josab-18-8-1174 (1179840)
Текст из файла
1174J. Opt. Soc. Am. B / Vol. 18, No. 8 / August 2001Chun-Fang Li and Qi WangDuration of tunneling photons in a frustratedtotal-internal-reflection structureChun-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, ChinaQi WangDepartment of Physics, Shanghai University, 99 Shangda Road, Baoshan, Shanghai 200436, ChinaReceived February 2, 2000; revised manuscript received September 5, 2000A 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 AmericaOCIS codes: 240.7040, 000.1600.1. INTRODUCTION1In 1931, Condon advanced the problem of traversal timeof a wave packet tunneling through a potential barrier.MacColl2 found in the next year that ‘‘there is no appreciable 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 literature.
Büttiker’s and Landauer’s 1982 paper4 rekindled 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 superluminality, even though the tunneling process is found not toviolate Einstein’s causality principle.7–10Thus 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 superluminality of tunneling times is not due to the nonrelativistic nature of Schrödinger’s equation.5The purpose of this paper is to introduce another tunneling time under the assumption that the tunnelingspeed of light is the energy-transfer speed in the tunneling 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,11d ⫽1j in冕a dx,(1)0where stands for the probability density in quantummechanics or the energy density in electromagnetism120740-3224/2001/081174-06$15.00within the barrier (assumed to extend from 0 to a), j instands for the incident probability flux density in quantum mechanics or the incident energy-flux density in electromagnetism in the tunneling direction. It should bementioned that the entire wave function in the barrier region is believed5,13 to consist of transmitted and reflectedportions; thus both of them contribute to the entire tunneling dwell time.
Therefore attempts are made to distinguish the transmission time from the reflection one.As will be pointed out in the following, it is the interference between the forward and the backward waves thatproduces tunneling flux14 (the meaning of forward andbackward waves is explained in the following). The forward 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 incident flux j in that makes the dwell time [Eq. (1)] superluminal.The new tunneling time under the assumption that thetunneling speed of light in frustrated total internal reflection (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 describing 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 ⑀ the dielectric constant of glassprisms. Let the incident light beam come to the vacuum© 2001 Optical Society of AmericaChun-Fang Li and Qi WangVol. 18, No. 8 / August 2001 / J.
Opt. Soc. Am. BH共 x, t 兲 ⫽再ky0关 C exp共 ⫺ x 兲 ⫹ D exp共 x 兲兴 x̂j⫹11750关 C exp共 ⫺ x 兲 ⫺ D exp共 x 兲兴 ŷ冎⫻ exp关 j 共 t ⫺ k y y 兲兴 .Fig. 1. Schematic diagram of the tunneling process in afrustrated-total-internal-reflection structure.gap from the left at incidence angle beyond the criticalangle sin⫺1(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 tobeEin(x, t) ⫽ exp关 j (t ⫺ k • x) 兴 ẑ,wherek⫽ (k x , k y , k z ) ⫽ (k cos , k sin , 0) and k ⫽ ( ⑀ 0 2 ) 1/2,the electric field in the tunneling region is proven to beE共 x, t 兲 ⫽ 关 Cexp共 ⫺ x 兲 ⫹ D exp共 x 兲兴 exp关 j 共 t ⫺ k y y 兲兴 ẑ,The first and the second terms, exp(⫺ x)exp关 j(t⫺ ky y)兴 ẑ and exp( x)exp关 j (t ⫺ ky y)兴 ẑ, in wave function(2) are called forward and backward waves, respectively.According to the definition of energy-flux density, j⫽ Re(E ⫻ H* /2), it is easy to show that the forwardwave as well as the backward wave alone produces no energy 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 energy flux in the x direction is produced by the interferencebetween the forward and the backward waves and has theform冉冊12k x cos2 ␦j x ⫽ Re ⫺ E z H y* ⫽Im共 CD * 兲 ⫽sin2 ␦ .20 0f 2(7)The energy flux in the y direction in the tunneling regionisj y ⫽ Re(2)⫽where ⫽ 共 k 2y ⫺ k 02 兲 1/2,C⫽D⫽exp共 a 兲 cos ␦fk 02 ⫽ ⑀ 0 0 2 ,12E z H x*2k y cos2 ␦ 0f 2冊关 sinh2 共 x ⫺ a 兲 ⫹ sin2 ␦ 兴 .S x ⫽ ⫺Re共 E z 兲 Re共 H y 兲 ⫽fexp共 ⫺j 兲 exp关 j 共 ␦ ⫺ /2兲兴 .⫹Parameters ␦ and ⌬ are defined byk x ⫹ j ⫽ ⌬exp共 j ␦ 兲 ,where ⌬ ⫽ (k 2 ⫺ k 02 ) 1/2.by(3)Parameters and f are definedcosh a sin 2 ␦ ⫹ j sinh a cos 2 ␦ ⫽ fexp共 j 兲 .(4)The transmitted electric field is Etr(x, t) ⫽ F exp关 j (t⫺ k • x) 兴 ẑ, where the transmission coefficient isF⫽冉sin 2 ␦fexp共 ⫺j 兲 exp共 jk x a 兲 .(8)It should be emphasized that tunneling energy flux (7)is the time average of the following time-dependent energy flux in the x direction:exp共 ⫺j 兲 exp关 ⫺j 共 ␦ ⫺ /2兲兴 ,exp共 ⫺ a 兲 cos ␦(6)(5)The magnetic fields in the respective regions can be obtained easily from Maxwell equation H ⫽ ( j/ 0 )ⵜ⫻ E.
Specifically, the magnetic field in the tunnelingregion is⫺20200兩 C 兩兩 D 兩 sin共 C ⫺ D 兲兩 C 兩 2 exp共 ⫺2 x 兲 sin 2 共 t ⫺ k y y ⫹ C 兲兩 D 兩 2 exp共 2 x 兲 sin 2 共 t ⫺ k y y ⫹ D 兲 ,(9)where C ⫽ arg(C) and D ⫽ 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, either. 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 directions and cannot cancel each other.
From these observations, we may conclude that tunneling flux (7) resulting1176J. Opt. Soc. Am. B / Vol. 18, No. 8 / August 2001Chun-Fang Li and Qi Wangfrom the interference between the forward and the backward waves is not the average of some right-going andsome left-going flux.With the energy density in the tunneling region,⫽⑀040兩 E兩 2 ⫹兩 H兩 2 ⫽4cos2 ␦ 2 0f 2关 2k 2y sinh2 共 x ⫺ a 兲⫹ 共 k 02 ⫹ k 2 兲 sin2 ␦ 兴 ,(10)one can easily find the traversal tunneling speed of electromagnetic energy in the tunneling region,vx ⫽jx⫽2k x sin2 ␦sinh 共 x ⫺ a 兲 ⫹2k 2y2共 k 02⫹ k 兲 sin ␦22,and the speed of energy flow in the y direction in the tunneling region,vy ⫽jy⫽2k y 关 sinh 共 x ⫺ a 兲 ⫹ sin ␦ 兴2k 2y2sinh2 共 x ⫺ a 兲 ⫹ 共 k 02 ⫹ k 2 兲 sin2 ␦At x ⫽ a the traversal tunneling speed v x reaches itsmaximum, v x max ⫽ 2kx /(k02 ⫹ k2), which is less than /k 0 , the vacuum speed of light in free space.
In fact,because of the following three relations, k y ⬎ k 0 , k 02⫹ k 2 ⬎ 2k 0 k, and k 02 ⫹ k 2 ⬎ 2k 02 , the total speed v⫽ (v 2x ⫹ v 2y ) 1/2 of the energy flow in the tunneling regionis less than /k 0 , which means that no superluminalityproblem arises. Notice that the tunneling speed v y in they direction is in general position dependent. Only when ⫽ k x is it uniform and has the value /k y .
A similarenergy-transport velocity was introduced in dispersivemedia.20Having traversal tunneling speed (11), we are ready tocalculate the tunneling time,冕a0⫽dx⫽vxk 2y a sin2 ␦1jx冉冕 k xf 2冉sinh2 a2a⫺k 02k 2y冊cos2 ␦ cos2 ␦ ,(14)1j inx冕a dx ⫽冉k 2y asin 2 ␦ sinh2 af 202a⫺k 02k 2y冊cos 2 ␦ ,(15)by the substitution of traversal tunneling flux j x for incident flux j inx ⫽ k x /2 0 in the x direction, so that.(12)⫽2k 2y aas 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 Ⰶ 1/ , the tunneling time is propor ⬇ 关 (k 2y ⫺ k 02tionaltothebarrierwidth,⫻ cos 2␦ )/ sin 2␦兴 a.(d) Tunneling time (13) is different from the dwell time,d ⫽(11)2 ⫽d⫽sin2 2 ␦f2⫽ 兩F兩2,(16)a dx0sinh2 a2a⫺k 02k 2y冊cos 2 ␦ .(13)Fig.
Характеристики
Тип файла PDF
PDF-формат наиболее широко используется для просмотра любого типа файлов на любом устройстве. В него можно сохранить документ, таблицы, презентацию, текст, чертежи, вычисления, графики и всё остальное, что можно показать на экране любого устройства. Именно его лучше всего использовать для печати.
Например, если Вам нужно распечатать чертёж из автокада, Вы сохраните чертёж на флешку, но будет ли автокад в пункте печати? А если будет, то нужная версия с нужными библиотеками? Именно для этого и нужен формат PDF - в нём точно будет показано верно вне зависимости от того, в какой программе создали PDF-файл и есть ли нужная программа для его просмотра.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
