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 Problemsn~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