PHYSICAL REVIEW E VOLUME 55. NUMBER 1 JANUA Propagation of electromagnetic energy and momentum through an absorbing dielectric R. Loudon and L. allen Department of Physics, Essex University, Colchester CO4 3S0, England D. F Nelson Department of Physics, Worcester Polytechnic Institute, Worcester, Massachusetts 01609 We calculate the energy and momentum densities and currents associated with electromagnetic wave propa- gation through an absorbing and dispersive diatomic dielectric, which is modeled by a single- resonance Lor- entz oscillator. The relative and center-of-mass coordinates of the dielectric sublattices and the electromagnetic field vectors are treated as dynamical variables, while the dielectric loss is modeled by a phenomenological damping force. The characteristics of the energy propagation agree with previous work, including the form of the energy velocity. The treatment of momentum propagation extends previous work to lossy media, and it is found that the damping plays an important role in the transfer of momentum from the electromagnetic field to the center of mass of the dielectric. We discuss the significances of the momentum, the pseudomomentum, and their sum, the wave momentum. For each of these quantities we derive the density, the current density, and the appropriate conservation or continuity equation. The general expressions are illustrated by applications to a steady-state monochromatic wave and to an excitation in the form of a localized Gaussian pulse. The velocities associated with propagation of the various kinds of momentum are derived and discussed PACS number(s):4225Bs,03.40.-t,03.50.De,41.20.Jb . INTRODUCTION where the momentum density is a vector quantity propor- tional to the energy current, G=S/c, and the momentum The nature of electromagnetic energy and the characteris- current density is a second rank tensor, or 3X3 matrix, re- tics of its propagation through dielectric media have been lated to the Maxwell stress tensor [1, 2, 9). For the electro- studied since the early years of electromagnetic theory. For magnetic momentum in material media, it is necessary to propagation through the simplest kind of linear, isotropic, take account of contributions from both the electromagnetic and homogeneous medium, the energy density W and energy field and the dielectric medium. The momentum current in a current density, or Poynting vector S are routinely treated in lossless dielectric was obtained by this approach as a modi standard texts [1, 2]. The forms of these energy densities and fied form of the Maxwell stress tensor. In addition, the na- their conservation law have also been evaluated for much tures of the momentumlike quantities that have been defined more general dielectric media [3]. For propagation through for the coupled system of electromagnetic field and dielectric absorbing or scattering materials, the classic treatment of material, including the densities proposed by Abraham and electromagnetic wave propagation, and particularly the iden- Minkowski, were identified [6] tification of the several distinct velocities that are associated The controversy has always revolved around a linear light with an optical pulse, was provided by Sommerfeld and Bril- wave for which deformation of the dielectric medium is ir- louin [4]. The detailed theory for lossy dielectrics is quite relevant, but a key ingredient of its recent resolution is the complicated, but the main features of the energy density and inclusion of deformation of the medium. This necessitates current, and of energy propagation, are correctly predicted by the use of both spatial(Eulerian) and material (Lagrangian) a simple calculation [5], based on the standard model of coordinates, and it allows the deduction of conservation laws electromagnetic waves in a lorentzian dielectric with a from Noether's theorem Thus the momentum conservation single resonance. The essential feature of this theory is the law follows from invariance to displacements of the spatial inclusion of contributions to the total energy density W and coordinates(homogeneity of free space), and the pseudomo- energy current density S of the optical excitation from both mentum conservation law follows from invariance to dis the electromagnetic field and the dielectric medium placements of the material coordinates(homogeneity of the interesting to determine whether there is an analo- material medium). This approach [6] found the electromag gous theory for electromagnetic momentum propagation in a netic momentum density Gm to be EoEXB, close to, but in lossy medium, and this is the primary purpose of the present general different from, the Abraham form EoloEXH. It also paper. Such an inquiry is particularly topical because recent found the pseudomomentum density Gsm to be PXB plus a rk [6]on di but lossless dielectrics has G and a resolution of the long-standing Minkowski-Abraham con- pseudomomentum densities, which we call the wave momen- troversy concerning the correct expressions for the densities tum, is the generalization of the Minkowski momentum of electromagnetic momentum and current, denoted here by DXB to include dispersion. However, the Minkowski mo- G and T, respectively(see [7, 8]for reviews). These densities mentum was proposed as being the ordinary momentum are well understood for electromagnetic fields in free space, while this derivation shows instead that it is the sum of or- 1063-651X/97/55(1)/1071(l5)10.00 c 1997 The American Physical Society
Propagation of electromagnetic energy and momentum through an absorbing dielectric R. Loudon and L. Allen Department of Physics, Essex University, Colchester CO4 3SQ, England D. F. Nelson Department of Physics, Worcester Polytechnic Institute, Worcester, Massachusetts 01609 ~Received 22 August 1996! We calculate the energy and momentum densities and currents associated with electromagnetic wave propagation through an absorbing and dispersive diatomic dielectric, which is modeled by a single-resonance Lorentz oscillator. The relative and center-of-mass coordinates of the dielectric sublattices and the electromagnetic field vectors are treated as dynamical variables, while the dielectric loss is modeled by a phenomenological damping force. The characteristics of the energy propagation agree with previous work, including the form of the energy velocity. The treatment of momentum propagation extends previous work to lossy media, and it is found that the damping plays an important role in the transfer of momentum from the electromagnetic field to the center of mass of the dielectric. We discuss the significances of the momentum, the pseudomomentum, and their sum, the wave momentum. For each of these quantities we derive the density, the current density, and the appropriate conservation or continuity equation. The general expressions are illustrated by applications to a steady-state monochromatic wave and to an excitation in the form of a localized Gaussian pulse. The velocities associated with propagation of the various kinds of momentum are derived and discussed. @S1063-651X~97!04901-5# PACS number~s!: 42.25.Bs, 03.40.2t, 03.50.De, 41.20.Jb I. INTRODUCTION The nature of electromagnetic energy and the characteristics of its propagation through dielectric media have been studied since the early years of electromagnetic theory. For propagation through the simplest kind of linear, isotropic, and homogeneous medium, the energy density W and energy current density, or Poynting vector S are routinely treated in standard texts @1,2#. The forms of these energy densities and their conservation law have also been evaluated for much more general dielectric media @3#. For propagation through absorbing or scattering materials, the classic treatment of electromagnetic wave propagation, and particularly the identification of the several distinct velocities that are associated with an optical pulse, was provided by Sommerfeld and Brillouin @4#. The detailed theory for lossy dielectrics is quite complicated, but the main features of the energy density and current, and of energy propagation, are correctly predicted by a simple calculation @5#, based on the standard model of electromagnetic waves in a Lorentzian dielectric with a single resonance. The essential feature of this theory is the inclusion of contributions to the total energy density W and energy current density S of the optical excitation from both the electromagnetic field and the dielectric medium. It is interesting to determine whether there is an analogous theory for electromagnetic momentum propagation in a lossy medium, and this is the primary purpose of the present paper. Such an inquiry is particularly topical because recent work @6# on dispersive, but lossless, dielectrics has proposed a resolution of the long-standing Minkowski-Abraham controversy concerning the correct expressions for the densities of electromagnetic momentum and current, denoted here by G and T, respectively ~see @7,8# for reviews!. These densities are well understood for electromagnetic fields in free space, where the momentum density is a vector quantity proportional to the energy current, G5S/c2 , and the momentum current density is a second rank tensor, or 333 matrix, related to the Maxwell stress tensor @1,2,9#. For the electromagnetic momentum in material media, it is necessary to take account of contributions from both the electromagnetic field and the dielectric medium. The momentum current in a lossless dielectric was obtained by this approach as a modi- fied form of the Maxwell stress tensor. In addition, the natures of the momentumlike quantities that have been defined for the coupled system of electromagnetic field and dielectric material, including the densities proposed by Abraham and Minkowski, were identified @6#. The controversy has always revolved around a linear light wave for which deformation of the dielectric medium is irrelevant, but a key ingredient of its recent resolution is the inclusion of deformation of the medium. This necessitates the use of both spatial ~Eulerian! and material ~Lagrangian! coordinates, and it allows the deduction of conservation laws from Noether’s theorem. Thus the momentum conservation law follows from invariance to displacements of the spatial coordinates ~homogeneity of free space!, and the pseudomomentum conservation law follows from invariance to displacements of the material coordinates ~homogeneity of the material medium!. This approach @6# found the electromagnetic momentum density Gm to be «0E3B, close to, but in general different from, the Abraham form «0m0E3H. It also found the pseudomomentum density Gpsm to be P3B plus a dispersive term. Thus the sum G of the momentum and pseudomomentum densities, which we call the wave momentum, is the generalization of the Minkowski momentum D3B to include dispersion. However, the Minkowski momentum was proposed as being the ordinary momentum, while this derivation shows instead that it is the sum of orPHYSICAL REVIEW E VOLUME 55, NUMBER 1 JANUARY 1997 1063-651X/97/55~1!/1071~15!/$10.00 1071 © 1997 The American Physical Society 55
R LOUDON. L. ALLEN. AND D. F. NELSON dinary momentum and pseudomomentum. The name"wave nomentum'' was introduced for this reason v×E While the inclusion of material deformation has played ar essential role in the clarification of what is momentum and what is pseudomomentum, it also acts as a barrier to simple V×B=j+ (24) physical understanding. The aim of the present paper is thus to find simplified versions of the conservation laws for mo- mentum and pseudomomentum, even after generalization of where the fields are functions of position and time previous work to include loss. This is achieved by the sim- E=E(r, t), and so on. The bound charge and current densi with cubic isotropy, essentially the single-resonance Lorentz time; they can be expressed in terms of the dielectric po i plification of the dielectric to a nonmagnetic diatomic crystal ties, p and j, respectively, are also functions of position model. The ions are assumed to be coupled to the electro- Ization P as magnetic field only by an electric-dipole interaction Before proceeding to the main calculations, we present a V. P in Sec. Il. An improved version of previous calculations of and the energy continuity equations and the velocity of energy propagation [5] leads to essentially the same results as be- (26) fore, but our method facilitates parallel discussions of the momentum propagation characteristics. It is found, however that the propagation of momentum involves both the center The electric displacement is defined in the usual way, of mass and relative coordinates of the diatomic dielectric whose proper treatment requires a Lagrangian formulation D=EoE+P current obey a conservation law when the center-of-mass and the magnetic field is given by momentum is included, but that the pseudomomentum, and hence the wave momentum, suffer dissipation on account of (28) he dielectric loss. The various electromagnetic densities de- Note that in the view implicit in these equations, E and b are rived in Secs. Il and Ill are evaluated for a steady-state the fundamental electromagnetic fields, while P describes the monochromatic wave in Sec. IV. where the velocities of propagation of energy and wave momentum are derived, and response of the matter, and Eqs. (2.7)and(2. 8)are constitu- for an optical pulse in Sec. V. The results are discussed in tive equations for D and H We consider a polar, diatomic, cubic, crystal lattice in which the relative spatial displacement field of the two ions in the unit cell is denoted s=s(r, t). The long-wavelength Il. SIMPLE THEORY OF ELECTROMAGNETIC ENERGY optic modes of vibration have a basic threefold degeneracy AND MOMENTUM PROPAGATION which is lifted by the long-range electrical forces to form a wofold-degenerate transverse mode and a nondegenerate The present section is devoted to a derivation of some longitudinal mode[10]. Then, if the frequency of the trans basic results for electromagnetic fields in a dielectric material verse mode is denoted or and its damping rate is denoted T, reated in the Lorentz model. We present a simple derivation the standard form of the lorentz equation for the ith Carte- of the equations that describe the propagation of energy, and sian component of the internal coordinate of the ionic motion show that the corresponding description of momentum IS propagation cannot be obtained by so simple a theory. The detailed derivations of the equations that describe momen- ms,+mrs+mo tum propagation and the identification of the different char- acters of the momentumlike contributions are given careful Here m is the reduced mass density of the two ions, of consideration in Sec. Ill masses Mi and M2, in the primitive unit cell of volume n2 A. Basic equations M1M2 1=(M1+M2) (2.10) The fundamental energy and momentum properties electromagnetic fields in matter are governed by maxwells nd the chars equations and by the equations of motion for the matter. we is given by ity s associated with the internal motion consider a nonmagnetic dielectric material that has no free charges or currents. The Maxwell-Lorentz forms of the equa s=e/① (2.11) tions in conventional notation and Systeme International (SI nits are then where e and -e are the charges on the two kinds of ion. The VE=p/Eo, (2. 1 polarization is expressed in terms of the internal coordinate V.B=0, (2.12)
dinary momentum and pseudomomentum. The name ‘‘wave momentum’’ was introduced for this reason. While the inclusion of material deformation has played an essential role in the clarification of what is momentum and what is pseudomomentum, it also acts as a barrier to simple physical understanding. The aim of the present paper is thus to find simplified versions of the conservation laws for momentum and pseudomomentum, even after generalization of previous work to include loss. This is achieved by the simplification of the dielectric to a nonmagnetic diatomic crystal with cubic isotropy, essentially the single-resonance Lorentz model. The ions are assumed to be coupled to the electromagnetic field only by an electric-dipole interaction. Before proceeding to the main calculations, we present a simplified discussion of energy and momentum propagation in Sec. II. An improved version of previous calculations of the energy continuity equations and the velocity of energy propagation @5# leads to essentially the same results as before, but our method facilitates parallel discussions of the momentum propagation characteristics. It is found, however, that the propagation of momentum involves both the center of mass and relative coordinates of the diatomic dielectric, whose proper treatment requires a Lagrangian formulation. Thus it is shown in Sec. III that the momentum density and current obey a conservation law when the center-of-mass momentum is included, but that the pseudomomentum, and hence the wave momentum, suffer dissipation on account of the dielectric loss. The various electromagnetic densities derived in Secs. II and III are evaluated for a steady-state monochromatic wave in Sec. IV, where the velocities of propagation of energy and wave momentum are derived, and for an optical pulse in Sec. V. The results are discussed in Sec. VI. II. SIMPLE THEORY OF ELECTROMAGNETIC ENERGY AND MOMENTUM PROPAGATION The present section is devoted to a derivation of some basic results for electromagnetic fields in a dielectric material treated in the Lorentz model. We present a simple derivation of the equations that describe the propagation of energy, and show that the corresponding description of momentum propagation cannot be obtained by so simple a theory. The detailed derivations of the equations that describe momentum propagation and the identification of the different characters of the momentumlike contributions are given careful consideration in Sec. III. A. Basic equations The fundamental energy and momentum properties of electromagnetic fields in matter are governed by Maxwell’s equations and by the equations of motion for the matter. We consider a nonmagnetic dielectric material that has no free charges or currents. The Maxwell-Lorentz forms of the equations in conventional notation and Syste`me International ~SI! units are then “•E5r/«0 , ~2.1! “•B50, ~2.2! “3E52 ]B ]t , ~2.3! 1 m0 “3B5j1«0 ]E ]t , ~2.4! where the fields are functions of position and time, E[E~r,t!, and so on. The bound charge and current densities, r and j, respectively, are also functions of position and time; they can be expressed in terms of the dielectric polarization P as r52“•P ~2.5! and j5 ]P ]t . ~2.6! The electric displacement is defined in the usual way, D5«0E1P, ~2.7! and the magnetic field is given by H5B/m0 . ~2.8! Note that in the view implicit in these equations, E and B are the fundamental electromagnetic fields, while P describes the response of the matter, and Eqs. ~2.7! and ~2.8! are constitutive equations for D and H. We consider a polar, diatomic, cubic, crystal lattice in which the relative spatial displacement field of the two ions in the unit cell is denoted s[s~r,t!. The long-wavelength optic modes of vibration have a basic threefold degeneracy which is lifted by the long-range electrical forces to form a twofold-degenerate transverse mode and a nondegenerate longitudinal mode @10#. Then, if the frequency of the transverse mode is denoted vT and its damping rate is denoted G, the standard form of the Lorentz equation for the ith Cartesian component of the internal coordinate of the ionic motion is ms¨i1mGs˙i1mvT 2 si5§Ei . ~2.9! Here m is the reduced mass density of the two ions, of masses M1 and M2 , in the primitive unit cell of volume V, m5 M1M2 V~M11M2! ~2.10! and the charge density § associated with the internal motion is given by §5e/V, ~2.11! where e and 2e are the charges on the two kinds of ion. The polarization is expressed in terms of the internal coordinate by P5§s. ~2.12! 1072 R. LOUDON, L. ALLEN, AND D. F. NELSON 55
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 vector, given by Sem5E3B/m0 ~2.13! for the cubic isotropic material assumed here. It is straightforward 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 electromagnetic energy continuity equation ~2.14! is thus balanced 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 electromagnetic 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 Lorentzian oscillator or optic mode, thus makes no explicit contribution 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 electromagnetic 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 Maxwell 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 standard vector operator identities is ] ]ri ~Tem!ji1 ] ]t ~Gem!j52rEj2~j3B
1074 R LOUDON. L. ALLEN. AND D. F. NELSON and the relation between the frequency and wave vector has been taken in its free-space form, for the purpose of an order- of-magnitude estimate. Clearly it is important to include the where the charge and current densities are related to the di center-of-mass momentum of the crystal in any theory of electric polarization by Eqs. (2.5)and(2.6).However, when nomentum propagation through an absorbing dielectric the center-of-mass motion is included, the latter expression The transfer of momentum to the dielectric must be ac- should be augmented by inclusion of the Rontgen current ompanied by a growth in its kinetic energy density, whose [11], to give a total current density value for the momentum density given by Eq (2.23)is aP MR +V×(P×R (36) where r is again the continuum center-of-mass coordinate The rest-mass energy density Mc2 of the crystal is The interaction Lagrangian density(3. 5)can be converted ery much larger than the initial energy density Ni with the use of this expression to rystal kinetic energy is thus completely negligible cor to Nho. This justifies the neglect of center-of-mass motion C1=P(E+R×B) (3.7) in the theory of energy propagation given in Sec. II B, de spite its importance in the theory of momentum propagation. where some perfect space and time derivative terms, which make no contribution to the Lagrange equations of motion, IIL LAGRANGIAN THEORY OF ELECTROMAGNETIC have been discarded [10]. The material Lagrangian density MOMENTUM PROPAGATION for a rigid body is This section is devoted to a rigorous derivation of the CM=MR2+显ms2-显mos2 (38) various momentum densities associated with the propagation of electromagnetic waves through absorbing dielectrics. The where the dielectric parameters are as defined in Sec. IL. The basic dielectric model is the same as that used in Sec. II, but theory also needs to include a term that allows for damping it is necessary to generalize the model to include center-of- of the internal motion at a rate proportional to T. This mass motion in order to describe momentum propagation. It conveniently implemented by a Rayleigh dissipation func- is also necessary to distinguish the contributions of momen- tion of the form tum and pseudomomentum. The continuum mechanics back ground to the calculations is described in detail in Ref [10] R=m2, (3.9) It is assumed throughout that the dielectric material fills all of space the effects of crystal boundaries are excluded from which is incorporated into the Euler-Lagrange equations by the calculations an appropriate additional term [12] The equations of motion for the electromagnetic and ma- A. Lagrangian formulation terial field variables are obtained by the standard Lagrangian The system of dielectric material (M) and electron procedures. Thus the Maxwell-Lorentz equations(2. 1)and netic field (F) coupled by an electric-dipole interaction(n) (2. 4)are rederived straightforwardly, while Eqs.(2.2)and described by a Lagrangian density (2.3)are satisfied automatically from the definitions (3.3) and (3. 4)of the fields in terms of the potentials. It should L=LMt Lte (3. 1) however be noted that the Rontgen term in the current den- sity(3. 6)causes a generalization of relation(2.8)between where the Lagrangian itself is formed by integration over the magnetic field and magnetic induction to [see, for example, Lagrangian density in the usual way Eq(76. 11)of Ref [21 The Lagrangian density of the electromagnetic field is (3.10) The new term is a function of both the internal relative- where the electric and magnetic fields are determined by the displacement coordinate and the center-of-mass coordinate calar potential and the vector potential A in the usual of the dielectric material For the dielectric spatial displacement variables, the equa- tion of motion for the relative position of the two ions in the ⅴd-at (3.3) unit cell is obtained with the use of Eqs. (3. 7)-(3.9)as 5+mIs;+mo3s;=s(E1+(R×B)),(3,1) which is identical to Eq.(2.9)except for the addition of the B=V×A (3.4) term proportional to the center-of-mass velocity R. The equation of motion for the continuum center-of-mass coordi- The interaction Lagrangian density is is obtained similarly
and the relation between the frequency and wave vector has been taken in its free-space form, for the purpose of an orderof-magnitude estimate. Clearly it is important to include the center-of-mass momentum of the crystal in any theory of momentum propagation through an absorbing dielectric. The transfer of momentum to the dielectric must be accompanied by a growth in its kinetic energy density, whose value for the momentum density given by Eq. ~2.23! is MR˙ 2 2 5N\v N\v 2M c2 . ~2.25! The rest-mass energy density M c2 of the crystal is always very much larger than the initial energy density N\v. The crystal kinetic energy is thus completely negligible compared to N\v. This justifies the neglect of center-of-mass motion in the theory of energy propagation given in Sec. II B, despite its importance in the theory of momentum propagation. III. LAGRANGIAN THEORY OF ELECTROMAGNETIC MOMENTUM PROPAGATION This section is devoted to a rigorous derivation of the various momentum densities associated with the propagation of electromagnetic waves through absorbing dielectrics. The basic dielectric model is the same as that used in Sec. II, but it is necessary to generalize the model to include center-ofmass motion in order to describe momentum propagation. It is also necessary to distinguish the contributions of momentum and pseudomomentum. The continuum mechanics background to the calculations is described in detail in Ref. @10#. It is assumed throughout that the dielectric material fills all of space; the effects of crystal boundaries are excluded from the calculations. A. Lagrangian formulation The system of dielectric material (M) and electromagnetic field (F) coupled by an electric-dipole interaction (I) is described by a Lagrangian density L5LM1LI1LF , ~3.1! where the Lagrangian itself is formed by integration over the Lagrangian density in the usual way. The Lagrangian density of the electromagnetic field is LF5«0 2 E22 1 2m0 B2, ~3.2! where the electric and magnetic fields are determined by the scalar potential f and the vector potential A in the usual way, E52“f2 ]A ]t ~3.3! and B5“3A. ~3.4! The interaction Lagrangian density is LI5j•A2rf, ~3.5! where the charge and current densities are related to the dielectric polarization by Eqs. ~2.5! and ~2.6!. However, when the center-of-mass motion is included, the latter expression should be augmented by inclusion of the Ro¨ntgen current @11#, to give a total current density j5 ]P ]t 1“3~P3R˙ !, ~3.6! where R is again the continuum center-of-mass coordinate. The interaction Lagrangian density ~3.5! can be converted with the use of this expression to Ll5P•~E1R˙ 3B!, ~3.7! where some perfect space and time derivative terms, which make no contribution to the Lagrange equations of motion, have been discarded @10#. The material Lagrangian density for a rigid body is LM5 1 2 MR˙ 21 1 2 ms˙22 1 2 mvT 2 s 2, ~3.8! where the dielectric parameters are as defined in Sec. II. The theory also needs to include a term that allows for damping of the internal motion at a rate proportional to G. This is conveniently implemented by a Rayleigh dissipation function of the form R5 1 2 mGs˙2, ~3.9! which is incorporated into the Euler-Lagrange equations by an appropriate additional term @12#. The equations of motion for the electromagnetic and material field variables are obtained by the standard Lagrangian procedures. Thus the Maxwell-Lorentz equations ~2.1! and ~2.4! are rederived straightforwardly, while Eqs. ~2.2! and ~2.3! are satisfied automatically from the definitions ~3.3! and ~3.4! of the fields in terms of the potentials. It should however be noted that the Ro¨ntgen term in the current density ~3.6! causes a generalization of relation ~2.8! between magnetic field and magnetic induction to @see, for example, Eq. ~76.11! of Ref. @2## H5 B m0 2P3R˙ . ~3.10! The new term is a function of both the internal relativedisplacement coordinate and the center-of-mass coordinate of the dielectric material. For the dielectric spatial displacement variables, the equation of motion for the relative position of the two ions in the unit cell is obtained with the use of Eqs. ~3.7!–~3.9! as ms¨i1mGs˙i1mvT 2 si5§„Ei1~R˙ 3B!i…, ~3.11! which is identical to Eq. ~2.9! except for the addition of the term proportional to the center-of-mass velocity R˙ . The equation of motion for the continuum center-of-mass coordinate is obtained similarly as 1074 R. LOUDON, L. ALLEN, AND D. F. NELSON 55
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 a
MR¨ j2§si ] ]rj „Ei1~R˙ 3B!i…2§ d dt ~s3B!j50. ~3.12! A more convenient form of this equation is found after considerable 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 continue to hold. The derivations that follow consider the more subtle problems and distinctions associated with the propagation of momentum and pseudomomentum through the absorbing 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, demonstrating the action and reaction of the forces between the electromagnetic 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 fieldmaterial 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 momentum 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 energy 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 coordinate 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 dissipation, 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 contribution «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 electromagnetic 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 electromagnetic 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 motion changes with time as, for example, in the propagation of a pulse of excitation through the crystal. Although the material motion makes an important contribution to the momentum density, its contribution to the momentum 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 quantity in quantum mechanics. Quantum-mechanical treatments of excitations in solids have often called \k the pseudomomentum 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 conserved 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 pseudomomentum conservation law for a homogeneous body. This is a 55 PROPAGATION OF ELECTROMAGNETIC ENERGY AND... 1075
R LOUDON. L. ALLEN. AND D. F. NELSON body of infinite, or at least large, extent compared to the D Wave momentum interaction volume considered, so that any boundary effects erve the pseudomo- can be ignored. The invariance used in the application of the mentum alone. Any interaction in a large homogeneous body theorem to pseudomomentum is with respect to an arbitrary involves a conserved pseudomomentum but, because the un- infinitesimal displacement of the material coordinates of a derlying vacuum space is also homogeneous, the interaction well. As simil analogous to the arbitrary infinitesimal displacement of the spatial coordinates, namely, the position of a point with re- momentum and pseudomomentum should combine togethe spect to the free-space vacuum, which is used with Noethers in summation to form the wave momentum. Such a summa- theorem to obtain the momentum conservation law. Based on tion is valid when the deformation of the material body is these definitions, it has been proved [6] that hik is a quantum negligibly small. Thus the experiments of Jones and co- of the wave momentum, that is, of the sum of momentum workers [1] on liquid dielectrics and of Gibson et al. [14] and pseudomomentum. The pseudomomentum conservation on semiconductors observed momentum transfers propor- tional to the refractive index, in agreement with the form of motion, without the use of Noether's theorem However, No- the unit of wave momentum hk in a nonabsorbing medium ether's theorem relates the conservation law to the invariance The wave momentum current density and momentum property that gives rise to it and thus provides a firm identi- density are denoted T and G, respectively. Dielectric defor fication of the conserved quantity mation is negligible in the vicinity of the optic mode fre- The pseudomomentum of the system considered here is quency, and we may add the momentum conservation equa- not in fact a conserved quantity, because of the loss intro- tion(3.14)to the pseudomomentum continuity equation duced by the optic mode damping. The conservation equa- (3.19), to form the wave momentum continuity equation tion is thus replaced by a continuity equation. To find it from the equations of motion, we need to combine only the inter nal coordinate equation(3. 11) and the version (3. 12)of the dt (323) center-of-mass continuum equation, that is, the material equations. The electromagnetic field equations, which essen- tially describe vacuum-based rather than material-based of the approximations (3. 18)and(3 22)for small center-of- after multiplication by as /ari, to Eq.(3. 12) gives a result mass velocity(R<c)as that can be written in the form of the continuity equation (Tsm),+-(Gpsm//mls.cs d THi=,Di+iEDSir BB +5(m52-mo22+EP) (324) ere and the expression for G is obtained by addition of the (Tpsm)i=im(52-0752)(E+RX B).P)8 general results(3.16)and(3.21)as G=(DX is the pseudomomentum current density, with a term of the same magnitude as the center-of-mass kinetic energy again neglected. The pseudomomentum density is given by where D is the electric displacement defined in Eq (2.7). It is en that the center-of-mass momentu dIs in the fo mation of the wave momentum. The term on the right of Eq MR -ms. ar+(PX B)).(3.21)(3.23)represents the rate of loss of wave momentum from he coupled field-lattice system caused by the optic mode The term on the right-hand side of Eq.(3. 19)represents the damping The first four terms in the current density (3. 24) have the rate of loss of pseudomomentum caused by damping of the form of the negative of the Maxwell stress tensor in a dielec- optic mode. The total time derivative occurs in accordance with the role of Gpsm as a momentum defined relative to the tric [1]. The remaining bracketted terms are additional, but loving medium heir cycle-averaged contributions vanish in the examples of The pseudomomentum current density simplifies when monochromatic and pulsed excitations of the system consid- the term that includes enter-of-mass velocity R is ne ered in Secs. IV and V, respectively(see also [6]). The glected, and Eq(3.20)reduces to momentum density (3.25)is the same as the Minkowski ex pression plus a dispersive term. However, the Minkowski (Tpsm)ji=(2 m(52-0732)+EP)S;i.(3. 22) tum, not for the sum of momentum and pseudomomentum as embodied in the wave momentum. It is seen from the above However, the center-of-mass term in the pseudomomentum expressions that, unlike the energy densities (2. 18)and density(3. 21)is comparable to the other terms, and it must (2.19), neither the momentum current density(3.24)nor the momentum density (3.25) separates into distinct electromag-
body of infinite, or at least large, extent compared to the interaction volume considered, so that any boundary effects can be ignored. The invariance used in the application of the theorem to pseudomomentum is with respect to an arbitrary infinitesimal displacement of the material coordinates of a point that moves with the dielectric. This is completely analogous to the arbitrary infinitesimal displacement of the spatial coordinates, namely, the position of a point with respect to the free-space vacuum, which is used with Noether’s theorem to obtain the momentum conservation law. Based on these definitions, it has been proved @6# that \k is a quantum of the wave momentum, that is, of the sum of momentum and pseudomomentum. The pseudomomentum conservation law derived below follows directly from the equations of motion, without the use of Noether’s theorem. However, Noether’s theorem relates the conservation law to the invariance property that gives rise to it and thus provides a firm identi- fication of the conserved quantity. The pseudomomentum of the system considered here is not in fact a conserved quantity, because of the loss introduced by the optic mode damping. The conservation equation is thus replaced by a continuity equation. To find it from the equations of motion, we need to combine only the internal coordinate equation ~3.11! and the version ~3.12! of the center-of-mass continuum equation, that is, the material equations. The electromagnetic field equations, which essentially describe vacuum-based rather than material-based quantities, do not contribute. Thus addition of Eq. ~3.11!, after multiplication by ]si/]rj , to Eq. ~3.12! gives a result that can be written in the form of the continuity equation ] ]ri ~Tpsm!ji1 d dt ~Gpsm!j5mGs˙ • ]s ]rj , ~3.19! where ~Tpsm!ji5$ 1 2 m~s˙ 22vT 2 s2!1~E1R˙ 3B!•P%d ji ~3.20! is the pseudomomentum current density, with a term of the same magnitude as the center-of-mass kinetic energy again neglected. The pseudomomentum density is given by ~Gpsm!j52MR˙ j2ms˙• ]s ]rj 1~P3B!j . ~3.21! The term on the right-hand side of Eq. ~3.19! represents the rate of loss of pseudomomentum caused by damping of the optic mode. The total time derivative occurs in accordance with the role of Gpsm as a momentum defined relative to the moving medium. The pseudomomentum current density simplifies when the term that includes the center-of-mass velocity R ˙ is neglected, and Eq. ~3.20! reduces to ~Tpsm!ji5$ 1 2 m~s˙ 22vT 2 s2!1E•P%d ji . ~3.22! However, the center-of-mass term in the pseudomomentum density ~3.21! is comparable to the other terms, and it must be retained. D. Wave momentum It is difficult, if not impossible, to observe the pseudomomentum alone. Any interaction in a large homogeneous body involves a conserved pseudomomentum but, because the underlying vacuum space is also homogeneous, the interaction involves a conserved momentum as well. As similar quantities with identical dimensions, it is not surprising that the momentum and pseudomomentum should combine together in summation to form the wave momentum. Such a summation is valid when the deformation of the material body is negligibly small. Thus the experiments of Jones and coworkers @13# on liquid dielectrics and of Gibson et al. @14# on semiconductors observed momentum transfers proportional to the refractive index, in agreement with the form of the unit of wave momentum \k in a nonabsorbing medium. The wave momentum current density and momentum density are denoted T and G, respectively. Dielectric deformation is negligible in the vicinity of the optic mode frequency, and we may add the momentum conservation equation ~3.14! to the pseudomomentum continuity equation ~3.19!, to form the wave momentum continuity equation ]Tji ]ri 1 ]Gj ]t 5mGs˙• ]s ]rj . ~3.23! In this relation, the expression for Tji is obtained by addition of the approximations ~3.18! and ~3.22! for small center-ofmass velocity (R ˙ !c) as Tji52EjDi1 1 2 E•Dd ji2 BjBi m0 1 B2 2m0 d ji 1 1 2 ~ms˙ 22mvT 2 s21E•P!d ji , ~3.24! and the expression for Gj is obtained by addition of the general results ~3.16! and ~3.21! as Gj5~D3B!j2ms˙• ]s ]rj , ~3.25! where D is the electric displacement defined in Eq. ~2.7!. It is seen that the center-of-mass momentum cancels in the formation of the wave momentum. The term on the right of Eq. ~3.23! represents the rate of loss of wave momentum from the coupled field-lattice system caused by the optic mode damping. The first four terms in the current density ~3.24! have the form of the negative of the Maxwell stress tensor in a dielectric @1#. The remaining bracketted terms are additional, but their cycle-averaged contributions vanish in the examples of monochromatic and pulsed excitations of the system considered in Secs. IV and V, respectively ~see also @6#!. The wave momentum density ~3.25! is the same as the Minkowski expression plus a dispersive term. However, the Minkowski momentum was proposed as an expression for the momentum, not for the sum of momentum and pseudomomentum as embodied in the wave momentum. It is seen from the above expressions that, unlike the energy densities ~2.18! and ~2.19!, neither the momentum current density ~3.24! nor the momentum density ~3.25! separates into distinct electromag- 1076 R. LOUDON, L. ALLEN, AND D. F. NELSON 55
PROPAGATION OF ELECTROMAGNETIC ENERGY AND 1077 netic and material contributions, as the polarization P, ex A. Dielectric function pressed in the form(2. 12), is a material variable Consider a plane wave of frequency o and wave vector k The final results(3.23)-(3. 25)can also be obtained di- that is propagated parallel to the axis with its electric and ctly from the simple theory of Sec. I. Thus multiplication magnetic vectors oriented in the directions of the x and y of Eq(2.)by as /ari gives axes, respectively. The real electric field is written conven- tionally as a sum of positive-and negative-frequency contri- ds E(=,t)=E+(x,t)+E-(=,t) where Eq.(2.12)has been used. Subtraction of Eq. (3.26) Et(aexp(-iot+ik=)+E(exp(iot-ikz) from Eg.(2.21)gives (4.1) Here Et(o) is the complex amplitude at ==0, t=0 o{(Tm)n+m(s2-0s2)6+E E-(o)=[E+(a)]*, (Gcm)-ms}+pE+(j×B)=ms可 and a similar notation is used for the other fields The am- plitude of the field at ==0 is assumed to be time indepen- (3.27) dent, and the model therefore provides for a constant supply It is not difficult to show with the use of Eqs. ( 2.3),(2.5), dielectric material is still assumed to fill all of space, and the boundary condition at ==0 does not imply the existence of (2.6)and standard vector operator identities that It follows from Eq. (2.9)that m2+pE,+(xB),=2 (P×B)+r-(E SE(o/ (E,P) (3. 28) The electric displacement D(o) and the dielectric function (o) are defined by Thus Eq.(3.27)can be written in the form of the continuity equation(3. 23)with the same definitions (3.24)and(3.25) DT(o)=EoE(oET(O)=EoE()+Pt(o),(4.4) the wave momentum densities. However, in contrast to this and use of Eqs. (2. 12)and(4.3)leads to the explicit expres direct derivation, the lagrangian formulation establishes the sion nature of the wave momentum. Its two distinct contributions arising from the conserved momentum and the dissipating pseudomomentum, are unambiguously identified. Their sepa- ()=1+ (4.5) rate conservation and continuity properties are expressed by Eqs. (3. 14)and(3.19)respectively The refractive index no) and extinction coefficient K(a) are defined in the usual way by IV MONOCHROMATIC WAVE e(u)=[m(o)+ik(a)]2 (46) No assumptions have so far been made about the time dependences of the fields. We now evaluate the various den- and it follows from Eq. (4.5)that sities that have been derived in the previous two sections for the simple example of a monochromatic plane wave. The (4.7) cycle averages of the various energy and momentum densi- ties associated with the electromagnetic fields and the inter- and time in this example, and they have exponentially decaying spatial dependences. The electromagnetic and internal parts 2n(o)k(a)= (48) of the system are thus subjected to a steady-state excitation Eom(0r-02))2+62r2 and the monochromatic case usefully displays the full fre- fluency dependences of the energy and momentum densities. The wave vector is given by the usual expression, k=ln(o)+ix(oJo/ which maintains the steady state, results in a center-of-mass momentum density that grows linearly with the time, pro- and the complex magnetic and electric field amplitudes are vided that the velocity remains nonrelativistic. The results related by derived in the present section are valid for arbitrarily strong B(o=Ln()+iK(o)JE(o/c (410)
netic and material contributions, as the polarization P, expressed in the form ~2.12!, is a material variable. The final results ~3.23!–~3.25! can also be obtained directly from the simple theory of Sec. II. Thus multiplication of Eq. ~2.9! by ]si/]rj gives ms¨i ]si ]rj 1mGs˙i ]si ]rj 1mvT 2 si ]si ]rj 5E• ]P ]rj , ~3.26! where Eq. ~2.12! has been used. Subtraction of Eq. ~3.26! from Eq. ~2.21! gives ] ]ri $~Tem!ji1 1 2 m~s˙ 22vT 2 s2!d ji%1E• ]P ]rj 1 ] ]t H ~Gem!j2ms˙• ]s ]rj J 1rEj1~j3B!j5mGs˙• ]s ]rj . ~3.27! It is not difficult to show with the use of Eqs. ~2.3!, ~2.5!, ~2.6! and standard vector operator identities that E• ]P ]rj 1rEj1~j3B!j5 ] ]t ~P3B!j1 ] ]rj ~E•P! 2 ] ]ri ~EjPi!. ~3.28! Thus Eq. ~3.27! can be written in the form of the continuity equation ~3.23! with the same definitions ~3.24! and ~3.25! of the wave momentum densities. However, in contrast to this direct derivation, the Lagrangian formulation establishes the nature of the wave momentum. Its two distinct contributions, arising from the conserved momentum and the dissipating pseudomomentum, are unambiguously identified. Their separate conservation and continuity properties are expressed by Eqs. ~3.14! and ~3.19! respectively. IV. MONOCHROMATIC WAVE No assumptions have so far been made about the time dependences of the fields. We now evaluate the various densities that have been derived in the previous two sections for the simple example of a monochromatic plane wave. The cycle averages of the various energy and momentum densities associated with the electromagnetic fields and the internal motion of the crystal lattice are all independent of the time in this example, and they have exponentially decaying spatial dependences. The electromagnetic and internal parts of the system are thus subjected to a steady-state excitation, and the monochromatic case usefully displays the full frequency dependences of the energy and momentum densities. However, the constant supply of electromagnetic energy, which maintains the steady state, results in a center-of-mass momentum density that grows linearly with the time, provided that the velocity remains nonrelativistic. The results derived in the present section are valid for arbitrarily strong damping. A. Dielectric function Consider a plane wave of frequency v and wave vector k that is propagated parallel to the z axis with its electric and magnetic vectors oriented in the directions of the x and y axes, respectively. The real electric field is written conventionally as a sum of positive- and negative-frequency contributions, E~z,t!5E1~z,t!1E2~z,t! 5E1~v!exp~2ivt1ikz!1E2~v!exp~ivt2ikz!. ~4.1! Here E1~v! is the complex amplitude at z50, t50, E2~v!5@E1~v!#*, ~4.2! and a similar notation is used for the other fields. The amplitude of the field at z50 is assumed to be time independent, and the model therefore provides for a constant supply of energy at the coordinate origin. It is emphasized that the dielectric material is still assumed to fill all of space, and the boundary condition at z50 does not imply the existence of any real boundary. It follows from Eq. ~2.9! that si 1~v!5 §Ei 1~v!/m vT 22v22ivG . ~4.3! The electric displacement D~v! and the dielectric function «~v! are defined by Di 1~v!5«0«~v!Ei 1~v!5«0Ei 1~v!1Pi 1~v!, ~4.4! and use of Eqs. ~2.12! and ~4.3! leads to the explicit expression «~v!511 §2 «0m 1 vT 22v22ivG . ~4.5! The refractive index h~v! and extinction coefficient k~v! are defined in the usual way by «~v!5@h~v!1ik~v!# 2, ~4.6! and it follows from Eq. ~4.5! that h~v! 22k~v! 2511 §2 «0m vT 22v2 ~vT 22v2! 21v2G2 ~4.7! and 2h~v!k~v!5 §2 «0m vG ~vT 22v2! 21v2G2 . ~4.8! The wave vector is given by the usual expression, k5@h~v!1ik~v!#v/c, ~4.9! and the complex magnetic and electric field amplitudes are related by B1~v!5@h~v!1ik~v!#E1~v!/c. ~4.10! 55 PROPAGATION OF ELECTROMAGNETIC ENERGY AND... 1077
1078 R LOUDON. L. ALLEN. AND D. F. NELSON It is convenient to simplify the expressions that occur in the remainder of the section by removal of explicit o depen dence from the notation for the dielectric properties and field amplitudes B Energy propagation The total-energy current density(2. 18)has only a nonzero z component for the geometry assumed here, and its cycle average is Eoc FIG. 1. Frequency dependence of the energy velocity (4. 16)for the dielectric function (4.5)with seom=0.136 this gives the L=c/20K 4.12) ratio of longitudinal to transverse optic mode frequencies WL/o=1066 found in GaAs. The curves are labeled with the ap- is the attenuation length, the distance after which the inten- propriate values of T/or sity of an electromagnetic wave in the dielectric decays to l/e of its initial value. The total-energy density(2. 19)has a cycle average Up LN and we note that it is not possible to express this quantity entirely in terms of macroscopic electromagnetic functions, is the phase velocity. Figure 1 shows the frequency depen- independent of the parameters of the optic mode. These ex- dence of the energy velocity in the vicinity of the transverse pressions for the total energy densities agree with a previous resonance for several values of the dampin derivation [5]. The cycle average of the energy dissipation Although derived for a specific model, relation (4. 17)be rate on the right of Eq.(2.17)is tween energy and phase velocities, decay length and damp- ing rate is found to apply to a wide range of systems, includ- -(mT5)=-4E0onKEPe-il=- 2EoCn1E+Pe. dielectrics in regions of resonant absorption [ 15, 16] and ir ing the propagation of pulsed optical signals through (4.14) regions of resonant amplification [17], self-induced transpar ency in two-level atoms [18], and energy transport in media 2. 17 cycle average of the energy continuity equation containing randomly distributed scatterers [19]. A similar re- takes the form lation is also valid for the propagation of ultrasonic signals [20]. Each of these systems has a detailed theoretical treat- 0(S2) ment for the relevant attenuation or amplification process, =-(m32) (4.15) but the derived and measured propagation velocities gene ally agree with the common form of energy velocity given and it is readily verified from Eqs. (4.11), (4. 12)and (4. 14) by Eq. (4. I7) that this relation is indeed satisfied. The energy that is con stantly supplied at ==0 in the example considered here tion,until none is left for propagation distances =>L. The (3. 18)and(2.20) has three nonzero components for the 9s steadily drains into the reservoir associated with the dissipa- Momentum. The momentum current density given by Eqs kinetic energy delivered to the dielectric material also grows ometry assumed here. The two transverse components have steadily in this example, but the material velocity is assumed cycle-averaged values to be always sufficiently small that the accumulated kinetic energy is negligible (Tm)x)=E0(1-2+3k2)E+Pe-m(419) Just as the ratio of the values of(S-) and(w)in a lossless dielectric gives the ray or energy velocity [10], so the ratio of and the energy densities is taken to define the velocity ve of energy transport through the absorbing dielectric as (Tm)y)=80(1-2-k2)E+P2e-.(4.20) 〈S2) W)刀+(2ok/T) (4.16) The cycle-averaged longitudinal component Is ((Tm)==((Tem) This can be rearranged in the form (421)
It is convenient to simplify the expressions that occur in the remainder of the section by removal of explicit v dependence from the notation for the dielectric properties and field amplitudes. B. Energy propagation The total-energy current density ~2.18! has only a nonzero z component for the geometry assumed here, and its cycle average is ^Sz&52«0chuE1u 2e22vkz/c 52«0chuE1u 2e2z/L, ~4.11! where L5c/2vk ~4.12! is the attenuation length, the distance after which the intensity of an electromagnetic wave in the dielectric decays to 1/e of its initial value. The total-energy density ~2.19! has a cycle average ^W&52«0S h21 2vhk G DuE1u 2e2z/L, ~4.13! and we note that it is not possible to express this quantity entirely in terms of macroscopic electromagnetic functions, independent of the parameters of the optic mode. These expressions for the total energy densities agree with a previous derivation @5#. The cycle average of the energy dissipation rate on the right of Eq. ~2.17! is 2^mGs˙ 2 &524«0vhkuE1u 2e2z/L52 2«0ch L uE1u 2e2z/L. ~4.14! The cycle average of the energy continuity equation ~2.17! takes the form ]^Sz& ]z 52^mGs˙ 2 &, ~4.15! and it is readily verified from Eqs. ~4.11!, ~4.12! and ~4.14! that this relation is indeed satisfied. The energy that is constantly supplied at z50 in the example considered here steadily drains into the reservoir associated with the dissipation, until none is left for propagation distances z@L. The kinetic energy delivered to the dielectric material also grows steadily in this example, but the material velocity is assumed to be always sufficiently small that the accumulated kinetic energy is negligible. Just as the ratio of the values of ^Sz& and ^W& in a lossless dielectric gives the ray or energy velocity @10#, so the ratio of the energy densities is taken to define the velocity ve of energy transport through the absorbing dielectric as ve5^Sz& ^W& 5 c h1~2vk/G! . ~4.16! This can be rearranged in the form 1 ve 5 1 vp 1 1 LG , ~4.17! where vp5c/h, ~4.18! is the phase velocity. Figure 1 shows the frequency dependence of the energy velocity in the vicinity of the transverse resonance for several values of the damping. Although derived for a specific model, relation ~4.17! between energy and phase velocities, decay length and damping rate is found to apply to a wide range of systems, including the propagation of pulsed optical signals through dielectrics in regions of resonant absorption @15,16# and in regions of resonant amplification @17#, self-induced transparency in two-level atoms @18#, and energy transport in media containing randomly distributed scatterers @19#. A similar relation is also valid for the propagation of ultrasonic signals @20#. Each of these systems has a detailed theoretical treatment for the relevant attenuation or amplification process, but the derived and measured propagation velocities generally agree with the common form of energy velocity given by Eq. ~4.17!. C. Momentum propagation Momentum. The momentum current density given by Eqs. ~3.18! and ~2.20! has three nonzero components for the geometry assumed here. The two transverse components have cycle-averaged values ^~Tm!xx&5«0~12h213k2!uE1u 2e2z/L ~4.19! and ^~Tm!yy&5«0~12h22k2!uE1u 2e2z/L. ~4.20! The cycle-averaged longitudinal component is ^~Tm!zz&5^~Tem!zz&5«0~11h21k2!uE1u 2e2z/L. ~4.21! FIG. 1. Frequency dependence of the energy velocity ~4.16! for the dielectric function ~4.5! with §2 /«0mv T 250.136; this gives the ratio of longitudinal to transverse optic mode frequencies vL/vT51.066 found in GaAs. The curves are labeled with the appropriate values of G/vT . 1078 R. LOUDON, L. ALLEN, AND D. F. NELSON 55
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+|2
The momentum density is given by Eqs. ~3.16! and ~2.22!, and only the z component is nonzero, with the cycleaveraged 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 cycleaveraged 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 crosssectional 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 conservation 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 momentum. Pseudomomentum. The pseudomomentum current density given by Eq. ~3.22! is the same for all three diagonal components, 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 equation 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 results 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 frequency 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
R LOUDON. L. ALLEN. AND D. F. NELSON ((Tm)2 (Gm)= (442) 0.25 With no motion of the center of mass, it is possible to define 1n7m){1,=3g (Tm)=2) FIG. 2. Frequency dependence of the difference between wave momentum and energy velocities for the same parameters as Fi The momentum densities both take their usual fre values when the phase velocity is set equal to c, and the D. Limit of zero damping momentum velocity also becomes equal to c It is instructive to consider the forms of the various de The pseudomomentum densities(4.26)and (4. 27)become sities defined above in the limit of zero damping [→0,k(ω)→0]l, when the refractive index is obtained from (m2=80-1+)E+P2 444 Eq. (4.7)as (4.36 G EP2,(445 Thus, for a lossless dielectric, the cycle-average energy cur- rent density(4.11) can be simply expressed in terms of the hase velocity(4.18) where the limit given in Eq(4.39)is used in the latter. A pseudomomentum velocity can thus be defined by ((Tpsm)2e) °pm(Gpm)2)=c2-2n The group velocity is defined by The pseudomomentum densities both vanish in free space The cycle-averaged wave momentum densities (4.31)and (4.32) become (om)=—+ A2_212 S2) E (447) and it is easily verified with the use of Eqs.(4.7)and(4.8) (4.39) 2 I→0 E+E The lossless limit of the cycle-average energy density(4.13) The cycle-averaged dissipation rate (4. 28)vanishes in the limit of zero damping, and the wave momentum velocity (4. 34)reduces, like the energy velocity, to the group velocity ER This velocity can be expressed in the fo The cycle-averaged dissipation rate (4.14)of course van ishes, and the energy velocity (4.16) becomes the same as ((Gm)= m+((Gpsm)-)u he group velocity in the limit of zero damping. This is ((Gm)-+((Nsm)-) shown as the T=0 curve in Fig. 1 In the absence of any material boundaries, and hence of of a sum of the momentum and pseudomomentum any reflection of the incident electromagnetic wave, no mo- weighted by their respective densities. However, mentum is transferred from the electromagnetic field to the the wave momentum velocity has the well-behaved form center-of-mass motion of a lossless dielectric. Thus R can shown by the T=0 curve in Fig. 1, the momentum velocity everywhere be set equal to zero. The cycle-average momen- (4.41)diverges at both the transverse and longitudinal fre- tum densities(4.21)and(4.22) become quencies while the pseudomomentum velocity(4. 46)di
D. Limit of zero damping It is instructive to consider the forms of the various densities defined above in the limit of zero damping @G→0,k~v!→0#, when the refractive index is obtained from Eq. ~4.7! as h2511 §2 «0m 1 vT 22v2 . ~4.36! Thus, for a lossless dielectric, the cycle-average energy current density ~4.11! can be simply expressed in terms of the phase velocity ~4.18! as ^Sz&5 2«0c2 vp uE1u 2. ~4.37! The group velocity is defined by c vg 5 ] ]v ~vh!5 c vp 1 vp c §2 «0m v2 ~vT 22v2! 2 , ~4.38! and it is easily verified with the use of Eqs. ~4.7! and ~4.8! that Lt G→0 S h21 2vhk G D 5h ] ]v ~vh!5 c2 vpvg . ~4.39! The lossless limit of the cycle-average energy density ~4.13! is thus ^W&5 2«0c2 vpvg uE1u 2. ~4.40! The cycle-averaged dissipation rate ~4.14! of course vanishes, and the energy velocity ~4.16! becomes the same as the group velocity in the limit of zero damping. This is shown as the G50 curve in Fig. 1. In the absence of any material boundaries, and hence of any reflection of the incident electromagnetic wave, no momentum is transferred from the electromagnetic field to the center-of-mass motion of a lossless dielectric. Thus R ˙ can everywhere be set equal to zero. The cycle-average momentum densities ~4.21! and ~4.22! become ^~Tm!zz&5«0S 11 c2 vp 2 D uE1u 2 ~4.41! and ^~Gm!z&5 2«0 vp uE1u 2. ~4.42! With no motion of the center of mass, it is possible to define a momentum velocity as vm5^~Tm!zz& ^~Gm!z& 5S 11 c2 vp 2 D vp 2 . ~4.43! The momentum densities both take their usual free-space values when the phase velocity is set equal to c, and the momentum velocity also becomes equal to c. The pseudomomentum densities ~4.26! and ~4.27! become ^~Tpsm!zz&5«0S 211 c2 vp 2 DuE1u 2 ~4.44! and ^~Gpsm!z&5 2«0 vp S 211 c2 vpvg DuE1u 2, ~4.45! where the limit given in Eq. ~4.39! is used in the latter. A pseudomomentum velocity can thus be defined by vpsm5^~Tpsm!zz& ^~Gpsm!z& 5 c22vp 2 c22vgvp vg 2 . ~4.46! The pseudomomentum densities both vanish in free space. The cycle-averaged wave momentum densities ~4.31! and ~4.32! become ^Tzz&5 2«0c2 vp 2 uE1u 25^Sz& vp ~4.47! and ^Gz&5 2«0c2 vp 2 vg uE1u 25^W& vp . ~4.48! The cycle-averaged dissipation rate ~4.28! vanishes in the limit of zero damping, and the wave momentum velocity ~4.34! reduces, like the energy velocity, to the group velocity vwm5ve5vg . ~4.49! This velocity can be expressed in the form vwm5^~Gm!z&vm1^~Gpsm!z&vpsm ^~Gm!z&1^~Gpsm!z& ~4.50! of a sum of the momentum and pseudomomentum velocities weighted by their respective densities. However, although the wave momentum velocity has the well-behaved form shown by the G50 curve in Fig. 1, the momentum velocity ~4.41! diverges at both the transverse and longitudinal frequencies while the pseudomomentum velocity ~4.46! diFIG. 2. Frequency dependence of the difference between wave momentum and energy velocities for the same parameters as Fig. 1. 1080 R. LOUDON, L. ALLEN, AND D. F. NELSON 55
