PROPAGATION OF ELECTROMAGNETIC ENERGY AN 1073 B. Energy propagation B. B The flow of electromagnetic energy through the dielectric (Tem)ji=-EoE E,+iEoE-Sjn or is determined by the energy current density, or Poynting vec- This quantity is usually identified as the negative of the Max (2.13) well stress tensor [1, 2], but occasionally as the Maxwell stress tensor [9]. The momentum continuity equation for the for the cubic isotropic material assumed here. It is straight- electromagnetic field is obtained from Maxwell's equations forward to show with the use of Eqs. (2.3)and(2.4)that products of EoE with Eq. (2.1), B/Ho Eq.(2.3), and B with Eq(2. 4), and (S-m)计+Wcm=-Ej (2.14) then adding the equations. The result after use of stan- dard vector operator identities is here the repeated index i is summed over the Cartesian coordinates x,y, and : and dr: (Tem)yi+ at gem) --PEj-GXB=-Fy, is the usual electromagnetic energy density. Equation(2.14)where sses the continuity of the electromagnetic energy, and the term on the right represents the rate of loss of energy Gcm=E0E×B from the field by transfer to the dielectric. Multiplication of (2.9)by s, gives is the electromagnetic momentum density. Equation (2.21) expresses continuity of electromagnetic momentum. The ms,+mIs:+moTS S = sE 5S, =Ej, ( 2.16) terms on the right represent the rate of loss of momentum from the field by transfer to the dielectric, in the form of where Eqs.(2.6)and(2. 12) have been used, similar to a minus the usual Lorentz force density, denoted Fi calculation in [4]. The rate of loss on the right of the elec- The transfer of momentum from the electromagnetic field tromagnetic energy continuity equation(2.14)is thus bal- implies that the dielectric as a whole is set into motion.The anced by the rate of gain of energy represented by the term internal relative displacement field s is itself invariant under on the right of Eq(2. 16)for the dielectric lattice The a uniform displacement of the crystal, and cannot therefore sum of Eqs. (2. 14)and(2. 16)can be written in the form of carry momentum. The dielectric momentum is carried by the an energy continuity equation for the coupled electromag- motion of the spatial displacement field R=R(r, t) defined by netic field and dielectric lattice he position of the center of mass of the two ions in the unit cell. A treatment of the propagation of momentum through as an (2. 17) cludes both the relative and center-of-mass coordinates s and R; this is provided by the Lagrangian formalism presented in here the total-energy current density Sec. ll S=Scm=E×B/0 The effect of the dissipation term in the internal equation (2.18) of motion(2.9) is to remove energy from the optic modes of is the same as the electromagnetic current density(2.13),but vibration. The sink for this energy is provided by a reservoir, the total energy density is whose nature is determined by the microscopic mechanism of the dissipation. For example, anharmonic forces in the W=IE0E+uoH2+ms2+mas2).(2.19) lattice transfer the optic-mode energy into continuous distri butions of other vibrational modes which are not directly The excitation of the dielectric lattice. that is of the Lorent- coupled to the electromagnetic field. Thus an initial excita- zian oscillator or optic mode, thus makes no explicit contri- tion of the coupled electromagnetic field and optic modes bution to the energy current density. The lattice does, decays to a steady state in which all of the energy is trans- though, have an implicit effect via the scaling of the ratio of ferred to the reservoir. This transfer has implications for both the magnetic and electric fields by the complex refractive the momentum and the kinetic energy associated with the index of the medium [see 0)]. However, the energy motion of the dielectric cryst: density explicitly contains the kinetic and potential energies Suppose that the initial excitation has N quanta of wave of the optic vibrational mode in addition to the electromag- vector k and frequency w per unit volume. The magnitude of netic energy density(2.15). The term on the right of Eq. the momentum density acquired by the dielectric crystal as a coupled field-lattice system by the optic mode sity from the whole, when all of the energy has been transferred to the (2. 17)represents the rate of loss of energy de damping reservoIr. IS C Momentum propagation MR= Nhk=Nho/c 2.23 The flow of electromagnetic momentum is determined by where M is the dielectric mass dens the momentum current density, whose components are given by[12,6,9,10] M=(M1+M2)/, (224)B. Energy propagation The flow of electromagnetic energy through the dielectric is determined by the energy current density, or Poynting vec￾tor, given by Sem5E3B/m0 ~2.13! for the cubic isotropic material assumed here. It is straight￾forward to show with the use of Eqs. ~2.3! and ~2.4! that ] ]ri ~Sem!i1 ] ]t Wem52E•j, ~2.14! where the repeated index i is summed over the Cartesian coordinates x, y, and z, and Wem5 1 2 «0E21 1 2 m0H2 ~2.15! is the usual electromagnetic energy density. Equation ~2.14! expresses the continuity of the electromagnetic energy, and the term on the right represents the rate of loss of energy from the field by transfer to the dielectric. Multiplication of ~2.9! by s˙i gives ms¨is˙i1mGs˙i 2 1mvT 2 sis˙i5§Eis˙i5E•j, ~2.16! where Eqs. ~2.6! and ~2.12! have been used, similar to a calculation in @4#. The rate of loss on the right of the elec￾tromagnetic energy continuity equation ~2.14! is thus bal￾anced by the rate of gain of energy represented by the term on the right of Eq. ~2.16! for the dielectric lattice mode. The sum of Eqs. ~2.14! and ~2.16! can be written in the form of an energy continuity equation for the coupled electromag￾netic field and dielectric lattice, ]Si ]ri 1 ]W ]t 52mGs˙ 2, ~2.17! where the total-energy current density S5Sem5E3B/m0 ~2.18! is the same as the electromagnetic current density ~2.13!, but the total energy density is W5 1 2 $«0E21m0H21ms˙ 21mvT 2 s2 %. ~2.19! The excitation of the dielectric lattice, that is of the Lorent￾zian oscillator or optic mode, thus makes no explicit contri￾bution to the energy current density. The lattice does, though, have an implicit effect via the scaling of the ratio of the magnetic and electric fields by the complex refractive index of the medium @see Eq. ~4.10!#. However, the energy density explicitly contains the kinetic and potential energies of the optic vibrational mode in addition to the electromag￾netic energy density ~2.15!. The term on the right of Eq. ~2.17! represents the rate of loss of energy density from the coupled field-lattice system by the optic mode damping. C. Momentum propagation The flow of electromagnetic momentum is determined by the momentum current density, whose components are given by @1,2,6,9,10# ~Tem!ji52«0EjEi1 1 2 «0E2d ji2 BjBi m0 1 B2 2m0 d ji . ~2.20! This quantity is usually identified as the negative of the Max￾well stress tensor @1,2#, but occasionally as the Maxwell stress tensor @9#. The momentum continuity equation for the electromagnetic field is obtained from Maxwell’s equations by forming the vector products of «0E with Eq. ~2.1!, B/m0 with Eq. ~2.2!, E with Eq. ~2.3!, and B with Eq. ~2.4!, and then adding the four equations. The result after use of stan￾dard vector operator identities is ] ]ri ~Tem!ji1 ] ]t ~Gem!j52rEj2~j3B![2Fj , ~2.21! where Gem5«0E3B ~2.22! is the electromagnetic momentum density. Equation ~2.21! expresses continuity of electromagnetic momentum. The terms on the right represent the rate of loss of momentum from the field by transfer to the dielectric, in the form of minus the usual Lorentz force density, denoted Fj . The transfer of momentum from the electromagnetic field implies that the dielectric as a whole is set into motion. The internal relative displacement field s is itself invariant under a uniform displacement of the crystal, and cannot therefore carry momentum. The dielectric momentum is carried by the motion of the spatial displacement field R[R~r,t! defined by the position of the center of mass of the two ions in the unit cell. A treatment of the propagation of momentum through the dielectric thus requires a theoretical framework that in￾cludes both the relative and center-of-mass coordinates s and R; this is provided by the Lagrangian formalism presented in Sec. III. The effect of the dissipation term in the internal equation of motion ~2.9! is to remove energy from the optic modes of vibration. The sink for this energy is provided by a reservoir, whose nature is determined by the microscopic mechanism of the dissipation. For example, anharmonic forces in the lattice transfer the optic-mode energy into continuous distri￾butions of other vibrational modes which are not directly coupled to the electromagnetic field. Thus an initial excita￾tion of the coupled electromagnetic field and optic modes decays to a steady state in which all of the energy is trans￾ferred to the reservoir. This transfer has implications for both the momentum and the kinetic energy associated with the motion of the dielectric crystal. Suppose that the initial excitation has N quanta of wave vector k and frequency v per unit volume. The magnitude of the momentum density acquired by the dielectric crystal as a whole, when all of the energy has been transferred to the reservoir, is MR˙ 5N\k5N\v/c, ~2.23! where M is the dielectric mass density, M5~M11M2!/V, ~2.24! 55 PROPAGATION OF ELECTROMAGNETIC ENERGY AND... 1073