PROPAGATION OF ELECTROMAGNETIC ENERGY AN The momentum density is given by Eqs. (3.16)and(2.22), The cycle average of the pseudomomentum dissipation rate and only the component is nonzero, with the cycle- that appears on the right-hand side of Eq. (3. 19) also has averaged value only the component 2 4 800n-K (Gm)2)=(MR)+(Gcm)2)=(MR)+=|E+|2en nI JE2e-ll The cycle averages of the continuity equation(2.21)for LE +|2 the electromagnetic momentum and of the conservation equation (3. 14) for the momentum lead to the equalities The cycle average of the pseudomomentum continuity equi tion Eq. (3. 19)thus takes the form F)=-(7m)=)=- ((Tm)e =s(MR:) ((Tpsm)=)->(MR: )=(mrs (429) (423) for the monochromatic wave excitation considered here. It is and this is seen to agree with Eqs. (4.23)and(4.24)when the readily verified that the explicit expression for the cycle- cycle averages(4. 26) and(4.28)are substituted averaged Lorentz force density, obtained from Eq.(2.21)as Wave momentum. The cycle averages of the various wave UB), agrees with that obtained from the momentum current momentum densities are now obtained by summation of the density component given in Eq (4.21). The time dependence momentum and pseudomomentum contributions, and the re- of the center-of-mass momentum density is thus obtained by sults are integration of Eq.(4.23)as (Tx)=-(Ty)=2E0k2|E+P2e-1(430) (MR(=,1)=(E0tL)(1+2+k2)E+P2e-m, (4.24) and and this quantity vanishes in the limit of a lossless dielectric (7)=20m2|E+2e- (431) as L-0. The dielectric material is here assumed to be a rigid for the wave momentum current density and body, and the total momentum transferred to the unit cross sectional area at time t is obtained by integration of Eq (4.24)as (G)= E07 nK E+|2e-=(4.32) d=(MR(-,1)=εo1(1+n2+k2)E2.(425)forthewavemomentumdensiy.Theycleaverageofthe wave momentum continuity equation (3. 23)takes the form The material center-of-mass momentum thus grows linearly 0(T) with the time as momentum is steadily transferred from field mls (43) to dielectric. The total momentum transfer vanishes in the limit of a lossless dielectric, and the apparent nonzero result and it is readily verified from Eqs.(4. 12),(4.28),and(4.31) btained from Eq.(4.25) for K-0 is an artifact of the prior that this relation is indeed satisfied integration over an infinite extent of the medium. The con ole to de servation law for the momentum density given in Eq. (3, 17 a velocity Uwm of wave momentum transport through the does not hold for the open system considered here, where absorbing dielectric in the direction of the z axis as there is a steady input of electromagnetic energy and mo- Pseudomomentum. The pseudomomentum current density given by Eq.(3.22) is the same for all three diagonal com (G2)-n2+k2+(2onk/T) (434) nents,and its cycle average is This can be rearranged in the form (Tpsm))=(-1+n2-k2)E+2e-m1, i=x,y (4.26) (435) The cycle average of the pseudomomentum density given by Eq(3.21) has only the component with a term additional to expression (4. 17)for the energy velocity. The wave momentum velocity is thus in general smaller than the energy velocity, and Fig. 2 shows the fre- (Gpm)=-(MR2)+ 1+n2+K2+ quency dependence of the difference between the two. The differences are small for the chosen parameters, but they ×|E+|2The momentum density is given by Eqs. ~3.16! and ~2.22!, and only the z component is nonzero, with the cycle￾averaged value ^~Gm!z&5^MR˙ z&1^~Gem!z&5^MR˙ z&1 2«0h c uE1u 2e2z/L. ~4.22! The cycle averages of the continuity equation ~2.21! for the electromagnetic momentum and of the conservation equation ~3.14! for the momentum lead to the equalities ^Fz&52 ] ]z ^~Tem!zz&52 ] ]z ^~Tm!zz&5 ] ]t ^MR˙ z& ~4.23! for the monochromatic wave excitation considered here. It is readily verified that the explicit expression for the cycle￾averaged Lorentz force density, obtained from Eq. ~2.21! as ^jB&, agrees with that obtained from the momentum current density component given in Eq. ~4.21!. The time dependence of the center-of-mass momentum density is thus obtained by integration of Eq. ~4.23! as ^MR˙ z~z,t!&5~«0t/L!~11h21k2!uE1u 2e2z/L, ~4.24! and this quantity vanishes in the limit of a lossless dielectric as L→`. The dielectric material is here assumed to be a rigid body, and the total momentum transferred to the unit cross￾sectional area at time t is obtained by integration of Eq. ~4.24! as E 0 ` dz^MR˙ z~z,t!&5«0t~11h21k2!uE1u 2. ~4.25! The material center-of-mass momentum thus grows linearly with the time, as momentum is steadily transferred from field to dielectric. The total momentum transfer vanishes in the limit of a lossless dielectric, and the apparent nonzero result obtained from Eq. ~4.25! for k→0 is an artifact of the prior integration over an infinite extent of the medium. The con￾servation law for the momentum density given in Eq. ~3.17! does not hold for the open system considered here, where there is a steady input of electromagnetic energy and mo￾mentum. Pseudomomentum. The pseudomomentum current density given by Eq. ~3.22! is the same for all three diagonal com￾ponents, and its cycle average is ^~Tpsm!ii&5«0~211h22k2!uE1u 2e2z/L, i5x,y,z. ~4.26! The cycle average of the pseudomomentum density given by Eq. ~3.21! has only the z component ^~Gpsm!z&52^MR˙ z&1 2«0h c S 211h21k21 2vhk G D 3uE1u 2e2z/L. ~4.27! The cycle average of the pseudomomentum dissipation rate that appears on the right-hand side of Eq. ~3.19! also has only the z component K mGs˙ ]s ]z L 52 4«0vh2k c uE1u 2e2z/L 52 2«0h2 L uE1u 2e2z/L. ~4.28! The cycle average of the pseudomomentum continuity equa￾tion Eq. ~3.19! thus takes the form ] ]z ^~Tpsm!zz&2 ] ]t ^MR˙ z&5K mGs˙ ]s ]z L , ~4.29! and this is seen to agree with Eqs. ~4.23! and ~4.24! when the cycle averages ~4.26! and ~4.28! are substituted. Wave momentum. The cycle averages of the various wave momentum densities are now obtained by summation of the momentum and pseudomomentum contributions, and the re￾sults are ^Txx&52^Tyy&52«0k2 uE1u 2e2z/L ~4.30! and ^Tzz&52«0h2 uE1u 2e2z/L ~4.31! for the wave momentum current density and ^Gz&5 2«0h c H h21k21 2vhk G J E1u 2e2z/L ~4.32! for the wave momentum density. The cycle average of the wave momentum continuity equation ~3.23! takes the form ]^Tzz& ]z 5K mGs˙ ]s ]z L , ~4.33! and it is readily verified from Eqs. ~4.12!, ~4.28!, and ~4.31! that this relation is indeed satisfied. As with the energy velocity ~4.16!, it is possible to define a velocity vwm of wave momentum transport through the absorbing dielectric in the direction of the z axis as vwm5^Tzz& ^Gz& 5 ch h21k21~2vhk/G! . ~4.34! This can be rearranged in the form 1 vwm 5 1 vp 1 1 LG 1 k2 ch 5 1 ve 1 k2 ch , ~4.35! with a term additional to expression ~4.17! for the energy velocity. The wave momentum velocity is thus in general smaller than the energy velocity, and Fig. 2 shows the fre￾quency dependence of the difference between the two. The differences are small for the chosen parameters, but they could be significant for larger damping. 55 PROPAGATION OF ELECTROMAGNETIC ENERGY AND... 1079