1174 J. Opt. Soc. Am. B/Vol. 18, No. 8 /August 2001 Chun-Fang Li and Qi Wang

Duration of tunneling photons in a frustrated-total-internal-reflection structure

Chun-Fang Li*

China Center of Advanced Science and Technology (World Laboratory), P.O. Box 8730, Beijing 100080, China, andDepartment of Physics, Shanghai University, 99 Shangda Road, Baoshan, Shanghai 200436, China

Qi Wang

Department of Physics, Shanghai University, 99 Shangda Road, Baoshan, Shanghai 200436, China

Received February 2, 2000; revised manuscript received September 5, 2000

A new definition for tunneling time of photons in a frustrated-total-internal-reflection structure is introducedunder the assumption that the tunneling speed of photons is the energy-transfer speed of light in the tunnelingregion. This definition eliminates the problem of superluminality, and the suggested tunneling speed definesan orbit equation that gives a lateral shift of tunneling photons. © 2001 Optical Society of America

OCIS codes: 240.7040, 000.1600.

1. INTRODUCTIONIn 1931, Condon1 advanced the problem of traversal timeof a wave packet tunneling through a potential barrier.MacColl2 found in the next year that ‘‘there is no appre-ciable delay in the transmission of the packet through thebarrier.’’ This conclusion was confirmed in 1962 byHartman.3 Such a phenomenon, that the time taken fora wave packet to tunnel through a potential barrier issmaller than the barrier thickness divided by the vacuumspeed of light, is referred to as superluminality in the lit-erature. Buttiker’s and Landauer’s 1982 paper4 re-kindled the tunneling–time debate. Since then, a varietyof definitions for tunneling times have been proposed, anda large number of optical experiments on tunneling timeshave been performed.5,6 One common feature sharedamong those proposed tunneling times is the superlumi-nality, even though the tunneling process is found not toviolate Einstein’s causality principle.7–10

Thus we are concerned with the question, what is thetraversal time that corresponds to the velocity limited bythe causality principle? In other words, which velocitydescribes the traversal motion of particles that does notexceed the vacuum speed of light? Before answering thisquestion, one thing that we should mention is that the su-perluminality of tunneling times is not due to the nonrel-ativistic nature of Schrodinger’s equation.5

The purpose of this paper is to introduce another tun-neling time under the assumption that the tunnelingspeed of light is the energy-transfer speed in the tunnel-ing region. Detailed analysis shows that the definition ofthis tunneling time is similar to but different from that ofthe dwell time, a time first advanced by Smith,11

td 51

j inE

0

a

rdx, (1)

where r stands for the probability density in quantummechanics or the energy density in electromagnetism12

0740-3224/2001/081174-06$15.00 ©

within the barrier (assumed to extend from 0 to a), j instands for the incident probability flux density in quan-tum mechanics or the incident energy-flux density in elec-tromagnetism in the tunneling direction. It should bementioned that the entire wave function in the barrier re-gion is believed5,13 to consist of transmitted and reflectedportions; thus both of them contribute to the entire tun-neling dwell time. Therefore attempts are made to dis-tinguish the transmission time from the reflection one.As will be pointed out in the following, it is the interfer-ence between the forward and the backward waves thatproduces tunneling flux14 (the meaning of forward andbackward waves is explained in the following). The for-ward wave as well as the backward wave alone producesno flux in the barrier. From the point of view of the flux,the entire wave function in the barrier region representsonly the tunneling particles. Furthermore, it is the inci-dent flux j in that makes the dwell time [Eq. (1)] superlu-minal.

The new tunneling time under the assumption that thetunneling speed of light in frustrated total internal reflec-tion (FTIR) is the energy-transfer speed in the tunnelingregion has no problem of the well-known superluminality.Also under the mentioned assumption, a trajectory in thetunneling region is found that may be interpreted as de-scribing the motion orbit of tunneling photons.

2. TUNNELING SPEED AND TIMEThe tunneling problem in FTIR has been studiedextensively.15–19 The situation considered here is similarto that of Ref. 17. Two right-angle glass prisms of indexn are placed with their hypotenuses in close proximity,forming a vacuum gap of width a between them (as shownin Fig. 1). We assume all three regions are nonmagneticso that their magnetic permeabilities are all equal tounity, and we denote by e the dielectric constant of glassprisms. Let the incident light beam come to the vacuum

2001 Optical Society of America

Chun-Fang Li and Qi Wang Vol. 18, No. 8 /August 2001 /J. Opt. Soc. Am. B 1175

gap from the left at incidence angle u beyond the criticalangle sin21(1/n). In the case of TE polarization (the caseof TM polarization can also be discussed in the same waywith similar results), taking the incident electric field tobe Ein(x, t) 5 exp@ j (vt 2 k • x)# z, where k5 (kx , ky , kz) 5 (k cos u, k sin u, 0) and k 5 (em0v2)1/2,the electric field in the tunneling region is proven to be

E~x, t ! 5 @Cexp~2kx ! 1 D exp~kx !#exp@ j~vt 2 kyy !# z,

(2)

where

k 5 ~ky2 2 k0

2!1/2, k02 5 e0m0v2,

C 5exp~ka !cos d

fexp~2jw!exp@2j~d 2 p/2!#,

D 5exp~2ka !cos d

fexp~2jw!exp@ j~d 2 p/2!#.

Parameters d and D are defined by

kx 1 jk 5 Dexp~ jd!, (3)

where D 5 (k2 2 k02)1/2. Parameters w and f are defined

by

cosh ka sin 2d 1 j sinh ka cos 2d 5 fexp~ jw!. (4)

The transmitted electric field is Etr(x, t) 5 F exp@ j (vt2 k • x)# z, where the transmission coefficient is

F 5sin 2d

fexp~2jw!exp~ jkxa !. (5)

The magnetic fields in the respective regions can be ob-tained easily from Maxwell equation H 5 ( j/vm0)¹3 E. Specifically, the magnetic field in the tunnelingregion is

Fig. 1. Schematic diagram of the tunneling process in afrustrated-total-internal-reflection structure.

H~x, t ! 5 H ky

vm0@C exp~2kx ! 1 D exp~kx !#x

1jk

vm0@C exp~2kx ! 2 D exp~kx !# yJ

3 exp@ j~vt 2 kyy !#. (6)

The first and the second terms, exp(2k x)exp@ j(vt2 ky y)# z and exp(k x)exp@ j (vt 2 ky y)# z, in wave function(2) are called forward and backward waves, respectively.According to the definition of energy-flux density, j5 Re(E 3 H* /2), it is easy to show that the forwardwave as well as the backward wave alone produces no en-ergy flux. In other words, neither the forward nor thebackward wave represents the tunneling electromagneticenergy from the point of view of energy flow. It is theircombination, the entire wave function (2), that representsthe tunneling electromagnetic energy. The tunneling en-ergy flux in the x direction is produced by the interferencebetween the forward and the backward waves and has theform

jx 5 ReS 21

2EzHy* D 5

k

vm0Im~CD* ! 5

2kx cos2 d

vm0f 2 sin2 d.

(7)

The energy flux in the y direction in the tunneling regionis

jy 5 ReS 1

2EzHx* D

52ky cos2 d

vm0f 2 @sinh2 k~x 2 a ! 1 sin2 d#. (8)

It should be emphasized that tunneling energy flux (7)is the time average of the following time-dependent en-ergy flux in the x direction:

Sx 5 2Re~Ez!Re~Hy! 5k

vm0uCuuDusin~fC 2 fD!

1k

2vm0uCu2exp~22kx !sin 2~vt 2 kyy 1 fC!

2k

2vm0uDu2exp~2kx !sin 2~vt 2 kyy 1 fD!,

(9)

where fC 5 arg(C) and fD 5 arg(D). The first term iscontributed by the interference between the forward andthe backward waves, which is time independent and isjust tunneling flux (7). The second and the third termsare contributed by the forward and the backward waves,respectively. Their time averages are equal to zero. Butthe second term cannot be regarded as a right-going fluxbecause it can be negative as well as positive. Similarly,the third term cannot be regarded as a left-going flux, ei-ther. Furthermore, the second and the third terms donot always have values of opposite signs at the same time.In a word, they do not represent currents of opposite di-rections and cannot cancel each other. From these obser-vations, we may conclude that tunneling flux (7) resulting

1176 J. Opt. Soc. Am. B/Vol. 18, No. 8 /August 2001 Chun-Fang Li and Qi Wang

from the interference between the forward and the back-ward waves is not the average of some right-going andsome left-going flux.

With the energy density in the tunneling region,

r 5e0

4uEu2 1

m0

4uHu2 5

cos2 d

v2m0f 2 @2ky2 sinh2k~x 2 a !

1 ~k02 1 k2!sin2d#, (10)

one can easily find the traversal tunneling speed of elec-tromagnetic energy in the tunneling region,

vx 5jx

r5

2kxv sin2d

2ky2 sinh2k~x 2 a ! 1 ~k0

2 1 k2!sin2d,

(11)

and the speed of energy flow in the y direction in the tun-neling region,

vy 5jy

r5

2kyv@sinh2 k~x 2 a ! 1 sin2d#

2ky2 sinh2k~x 2 a ! 1 ~k0

2 1 k2!sin2 d.

(12)

At x 5 a the traversal tunneling speed vx reaches itsmaximum, vx max 5 2vkx /(k0

2 1 k2), which is less thanv/k0 , the vacuum speed of light in free space. In fact,because of the following three relations, ky . k0 , k0

2

1 k2 . 2k0k, and k02 1 k2 . 2k0

2, the total speed v5 (vx

2 1 vy2)1/2 of the energy flow in the tunneling region

is less than v/k0 , which means that no superluminalityproblem arises. Notice that the tunneling speed vy in they direction is in general position dependent. Only whenk 5 kx is it uniform and has the value v/ky . A similarenergy-transport velocity was introduced in dispersivemedia.20

Having traversal tunneling speed (11), we are ready tocalculate the tunneling time,

t 5 E0

a dx

vx5

1

jxE

0

a

rdx

5ky

2a

vk sin2d S sinh2ka

2ka2

k02

ky2 cos 2d D . (13)

Note the following:

(a) Tunneling time (13) does not saturate as the barrierwidth increases. In fact, when a @ 1/k, it is superlin-early dependent on a,

t 'ky

2

2vksin 2dexp~2ka !a,

as is shown in Fig. 2, where the wavelength of the light istaken to be l 5 632.8 nm, the refractive index of theglass prisms n 5 1.52, and the incidence angle u5 0.72 rad. This behavior of t is different from that ofButtiker and Landauer’s semiclassical time, which is lin-early dependent on the barrier width for opaquebarriers.4,5

(b) The superlinear dependence of t is also comparedwith the saturation characteristic of the following phasetime18,19:

tw 52ky

2a

vkxf 2 S sinh2ka

2ka2

k02

ky2 cos2dcos2d D , (14)

as is seen explicitly from Fig. 3, where the conditions arethe same as in Fig. 2.

(c) When the barrier width is much smaller than thepenetration depth, a ! 1/k, the tunneling time is propor-tional to the barrier width, t ' @(ky

2 2 k02

3 cos 2d )/vk sin 2d# a.(d) Tunneling time (13) is different from the dwell time,

td 51

j inxE

0

a

rdx 5ky

2 asin 2d

vkf 2 S sinh2ka

2ka2

k02

ky2 cos 2d D ,

(15)

by the substitution of traversal tunneling flux jx for inci-dent flux j inx 5 kx/2vm0 in the x direction, so that

td

t5

sin2 2d

f 2 5 uFu2, (16)

Fig. 2. Dependence of tunneling time (13) on the barrier width,where l 5 632.8 nm, n 5 1.52, and u 5 0.72 rad.

Fig. 3. Dependence of phase time (14) on the barrier width, withthe same conditions as in Fig. 2.

Chun-Fang Li and Qi Wang Vol. 18, No. 8 /August 2001 /J. Opt. Soc. Am. B 1177

where F is the transmission coefficient [Eq. (5)]. The in-cident flux j inx makes the dwell time [Eq. (15)] superlumi-nal in the deep-tunneling limit (see the following discus-sion).

(e) The speeds vx and vy define an orbit equation in thetunneling region by relation dx/vx 5 dy/vy , which is

y~x ! 5ky

k sin 2dF sinh2k~x 2 a ! 1 sinh2ka

2k2 x cos 2dG ,

(17)

assuming that it goes through the point (0, 0). Figure 4shows an example of this orbit, where the wave length ofthe light is chosen to be l 5 632.8 nm, the refractive in-dex of the glass prisms n 5 1.52, the incidence angle u5 0.72 rad, and the width of the vacuum gap a5 500 nm.

(f) There is another orbit approach21,22 to thetunneling-time problem, which depends on Bohm’s trajec-tory interpretation of quantum mechanics.23,24

3. DISCUSSIONThe above physical quantities, tunneling speeds (11) and(12), tunneling time (13), and orbit equation (17) in thetunneling region, are obtained from the point of view ofenergy flow. Since the energy of the electromagnetic fieldis carried by photons in quantum theory, and a photonhas energy \v corresponding to a definite frequency, thesequantities may be interpreted as describing the behaviorsof tunneling photons. In this connection, orbit equation(17) in the tunneling region gives a lateral shift of tunnel-ing photons along the y direction,

y0 5 y~a ! 5kya

k sin 2dS sinh2ka

2ka2 cos 2d D . (18)

Note that tunneling time (13) can be rewritten as

t 5k cot 2d

va 1

ky

vy0 (19)

in terms of the barrier width a and the lateral shift y0 .Expression (19) for tunneling time does not mean that thetunneling photons first move at speed v/k cot 2d in the x

Fig. 4. Orbit of tunneling photons in the tunneling region,where l 5 632.8 nm, n 5 1.52, u 5 0.72 rad, and a 5 500 nm.

direction and then, after arriving at (x 5 a, y 5 0), moveat speed v/ky in the y direction. In fact, the photons inthe tunneling region move at a single position-dependentspeed v 5 (vx , vy). Lateral shift y0 can be experimen-tally measured, which provides a method to measure thetunneling time according to Eq. (19).

It should be pointed out that apart from lateral shift(18), which is associated with tunneling time (13), there isanother lateral shift17,18 that is associated with the phasetime. Lateral shift (18) describes the transfer of energy,and the phase-time-associated lateral shift describes themotion of a wave packet.

Next we discuss several physical limits.

(a) The deep-tunneling limit k @ kx : Near grazing in-cidence u → p/2, the x component kx of the incident wavevector is much smaller than the evanescent decay con-stant k; thus we have d → p/2. In this limit the tra-versal tunneling speed vx approaches zero according toEq. (11). The tunneling time thus diverges as is shown inFig. 5, where the wavelength of the light is taken to bel 5 632.8 nm, the refractive index of the glass prisms n5 1.52, and the width of the vacuum gap a 5 500 nm.This is expected from physical consideration. This result

Fig. 5. Dependence of tunneling time (13) on the incidenceangle, where l 5 632.8 nm, n 5 1.52, and a 5 500 nm.

Fig. 6. Dependence of dwell time (15) on the incidence angle,with the same conditions as in Fig. 5.

1178 J. Opt. Soc. Am. B/Vol. 18, No. 8 /August 2001 Chun-Fang Li and Qi Wang

is to be compared with the property of dwell time (15). Inthe deep-tunneling limit d → p/2 (that is, u → p/2), tdtends to zero, exhibiting superluminality, as is shown inFig. 6, where the conditions are the same as in Fig. 5.

(b) The critical limit k ! kx : When the incidenceangle u approaches the critical angle sin21(1/n), the eva-nescent decay constant approaches zero, k → 0, so thatwe have d → 0. In this limit, the traversal tunnelingspeed vx approaches 2Dv/@k0

2 1 k2 1 2k02D2(x 2 a)2#.

The tunneling time thus approaches Da/2v 1 (k02a/Dv)

3 (1 1 D2a2/3). Because the orbit of tunneling photonsin this limit behaves like y 5 (k0x/D)(D2x2/3 2 D2ax1 D2a2 1 1), which gives a lateral shift y0 5 (k0a/D)3 (1 1 D2a2/3), the tunneling time can be rewritten asDa/2v 1 k0y0 /v.

(c) Opaque limit ka → `: In this case the traversaltunneling speed vx vanishes except near point x 5 a.The tunneling time thus tends to infinity according to Eq.(13). This is similar to the deep-tunneling limit, as canbe seen by inspection of Figs. 2 and 5. In fact, in both ofthese two cases the transmission coefficient F vanishes.

4. CONCLUSIONSIn conclusion, we proposed another tunneling time in thetwo-dimensional optical frustrated-total-internal-reflection structure, assuming that the tunneling speed oflight is the energy-transfer speed in the tunneling region.This tunneling speed defines an orbit of tunneling pho-tons within the tunneling region and gives a lateral shiftof tunneling photons. The relation of the tunneling timewith the lateral shift was given. The comparisons withthe phase time and the dwell time were also made. Itwas shown that the proposed tunneling time eliminatesthe well-known superluminality. Last, several importantlimits were discussed, including the deep-tunneling limit,the critical limit, and the opaque limit.

The key point of the proposed tunneling time is thetunneling flux within the tunneling region. It was shownthat tunneling flux (7) is the time average of time-dependent energy flux (9), rather than the average ofsome right-going and some left-going flux. The tunnelingflux is contributed by the interference between the for-ward and the backward waves and is a result of the wavenature of the light.

Two kinds of experiments were performed for opticaltunneling in the FTIR structure. Carniglia’s and Man-del’s early experiments15,16 measured the phase shifts,which agreed well with the theoretical prediction that thephase shifts are substantially independent of the barrierwidth in the opaque limit. Although their research didnot directly address the tunneling-time problem, the re-sults were indirectly connected with the saturation of thephase time. Recently, Balcou and Dutriaux18 measuredsimultaneously two different kinds of tunneling times.They confirmed that the phase time saturates with in-creasing barrier width by measuring the phase-time-associated lateral shift and that the loss time (which ap-proaches Buttiker and Landauer’s semiclassical time foropaque barriers) increases linearly with this width bymeasuring the angular deflection of the transmitted wavepacket. To measure proposed tunneling time (13),

stationary-state light beams, rather than wave packets,are required in accordance with the discussion made here.The implications of the present tunneling-time definitionwill be further analyzed for the case of wave packets, par-ticularly in connection with the time average of time-dependent energy flux and energy density.

ACKNOWLEDGMENTSWe thank Ai-Min Yan for her help in drawing the figures.This study was supported in part by the National NaturalScience Foundation of China under grant 69877009 and agrant of the Science and Technology Committee of Shang-hai.

*Mailing address: Department of Physics, ShanghaiUniversity, 99 Shangda Road, Baoshan, Shang-hai 200436, China. E-mail: c-fli@online.sh.cn.

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