SJOERD STALLINGA PHYSICAL REVIEW E 73. 026606(2006) continuous reservoir of oscillators into which the wave dis- d[n(oo)ool sipates energy. The reservoir gains energy by dissipation and, because of causality, depends only on the electric fields dab Eo(c,m/ d[n(oo)wo]S doo previous times. The energy flux is directed along the z axis and has a ma proving that the energy of the wave packet travels at the S,=eOc D(o, c)[VE(o)+Ve(')].(84 (84) group velocity c/n,o)with the group refractive index g(o)dn(o)o (94) It turns out that the nonpropagating energy density satisfies do lim I P=0 (85) It appears that Eo(z, t)is a slowly varying function if Eo(O, t) is a slowly varying function, provided that z is sufficiently small, irrespective of whether oo is close to a resonance or lim P= dtW=E0_ 2T O)E(, o).(86) not [22]. In this transparency regime the group velocity can exceed the speed of light in vacuum and can even be nega- ive, as demonstrated experimentally in Ref [23]. The arrival It follows that the loss in the propagating part of the time of a pulse can be given a well-defined meaning, even in density over the total duration of the pulse is always these exotic regimes [24]. In turn, if z is sufficiently large as d(o)is positive for all o. This irreversibility is Eo(z, t)cannot be a slowly varying function, even if eo(o, 1) ment with expectations. The integral of the propagating en- is, and if wo is far away from a resonance. In this latter ergy density and energy flux density over the duration of the regime, the asymptotic regime, a different treatment is pulse follow as needed [25, 26] de,oo +|(o)E(o)P B Momentum The nondispersive (Minkowski) part of the momentum (87) density is given by dts,=eoc (8)820=DB≈0|Dao)aoa(0)+a(o)va(o) Consider now a pulse The dispersive part of the momentum density E(z, t)=Eo(z, t)exp(-ioot)+Eo(z, t)exp(iwor) dwod(oo)Fr d,0, Fr (89) where wo is a carrier frequency and where Eo(z, t)is a slowly |Daa)aVl(oa2+l(a)a arying envelope function for all z concerned. It then follows that the spectrum E(z, a)has narrow peaks at too so that the (96) dielectric function can be taken constant across the integra ion range. In this approximation the propagating energy, rate with of work, and energy flux density are given by Y(o, w) d[e(oo)wol J。d0(a-(+1)厘m-(-1y W"=2EoG oleO(z D) (91) (a+o')(a-a’+iy) S=Eo(oocEo(z, 12 +a)( d(o) d(w) If the absorption is small the nonpropagating contribution may be neglected and the total energy (which is equal to the propagating energy in this limit)may be approximated as The total momentum can be rewritten using 026606-8continuous reservoir of oscillators into which the wave dissipates energy. The reservoir gains energy by dissipation and, because of causality, depends only on the electric fields at previous times. The energy flux is directed along the z axis and has a magnitude Sz = 1 2 0c D,ˆ + ˆ * . 84 It turns out that the nonpropagating energy density satisfies lim t→− uNP = 0, 85 lim t→+ uNP = − dtW = 0 − d 2 ˆE ˆz,2. 86 It follows that the loss in the propagating part of the energy density over the total duration of the pulse is always positive, as ˆ is positive for all . This irreversibility is in agreement with expectations. The integral of the propagating energy density and energy flux density over the duration of the pulse follow as − dtuPR = 1 2 0 − d 2 dˆr d + ˆE ˆz,2, 87 − dtSz = 0c − d 2 nE ˆ2. 88 Consider now a pulse Exz,t = 1 2 E0z,texp− i0t + 1 2 E0z,t * expi0t, 89 where 0 is a carrier frequency and where E0z,t is a slowly varying envelope function for all z concerned. It then follows that the spectrum E ˆz, has narrow peaks at ±0 so that the dielectric function can be taken constant across the integration range. In this approximation the propagating energy, rate of work, and energy flux density are given by uPR = 1 4 0 dˆr00 d0 + ˆ0E0z,t2, 90 W = 1 2 0ˆ0E0z,t2, 91 Sz = 1 2 0n0cE0z,t2. 92 If the absorption is small the nonpropagating contribution may be neglected and the total energy which is equal to the propagating energy in this limit may be approximated as u = 1 2 0n0 dn00 d0 E0z,t2 = dn00 d0 Sz c , 93 proving that the energy of the wave packet travels at the group velocity c/ng with the group refractive index ng = dn d . 94 It appears that E0z,t is a slowly varying function if E00,t is a slowly varying function, provided that z is sufficiently small, irrespective of whether 0 is close to a resonance or not 22. In this transparency regime the group velocity can exceed the speed of light in vacuum and can even be negative, as demonstrated experimentally in Ref. 23. The arrival time of a pulse can be given a well-defined meaning, even in these exotic regimes 24. In turn, if z is sufficiently large E0z,t cannot be a slowly varying function, even if E00,t is, and if 0 is far away from a resonance. In this latter regime, the asymptotic regime, a different treatment is needed 25,26. B. Momentum The nondispersive Minkowski part of the momentum density is given by gz ND = DxBy = 0 2c D,ˆˆ * + ˆ *ˆ. 95 The dispersive part of the momentum density is gz DS = 20 0 d0ˆ0Fxt zFx = 0 2c D,Y,ˆ2 + ˆ * 2 , 96 with Y, = 2 0 d0ˆ0 1 0 2 − + i 2 0 2 − − i 2 = ˆ − ˆ * + − + i = 1 + − + i ˆr − ˆr + iˆ + ˆ . 97 The total momentum can be rewritten using SJOERD STALLINGA PHYSICAL REVIEW E 73, 026606 2006 026606-8