 ?m wave-
length of optical communications. Our results represent an
extension at optical wavelengths of similar experimental
achievements previously reported with microwaves ?9? and
provide a clear experimental evidence that, for opaque bar-
riers, the traversal time is independent of barrier distance
PHYSICAL REVIEW E, VOLUME 65, 046610
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?generalized Hartman effect?. The paper is organized as fol-
lows. In Sec. II, the basic model of tunneling in a DB rect-
angular FBG is reviewed and the quantum-mechanical anal-
ogy of electron tunneling is outlined. In Sec. III the
experimental measurements of tunneling times are presented.
Finally, in Sec. IV the main conclusions are outlined.
II. OPTICAL TUNNELING IN A DB FBG: BASIC
EQUATIONS AND QUANTUM-MECHANICAL 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, separated by a distance L, which realize a weak modula-
tion of the refractive index n along the fiber axis z according
to n(z)?n0?1?2V(z)cos(2?z/?)?, where n0is the average
refractive index of the structure, ? is the Bragg modulation
period, and V(z) is profiled to simulate a symmetric rectan-
gular 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 ? close to the Bragg resonance ?B
?c0?/(n0?) (c0is the speed of light in vacuum? occurs in
the grating regions, whereas multiple wave interference be-
tween the two barriers leads to Fabry-Perot resonances in the
transmission spectrum. The tunneling problem in the DB
FBG structure bears a close connection to that of nonrelativ-
istic electrons through a symmetric 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 electromagnetic case, a monochromatic field
E(z,t) at an optical frequency ? close to the Bragg fre-
quency ?Bpropagating inside the fiber can be written as a
superposition ofcounterpropagating
?u(z)exp(?i?t?ikBz)?v(z)exp(?i?t?ikBz)?c.c.,
kB??/? is the Bragg wave number; for a small index
modulation (V0?1), the envelopes u,v of counterpropagat-
ing waves satisfy the following coupled-mode equations
?24?:
waves,
E(z,t)
where
du
dz?i?u?ikBV?z?u,
?1a?
dv
dz??i?v?ikBV?z?v,
?1b?
where ??k?kB?n0(???B)/c0is the detuning parameter
between wave number k?n0?/c0 of counterpropagating
waves and Bragg wave number kB. The wave envelopes u
and v are oscillatory ?propagative? in the region L0?z?L0
?L, whereas they are exponential ?evanescent? inside the
gratings when ????kBV0. The spectral transmission of the
structure, given by t(?)??u(L)/u(0)?v(L)?0, can be ana-
lytically determined by standard transfer matrix methods
?24?. As an estimate of the tunneling time for a wave packet
crossing the structure, we use the group delay ?or phase time?
as calculated by the method of stationary phase, which is
given by ?25? ??Im?? ln(t)/???. A typical behavior of
power transmission T??t?2and group delay ? versus fre-
TABLE I. Analogies between tunneling of optical waves and
electrons in a symmetric rectangular DB potential.
PhotonsElectrons
Equations
du/dz?i?u?ikBV(z)v
d2?
dz2?2m
?2?E?V(z)???0
dv/dz??i?v?ikBV(z)u
DB Transmissiona?off-resonance?
T??t?2?1/cosh2(2kBV0L0)
Phase timea?off-resonance?
??Im?
?1??n0/(c0kBV0)?tanh(2kBV0L0)
?2?(n0L/c0)/cosh(2kBV0L0)
T??t?2?1/cosh2(2?L0)
? ln?t?
?????1??2
???Im?
?1??2/(?vg)?tanh(2?L0)
?2?(L/vg)/cosh(2?L0)
? ln?t?
?E???1??2
aFor electrons, calculations are made assuming a mean energy of
incident wave packet equal to half of the barrier height, i.e., E
?V0/2, and assuming off-resonance tunneling, i.e., ?L is an integer
multiple of ?/2, where ???mV0/? is the wave number of oscilla-
tory wave function between the two barriers. vg???/m is the
group velocity of free wave packet.
FIG. 1. Schematic of tunneling through a rectangular DB pho-
tonic 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 ?B?1.261?1015rad/s. Upper and
lower figures refer to measured and predicted spectral curves, re-
spectively.
S. LONGHI, P. LAPORTA, M. BELMONTE, AND E. RECAMIPHYSICAL REVIEW E 65 046610
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quency detuning ??(???B)/(2?), computed for one of
the DB structures used in the experiments, is shown in Fig.
2. Notice that, far from the sharp Fabry-Perot resonances, the
group delay is shorter than that for free propagation from
input to output planes, implying superluminal propagation.
At the center of the band gap (??0), simple analytical ex-
pressions for the power transmission and group delay can be
derived and read
T?
1
cosh2?2kBV0L0?,
?2?
???1??2,
?3?
where
?1?
n0
c0kBV0tanh?2kBV0L0???1?T
n0
c0kBV0
?4a?
?2?n0L
c0
1
cosh?2kBV0L0???T
n0L
c0
.
?4b?
Equations ?3? and ?4? clearly show that two distinct contri-
butions are involved in the expression for the group delay.
The former term ?1is independent of the barrier separation,
and coincides with the tunneling time of a single barrier of
width 2L0. For an opaque barrier (L0V0kB?1), ?1becomes
independent of barrier width and saturates to the value ?1
?n0/(c0kBV0) ?Hartman effect?. Conversely, the latter con-
tribution ?2is always shorter than the free-propagation time
over a length L and goes to zero for an opaque barrier, im-
plying that the tunneling time becomes independent of the
barrier distance ?generalized Hartman effect?. Similar results
are obtained for off-resonance tunneling of a nonrelativistic
electron through a rectangular DB potential V(z) assuming
that the incident wave packet has a below-barrier mean en-
ergy E half the barrier width V0; the corresponding expres-
sions 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 experi-
mental verifiable model for the quantum-mechanical case.
III. TUNNELING TIME MEASUREMENTS
We performed a series of tunneling time measurements
through DB FBG structures operating at around 1.5 ?m in
order to assess the independence of the peak pulse transit
times with barrier distance L. The FBGs used for the experi-
ments were manufactured by using standard writing tech-
niques, with an exposure time to UV laser beam and phase
mask length such as to realize a grating with sharp fall-off
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 an-
tiresonance 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 differ-
ent DB structures were realized with grating separation L of
18, 27, 35, 42, and 47 mm. For such structures, both trans-
mission 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 42-mm separation DB
FBG is shown in Fig. 2. Notice that, according to the theo-
retical curve shown in the same figure, far from the Fabry-
Perot resonances the group delay is superluminal, with ex-
pected time advancements of the order of 240–250 ps.
Notice also that the sharp Fabry-Perot resonances are not
fully resolved in the experimental curves due to bandwidth
limitations (?2 pm) of the measurement apparatus.
Direct time-domain measurements of tunneling delay
times were performed in transmission experiments using
probing optical pulse with ?1.3 ns duration, corresponding
to a spectral pulse bandwidth, which is less than the fre-
quency separation of Fabry-Perot resonances for all the five
DB structures. Since both transmission and group delay are
slowly varying functions of frequency far from Fabry-Perot
resonances ?see Fig. 2?, pulse advancement with a weak
pulse-shape distortion is thus expected for off-resonance
pulse transmission. The experimental setup for delay time
measurements is shown in Fig. 3. A pulse train, at a repeti-
tion frequency fm?300 MHz, was generated by external
modulation of a single-frequency continuous-wave tunable
laser diode ?Santec mod. ECL-200/210?, equipped with both
a coarse and a fine ?thermal? tuning control of frequency
emission with a resolution of ?100 MHz. The fiber-coupled
?10-mW output power emitted by the laser diode was am-
plified using a high-power erbium-doped fiber amplifier ?IPG
Mod. EAD-2-PM; EDFA1 in Fig. 3?, and then sent to a
LiNbO3-based Mach-Zehnder modulator, sinusoidally driven
at a frequency fm?300 MHz by a low-noise radio-
frequency ?rf? synthetizer. The bias point of the modulator
and the rf modulation power level were chosen to generate a
train with a pulse duration ?full width at half-maximum? of
?1.3 ns; the measured average output power of the pulse
train available for the transmission experiments was
FIG. 3. Schematic of the experimental setup.
LD, tunable laser diode; MZM, Mach-Zehnder
waveguide modulator; OC, optical circulator;
EDFA1 and EDFA2, erbium-doped fiber amplifi-
ers; rf, radio-frequency synthetizer.
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?130 mW. The pulse train was sent to the DB FBG through
a three-port optical circulator that enables both transmitted
and reflected signals to be simultaneously detected. The sig-
nal transmitted through the DB FBG was sent to a low-noise
erbium-doped fiber amplifier ?OptoCom Mod. OI LNPA;
EDFA2 in Fig. 3? with a low saturation power (?30 ?W at
1550 nm? that maintains the average power level of the out-
put 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 Fabry-Perot resonances or
antiresonances of the structure or outside the stop band, 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 sinusoidal rf signal that drives
the Mach-Zehnder modulator was used as an external trigger
for the oscilloscope, thus providing precise synchronism
among successive pulses. Off-resonance tunneling was
achieved by a careful tuning control of the laser spectrum,
which was detected by monitoring the reflected signal, avail-
able at port 3 of the optical circulator, using both an optical
spectrum analyzer ?Anritsu Mod. MS9710B? with a resolu-
tion of 0.07 nm, and a plane-plane scanning Fabry-Perot in-
terferometer ?Burleigh Model RC1101R? with a free-spectral
range of ?50 GHz and a measured finesse of ?180, which
permits to resolve the Fabry-Perot resonances of the DB
FBG structures. The reflectivity spectrum of the DB FBG
was first measured by the Fabry-Perot interferometer and re-
corded on a digital oscilloscope by disconnecting the laser
diode from the input port of EDFA1 and sending to the DB
FBG the broadband amplified spontaneous emission signal
of the optical amplifier. This trace is then used as a reference
to tune the pulse spectrum at the center of the off-resonance
plateau between the two central Fabry-Perot 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 off-
resonance 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 stop band of the DB FBG structure. A com-
parison 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 ap-
preciable pulse distortion nor any measurable time delay de-
pendence on the amplification level. Time delay measure-
ments 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 separation L for pulses
tuned far away from the band gap 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 veloc-
ity c0/n0. The solid line is, in turn, the expected transit time
for off-resonance 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 tunnel-
ing transit time. Notice that, within the experimental errors,
the agreement between the 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 can be qualita-
tively explained as the result of two simultaneous effects that
are a signature of the wave nature of the tunneling processes.
On the one hand, following Refs. ?2,11,12?, peak pulse ad-
vancement 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 the tunneling time on the
barrier distance can be explained, following Ref. ?19?, as an
FIG. 4. Pulse traces recorded on the sampling oscilloscope cor-
responding to the transmitted pulse for off-resonance tunneling
?curve 1? and reference pulse propagating outside the stop band of
the structure ?curve 2? for the 42-mm separation DB FBG.
FIG. 5. Off-resonance tunneling time versus barrier separation L
for a rectangular symmetric 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 measure-
ments; 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.
S. LONGHI, P. LAPORTA, M. BELMONTE, AND E. RECAMIPHYSICAL REVIEW E 65 046610
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effective ‘‘acceleration’’ of the forward traveling waves in
the intermediate classically allowed region that arises in con-
sequence of the destructive interference between the two bar-
riers. Though the observation of superluminal tunneling
times does not constitute any violation of the naive Einstein’s
causality ?28?, it is nevertheless remarkable that the reshap-
ing phenomenon that occurs inside the FBG structure leads
to an undistorted replica, albeit attenuated, of the original
pulse shape.
IV. CONCLUSIONS
In this paper off-resonant tunneling of optical pulses has
been experimentally investigated in fiber Bragg photonic
barriers. Tunneling time measurements have been shown to
be in good agreement with theoretical predictions based on
phase-time analysis and have confirmed that, for opaque bar-
riers, the tunneling time is independent not only of the bar-
rier width, as previously shown in Ref. ?12?, but even of the
barrier separation. Our results extend to the optical region
previous experimental achievements performed with micro-
waves ?9?, and may be of interest in the field of optical
tunneling and the related issue of superluminal propagation.
We also envisage that the experimental demonstration of su-
perluminal off-resonance tunneling in FBG structures may
be of potential interest in optoelectronic applications when-
ever a precise control of the group delay of an optical signal
is required ?see, e.g., ?29??.
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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of photon tunneling times presented here follows a different
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?27? We mention that superluminal group velocities as large as
?10c0were observed by G. Nimtz and co-workers, but in
microwave transmission experiments.
?28? We remark that the front velocity of a discontinuous step-wise
signal that propagates through the DB FBG structure is always
equal to c0, so that there is no violation of Einstein’s causality
andG. Salesi,e-print
MEASUREMENT OF SUPERLUMINAL OPTICAL . . .PHYSICAL REVIEW E 65 046610
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in our experimental findings. This circumstance follows tech-
nically from the analytic properties of spectral transmission
t(?) and has been discussed in Ref. ?21?. It should never-
theless be pointed out that the existence and the physical rel-
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that superluminal signal propagation is not possible, has been
the subject of controversial debate; for a general discussion on
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?29? G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, J.
Lightwave Technol. 37, 525 ?2001?.
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