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SJOERD STALLINGA PHYSICAL REVIEW E 73. 026606(2006) function of the electric field) need to be taken into account diT 1-2[e, ()+ E()IE(z, )12 [27-29]. This would also give rise to separate conservation laws for momentum and pseudomomentum as then there are 广c two independent continuous translation symmetries, one re- O)2E(z,) (110) flecting uniformity of space and one reflecting homogeneity of matter in the undeformed reference state Similar to the dissipation of energy it follows that each Fou Instead of such a first principles approach we may also rier component of the integrated density and flux density of introduce inhomogeneity in an ad hoc manner by making the the propagating momentum is a factor n(a)/c times the Fou- conduction function space dependent, i.e., by replacing Gf() rier component of the integrated density and flux density of by G(r, o)everywhere. This does not alter the equations of the propagating energy. It is this relation between energy and motion of the model, nor the expression of the conservation momentum that has motivated the choice of the density and of energy. The total momentum is no longer conserved be flux density of momentum given by Eqs.(49)and(50)over cause of the broken translational symmetry(the Lagrangian the forms given in Eqs. (51)and(52). It is mentioned that the density depends explicitly on the spatial coordinates). It turns dispersive momentum flux density does not contribute to the out that now time integral of the total momentum flux density, the only nonzero contribution comes from the nondispersive 0,ga+aBab= (Minkowski) stress tensor. where the dissipation of momentum due to the inhomogene For the narrow-band pulse it is found that ity is given by 2=14+1E fint=-Eol dud d(r, o)[(G,F)2-0]2+2F.E] (111) The implication is that inhomogeneities are accompanied by f"=-doln(wo)Eo(z, 1)P (112) forces on the system. As a consequence, the total field-plus matter system considered so far must be an open system, as external forces are needed to maintain the static inhomoge =2=(0akx (113) neity of the system when an electromagnetic field is applied These external forces can be identified as the mechanical forces that have been excluded from the description in the If the absorption is small the nonpropagating momentum beginning. The open character of the system has also been density may be neglected, and the total momentum may be noticed by Garrison and Chiao as important for the applica approximated as bility of the total momentum [4] 8:=;n(an)2n(o dn(oo)]. An explicit expression for the force density may be found dao Eo(z, in the Fourier domain(similar to the expressions derived for the wave packets studied in Sec. IV) (114) proving that the momentum of the wave packet travels at a --2eo D(r, o, o)a el(r, so) o'+e(r, o"y'o o+a speed equal to the group velocity, just as the energy. The ratios of the density, Alux density, and dissipation of energy (118) and the density, flux density, and dissipation of momentum where the shorthand D(r, a, a')is defined by are all equal to dodo R INP S D(r,o,o')f(a,3)= DE f(o, w) (115 (2丌) 8 rk 8 tx f n(oo) i.e., the ratio between total energy and total momentum of XE(r, o) exp (a-t he wave packet is equal to the phase velocity 119) This gives the time integral V DISCUSSION The momentum conservation law that has been derived dtf= E(r,o)P2ae,(r,o),(120) this paper applies to homogeneous dielectrics only, as defor- mations of the medium are excluded from the start. A more so that the dissipation of momentum integrated over the du- general theory should address the deformability of the mate- ration of the interaction between the medium and the elec rial medium. Then the kinetic energy, kinetic momentum, tromagnetic field is proportional to the spectral average of hydrostatic forces(for fluids), and elastic forces(for solids), the product of the square of the electric field and the gradient and effects such as electrostriction (change in density as a of the real part of the dielectric function. This agrees with the 026606-10 − dtTzz = 1 2 0 − d 2 ˆr + ˆ  E ˆz, 2 = 0 − d 2 n 2 E ˆz, 2. 110 Similar to the dissipation of energy it follows that each Fou￾rier component of the integrated density and flux density of the propagating momentum is a factor n/c times the Fou￾rier component of the integrated density and flux density of the propagating energy. It is this relation between energy and momentum that has motivated the choice of the density and flux density of momentum given by Eqs. 49 and 50 over the forms given in Eqs. 51 and 52. It is mentioned that the dispersive momentum flux density does not contribute to the time integral of the total momentum flux density, the only nonzero contribution comes from the nondispersive Minkowski stress tensor. For the narrow-band pulse it is found that gz PR = 1 2 0 c  dˆr00 d0 + ˆ0 n0 E0z,t 2, 111 fz = 1 2 0 c ˆ0n0 E0z,t 2, 112 Tzz = 1 2 0n0 2 E0z,t 2. 113 If the absorption is small the nonpropagating momentum density may be neglected, and the total momentum may be approximated as gz = 1 2 0n0 2 dn00 d0 E0z,t 2 = dn00 d0 Tzz c , 114 proving that the momentum of the wave packet travels at a speed equal to the group velocity, just as the energy. The ratios of the density, flux density, and dissipation of energy and the density, flux density, and dissipation of momentum are all equal to uPR gz PR = uNP gz NP = Sz Tzz = W fz = c n0 , 115 i.e., the ratio between total energy and total momentum of the wave packet is equal to the phase velocity. V. DISCUSSION The momentum conservation law that has been derived in this paper applies to homogeneous dielectrics only, as defor￾mations of the medium are excluded from the start. A more general theory should address the deformability of the mate￾rial medium. Then the kinetic energy, kinetic momentum, hydrostatic forces for fluids, and elastic forces for solids, and effects such as electrostriction change in density as a function of the electric field need to be taken into account 27–29. This would also give rise to separate conservation laws for momentum and pseudomomentum as then there are two independent continuous translation symmetries, one re- flecting uniformity of space and one reflecting homogeneity of matter in the undeformed reference state. Instead of such a first principles approach we may also introduce inhomogeneity in an ad hoc manner by making the conduction function space dependent, i.e., by replacing ˆ by ˆr, everywhere. This does not alter the equations of motion of the model, nor the expression of the conservation of energy. The total momentum is no longer conserved be￾cause of the broken translational symmetry the Lagrangian density depends explicitly on the spatial coordinates. It turns out that now t g + T = − f inh, 116 where the dissipation of momentum due to the inhomogene￾ity is given by f inh = − 0  0 dˆr,t F 2 − 2 F2 + 2F · E. 117 The implication is that inhomogeneities are accompanied by forces on the system. As a consequence, the total field-plus￾matter system considered so far must be an open system, as external forces are needed to maintain the static inhomoge￾neity of the system when an electromagnetic field is applied. These external forces can be identified as the mechanical forces that have been excluded from the description in the beginning. The open character of the system has also been noticed by Garrison and Chiao as important for the applica￾bility of the total momentum 4. An explicit expression for the force density may be found in the Fourier domain similar to the expressions derived for the wave packets studied in Sec. IV f inh = − 1 2 0  Dr,, ˆr, + ˆr, *   +  , 118 where the shorthand Dr,, is defined by  Dr,,f,  −  − dd 2 2 f,E ˆ r, E ˆ r, * exp− i − t. 119 This gives the time integral  − dtf inh = − 1 2 0 − d 2 Er, 2 ˆrr,, 120 so that the dissipation of momentum integrated over the du￾ration of the interaction between the medium and the elec￾tromagnetic field is proportional to the spectral average of the product of the square of the electric field and the gradient of the real part of the dielectric function. This agrees with the SJOERD STALLINGA PHYSICAL REVIEW E 73, 026606 2006 026606-10
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