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ENERGY AND MOMENTUM OF LIGHT IN DIELECTRIC PHYSICAL REVIEW E 73. 026606(2006) E(a)a2+VE(a)°o2 (a+a3)(a-a'+iy) r2=--|do0(ao(aF)2-02F2+FE VE()o+VE(a)o' vE(o)-VE(o')w' D(o,oQ(o, a'). (105) (98) so that a division of total momentum into propagating and with nonpropagating parts 8 =8 tg can be made such that BPR=E D(o, c E(o)E(aIVE()+et() a2-(+iy)2∞2-(o+iy2 E(o)-e, ( o"l(a o Ve(),'1 (a+iy)2][n2-(a'-iy)2 e(u)-8(u) E(o-E(o (ol-V(a1)o].(100 The zz component of the total stress tensor then follows as “py XIvE(o)o+vE(o)o'] Eo ilo(o)+G(o) (106) e(ω)+V(o) (( (101) The nonpropagating momentum density satisfies The nonpropagating momentum density may be written as lim NP=0 (107) the integral of the force density 82P=dr"(e). (108) I=4c D(wu, w )(G( o)+d()IVE()+ve()] from which it may be concluded that the dissipation of propagating momentum integrated over the entire pulse is ach Fourier component of th (103) momentum dissipation is a factor n(o)/c times the Fourier The flow of z momentum in the z direction is given by the omponent of the integrated energy dissipation. The sum of the nondispersive(Minkowski) stress tensor compo. tegral of the propagating momentum density and flux density nent dE (o) TWD=E, D,+3H, By 广_=广 EEo D(o, a')E(o)+E(a')+2ve(o)e(') Xn(o)E(z, o)2 do d (eo)w (104) +|(a)n(o)E(z,o)2 and the dispersive stress tensor component (109)2 ˆ2 + ˆ * 2 + − + i = ˆ + ˆ * − + i + ˆ − ˆ * + , 98 so that a division of total momentum into propagating and nonpropagating parts gz = gz PR + gz NP, 99 can be made, such that gz PR = 0 4c D,2 ˆˆ * ˆ + ˆ * + ˆr − ˆr − ˆ + ˆ * + ˆ − ˆ * + ˆ − ˆ * , 100 gz NP = 0 4c D, iˆ/ + ˆ/ − + i ˆ + ˆ * = 0 4c D, iˆ + ˆ − + i ˆ + ˆ * . 101 The nonpropagating momentum density may be written as the integral of the force density gz NP = − t dtfz t, 102 with fz = 0 4c D,ˆ + ˆ ˆ + ˆ * . 103 The flow of z momentum in the z direction is given by the sum of the nondispersive Minkowski stress tensor component Tzz ND = 1 2 ExDx + 1 2 HyBy = 1 4 0 D,ˆ + ˆ * + 2 ˆˆ * , 104 and the dispersive stress tensor component Tzz DS = − 0 0 d0ˆ0t Fx 2 − 0 2 Fx 2 + FxEx = 0 2 D,Q,, 105 with Q, = 2 0 d0ˆ02 + 0 2 0 2 − + i 2 0 2 − − i 2 − 1 0 2 − + i 2 − 1 0 2 − + i 2 =− 2 0 d0ˆ0 − 2 0 2 − + i 2 0 2 − − i 2 = − − ˆ − ˆ * + . The zz component of the total stress tensor then follows as Tzz = 1 4 0 D,ˆ + ˆ * + 2 ˆˆ * − 2 − ˆ − ˆ * + . 106 The nonpropagating momentum density satisfies lim t→− gz NP = 0, 107 lim t→+ gz NP = − dtfz = 0 c − d 2 ˆn E ˆz, 2, 108 from which it may be concluded that the dissipation of propagating momentum integrated over the entire pulse is always positive. Each Fourier component of the integrated momentum dissipation is a factor n/c times the Fourier component of the integrated energy dissipation. The time integral of the propagating momentum density and flux density are − dtgz PR = 0 c − d 2 ˆ − 2 + 1 2 dˆr d n E ˆz, 2 = 0 2c − d 2 dˆr d + ˆ n E ˆz, 2, 109 ENERGY AND MOMENTUM OF LIGHT IN DIELECTRIC... PHYSICAL REVIEW E 73, 026606 2006 026606-9