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 ProblemsD. 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