Negative group velocity Kirk T Mcdonalda Received 21 August 2000; accepted 21 September 2000) [DO:10.1119/1.1331304] . PROBLEM C. Fourier analysis Consider a variant on the physical situation of slow pply the transformations between an incident monochro- ight"1, in which two closely spaced spectral lines are now matic wave and the wave in and beyond the medium to the lis both optically pumped to show that the group velocity can be Fourier analysis of an incident pulse of form f(Elc-i) negative at the central frequency, which leads to apparent superluminal behavior D. Propagation of a sharp wave front In the approximation that on varies linearly with a, de- A Negative group velocity duce the waveforms in the regions 0<=<a and a<= for an incident pulse S(E/c-t), where S is the Dirac delta function In more detail, consider a classical model of matter in Show that the pulse emerges out of the gain region at ==a at which spectral lines are associated with oscillators. In par- ticular, consider a gas with two closely spaced spectral lines time t=alug, which appears to be earlier than when it enters of angular frequencies 12=(+A/2, where A< oo. Each this region if the group velocity is negative. Show also that line has the same damping constant(and spectral width)y inside the negative group velocity medium a pulse propa- Ordinarily, the gas would exhibit strong absorption of gates backwards from ==a at time t=alUg <0 to ==0 at t light in the vicinity of the spectral lines. But suppose that 0, at which time it appears to annihilate the incident pulse lasers of frequencies a and pump both oscillators into inverted populations. This can be described classically by E. Propagation of a Gaussian pulse assigning negative oscillator strengths to these oscillators Deduce an expression for the group velocity u (wo)of a As a more physical example, deduce the waveforms in the pulse of light centered on frequency wo in terms of the(uni- regions 0<- <a and as: for a gaussian incident pulse valent) plasma frequency p of the medium, given by Eoe-(/c-n'nreloo(=/c-n). Carry the frequency expansion of on(o)to second order to obtain conditions of validity of the 4丌N analysis such as maximum pulse width T, maximum length a 1) of the gain region, and maximum time of advance of the where N is the number density of atoms, and e and m are the in a negative group velocity medium can lead to superlumi- separation A compared to the linewidth y such that the group nal signal propagation mass of an velocity u,(oo) is negative In a recent experiment by Wang et al., a group velocity of IL. SOLUTION c/310, where c is the speed of light in vacuum, was The concept of group velocity appears to have been first demonstrated in cesium vapor using a pair of spectral lines enunciated by Hamilton in 1839 in lectures of which only with separation△/2丌≈2 MHz and linewidth y/2 abstracts were published. The first recorded observation of 0. 8 MHZ the group velocity of a(water)wave is due to Russell in 1844.However, widespread awareness of the group velocity dates from 1876 when Stokes used it as the topic of a B. Propagation of a monochromatic plane wave Smiths Prize examination paper. The early history of group velocity has been reviewed by Havelock H. Lamb credits A. Schuster with noting in 1904 that a Consider a wave with electric field Eoeloilc-n that is negative group velocity, i.e., a group velocity of opposite incident from -<0 on a medium that extends from - =0 to a Ignore reflection at the boundaries, as is reasonable if the sign to that of the phase velocity, is possible due to anoma- index of refraction n( o) is near unity. Particularly simple 1905. Lamb gave two examples of strings subject to external sumption that the on(o) vares linearly with frequency considerations assumed that in case of a wave with positive about a central frequency wo. Deduce a transformation that group and phase velocities incident on the anomalous me has a frequency-dependent part and a frequency-independent dium, energy would be transported into the medium with a part between the phase of the wave for -<0 to that of the positive group velocity, and so there would be waves with wave inside the medium, and to that of the wave in the negative phase velocity inside the medium. Such negative egion a<- phase velocity waves are formally consistent with Snells Am J. Phys. 69(5), May 2001 http://ojps.aiporg/ajp/ c 2001 American Association of Physics TeachersNegative group velocity Kirk T. McDonalda) Joseph Henry Laboratories, Princeton University, Princeton, New Jersey 08544 ~Received 21 August 2000; accepted 21 September 2000! @DOI: 10.1119/1.1331304# I. PROBLEM Consider a variant on the physical situation of ‘‘slow light’’ 1,2 in which two closely spaced spectral lines are now both optically pumped to show that the group velocity can be negative at the central frequency, which leads to apparent superluminal behavior. A. Negative group velocity In more detail, consider a classical model of matter in which spectral lines are associated with oscillators. In par￾ticular, consider a gas with two closely spaced spectral lines of angular frequencies v1,25v06D/2, where D!v0 . Each line has the same damping constant ~and spectral width! g. Ordinarily, the gas would exhibit strong absorption of light in the vicinity of the spectral lines. But suppose that lasers of frequencies v1 and v2 pump both oscillators into inverted populations. This can be described classically by assigning negative oscillator strengths to these oscillators.3 Deduce an expression for the group velocity vg(v0) of a pulse of light centered on frequency v0 in terms of the ~uni￾valent! plasma frequency vp of the medium, given by vp 2 54pNe2 m , ~1! where N is the number density of atoms, and e and m are the charge and mass of an electron. Give a condition on the line separation D compared to the linewidth g such that the group velocity vg(v0) is negative. In a recent experiment by Wang et al., 4 a group velocity of vg52c/310, where c is the speed of light in vacuum, was demonstrated in cesium vapor using a pair of spectral lines with separation D/2p'2 MHz and linewidth g/2p '0.8 MHz. B. Propagation of a monochromatic plane wave Consider a wave with electric field E0eiv(z/c2t) that is incident from z,0 on a medium that extends from z50 to a. Ignore reflection at the boundaries, as is reasonable if the index of refraction n(v) is near unity. Particularly simple results can be obtained when you make the ~unphysical! as￾sumption that the vn(v) varies linearly with frequency about a central frequency v0 . Deduce a transformation that has a frequency-dependent part and a frequency-independent part between the phase of the wave for z,0 to that of the wave inside the medium, and to that of the wave in the region a,z. C. Fourier analysis Apply the transformations between an incident monochro￾matic wave and the wave in and beyond the medium to the Fourier analysis of an incident pulse of form f(z/c2t). D. Propagation of a sharp wave front In the approximation that vn varies linearly with v, de￾duce the waveforms in the regions 0,z,a and a,z for an incident pulse d(z/c2t), where d is the Dirac delta function. Show that the pulse emerges out of the gain region at z5a at time t5a/vg , which appears to be earlier than when it enters this region if the group velocity is negative. Show also that inside the negative group velocity medium a pulse propa￾gates backwards from z5a at time t5a/vg,0 to z50 at t 50, at which time it appears to annihilate the incident pulse. E. Propagation of a Gaussian pulse As a more physical example, deduce the waveforms in the regions 0,z,a and a,z for a Gaussian incident pulse E0e2(z/c2t)2/2t 2 eiv0(z/c2t) . Carry the frequency expansion of vn(v) to second order to obtain conditions of validity of the analysis such as maximum pulse width t, maximum length a of the gain region, and maximum time of advance of the emerging pulse. Consider the time required to generate a pulse of rise time t when assessing whether the time advance in a negative group velocity medium can lead to superlumi￾nal signal propagation. II. SOLUTION The concept of group velocity appears to have been first enunciated by Hamilton in 1839 in lectures of which only abstracts were published.5 The first recorded observation of the group velocity of a ~water! wave is due to Russell in 1844.6 However, widespread awareness of the group velocity dates from 1876 when Stokes used it as the topic of a Smith’s Prize examination paper.7 The early history of group velocity has been reviewed by Havelock.8 H. Lamb9 credits A. Schuster with noting in 1904 that a negative group velocity, i.e., a group velocity of opposite sign to that of the phase velocity, is possible due to anoma￾lous dispersion. Von Laue10 made a similar comment in 1905. Lamb gave two examples of strings subject to external potentials that exhibit negative group velocities. These early considerations assumed that in case of a wave with positive group and phase velocities incident on the anomalous me￾dium, energy would be transported into the medium with a positive group velocity, and so there would be waves with negative phase velocity inside the medium. Such negative phase velocity waves are formally consistent with Snell’s 607 Am. J. Phys. 69 ~5!, May 2001 http://ojps.aip.org/ajp/ © 2001 American Association of Physics Teachers 607