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 610eivz/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