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 SocietyPropagation 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 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. @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 characteris￾tics 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 iden￾tification of the several distinct velocities that are associated with an optical pulse, was provided by Sommerfeld and Bril￾louin @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 analo￾gous 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 con￾troversy 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 propor￾tional to the energy current, G5S/c2 , and the momentum current density is a second rank tensor, or 333 matrix, re￾lated to the Maxwell stress tensor @1,2,9#. For the electro￾magnetic 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 na￾tures 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 ir￾relevant, 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 pseudomo￾mentum conservation law follows from invariance to dis￾placements of the material coordinates ~homogeneity of the material medium!. This approach @6# found the electromag￾netic 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 momen￾tum, is the generalization of the Minkowski momentum D3B to include dispersion. However, the Minkowski mo￾mentum was proposed as being the ordinary momentum, while this derivation shows instead that it is the sum of or￾PHYSICAL REVIEW E VOLUME 55, NUMBER 1 JANUARY 1997 1063-651X/97/55~1!/1071~15!/$10.00 1071 © 1997 The American Physical Society 55