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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

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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 of counterpropagating

?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.

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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. RECAMI PHYSICAL 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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?23? The analogy between tunneling of electrons and photons in

superlattice structures and closed-form solutions for the tun-

neling times have been recently reported, in the general case,

in P. Pereyra, Phys. Rev. Lett. 84, 1772 ?2000?. The derivation

of photon tunneling times presented here follows a different

and simpler approach, based on coupled-mode equations for

Bragg scattering, which is suited for FBG structures.

?24? See, for instance, T. Ergodan, J. Lightwave Technol. LT15,

1277 ?1997?.

?25? Among others, the phase time has been shown to be consistent

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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

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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

this point we refer the reader to Ref. ?4? and to G. Nimtz, Eur.

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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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