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2. Dependence of tunneling time (13) on the barrier width,where ⫽ 632.8 nm, n ⫽ 1.52, and ⫽ 0.72 rad.Note the following:(a) Tunneling time (13) does not saturate as the barrierwidth increases. In fact, when a Ⰷ 1/ , it is superlinearly dependent on a,⬇k 2y2 sin 2 ␦exp共 2 a 兲 a,as is shown in Fig. 2, where the wavelength of the light istaken to be ⫽ 632.8 nm, the refractive index of theglass prisms n ⫽ 1.52, and the incidence angle ⫽ 0.72 rad. This behavior of is different from that ofBüttiker and Landauer’s semiclassical time, which is linearly dependent on the barrier width for opaquebarriers.4,5(b) The superlinear dependence of is also comparedwith the saturation characteristic of the following phasetime18,19:Fig.
3. Dependence of phase time (14) on the barrier width, withthe same conditions as in Fig. 2.Chun-Fang Li and Qi WangVol. 18, No. 8 / August 2001 / J. Opt. Soc. Am. B1177direction and then, after arriving at (x ⫽ a, y ⫽ 0), moveat speed /k y in the y direction. In fact, the photons inthe tunneling region move at a single position-dependentspeed v ⫽ (v x , v y ). Lateral shift y 0 can be experimentally 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.Fig.
4. Orbit of tunneling photons in the tunneling region,where ⫽ 632.8 nm, n ⫽ 1.52, ⫽ 0.72 rad, and a ⫽ 500 nm.where F is the transmission coefficient [Eq. (5)]. The incident flux j inx makes the dwell time [Eq. (15)] superluminal in the deep-tunneling limit (see the following discussion).(e) The speeds v x and v y define an orbit equation in thetunneling region by relation dx/v x ⫽ dy/v y , which isy共 x 兲 ⫽ky sin 2 ␦冋sinh2 共 x ⫺ a 兲 ⫹ sinh2 a2(a) The deep-tunneling limit Ⰷ k x : Near grazing incidence → /2, the x component k x of the incident wavevector is much smaller than the evanescent decay constant ; thus we have ␦ → /2. In this limit the traversal tunneling speed v x approaches zero according toEq. (11).
The tunneling time thus diverges as is shown inFig. 5, where the wavelength of the light is taken to be ⫽ 632.8 nm, the refractive index of the glass prisms n⫽ 1.52, and the width of the vacuum gap a ⫽ 500 nm.This is expected from physical consideration. This result册⫺ x cos 2 ␦ ,(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 ⫽ 632.8 nm, the refractive index of the glass prisms n ⫽ 1.52, the incidence angle ⫽ 0.72 rad, and the width of the vacuum gap a⫽ 500 nm.(f) There is another orbit approach21,22 to thetunneling-time problem, which depends on Bohm’s trajectory interpretation of quantum mechanics.23,243.
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 ប 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 tunneling photons along the y direction,y0 ⫽ y共 a 兲 ⫽k ya sin 2 ␦冉sinh2 a2a冊⫺ cos 2 ␦ .Fig. 5. Dependence of tunneling time (13) on the incidenceangle, where ⫽ 632.8 nm, n ⫽ 1.52, and a ⫽ 500 nm.(18)Note that tunneling time (13) can be rewritten as⫽ cot 2 ␦a⫹kyy0(19)in terms of the barrier width a and the lateral shift y 0 .Expression (19) for tunneling time does not mean that thetunneling photons first move at speed / cot 2␦ in the xFig. 6.
Dependence of dwell time (15) on the incidence angle,with the same conditions as in Fig. 5.1178J. Opt. Soc. Am. B / Vol. 18, No. 8 / August 2001is to be compared with the property of dwell time (15). Inthe deep-tunneling limit ␦ → /2 (that is, → /2), dtends to zero, exhibiting superluminality, as is shown inFig. 6, where the conditions are the same as in Fig. 5.(b) The critical limit Ⰶ k x : When the incidenceangle approaches the critical angle sin⫺1(1/n), the evanescent decay constant approaches zero, → 0, so thatwe have ␦ → 0. In this limit, the traversal tunnelingspeed v x approaches 2⌬ / 关 k 02 ⫹ k 2 ⫹ 2k 02 ⌬ 2 (x ⫺ a) 2 兴 .The tunneling time thus approaches ⌬a/2 ⫹ (k 02 a/⌬ )⫻ (1 ⫹ ⌬ 2 a 2 /3).
Because the orbit of tunneling photonsin this limit behaves like y ⫽ (k 0 x/⌬)(⌬ 2 x 2 /3 ⫺ ⌬ 2 ax⫹ ⌬ 2 a 2 ⫹ 1), which gives a lateral shift y 0 ⫽ (k 0 a/⌬)⫻ (1 ⫹ ⌬ 2 a 2 /3), the tunneling time can be rewritten as⌬a/2 ⫹ k 0 y 0 / .(c) Opaque limit a → ⬁: In this case the traversaltunneling speed v x vanishes except near point x ⫽ 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.Chun-Fang Li and Qi Wangstationary-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, particularly in connection with the time average of timedependent 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 Shanghai.*Mailing address: Department of Physics, ShanghaiUniversity, 99 Shangda Road, Baoshan, Shanghai 200436, China.
E-mail: c-fli@online.sh.cn.REFERENCES1.4. CONCLUSIONSIn conclusion, we proposed another tunneling time in thetwo-dimensionalopticalfrustrated-total-internalreflection structure, assuming that the tunneling speed oflight is the energy-transfer speed in the tunneling region.This tunneling speed defines an orbit of tunneling photons 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 timedependent energy flux (9), rather than the average ofsome right-going and some left-going flux. The tunnelingflux is contributed by the interference between the forward 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 Mandel’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 results 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 increasing barrier width by measuring the phase-timeassociated lateral shift and that the loss time (which approaches Büttiker 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),2.3.4.5.6.7.8.9.10.11.12.13.14.15.16.17.18.E. U. Condon, ‘‘Quantum mechanics of collision processes,’’Rev. Mod.
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