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 Problemslaw11 ~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