PROPAGATION OF ELECTROMAGNETIC ENERGY ANI 1081 verges at the longitudinal frequency and is negative there and at all higher frequencies. The unphysical aspects of these 070+(a-o0) velocities are discussed in Sec. VI where mo=noo) and the group velocity U, is defined in Eq (4.38). The extinction coefficient is assumed to have negli- The example of a constantly sustained monochromate gible dispersion around its value Ko at frequency a wave treated in Sec. IV is somewhat untypical; the energy With these assumptions and approximations, and with k and momentum densities are independent of the time, except given by Eq (4.9), the positive-frequency field(5.1)can be for the center-of-mass momentum which increases linearly. put in the form We now consider the more realistic example of a finite op- tical pulse which is initiated at the origin of coordinates, and left to decay as the dissipation takes effect. It is difficult to E(=,)= treat the propagation of a general optical pulse. We here choose a Gaussian envelope whose parameters have relative magnitudes that are convenient both for evaluation of the various integrals that occur in the theory and for illustration of the effects of dissipation. Specifically, the frequency spread of the pulse is assumed to be much smaller than its 。dexp-il/、x)2 central frequency, and its spatial length is assumed to be much smaller than the optical attenuation length of the me- (56) dium. The effects of loss are included only in the decay of the optical pulse and in the eventual transfer of its initial where p=c/no is the phase velocity, and 1=0-fowith nomentum to the medium. The results derived in the present section are thus valid only for weak damping 2 20= (5.7) e, The coordinate axes are as defined in Sec. IV A. The real second term on the right of Eq. (5.7)is much smaller than ectric field associated with an optical pulse continues to the first, and with the inequality (5.3), the lower limit on the have the form in the first line of Eq (4.1), but its positive- integral in Eq.(5.6)is effectively -oo. The third and fourth frequency part is generalized to terms in the first exponent of Eq. (5.6)are correspondingly negligible compared to the second term. The positive dw et(, d) exp(-iot+ik=) frequency field thus takes the form Et(=1)=Ete-0---(r-- and the negative-frequency part is given by the complex con (58) jugate expression. For a Gaussian pulse, we choose to a very good approximation. It shows the familiar proper- E+(x,a)= v2 A2 (5.2) ties of a phase that propagates with the phase velocity, a magnitude that diminishes with the characteristic attenuation length 2Lo=c/aoKo and a peak that propagates with the quency is assumed to be very much larger than the frequency from the Maxwell equation(2.3)and, with the variou o where I is the spatial length of the pulse. Its central fre- group velocity. The corresponding magnetic field is obta equalities assumed above, it takes the approximate form B+(z,1)=E(=,D)/Up (5 and possible numerical values of these and other parameters A good approximation to the relative spatial displacement re considered in the Appendix. It is also assumed that the field of the two ions in the primitive cell is obtained from Eq refractive index and extinction coefficient satisfy the inequal-(2.9) (54) (二,t) E+(,t) quency spread c/l, so that it can be expanded around wo B Energy propagation ()ao列ab)+(a-b)-列(a) The cycle-average value of the total-energy cur ity obtained from Eq(2.18)with use of the fields(5.8)andverges at the longitudinal frequency and is negative there and at all higher frequencies. The unphysical aspects of these velocities are discussed in Sec. VI. V. OPTICAL PULSE The example of a constantly sustained monochromatic wave treated in Sec. IV is somewhat untypical; the energy and momentum densities are independent of the time, except for the center-of-mass momentum which increases linearly. We now consider the more realistic example of a finite op￾tical pulse which is initiated at the origin of coordinates, and left to decay as the dissipation takes effect. It is difficult to treat the propagation of a general optical pulse. We here choose a Gaussian envelope whose parameters have relative magnitudes that are convenient both for evaluation of the various integrals that occur in the theory and for illustration of the effects of dissipation. Specifically, the frequency spread of the pulse is assumed to be much smaller than its central frequency, and its spatial length is assumed to be much smaller than the optical attenuation length of the me￾dium. The effects of loss are included only in the decay of the optical pulse and in the eventual transfer of its initial momentum to the medium. The results derived in the present section are thus valid only for weak damping. A. Gaussian pulse The coordinate axes are as defined in Sec. IV A. The real electric field associated with an optical pulse continues to have the form in the first line of Eq. ~4.1!, but its positive￾frequency part is generalized to E1~z,t!5 1 A2p E 0 ` dv E1~z,v!exp~2ivt1ikz!, ~5.1! and the negative-frequency part is given by the complex con￾jugate expression. For a Gaussian pulse, we choose E1~z,v!5 lE1 &c expS 2 l 2 ~v2v0! 2 4c2 D , ~5.2! where l is the spatial length of the pulse. Its central fre￾quency is assumed to be very much larger than the frequency width, v0@c/l, ~5.3! and possible numerical values of these and other parameters are considered in the Appendix. It is also assumed that the refractive index and extinction coefficient satisfy the inequal￾ity h~v!@k~v!. ~5.4! The refractive index is assumed to vary slowly over the fre￾quency spread c/l, so that it can be expanded around v0 as vh~v!'v0h~v0!1~v2v0! ]„vh~v!… ]v U v0 5v0h 01~v2v0! c vg , ~5.5! where h0[h~v0! and the group velocity vg is defined in Eq. ~4.38!. The extinction coefficient is assumed to have negli￾gible dispersion around its value k0 at frequency v0 . With these assumptions and approximations, and with k given by Eq. ~4.9!, the positive-frequency field ~5.1! can be put in the form E1~z,t!5 lE1 2Apc expH 2iv0S t2 z vp D 2 v0k0z c 1 k 0 2 z2 l 2 1 2ick0z l 2 S t2 z vg D J 3 E 2V0 ` dV expH 2iVS t2 z vg D 2 l 2V2 4c2 J , ~5.6! where vp5c/h0 is the phase velocity, and V5v2V0 with V05v02 2ck0z l 2 . ~5.7! As is discussed in the Appendix, we may assume that the second term on the right of Eq. ~5.7! is much smaller than the first, and with the inequality ~5.3!, the lower limit on the integral in Eq. ~5.6! is effectively 2`. The third and fourth terms in the first exponent of Eq. ~5.6! are correspondingly negligible compared to the second term. The positive￾frequency field thus takes the form E1~z,t!5E1expH 2iv0S t2 z vp D 2 z 2L0 2 c2 l 2 S t2 z vg D 2 J ~5.8! to a very good approximation. It shows the familiar proper￾ties of a phase that propagates with the phase velocity, a magnitude that diminishes with the characteristic attenuation length 2L05c/v0k0 and a peak that propagates with the group velocity. The corresponding magnetic field is obtained from the Maxwell equation ~2.3! and, with the various in￾equalities assumed above, it takes the approximate form B1~z,t!5E1~z,t!/vp . ~5.9! A good approximation to the relative spatial displacement field of the two ions in the primitive cell is obtained from Eq. ~2.9! as s1~z,t!5 § m~vT 22v0 2 ! E1~z,t!. ~5.10! B. Energy propagation The cycle-average value of the total-energy current den￾sity obtained from Eq. ~2.18! with use of the fields ~5.8! and ~5.9! is 55 PROPAGATION OF ELECTROMAGNETIC ENERGY AND... 1081