PHYSICAL REVIEW E 73. 026606(2006) Energy and momentum of light in dielectric media Philips Research Laboratories, Professor Holstlaan 4, 561/ AA Eindhoven, The Netherland. Received 10 November 2005; published 9 February 2006) The conservation of energy, linear momentum, and angular momentum of the electromagnetic field in linear dielectric media with arbitrary dispersion and absorption is studied in the framework of an auxiliary field approach in which the electric and magnetic fields are complemented by a material field. This material field depends on a continuous variable o, and describes harmonic motions of the charges with eigenfrequency o. It carries an electric dipole moment and couples as such to the electric field. The equations of motion of the model are equivalent to Maxwell's equations in an arbitrary dispersive and absorbing dielectric and imply that several quantities are conserved. These quantities may be interpreted as the energy, momentum, and angular momentum of the total system, and can be viewed as the sum of the corresponding quantities of the field and matter subsystems. The total momentum turns out to be equal to the Minkowski momentum plus a dispersive contribution. The total energy and total momentum of a wave packet both travel with the group velocity, while the ratio of total momentum and total energy is given by the phase velocity DOI:10.1103/ PhysReve.73.026606 PACS number(s): 03.50.De, 42.25.Bs L INTRODUCTION pret the physical meaning of these conserved quantities. The starting point of this paper is an auxiliary field model The linear momentum of light in dielectric media is a for the description of electromagnetic fields in linear dielec- omplicated concept, as evidenced by the variety of views on tric media with arbitrary dispersion and absorption intro- the subject that can be found in the literature. Most of the duced by Tip [9, 10]. A similar model has later been proposed work focuses on the abraham and minkowski forms for the electromagnetic momentum(see [1] for a review). Differe y Figotin and Schenker [11]. The basic variables of the approaches to the problem can be found in the papers by an auxiliary field F representing the material degrees of free Gordon [2], Nelson [3]. Garrison and Chiao [4], Loudon and dom interacting with the electromagnetic field. The material co-workers [5,6]. Obukhov and Hehl [7], and Mansuripur field F effectively describes the harmonic motions of the 8]. This list of references is far from comprehensive, but charges inside the dielectric. a difference between the elec- gives a fair view of the different approache tromagnetic fields e and b and the material field F is that the Several aspects of the momentum concept are very subtle former depend on position r and time t only, whereas the and do not lend themselves to easy understanding. In this latter depends on a third continuous variable o as well. This paper,two of these aspects are studied in some detail. The third variable can be interpreted as the(angular)eigenfre- first is the role of dispersion and dissipation. The dynamics quency of the harmonic material motions. The electromag tools that are frequently used for conservative systems, in The coupling is proportional to a function d(o)which turns particular the canonical framework based on the use of out to be(the Fourier transform of) the conductivity, which Lagrangians and Hamiltonians. For that reason it is not clear for a dielectric may be defined as E,()o, where e ()is the how to define momentum, a conserved quantity, for dissipa imaginary part of (the Fourier transform of) the dielectric tive systems. The second aspect concerns the difference be- function. The strength of the model is that the equations of tween uniformity of space and homogeneity of matter. The motion are formally equivalent to the set of equations con- invariance for translations of the total system gives rise to sisting of Maxwell's equations and the constitutive relation conservation of momentum, the invariance for material dis- between the dielectric displacement D and E for an arbitrary placements of the dielectric gives rise to conservation of dispersive and absorbing medium. The equations of motion pseudomomentum. Depending on the experimental circum- can be derived from the standard variational principle based stances one or the other, or even a combination of both types upon the action being the integral over space and time of the of momenta seems useful. The difficulty in describing dissi- Lagrangian density pative systems can be overcome, at least in some cases, by making the system larger. Additional degrees of freedom that The canonical framework defined by the Lagrangian den- interact with the dissipative system can be introduced so that sity implies the existence of several conserved quantities, the total system is conservative. It is the goal of this paper to which may be interpreted as the energy, momentum, and entum of the find such an enlarged system description, investigate the at- the total system con sisting of the electromagnetic field and the material system. tendant conservation laws for the enlarged system, and inter- The conservation laws will be derived from the equations of motion of the model. Alternative proofs based on Noether's theorem are possible but will not be presented. For each Electronicaddresssjoerdstallinga@philips.com conserved quantity a density p and a flow v may be defined 1539-3755/200673(2)/026606(12)/$23.00 026606-1 @2006 The American Physical SocietyEnergy and momentum of light in dielectric media Sjoerd Stallinga* Philips Research Laboratories, Professor Holstlaan 4, 5611 AA Eindhoven, The Netherlands
Received 10 November 2005; published 9 February 2006 The conservation of energy, linear momentum, and angular momentum of the electromagnetic field in linear dielectric media with arbitrary dispersion and absorption is studied in the framework of an auxiliary field approach in which the electric and magnetic fields are complemented by a material field. This material field depends on a continuous variable
, and describes harmonic motions of the charges with eigenfrequency
. It carries an electric dipole moment and couples as such to the electric field. The equations of motion of the model are equivalent to Maxwell’s equations in an arbitrary dispersive and absorbing dielectric and imply that several quantities are conserved. These quantities may be interpreted as the energy, momentum, and angular momentum of the total system, and can be viewed as the sum of the corresponding quantities of the field and matter subsystems. The total momentum turns out to be equal to the Minkowski momentum plus a dispersive contribution. The total energy and total momentum of a wave packet both travel with the group velocity, while the ratio of total momentum and total energy is given by the phase velocity. DOI: 10.1103/PhysRevE.73.026606 PACS number
s: 03.50.De, 42.25.Bs I. INTRODUCTION The linear momentum of light in dielectric media is a complicated concept, as evidenced by the variety of views on the subject that can be found in the literature. Most of the work focuses on the Abraham and Minkowski forms for the electromagnetic momentum
see 1 for a review. Different approaches to the problem can be found in the papers by Gordon 2, Nelson 3, Garrison and Chiao 4, Loudon and co-workers 5,6, Obukhov and Hehl 7, and Mansuripur 8. This list of references is far from comprehensive, but gives a fair view of the different approaches. Several aspects of the momentum concept are very subtle and do not lend themselves to easy understanding. In this paper, two of these aspects are studied in some detail. The first is the role of dispersion and dissipation. The dynamics of dissipative systems cannot be described by the theoretical tools that are frequently used for conservative systems, in particular the canonical framework based on the use of Lagrangians and Hamiltonians. For that reason it is not clear how to define momentum, a conserved quantity, for dissipa￾tive systems. The second aspect concerns the difference be￾tween uniformity of space and homogeneity of matter. The invariance for translations of the total system gives rise to conservation of momentum, the invariance for material dis￾placements of the dielectric gives rise to conservation of pseudomomentum. Depending on the experimental circum￾stances one or the other, or even a combination of both types of momenta seems useful. The difficulty in describing dissi￾pative systems can be overcome, at least in some cases, by making the system larger. Additional degrees of freedom that interact with the dissipative system can be introduced so that the total system is conservative. It is the goal of this paper to find such an enlarged system description, investigate the at￾tendant conservation laws for the enlarged system, and inter￾pret the physical meaning of these conserved quantities. The starting point of this paper is an auxiliary field model for the description of electromagnetic fields in linear dielec￾tric media with arbitrary dispersion and absorption intro￾duced by Tip 9,10. A similar model has later been proposed by Figotin and Schenker 11. The basic variables of the theory are the electric field E and magnetic induction B and an auxiliary field F representing the material degrees of free￾dom interacting with the electromagnetic field. The material field F effectively describes the harmonic motions of the charges inside the dielectric. A difference between the elec￾tromagnetic fields E and B and the material field F is that the former depend on position r and time t only, whereas the latter depends on a third continuous variable
as well. This third variable can be interpreted as the
angular eigenfre￾quency of the harmonic material motions. The electromag￾netic and material fields interact through a dipole coupling. The coupling is proportional to a function ˆ

which turns out to be
the Fourier transform of the conductivity, which for a dielectric may be defined as ˆi


, where ˆi

is the imaginary part of
the Fourier transform of the dielectric function. The strength of the model is that the equations of motion are formally equivalent to the set of equations con￾sisting of Maxwell’s equations and the constitutive relation between the dielectric displacement D and E for an arbitrary dispersive and absorbing medium. The equations of motion can be derived from the standard variational principle based upon the action being the integral over space and time of the Lagrangian density. The canonical framework defined by the Lagrangian den￾sity implies the existence of several conserved quantities, which may be interpreted as the energy, momentum, and angular momentum of the total system, the total system con￾sisting of the electromagnetic field and the material system. The conservation laws will be derived from the equations of motion of the model. Alternative proofs based on Noether’s theorem are possible but will not be presented. For each *Electronic address: sjoerd.stallinga@philips.com conserved quantity a density  and a flow v may be defined PHYSICAL REVIEW E 73, 026606
2006 1539-3755/2006/73
2/026606
12/$23.00 ©2006 The American Physical Society 026606-1