R LOUDON. L. ALLEN. AND D. F. NELSON ((Tm)2 (Gm)= (442) 0.25 With no motion of the center of mass, it is possible to define 1n7m){1,=3g (Tm)=2) FIG. 2. Frequency dependence of the difference between wave momentum and energy velocities for the same parameters as Fi The momentum densities both take their usual fre values when the phase velocity is set equal to c, and the D. Limit of zero damping momentum velocity also becomes equal to c It is instructive to consider the forms of the various de The pseudomomentum densities(4.26)and (4. 27)become sities defined above in the limit of zero damping [→0,k(ω)→0]l, when the refractive index is obtained from (m2=80-1+)E+P2 444 Eq. (4.7)as (4.36 G EP2,(445 Thus, for a lossless dielectric, the cycle-average energy cur- rent density(4.11) can be simply expressed in terms of the hase velocity(4.18) where the limit given in Eq(4.39)is used in the latter. A pseudomomentum velocity can thus be defined by ((Tpsm)2e) °pm(Gpm)2)=c2-2n The group velocity is defined by The pseudomomentum densities both vanish in free space The cycle-averaged wave momentum densities (4.31)and (4.32) become (om)=—+ A2_212 S2) E (447) and it is easily verified with the use of Eqs.(4.7)and(4.8) (4.39) 2 I→0 E+E The lossless limit of the cycle-average energy density(4.13) The cycle-averaged dissipation rate (4. 28)vanishes in the limit of zero damping, and the wave momentum velocity (4. 34)reduces, like the energy velocity, to the group velocity ER This velocity can be expressed in the fo The cycle-averaged dissipation rate (4.14)of course van ishes, and the energy velocity (4.16) becomes the same as ((Gm)= m+((Gpsm)-)u he group velocity in the limit of zero damping. This is ((Gm)-+((Nsm)-) shown as the T=0 curve in Fig. 1 In the absence of any material boundaries, and hence of of a sum of the momentum and pseudomomentum any reflection of the incident electromagnetic wave, no mo- weighted by their respective densities. However, mentum is transferred from the electromagnetic field to the the wave momentum velocity has the well-behaved form center-of-mass motion of a lossless dielectric. Thus R can shown by the T=0 curve in Fig. 1, the momentum velocity everywhere be set equal to zero. The cycle-average momen- (4.41)diverges at both the transverse and longitudinal fre- tum densities(4.21)and(4.22) become quencies while the pseudomomentum velocity(4. 46)diD. Limit of zero damping It is instructive to consider the forms of the various den￾sities defined above in the limit of zero damping @G→0,k~v!→0#, when the refractive index is obtained from Eq. ~4.7! as h2511 §2 «0m 1 vT 22v2 . ~4.36! Thus, for a lossless dielectric, the cycle-average energy cur￾rent density ~4.11! can be simply expressed in terms of the phase velocity ~4.18! as ^Sz&5 2«0c2 vp uE1u 2. ~4.37! The group velocity is defined by c vg 5 ] ]v ~vh!5 c vp 1 vp c §2 «0m v2 ~vT 22v2! 2 , ~4.38! and it is easily verified with the use of Eqs. ~4.7! and ~4.8! that Lt G→0 S h21 2vhk G D 5h ] ]v ~vh!5 c2 vpvg . ~4.39! The lossless limit of the cycle-average energy density ~4.13! is thus ^W&5 2«0c2 vpvg uE1u 2. ~4.40! The cycle-averaged dissipation rate ~4.14! of course van￾ishes, and the energy velocity ~4.16! becomes the same as the group velocity in the limit of zero damping. This is shown as the G50 curve in Fig. 1. In the absence of any material boundaries, and hence of any reflection of the incident electromagnetic wave, no mo￾mentum is transferred from the electromagnetic field to the center-of-mass motion of a lossless dielectric. Thus R ˙ can everywhere be set equal to zero. The cycle-average momen￾tum densities ~4.21! and ~4.22! become ^~Tm!zz&5«0S 11 c2 vp 2 D uE1u 2 ~4.41! and ^~Gm!z&5 2«0 vp uE1u 2. ~4.42! With no motion of the center of mass, it is possible to define a momentum velocity as vm5^~Tm!zz& ^~Gm!z& 5S 11 c2 vp 2 D vp 2 . ~4.43! The momentum densities both take their usual free-space values when the phase velocity is set equal to c, and the momentum velocity also becomes equal to c. The pseudomomentum densities ~4.26! and ~4.27! become ^~Tpsm!zz&5«0S 211 c2 vp 2 DuE1u 2 ~4.44! and ^~Gpsm!z&5 2«0 vp S 211 c2 vpvg DuE1u 2, ~4.45! where the limit given in Eq. ~4.39! is used in the latter. A pseudomomentum velocity can thus be defined by vpsm5^~Tpsm!zz& ^~Gpsm!z& 5 c22vp 2 c22vgvp vg 2 . ~4.46! The pseudomomentum densities both vanish in free space. The cycle-averaged wave momentum densities ~4.31! and ~4.32! become ^Tzz&5 2«0c2 vp 2 uE1u 25^Sz& vp ~4.47! and ^Gz&5 2«0c2 vp 2 vg uE1u 25^W& vp . ~4.48! The cycle-averaged dissipation rate ~4.28! vanishes in the limit of zero damping, and the wave momentum velocity ~4.34! reduces, like the energy velocity, to the group velocity vwm5ve5vg . ~4.49! This velocity can be expressed in the form vwm5^~Gm!z&vm1^~Gpsm!z&vpsm ^~Gm!z&1^~Gpsm!z& ~4.50! of a sum of the momentum and pseudomomentum velocities weighted by their respective densities. However, although the wave momentum velocity has the well-behaved form shown by the G50 curve in Fig. 1, the momentum velocity ~4.41! diverges at both the transverse and longitudinal fre￾quencies while the pseudomomentum velocity ~4.46! di￾FIG. 2. Frequency dependence of the difference between wave momentum and energy velocities for the same parameters as Fig. 1. 1080 R. LOUDON, L. ALLEN, AND D. F. NELSON 55