PROPAGATION OF ELECTROMAGNETIC ENERGY AN 1075 with the same magnitude as the center-of-mass kinetic en- MR,-sS, dr:(E, +(RXB))-s dt (sx b))=0 rgy has been omitted from the momentum current density (3.12)(3.15)in accordance with the discussion that follows Eq A more convenient form of this equation is found after con- It should be noted that the equation of motion(3. 11)for siderable manipulation [10], using Eqs. 2.2),(2.5)and(3.6), the internal coordinate plays no role in the above derivation, to be on account of the inability of the relative coordinate to carry momentum [6]. As the dissipation described by Eq (3.9)acts s(E+(R×B)s}=pE+(j×B)=F only on the internal coordinate, the absence of this coordi nate from the derivation of Eqs. (3.14)-(3. 16)accounts for (3.13) the lack of damping terms in these results. The fulfilment of momentum conservation in the presence of energy dissipa- The significances of the terms on the left are clarified by tion, as described by the term on the right-hand side of ec their contributions to the momentum densities defined in Sec. (2.17), is of course a common occurrence in mechanics III B below As discussed in connection with Eq. (2.25), the kinetic contribution mr from the center-of-mass motion and a con- energy of the dielectric is negligible compared to the energy tribution EoEXB from the electromagnetic field. The latter densities associated with the electromagnetic field and the differs from the abraham form EoloEXH on account of the optic mode vibration of the lattice. The contribution of the generalized relation(3. 10) between the magnetic field and center-of-mass motion to the Lorentz force on the right-hand the induction. It is seen that, in contrast to the magnetic side of (3. 11)is also small, as RB is of order RE/c, which is induction B, which is purely a property of the electromag certainly much smaller than E. With terms in R removed, the netic field, the magnetic field H contains a contribution that Lagrangian theory reproduces the results of Sec. IB as Eq. depends on the material internal coordinate contained in P (3. 11)reduces to Eq.(2.9), and the Maxwell equations are The occurrence of b rather than h ensures that the electro. unchanged. The energy continuity equation (2.17)and the magnetic momentum density is properly independent of any energy densities given in Eqs.(2.18)and (2.19)thus con- material variables The total momentum is obtained by integration of Gn (r, t) subtle problems and distinctions associated with the propa- over all space, and this quantity is conserved for a closed sorbing dielectric Thus integration of Eq. (3. 14)over the effectively-infinite dielectric material gives B. Momentum conservation The law of momentum conservation is a consequence of (3.17) the invariance of the laws of physics to arbitrary infinitesimal dt dr gm(r, 0=0, displacements of the spatial coordinates. The momentum is thus defined with respect to the Maxwell equations(2.1)- provided that Tm(r, t)vanishes at r=oo. The total momentum (2.4)and the center-of-mass equation (3. 13). The continuty electromagnetic field and the dielectric center-of-mass mo- equation(2.21)for the electromagnetic momentum, which is based entirely on Maxwell's equations, therefore remains tion changes with time as, for example, in the propagation of valid. The Lorentz force densities F, on the right-hand sides a pulse of excitation through the crystal Although the material motion makes an important contri of Eqs.(2.21)and(3. 13)are equal and opposite, demonstrat- bution to the momentum density, its contribution to the mo- magnetic and material parts of the coupled system. Addition mentum current density(3. 15)is generally less important of these equations, using definition(2. 12)of the polarization, E, and, with the corresponding term removed, Eq. (3.15) reduces to (3.14) (Tm=-E,P+(T which is the conservation law for the momentum of the field- C. Pseudomomentum aterial system. Here Pseudomomentum has been a much neglected quantity in (Tm)ji=-(E+(RX B)))P, +(Tem)ji (3. 15) continuum mechanics, and a regularly misin erpreted quar tity in quantum mechanics. Quantum-mechanical treatments is the momentum current density, and of excitations in solids have often called hk the pseudomo- mentum of an excitation quantum. This was shown to be Gm=Mr+ gem (3. 16) wrong [6] on the basis of an unambiguous definition of the is the momentum density of the coupled field and material. served by virtue of the homogeneity of the material dog a pseudomomentum as the momentumlike quantity that is ce Expressions for the electromagnetic contributions to the Noether's theorem [10] can be used with a Lagrangian mentum current density and the momentum density are given formulation to obtain a rigorous derivation of the pseudomo- in Eqs. (2.20)and(2.22), respectively. A tensor contribution mentum conservation law for a homogeneous body. This is aMR¨ j2§si ] ]rj „Ei1~R˙ 3B!i…2§ d dt ~s3B!j50. ~3.12! A more convenient form of this equation is found after con￾siderable manipulation @10#, using Eqs.~ 2.2!, ~2.5! and ~3.6!, to be MR¨ j2 ] ]ri $§„Ej1~R˙ 3B!j…si%5rEj1~j3B!j5Fj . ~3.13! The significances of the terms on the left are clarified by their contributions to the momentum densities defined in Sec. III B below. As discussed in connection with Eq. ~2.25!, the kinetic energy of the dielectric is negligible compared to the energy densities associated with the electromagnetic field and the optic mode vibration of the lattice. The contribution of the center-of-mass motion to the Lorentz force on the right-hand side of ~3.11! is also small, as R ˙ B is of order R ˙ E/c, which is certainly much smaller than E. With terms in R˙ removed, the Lagrangian theory reproduces the results of Sec. II B as Eq. ~3.11! reduces to Eq. ~2.9!, and the Maxwell equations are unchanged. The energy continuity equation ~2.17! and the energy densities given in Eqs. ~2.18! and ~2.19! thus con￾tinue to hold. The derivations that follow consider the more subtle problems and distinctions associated with the propa￾gation of momentum and pseudomomentum through the ab￾sorbing dielectric. B. Momentum conservation The law of momentum conservation is a consequence of the invariance of the laws of physics to arbitrary infinitesimal displacements of the spatial coordinates. The momentum is thus defined with respect to the Maxwell equations ~2.1!– ~2.4! and the center-of-mass equation ~3.13!. The continuity equation ~2.21! for the electromagnetic momentum, which is based entirely on Maxwell’s equations, therefore remains valid. The Lorentz force densities Fj on the right-hand sides of Eqs. ~2.21! and ~3.13! are equal and opposite, demonstrat￾ing the action and reaction of the forces between the electro￾magnetic and material parts of the coupled system. Addition of these equations, using definition ~2.12! of the polarization, gives ] ]ri ~Tm!ji1 ] ]t ~Gm!j50, ~3.14! which is the conservation law for the momentum of the field￾material system. Here ~Tm!ji52„Ej1~R˙ 3B!j…Pi1~Tem!ji ~3.15! is the momentum current density, and Gm5MR˙ 1Gem ~3.16! is the momentum density of the coupled field and material. Expressions for the electromagnetic contributions to the mo￾mentum current density and the momentum density are given in Eqs. ~2.20! and ~2.22!, respectively. A tensor contribution with the same magnitude as the center-of-mass kinetic en￾ergy has been omitted from the momentum current density ~3.15! in accordance with the discussion that follows Eq. ~2.25!. It should be noted that the equation of motion ~3.11! for the internal coordinate plays no role in the above derivation, on account of the inability of the relative coordinate to carry momentum @6#. As the dissipation described by Eq. ~3.9! acts only on the internal coordinate, the absence of this coordi￾nate from the derivation of Eqs. ~3.14!–~3.16! accounts for the lack of damping terms in these results. The fulfilment of momentum conservation in the presence of energy dissipa￾tion, as described by the term on the right-hand side of Eq. ~2.17!, is of course a common occurrence in mechanics. The momentum density ~3.16! is clearly separated into a contribution MR˙ from the center-of-mass motion and a con￾tribution «0E3B from the electromagnetic field. The latter differs from the Abraham form «0m0E3H on account of the generalized relation ~3.10! between the magnetic field and the induction. It is seen that, in contrast to the magnetic induction B, which is purely a property of the electromag￾netic field, the magnetic field H contains a contribution that depends on the material internal coordinate contained in P. The occurrence of B rather than H ensures that the electro￾magnetic momentum density is properly independent of any material variables. The total momentum is obtained by integration of Gm~r,t! over all space, and this quantity is conserved for a closed system with no flow of momentum through its boundaries. Thus integration of Eq. ~3.14! over the effectively-infinite dielectric material gives ] ]t E dr Gm~r,t!50, ~3.17! provided that Tm~r,t! vanishes at r5`. The total momentum is therefore conserved, and only its division between the electromagnetic field and the dielectric center-of-mass mo￾tion changes with time as, for example, in the propagation of a pulse of excitation through the crystal. Although the material motion makes an important contri￾bution to the momentum density, its contribution to the mo￾mentum current density ~3.15! is generally less important. Thus R ˙ B is again of order R ˙ E/c, which is much smaller than E, and, with the corresponding term removed, Eq. ~3.15! reduces to ~Tm!ji52EjPi1~Tem!ji . ~3.18! C. Pseudomomentum Pseudomomentum has been a much neglected quantity in continuum mechanics, and a regularly misinterpreted quan￾tity in quantum mechanics. Quantum-mechanical treatments of excitations in solids have often called \k the pseudomo￾mentum of an excitation quantum. This was shown to be wrong @6# on the basis of an unambiguous definition of the pseudomomentum as the momentumlike quantity that is con￾served by virtue of the homogeneity of the material body. Noether’s theorem @10# can be used with a Lagrangian formulation to obtain a rigorous derivation of the pseudomo￾mentum conservation law for a homogeneous body. This is a 55 PROPAGATION OF ELECTROMAGNETIC ENERGY AND... 1075