E(1=Eoe-(//ce -igf (=<0) (30) where A is evaluated at ==a here. As expected, the forms (36) and (38)revert to those of Eq. (31)when Inserting this in Eq. (28)we find d-(on(oo))/de E So long as the factor A(a) is not greatly different from unity, the pulse emerges from the medium essentially undis eoe (0<二<a) torted, which requires E(=,t)2e 1△2△ T Xelewoflc-n)(a<=) (31) using Egs.(18)and(37). In the experiment of Wang et al The factor elega m( oo)-Ic in Eq. (31)for a<: becomes this condition is that a/cT< 1/120, which was well satisfied e"pyala using Eq(11), and represents a small gain due to with a=6 cm and cT=300m As in the case of the delta function the centroid of a traversing the negative group velocity medium. In the experI- Gaussian pulse emerges from a negative group velocity me- nent of Wang et al, this factor was only 1.16 We have already noted in the previous section that the dium at time ear approximation to n(o) is only good over a fre quency interval about wo of order A, and so Eq. (31) for the pulse after the gain medium applies only for pulse widths which is earlier than the time t=0 when the centroid enters vance of the pulse was a/v I-=300/c≈6×10-8s Further constraints on the validity of Eq. (31) eB(32) the medium. In the experiment of Wang et al., the time ≈0.067 be ob- If one attempts to observe the negative group velocity tained using the expansion of on( o)to second order. For pulse inside the medium, the incident wave would be per- this, we repeat the derivation of Eq. (31) in slightly more turbed and the backwards-movi composition(27) tected. Rather, one must deduce the effect of the negative group velocity medium by observation of the pulse that emerges into the region =>a beyond that medium, where the E。() (o-wonele=lc(=<0 (33) significance of the time advance(40)is the main issue The time advance caused by a negative group velocity We again extrapolate the Fourier component at frequency o medium is larger when lug is smaller. It is possible that into the region :>0 using Eq (20), which yields JUd>c, but this gives a smaller time advance than when the negative group velocity is such that v I<c. Hence, there is E(E)=5Eoe-2(o-wopeianslc (0<=<a).(34) no special concern as to the meaning of negative group ve- The maximum possible time advance fmax by this tech We now appro he factor on(o)by its Taylor ex- nique can be estimated from Eqs.(17), (39), and(40)as order: 1△ The pulse can advance by at most a few rise times due to I d(on) (35) passage through the negative group velocity medium While this aspect of the pulse propagation appears to be uperluminal, it does not imply superluminal signal propaga- With this, we find from Eqs. (26)and(34)that In accounting for signal propagation time, the time needed E(e-6-nn/mole- E to generate the signal must be included as well. A pulse with a finite frequency bandwidth A takes at least time T 1/4 to be generated, and so is delay ed by a time of order of its rise (36) time compared to the case of an idealized sharp wave front here Thus, the advance of a pulse front in a negative group veloc- ity medium by sr can at most compensate for the original A2(z)=1- d(on) delay in generating that pulse. The signal velocity, as defined (7) by the path length between the source and detector divided by the overall time from onset of signal generation to signal The waveform for =>a is obtained from that for 0<=<a by detection, remains bounded by c the substitutions(22)with the result As has been emphasized by Garrett and Mc Cumbe Chiao,8,I% the time advance of a pulse emerging from a gain medium Is poss (二,1) Todo(6o)-1)/ce-(elc-a(lc-1/ug)-0224222 pulse gives advance warning of the later arrival of the peak The leading edge of the pulse can be amplified by the gain Xe0-e-io'(a<- (38) medium, which gives the appearance of superluminal pulse Am J. Phys., Vol. 69, No. 5, May 2001 New Problems 612E~z,t!5E0e2~z/c2t! 2/2t 2 eiv0z/c e2iv0t ~z,0!. ~30! Inserting this in Eq. ~28! we find E~z,t!5 5 E0e2~z/c2t! 2/2t 2 eiv0~z/c2t! ~z,0! E0e2~z/vg2t! 2/2t 2 eiv0~n~v0!z/c2t! ~0,z,a! E0eiv0a~n~v0!21!/c e2~z/c2a~1/c21/vg!2t! 2/2t 2 3eiv0~z/c2t! ~a,z!. ~31! The factor eiv0a(n(v0)21)/c in Eq. ~31! for a,z becomes ev p 2 ga/D2c using Eq. ~11!, and represents a small gain due to traversing the negative group velocity medium. In the experi￾ment of Wang et al., this factor was only 1.16. We have already noted in the previous section that the linear approximation to vn(v) is only good over a fre￾quency interval about v0 of order D, and so Eq. ~31! for the pulse after the gain medium applies only for pulse widths t* 1 D . ~32! Further constraints on the validity of Eq. ~31! can be ob￾tained using the expansion of vn(v) to second order. For this, we repeat the derivation of Eq. ~31! in slightly more detail. The incident Gaussian pulse ~30! has the Fourier de￾composition ~27!, Ev~z!5 t A2p E0e2t 2~v2v0! 2/2eivz/c ~z,0!. ~33! We again extrapolate the Fourier component at frequency v into the region z.0 using Eq. ~20!, which yields Ev~z!5 t A2p E0e2t 2~v2v0! 2/2eivnz/c ~0,z,a!. ~34! We now approximate the factor vn(v) by its Taylor ex￾pansion through second order: vn~v!'v0n~v0!1 c vg ~v2v0! 1 1 2 d2 ~vn! dv2 U v0 ~v2v0! 2. ~35! With this, we find from Eqs. ~26! and ~34! that E~z,t!5 E0 A e2~z/vg2t! 2/2A2t 2 eiv0n~v0!z/c e2iv0t ~0,z,a!. ~36! where A2 ~z!512i z ct2 d2 ~vn! dv2 U v0 . ~37! The waveform for z.a is obtained from that for 0,z,a by the substitutions ~22! with the result E~z,t!5 E0 A eiv0a~n~v0!21!/c e2~z/c2a~1/c21/vg!2t! 2/2A2t 2 3eiv0z/c e2iv0t ~a,z!, ~38! where A is evaluated at z5a here. As expected, the forms ~36! and ~38! revert to those of Eq. ~31! when d2(vn(v0))/dv250. So long as the factor A(a) is not greatly different from unity, the pulse emerges from the medium essentially undis￾torted, which requires a ct ! 1 24 D2 vp 2 D g Dt, ~39! using Eqs. ~18! and ~37!. In the experiment of Wang et al., this condition is that a/ct!1/120, which was well satisfied with a56 cm and ct5300 m. As in the case of the delta function, the centroid of a Gaussian pulse emerges from a negative group velocity me￾dium at time t5 a vg ,0, ~40! which is earlier than the time t50 when the centroid enters the medium. In the experiment of Wang et al., the time ad￾vance of the pulse was a/uvgu'300a/c'631028 s '0.06t. If one attempts to observe the negative group velocity pulse inside the medium, the incident wave would be per￾turbed and the backwards-moving pulse would not be de￾tected. Rather, one must deduce the effect of the negative group velocity medium by observation of the pulse that emerges into the region z.a beyond that medium, where the significance of the time advance ~40! is the main issue. The time advance caused by a negative group velocity medium is larger when uvgu is smaller. It is possible that uvgu.c, but this gives a smaller time advance than when the negative group velocity is such that uvgu,c. Hence, there is no special concern as to the meaning of negative group ve￾locity when uvgu.c. The maximum possible time advance tmax by this tech￾nique can be estimated from Eqs. ~17!, ~39!, and ~40! as tmax t ' 1 12 D g Dt'1. ~41! The pulse can advance by at most a few rise times due to passage through the negative group velocity medium. While this aspect of the pulse propagation appears to be superluminal, it does not imply superluminal signal propaga￾tion. In accounting for signal propagation time, the time needed to generate the signal must be included as well. A pulse with a finite frequency bandwidth D takes at least time t'1/D to be generated, and so is delayed by a time of order of its rise time t compared to the case of an idealized sharp wave front. Thus, the advance of a pulse front in a negative group veloc￾ity medium by &t can at most compensate for the original delay in generating that pulse. The signal velocity, as defined by the path length between the source and detector divided by the overall time from onset of signal generation to signal detection, remains bounded by c. As has been emphasized by Garrett and McCumber13 and by Chiao,18,19 the time advance of a pulse emerging from a gain medium is possible because the forward tail of a smooth pulse gives advance warning of the later arrival of the peak. The leading edge of the pulse can be amplified by the gain medium, which gives the appearance of superluminal pulse 612 Am. J. Phys., Vol. 69, No. 5, May 2001 New Problems 612