《电动力学》课程参考文献：Negative group velocity

Negative group velocity Kirk T Mcdonalda Received 21 August 2000; accepted 21 September 2000) [DO:10.1119/1.1331304] . PROBLEM C. Fourier analysis Consider a variant on the physical situation of slow pply the transformations between an incident monochro- ight"1, in which two closely spaced spectral lines are now matic wave and the wave in and beyond the medium to the lis both optically pumped to show that the group velocity can be Fourier analysis of an incident pulse of form f(Elc-i) negative at the central frequency, which leads to apparent superluminal behavior D. Propagation of a sharp wave front In the approximation that on varies linearly with a, de- A Negative group velocity duce the waveforms in the regions 0<=<a and a<= for an incident pulse S(E/c-t), where S is the Dirac delta function In more detail, consider a classical model of matter in Show that the pulse emerges out of the gain region at ==a at which spectral lines are associated with oscillators. In par- ticular, consider a gas with two closely spaced spectral lines time t=alug, which appears to be earlier than when it enters of angular frequencies 12=(+A/2, where A< oo. Each this region if the group velocity is negative. Show also that line has the same damping constant(and spectral width)y inside the negative group velocity medium a pulse propa- Ordinarily, the gas would exhibit strong absorption of gates backwards from ==a at time t=alUg <0 to ==0 at t light in the vicinity of the spectral lines. But suppose that 0, at which time it appears to annihilate the incident pulse lasers of frequencies a and pump both oscillators into inverted populations. This can be described classically by E. Propagation of a Gaussian pulse assigning negative oscillator strengths to these oscillators Deduce an expression for the group velocity u (wo)of a As a more physical example, deduce the waveforms in the pulse of light centered on frequency wo in terms of the(uni- regions 0<- <a and as: for a gaussian incident pulse valent) plasma frequency p of the medium, given by Eoe-(/c-n'nreloo(=/c-n). Carry the frequency expansion of on(o)to second order to obtain conditions of validity of the 4丌N analysis such as maximum pulse width T, maximum length a 1) of the gain region, and maximum time of advance of the where N is the number density of atoms, and e and m are the in a negative group velocity medium can lead to superlumi- separation A compared to the linewidth y such that the group nal signal propagation mass of an velocity u,(oo) is negative In a recent experiment by Wang et al., a group velocity of IL. SOLUTION c/310, where c is the speed of light in vacuum, was The concept of group velocity appears to have been first demonstrated in cesium vapor using a pair of spectral lines enunciated by Hamilton in 1839 in lectures of which only with separation△/2丌≈2 MHz and linewidth y/2 abstracts were published. The first recorded observation of 0. 8 MHZ the group velocity of a(water)wave is due to Russell in 1844.However, widespread awareness of the group velocity dates from 1876 when Stokes used it as the topic of a B. Propagation of a monochromatic plane wave Smiths Prize examination paper. The early history of group velocity has been reviewed by Havelock H. Lamb credits A. Schuster with noting in 1904 that a Consider a wave with electric field Eoeloilc-n that is negative group velocity, i.e., a group velocity of opposite incident from -<0 on a medium that extends from - =0 to a Ignore reflection at the boundaries, as is reasonable if the sign to that of the phase velocity, is possible due to anoma- index of refraction n( o) is near unity. Particularly simple 1905. Lamb gave two examples of strings subject to external sumption that the on(o) vares linearly with frequency considerations assumed that in case of a wave with positive about a central frequency wo. Deduce a transformation that group and phase velocities incident on the anomalous me has a frequency-dependent part and a frequency-independent dium, energy would be transported into the medium with a part between the phase of the wave for -<0 to that of the positive group velocity, and so there would be waves with wave inside the medium, and to that of the wave in the negative phase velocity inside the medium. Such negative egion a<- phase velocity waves are formally consistent with Snells Am J. Phys. 69(5), May 2001 http://ojps.aiporg/ajp/ c 2001 American Association of Physics Teachers

Negative group velocity Kirk T. McDonalda) Joseph Henry Laboratories, Princeton University, Princeton, New Jersey 08544 ~Received 21 August 2000; accepted 21 September 2000! @DOI: 10.1119/1.1331304# I. PROBLEM Consider a variant on the physical situation of ‘‘slow light’’ 1,2 in which two closely spaced spectral lines are now both optically pumped to show that the group velocity can be negative at the central frequency, which leads to apparent superluminal behavior. A. Negative group velocity In more detail, consider a classical model of matter in which spectral lines are associated with oscillators. In par￾ticular, consider a gas with two closely spaced spectral lines of angular frequencies v1,25v06D/2, where D!v0 . Each line has the same damping constant ~and spectral width! g. Ordinarily, the gas would exhibit strong absorption of light in the vicinity of the spectral lines. But suppose that lasers of frequencies v1 and v2 pump both oscillators into inverted populations. This can be described classically by assigning negative oscillator strengths to these oscillators.3 Deduce an expression for the group velocity vg(v0) of a pulse of light centered on frequency v0 in terms of the ~uni￾valent! plasma frequency vp of the medium, given by vp 2 54pNe2 m , ~1! where N is the number density of atoms, and e and m are the charge and mass of an electron. Give a condition on the line separation D compared to the linewidth g such that the group velocity vg(v0) is negative. In a recent experiment by Wang et al., 4 a group velocity of vg52c/310, where c is the speed of light in vacuum, was demonstrated in cesium vapor using a pair of spectral lines with separation D/2p'2 MHz and linewidth g/2p '0.8 MHz. B. Propagation of a monochromatic plane wave Consider a wave with electric field E0eiv(z/c2t) that is incident from z,0 on a medium that extends from z50 to a. Ignore reflection at the boundaries, as is reasonable if the index of refraction n(v) is near unity. Particularly simple results can be obtained when you make the ~unphysical! as￾sumption that the vn(v) varies linearly with frequency about a central frequency v0 . Deduce a transformation that has a frequency-dependent part and a frequency-independent part between the phase of the wave for z,0 to that of the wave inside the medium, and to that of the wave in the region a,z. C. Fourier analysis Apply the transformations between an incident monochro￾matic wave and the wave in and beyond the medium to the Fourier analysis of an incident pulse of form f(z/c2t). D. Propagation of a sharp wave front In the approximation that vn varies linearly with v, de￾duce the waveforms in the regions 0,z,a and a,z for an incident pulse d(z/c2t), where d is the Dirac delta function. Show that the pulse emerges out of the gain region at z5a at time t5a/vg , which appears to be earlier than when it enters this region if the group velocity is negative. Show also that inside the negative group velocity medium a pulse propa￾gates backwards from z5a at time t5a/vg,0 to z50 at t 50, at which time it appears to annihilate the incident pulse. E. Propagation of a Gaussian pulse As a more physical example, deduce the waveforms in the regions 0,z,a and a,z for a Gaussian incident pulse E0e2(z/c2t)2/2t 2 eiv0(z/c2t) . Carry the frequency expansion of vn(v) to second order to obtain conditions of validity of the analysis such as maximum pulse width t, maximum length a of the gain region, and maximum time of advance of the emerging pulse. Consider the time required to generate a pulse of rise time t when assessing whether the time advance in a negative group velocity medium can lead to superlumi￾nal signal propagation. II. SOLUTION The concept of group velocity appears to have been first enunciated by Hamilton in 1839 in lectures of which only abstracts were published.5 The first recorded observation of the group velocity of a ~water! wave is due to Russell in 1844.6 However, widespread awareness of the group velocity dates from 1876 when Stokes used it as the topic of a Smith’s Prize examination paper.7 The early history of group velocity has been reviewed by Havelock.8 H. Lamb9 credits A. Schuster with noting in 1904 that a negative group velocity, i.e., a group velocity of opposite sign to that of the phase velocity, is possible due to anoma￾lous dispersion. Von Laue10 made a similar comment in 1905. Lamb gave two examples of strings subject to external potentials that exhibit negative group velocities. These early considerations assumed that in case of a wave with positive group and phase velocities incident on the anomalous me￾dium, energy would be transported into the medium with a positive group velocity, and so there would be waves with negative phase velocity inside the medium. Such negative phase velocity waves are formally consistent with Snell’s 607 Am. J. Phys. 69 ~5!, May 2001 http://ojps.aip.org/ajp/ © 2001 American Association of Physics Teachers 607

law(since 0,=sin[(n /n )sin 0] can be in either the first 0.000002 or second quadrant), but they seemed physically implausible Re(n·1) and the topic was largely dropped Present interest in negative group velocity is based nomalous dispersion in a gain medium, where the sign of the phase velocity is the same for incident and transmitted 0 and energy flows inside the gain medium in the op- posite direction to the incident energy flow in vacuum The propagation of electromagnetic waves at frequencies near those of spectral lines of a medium was first extensively -0.000002 discussed by Sommerfeld and Brillouin, with emphasis on frequency the distinction between signal velocity and group velocity hen the latter exceeds c. The solution presented here is Fig. 1. The real and imaginary parts of the index of refraction in a medium based on the work of Garrett and McCumber 3 as extended lines are separated by angular frequency A and have widths y=0.44.The with two spectral lines that have been pumped to inverted populations by Chiao et al. A discussion of negative group velocity in electronic circuits has been given by mitchell and chiao force -myi, where the dot indicates differentiation with A Negative group velocity respect to time. The equation of motion in the presence of an electromagnetic wave of frequency o is In a medium of index of refraction n(o), the dispersion elation can be written xty r+ox= (2)Hence where k is the wave number. The group velocity is then 6 2+iyo (7) Ug=Rel dk"Reldk/do and the polarizability m (ol-)+xo Reld(on)/do n+oReldn/dol (3) In the present problem we have two spectral lines,012 We see from Eq. (3) that if the index of refraction de- (o+A/2, both of oscillator strength -I to indicate that the creases rapidly enough with frequency, the group velocity populations of both lines are inverted, with damping con- can be negative. It is well known that the index of refraction stants y1=y2=y. In this case, the polarizability is given by decreases rapidly with frequency near an absorption line e2(a0-△/2)2-a2+iyo where"anomalous'" wave propagation effects can occur. However, the absorption makes it difficult to study these (-△/2)2-2)2+ effects. The insight of Garrett and Mc Cumber> and of Chiao et al. 4, 5, 17-19is that demonstrations of negative group ve locity are possible in media with inverted populations that gain rather than absorption occurs at the frequencies of interest. This was dramatically realized in the experiment of perhaps first suggested by Steinberg and Chiao Tn lines, as Wang et al. by use of a closely spaced pair of gar m(o6-△oo-a2)2+y e2o3+2△oo-a2+iyo We use a classical oscillator model for the index of refrac- 2,22,22 tion. The index n is the square root of the dielectric constant e, which is in turn related to the atomic polarizability a ac- where the approximation is obtained by the neglect of terms cording to D=E=E+4丌P=E(1+4丌Na) (4) For a probe beam at frequency o, the index of refraction n Gaussian units), where D is the electric displacement, E is (5)has the form he electric field, and P is the polarization density. Then, the index of refraction of a dilute gas The polarizability a is obtained from the electric dipole moment p=ex=aE induced by electric field E In the ca of a single spectral line of frequency oj, we say that an where ap is the plasma frequency given by Eq (1). This is electron is bound to the(fixed) nucleus by a spring of con- illustrated in Fig. 1 stant K=mof, and that the motion is subject to the damping The index at the central frequency oo is Am J. Phys., Vol. 69, No. 5, May 2001 New Problems

law11 ~since u t5sin21 @(ni /nt )sin ui# can be in either the first or second quadrant!, but they seemed physically implausible and the topic was largely dropped. Present interest in negative group velocity is based on anomalous dispersion in a gain medium, where the sign of the phase velocity is the same for incident and transmitted waves, and energy flows inside the gain medium in the op￾posite direction to the incident energy flow in vacuum. The propagation of electromagnetic waves at frequencies near those of spectral lines of a medium was first extensively discussed by Sommerfeld and Brillouin,12 with emphasis on the distinction between signal velocity and group velocity when the latter exceeds c. The solution presented here is based on the work of Garrett and McCumber,13 as extended by Chiao et al.14,15 A discussion of negative group velocity in electronic circuits has been given by Mitchell and Chiao.16 A. Negative group velocity In a medium of index of refraction n(v), the dispersion relation can be written k5 vn c , ~2! where k is the wave number. The group velocity is then given by vg5ReF dv dk G 5 1 Re@dk/dv# 5 c Re@d~vn!/dv# 5 c n1v Re@dn/dv# . ~3! We see from Eq. ~3! that if the index of refraction de￾creases rapidly enough with frequency, the group velocity can be negative. It is well known that the index of refraction decreases rapidly with frequency near an absorption line, where ‘‘anomalous’’ wave propagation effects can occur.12 However, the absorption makes it difficult to study these effects. The insight of Garrett and McCumber13 and of Chiao et al.14,15,17–19 is that demonstrations of negative group ve￾locity are possible in media with inverted populations, so that gain rather than absorption occurs at the frequencies of interest. This was dramatically realized in the experiment of Wang et al.4 by use of a closely spaced pair of gain lines, as perhaps first suggested by Steinberg and Chiao.17 We use a classical oscillator model for the index of refrac￾tion. The index n is the square root of the dielectric constant e, which is in turn related to the atomic polarizability a ac￾cording to D5eE5E14pP5E~114pNa! ~4! ~in Gaussian units!, where D is the electric displacement, E is the electric field, and P is the polarization density. Then, the index of refraction of a dilute gas is n5Ae'112pNa. ~5! The polarizability a is obtained from the electric dipole moment p5ex5aE induced by electric field E. In the case of a single spectral line of frequency vj , we say that an electron is bound to the ~fixed! nucleus by a spring of con￾stant K5mv j 2 , and that the motion is subject to the damping force 2mg jx˙, where the dot indicates differentiation with respect to time. The equation of motion in the presence of an electromagnetic wave of frequency v is x¨1g jx˙1v j 2 x5 eE m 5 eE0 m eivt . ~6! Hence, x5 eE m 1 v j 2 2v22ig jv5 eE m v j 2 2v21ig jv ~v j 2 2v2! 21g j 2 v2 , ~7! and the polarizability is a5 e2 m v j 2 2v21ig jv ~v j 2 2v2! 21g j 2 v2 . ~8! In the present problem we have two spectral lines, v1,2 5v06D/2, both of oscillator strength 21 to indicate that the populations of both lines are inverted, with damping con￾stants g15g25g. In this case, the polarizability is given by a52 e2 m ~v02D/2! 22v21igv ~ ~v02D/2! 22v2! 21g2v2 2 e2 m ~v01D/2! 22v21igv ~ ~v01D/2! 22v2! 21g2v2 '2 e2 m v0 2 2Dv02v21igv ~v0 2 2Dv02v2! 21g2v2 2 e2 m v0 2 12Dv02v21igv ~v0 2 1Dv02v2! 21g2v2 , ~9! where the approximation is obtained by the neglect of terms in D2 compared to those in Dv0 . For a probe beam at frequency v, the index of refraction ~5! has the form n~v!'12 vp 2 2 F v0 2 2Dv02v21igv ~v0 2 2Dv02v2! 21g2v2 1 v0 2 1Dv02v21igv ~v0 2 1Dv02v2! 21g2v2G , ~10! where vp is the plasma frequency given by Eq. ~1!. This is illustrated in Fig. 1. The index at the central frequency v0 is Fig. 1. The real and imaginary parts of the index of refraction in a medium with two spectral lines that have been pumped to inverted populations. The lines are separated by angular frequency D and have widths g50.4D. 608 Am. J. Phys., Vol. 69, No. 5, May 2001 New Problems 608

r/ap (17) 75,0.25) A value of ug -c/10 as in the experiment of Wang cor- △3/ responds to A/o,1/12. In this case, the gain length For later use we record the second derivative ny(3△ ≈24i (18) where the second approxi holds if y≤Δ Fie egallvewed region(14) in(4 r) space such that the group ve. B. Propagation of a monochromatic plane wave To illustrate the optical properties of a medium with ive group velocity, we consider the propagation of an elec tromagnetic wave through it. The medium extends from z y n(0)≈1-i 0 to a, and is surrounded by vacuum. Because the index of refraction (10) is near unity in the frequency range of inter- where the second approximation holds when y<A. The est, we ignore reflections at the boundaries of the medium electric A monochromatic plane wave of frequency o and incident field of a continuous probe wave then propagates from :<0 propagates with phase velocity c in vacuum. Its according to electric field can be written E(, t=ei(k=-oon)=elo(n(wo=/c-n E(=, n=Eoe-e-or(=<0 ≈e4clya2p)eao(=-t) 12) Inside the medium this wave propagates with phase velocity From this we see that at frequency ao the phase velocity is c cIn(o)according to and the medium has an amplitude gain length 42c/yo E(, t)=Eoelonilce-iot (0<:<a) To obtain the group velocity(3)at fre the derivative where the amplitude is unchanged since we neglect the small reflection at the boundary ==0. When the wave emerges into d(on) 2o(△ vacuum at ==a, the phase velocity is again c, but it has (13) accumulated a phase lag of (o/c)(n-1)a, and so appears a E(, t=eoe where we have neglected terms in A and y compared to wo if n Eq (3), we see that the group velocity can be negative F1 E (21) It is noteworthy that a monochromatic wa >a has the same form as that inside the medium make the (14) frequency-independent substitutions The boundary of the allowed region(14)in(42, Y)space is (22) a parabola whose axis is along the line y? in Fig. 2. For the physical region y=0, the boundary is as shown Since an arbitrary wave form can be expressed in terms of given by monochromatic plane waves via Fourier analysis, we can use these substitutions to convert any wave in the region 0<= (15)a to its continuation in the A general relation can be deduced in the case where the second and higher derivatives of on(o) are very small.We Thus, to have a negative group velocity, we must have can then write which limit is achieved when y=0; the maximum value of y is0.5 (. when△=08660 where ug is the group velocity for a pulse with central fre- Near the boundary of the negative group velocity region, quency o. Using this in Eq(20), we have Ugl exceeds c, which alerts us to concerns of superluminal behavior. However. as will be seen in the following sections E(E,tsEoelog-((uollc-lgelocluge -iot (0<x<a the effect of a negative group velocity is more dramatic when (24) u, is small rather than large In this approximation, the Fourier component E(=)at fre- The region of recent experimental interest is y<A<op, quency w of a wave inside the gain medium is related to that for which Eqs. (3)and (13)predict that of the incident wave by replacing the frequency dependence Am. J. Phys., Vol. 69, No. 5, May 2001 New Problems

n~v0!'12i vp 2 g ~D21g2!v0 '12i vp 2 D2 g v0 , ~11! where the second approximation holds when g!D. The electric field of a continuous probe wave then propagates according to E~z,t!5ei~kz2v0t! 5eiv~n~v0!z/c2t! 'ez/@D2c/gv~2/p!# eiv0~z/c2t! . ~12! From this we see that at frequency v0 the phase velocity is c, and the medium has an amplitude gain length D2c/gvp 2 . To obtain the group velocity ~3! at frequency v0 , we need the derivative d~vn! dv U v0 '12 2vp 2 ~D22g2! ~D21g2! 2 , ~13! where we have neglected terms in D and g compared to v0 . From Eq. ~3!, we see that the group velocity can be negative if D2 vp 22 g2 vp 2 > 1 2 S D2 vp 2 1 g2 vp 2 D 2 . ~14! The boundary of the allowed region ~14! in (D2,g2) space is a parabola whose axis is along the line g252D2, as shown in Fig. 2. For the physical region g2>0, the boundary is given by g2 vp 2 5A114 D2 vp 2212 D2 vp 2 . ~15! Thus, to have a negative group velocity, we must have D<&vp , ~16! which limit is achieved when g50; the maximum value of g is 0.5vp when D50.866vp . Near the boundary of the negative group velocity region, uvgu exceeds c, which alerts us to concerns of superluminal behavior. However, as will be seen in the following sections, the effect of a negative group velocity is more dramatic when uvgu is small rather than large. The region of recent experimental interest is g!D!vp , for which Eqs. ~3! and ~13! predict that vg'2 c 2 D2 vp 2 . ~17! A value of vg'2c/310 as in the experiment of Wang cor￾responds to D/vp'1/12. In this case, the gain length D2c/gvp 2 was approximately 40 cm. For later use we record the second derivative, d2 ~vn! dv2 U v0 '8i vp 2 g~3D22g2! ~D21g2! 3 '24i vp 2 D2 g D2 , ~18! where the second approximation holds if g!D. B. Propagation of a monochromatic plane wave To illustrate the optical properties of a medium with nega￾tive group velocity, we consider the propagation of an elec￾tromagnetic wave through it. The medium extends from z 50 to a, and is surrounded by vacuum. Because the index of refraction ~10! is near unity in the frequency range of inter￾est, we ignore reflections at the boundaries of the medium. A monochromatic plane wave of frequency v and incident from z,0 propagates with phase velocity c in vacuum. Its electric field can be written Ev~z,t!5E0eivz/c e2ivt ~z,0!. ~19! Inside the medium this wave propagates with phase velocity c/n(v) according to Ev~z,t!5E0eivnz/c e2ivt ~0,z,a!, ~20! where the amplitude is unchanged since we neglect the small reflection at the boundary z50. When the wave emerges into vacuum at z5a, the phase velocity is again c, but it has accumulated a phase lag of (v/c)(n21)a, and so appears as Ev~z,t!5E0eiva~n21!/c eivz/c e2ivt 5E0eivan/c e2iv~t2~z2a!/c! ~a,z!. ~21! It is noteworthy that a monochromatic wave for z.a has the same form as that inside the medium if we make the frequency-independent substitutions z→a, t→t2 z2a c . ~22! Since an arbitrary waveform can be expressed in terms of monochromatic plane waves via Fourier analysis, we can use these substitutions to convert any wave in the region 0,z ,a to its continuation in the region a,z. A general relation can be deduced in the case where the second and higher derivatives of vn(v) are very small. We can then write vn~v!'v0n~v0!1 c vg ~v2v0!, ~23! where vg is the group velocity for a pulse with central fre￾quency v0 . Using this in Eq. ~20!, we have Ev~z,t!'E0eiv0z~n~v0!/c21/vg! eivz/vge2ivt ~0,z,a!. ~24! In this approximation, the Fourier component Ev(z) at fre￾quency v of a wave inside the gain medium is related to that of the incident wave by replacing the frequency dependence Fig. 2. The allowed region ~14! in (D2,g2) space such that the group ve￾locity is negative. 609 Am. J. Phys., Vol. 69, No. 5, May 2001 New Problems 609

A八 Vacuum Negative group velocity medium Fig 3. A snapshot of three Fourier components of a pulse in the vicinity of a negative group velocity medium. The com is unaltered by the medium, but the wavelength of a longer wavelength component is shortened, and that of a shorter wave Then, even when the incident pulse has not yet reached the medu cuum region beyond the medium at which the Fourier components are again all in phase, and a third peak appears. The peaks in the vacuum regions move with group velocity u, =c, but the peak inside the medium moves with a negative group elocity, shown as ux=-cl2. The phase velocity Up is c in vacuum, and close to c in the medium g,1.e. eplacin Elc by =/ug, and multi- to Eq(19), we see that the peak occurs when ==ct. As plying by the frequency-independent phase factor usual, we say that the group velocity of this wave is c in elog(n(oo)/c-l/g). Then, using transformation(22), the wave vacuum at emerges into vacuum beyond the medium Is Inside the medium, Eq(24)describes the phases of th E(E, n)sEoelooa(n(ao)/c-l/ug) components, which all have a common frequency independent phase wo=(n(oo)/c-1/ug) at a given =, as well Xe o(lc-a(le- -ior (a<= (25) as a frequency-dependent part o(=/vg -t). The peak of the The wave beyond the medium is related to the incident wave pulse occurs when all the frequency-dependent phases van- by multiplying by a frequency-independent phase, and by ish, the overall frequency-independent phase does not affect placing :/c by =/c-a(l/c-1/ug) in the frequency the pulse size. Thus, the peak of the p agates within Eqs.(24)and(25)has been called"rephasing. "1 1bed by vg, the group velocity of he et. The velocity of the pea dependent part of the phase the medium according to The "rephasing"(24)within the medium changes the C. Fourier analysis and"rephasing wavelengths of the component waves. Typically the wave length increases, and by greater amounts at longer wave The transformations between the monochromatic incident lengths A longer time is required before the phases of the wave(19)and its continuation in and beyond the medium, waves all become the same at some point inside the me- (24)and (25), imply that an incident wave dium, so in a normal medium the velocity of the peak ap- E(,n)=f(=lc-1)=E()e iot do (<0),(26) pears to be slowed down. But in a negative group velocity medium, wavelengths short compared to A lengthen whose Fourier components are given by long waves are shortened, and the velocity of the peak ap- pears to be reversed By a similar argument, Eq (25)tells us that in the vacuum Ea(=) E(, er dt (27) region beyond the medium the peak of the pulse propagates according to ==ct+a(llc-llug). The group velocity is again c, but the"rephasing within the medium results in a f(-/c-1)(<0) shift of the position of the peak by the amount a(1/c normal medium where 0<usc the shift is 2f(-/ug-1)(0<:<a) E(,1)≈ elooa(n(o/c-l/gf(Elc-t-a( llc-llvg))(<8) negative; the pulse appears to have been delayed during its passage through the medium. But after a negative group ve locity medium, the pulse appears to have advanced This advance is possible because, in the Fourier view follows. Fouras tion of Eq(28)in terms of"rephasing is as each component wave extends over all space, even if the An interpre tude of a pulse made of waves of many frequencies, each of negative group velocity medium shifts the phases of the fre- he form E(,t=Eo(o)els(o)=Eo with Eo>0, is determined by adding the amplitudes Eo(o). the nominal peak such that the phases all coincide, and a This maximum is achieved only if there exist points(, n) peak is observed, at times earlier than expected at points such that all phases (o) have the same value beyond the medium frequencies vanish, as shown at the left of Fig. 3. Referring neous appearance of peaks in all three regione examples For example, we consider a pulse in the region :<0 As shown in Fig. 3 and further illustrated in the whose maximum occurs when the phases of all component in the following, the"rephasing can result in the simulta- 610 Am. J. Phys., Vol. 69, No. 5, May 2001 New Problems 610

eivz/c by eivz/vg, i.e., by replacing z/c by z/vg , and multi￾plying by the frequency-independent phase factor eiv0z(n(v0)/c21/vg) . Then, using transformation ~22!, the wave that emerges into vacuum beyond the medium is Ev~z,t!'E0eiv0a~n~v0!/c21/vg! 3eiv~z/c2a~1/c21/vg! !e2ivt ~a,z!. ~25! The wave beyond the medium is related to the incident wave by multiplying by a frequency-independent phase, and by replacing z/c by z/c2a(1/c21/vg) in the frequency￾dependent part of the phase. The effect of the medium on the wave as described by Eqs. ~24! and ~25! has been called ‘‘rephasing.’’ 4 C. Fourier analysis and ‘‘rephasing’’ The transformations between the monochromatic incident wave ~19! and its continuation in and beyond the medium, ~24! and ~25!, imply that an incident wave E~z,t!5 f~z/c2t!5 E 2` ` Ev~z!e2ivt dv ~z,0!, ~26! whose Fourier components are given by Ev~z!5 1 2p E 2` ` E~z,t!eivt dt, ~27! propagates as E~z,t!' 5 f~z/c2t! ~z,0! eiv0z~n~v0!/c21/vg! f~z/vg2t! ~0,z,a! eiv0a~n~v0!/c21/vg! f~z/c2t2a~1/c21/vg! ! ~a,z!. ~28! An interpretation of Eq. ~28! in terms of ‘‘rephasing’’ is as follows. Fourier analysis tells us that the maximum ampli￾tude of a pulse made of waves of many frequencies, each of the form Ev(z,t)5E0(v)eif(v) 5E0(v)ei(k(v)z2vt1f0(v)) with E0>0, is determined by adding the amplitudes E0(v). This maximum is achieved only if there exist points ~z,t! such that all phases f~v! have the same value. For example, we consider a pulse in the region z,0 whose maximum occurs when the phases of all component frequencies vanish, as shown at the left of Fig. 3. Referring to Eq. ~19!, we see that the peak occurs when z5ct. As usual, we say that the group velocity of this wave is c in vacuum. Inside the medium, Eq. ~24! describes the phases of the components, which all have a common frequency￾independent phase v0z(n(v0)/c21/vg) at a given z, as well as a frequency-dependent part v(z/vg2t). The peak of the pulse occurs when all the frequency-dependent phases van￾ish; the overall frequency-independent phase does not affect the pulse size. Thus, the peak of the pulse propagates within the medium according to z5vgt. The velocity of the peak is vg , the group velocity of the medium, which can be nega￾tive. The ‘‘rephasing’’ ~24! within the medium changes the wavelengths of the component waves. Typically the wave￾length increases, and by greater amounts at longer wave￾lengths. A longer time is required before the phases of the waves all become the same at some point z inside the me￾dium, so in a normal medium the velocity of the peak ap￾pears to be slowed down. But in a negative group velocity medium, wavelengths short compared to l0 are lengthened, long waves are shortened, and the velocity of the peak ap￾pears to be reversed. By a similar argument, Eq. ~25! tells us that in the vacuum region beyond the medium the peak of the pulse propagates according to z5ct1a(1/c21/vg). The group velocity is again c, but the ‘‘rephasing’’ within the medium results in a shift of the position of the peak by the amount a(1/c 21/vg). In a normal medium where 0,vg<c the shift is negative; the pulse appears to have been delayed during its passage through the medium. But after a negative group ve￾locity medium, the pulse appears to have advanced! This advance is possible because, in the Fourier view, each component wave extends over all space, even if the pulse appears to be restricted. The unusual ‘‘rephasing’’ in a negative group velocity medium shifts the phases of the fre￾quency components of the wave train in the region ahead of the nominal peak such that the phases all coincide, and a peak is observed, at times earlier than expected at points beyond the medium. As shown in Fig. 3 and further illustrated in the examples in the following, the ‘‘rephasing’’ can result in the simulta￾neous appearance of peaks in all three regions. Fig. 3. A snapshot of three Fourier components of a pulse in the vicinity of a negative group velocity medium. The component at the central wavelength l0 is unaltered by the medium, but the wavelength of a longer wavelength component is shortened, and that of a shorter wavelength component is lengthened. Then, even when the incident pulse has not yet reached the medium, there can be a point inside the medium at which all components have the same phase, and a peak appears. Simultaneously, there can be a point in the vacuum region beyond the medium at which the Fourier components are again all in phase, and a third peak appears. The peaks in the vacuum regions move with group velocity vg5c, but the peak inside the medium moves with a negative group velocity, shown as vg52c/2. The phase velocity vp is c in vacuum, and close to c in the medium. 610 Am. J. Phys., Vol. 69, No. 5, May 2001 New Problems 610

D. Propagation of a sharp wave front Gain To assess the effect of a medium with negative group ve- locity on the propagation of a signal, we first consider a waveform with a sharp front, as recommended by Sommer- feld and brillouin As an extreme but convenient example, we take the inci- dent pulse to be a Dirac delta function, E(=, t)=Eod(/c 1). Inserting this in Eq.(28), which is based on the line pproximation(23), we find Gain E0b(/c-1)(x<0) E(2((o-b/g-0)(0<=≤a) -t-a(llc-1lug) Gain Gain According to Eq(29), the delta-function pulse emerges from the medium at ==a at time t=a/ug. If the group ve- locity is negative, the pulse emerges from the medium before it enters at t=0! e am ple histons de ( ssia e use r opeagatin is d lum. an(anti)pulse propagates backwards in space from ==a time t=alux<0 to ==0 at time t=0, at which point it pears to annihilate the incident pulse This behavior is analogous to barrier penetration by a rela- vistic electron- in which an electron can emerge from the far side of the barrier earlier than it hits the near side. if the electron emission at the far side is accompanied by positron emission, and the positron propagates within the barrier so as to annihilate the incident electron at the near side. in the Wheeler-Feynman view, this process involves only a single electron which propagates backwards in time when inside the barrier. In this spirit, we might say that pulses propagate backwards in time(but forward in space) inside a negative group velocity medium Gain The Fourier components of the delta function are indepen dent of frequency, so the advanced appearance of the sharp wave front as described by Eq. (29)can occur only for a gain medium such that the index of refraction varies linearly at all frequencies. If such a medium existed with negative slope dn/do, then Eq.(29) would constitute superluminal signal ation owever, from Fig. I we see that a linear approximation to the index of refraction is reasonable in the negative group velocity medium only for o-woIsA/2. The sharpest wave front that can be supported within this bandwidth has char- 200·1501005005010015020025 acteristic rise time T=1/A For the experiment of Wang et al. where A/2T 10 Hz, Fig 4. Ten"snapshots"of a Gaussian pulse as it traverses a negative group an analysis based on Eq (23)would be valid only for pulses velocity region(O<=<50), according to Eq (5). The group velocity in the with r20.I us. Wang et al. used a pulse with ra l us, close to the minimum value for which Eq (23)is a reason- able approximation Since a negative group velocity can only be experienced E. Propagation of a Gaussian pulse over a limited bandwidth, very sharp wave fronts must be excluded from the discussion of signal propagation. How- We now consider a Gaussian pulse of temporal length T ever,it is well knownthat great care must be taken when centered on frequency wo(the carrier frequency), for which discussing the signal velocity if the waveform is not sharp. the incident waveform is Am. J. Phys., Vol. 69, No. 5, May 2001 New Problems

D. Propagation of a sharp wave front To assess the effect of a medium with negative group ve￾locity on the propagation of a signal, we first consider a waveform with a sharp front, as recommended by Sommer￾feld and Brillouin.12 As an extreme but convenient example, we take the inci￾dent pulse to be a Dirac delta function, E(z,t)5E0d(z/c 2t). Inserting this in Eq. ~28!, which is based on the linear approximation ~23!, we find E~z,t!' 5 E0d~z/c2t! ~z,0! E0eiv0z~n~v0!/c21/vg! d~z/vg2t! ~0,z,a! E0eiv0a~n~v0!/c21/vg! d~z/c2t2a~1/c21/vg! ! ~a,z!. ~29! According to Eq. ~29!, the delta-function pulse emerges from the medium at z5a at time t5a/vg . If the group ve￾locity is negative, the pulse emerges from the medium before it enters at t50! A sample history of ~Gaussian! pulse propagation is illus￾trated in Fig. 4. Inside the negative group velocity medium, an ~anti!pulse propagates backwards in space from z5a at time t5a/vg,0 to z50 at time t50, at which point it ap￾pears to annihilate the incident pulse. This behavior is analogous to barrier penetration by a rela￾tivistic electron20 in which an electron can emerge from the far side of the barrier earlier than it hits the near side, if the electron emission at the far side is accompanied by positron emission, and the positron propagates within the barrier so as to annihilate the incident electron at the near side. In the Wheeler–Feynman view, this process involves only a single electron which propagates backwards in time when inside the barrier. In this spirit, we might say that pulses propagate backwards in time ~but forward in space! inside a negative group velocity medium. The Fourier components of the delta function are indepen￾dent of frequency, so the advanced appearance of the sharp wave front as described by Eq. ~29! can occur only for a gain medium such that the index of refraction varies linearly at all frequencies. If such a medium existed with negative slope dn/dv, then Eq. ~29! would constitute superluminal signal propagation. However, from Fig. 1 we see that a linear approximation to the index of refraction is reasonable in the negative group velocity medium only for uv2v0u&D/2. The sharpest wave front that can be supported within this bandwidth has char￾acteristic rise time t'1/D. For the experiment of Wang et al. where D/2p'106 Hz, an analysis based on Eq. ~23! would be valid only for pulses with t*0.1 ms. Wang et al. used a pulse with t'1 ms, close to the minimum value for which Eq. ~23! is a reason￾able approximation. Since a negative group velocity can only be experienced over a limited bandwidth, very sharp wave fronts must be excluded from the discussion of signal propagation. How￾ever, it is well known12 that great care must be taken when discussing the signal velocity if the waveform is not sharp. E. Propagation of a Gaussian pulse We now consider a Gaussian pulse of temporal length t centered on frequency v0 ~the carrier frequency!, for which the incident waveform is Fig. 4. Ten ‘‘snapshots’’ of a Gaussian pulse as it traverses a negative group velocity region (0,z,50), according to Eq. ~31!. The group velocity in the gain medium is vg52c/2, and c has been set to 1. 611 Am. J. Phys., Vol. 69, No. 5, May 2001 New Problems 611