《电动力学》课程参考文献：Energy and momentum of light in dielectric media

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 Society

Energy 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

SJOERD STALLINGA PHYSICAL REVIEW E 73. 026606(2006) satisfying a transport equation of the form a,p+Vv=0 (with and the dependence of F on(angular) eigenfrequency a obvious generalization to conserved quantities with a vecto- position r, and time t is suppressed, except when this com- rial character). Balance equations for two subsystems, for pact notation can give rise to ambiguity. Vector notation is example the"field!"and"matter"subsystems, have the form used if convenient and the tensor notation in all other cases p1+V.v1=-Q, (1) ne partial derivative with respect to time is denoted by an the partial derivative with respect to the spatial coordinates by da where a=x,y, z, and the Einstein summation conven- dp ,+v v2=Q (2) tion is used. Partial derivatives only apply to the quantity here p=p1+p2 and V=V+V2, and where Q represents the directly following the derivative unless brackets indicate oth dissipation of field energy, momentum or angular momentum erwise. The tensor Sap is the Kronecker tensor(SB=l if a from subsystem 1 to 2. It turns out that the exchange of =B and O otherwise), and the tensor eaBy is the Levi-Civita and material parts is such that the dissipation integrated over odd permutations, and O otherwise) the duration of the interaction is always positive. This irre- versibility is related to the coupling of the electromagnetic IL. EQUATIONS OF MOTION degrees of freedom to a continuum of harmonic oscillators. rather than to a finite number of degrees of freedom The action is the integral over time and space of the La- The split of the conservation laws into balance equations grangian density for the field and material subsystems is to some extent arbi trary, and various definitions will do. As a consequence the 1= darl dissipation of energy, momentum, and angular momentum of the field to matter are also ambiguous. a key point of inter- where the Lagrangian density is the sum of an electromag pretation is thus how to relate these quantities to the ab- sorbed heat, force, and torque on the medium that are actu- etc contribution, a contribution from the material field, and lly observed in experiment. It may therefore be the case that an interaction contribution different experimental circumstances require the application of different descriptions of momenta and forces. The answer C=E2 LB2+ do(o)[(F)2-2F2+2F·E] to the abraham- Minkowksi debate in this view is not a defi- 2 nition of "the" momentum of light in dielectric media but tum. An attempt is made in this paper to find out for which The function G(o)is positive for all nonzero o and defined physical situation the total field-plus-matter momentum of for negative angular frequencies by d(a)=G(o). The ab- he auxiliary field model is a useful quantity ence of free charges and currents implies that G(o)-0 if The main shortcoming of the auxiliary field model is that 0-0. It may be defined for complex o by analytical con- it does not take into account deformation or displacement of tinuation and is assumed to have no poles in the upper half the material medium. It is assumed that the position of each complex plane (in view of causality). The electromagnetic material point is kept fixed throughout the interaction with part of the Lagrangian density is just the vacuum electromag the electromagnetic field. This implies that the distinction netic Lagrangian density, the material part describes a con- between the space-fixed coordinate frame and the coordinate tinuous set of harmonic oscillators, and the interaction term frame fixed to the material points is lost so that a clear iden- describes the interaction of the electric field with a continu- tification as to which quantity is momentum and which quan- ous set of electric dipoles. The polarization P is thus entirely tity is pseudomomentum cannot be made. This indistinguis defined in terms of the material field F ability of uniformity of space and homogeneity of matter has the consequence that only one meaningful momentumlike dodo) conserved quantity exists within the model. This total system momentum corresponds to what is called pseudomomentum by Gordon [2], wave momentum(the sum of momentum and The dielectric displacement D and magnetic field H are then pseudomomentum)by Nelson [3] and canonical momentum defined by by Garrison and Chiao [4] The paper is organized as follows. In Sec. Il the equations of motion are derived and shown to be equivalent to Max E+2E0 wells equations in general linear dielectrics. The conserva tion laws are treated in Sec. ll and sec. lv focuses on the energy and momentum of a one-dimensional wave packet H B The paper is concluded in Sec. V with a discussion of the obtained results and an outlook on possibilities for future The scalar potential and vector potential A are introduced explorations Concerning the notation. it is mentioned that in the fol- lowing the dependence of E and B on position r and time t E=-Vd-aA 0266062

satisfying a transport equation of the form
t +
·v= 0
with obvious generalization to conserved quantities with a vecto￾rial character. Balance equations for two subsystems, for example the “field” and “matter” subsystems, have the form
t 1 +
· v1 = − Q,
1
t 2 +
· v2 = Q,
2 where =1+2 and v=v1+v2, and where Q represents the dissipation of field energy, momentum or angular momentum from subsystem 1 to 2. It turns out that the exchange of energy, momentum, and angular momentum between field and material parts is such that the dissipation integrated over the duration of the interaction is always positive. This irre￾versibility is related to the coupling of the electromagnetic degrees of freedom to a continuum of harmonic oscillators, rather than to a finite number of degrees of freedom. The split of the conservation laws into balance equations for the field and material subsystems is to some extent arbi￾trary, and various definitions will do. As a consequence the dissipation of energy, momentum, and angular momentum of the field to matter are also ambiguous. A key point of inter￾pretation is thus how to relate these quantities to the ab￾sorbed heat, force, and torque on the medium that are actu￾ally observed in experiment. It may therefore be the case that different experimental circumstances require the application of different descriptions of momenta and forces. The answer to the Abraham-Minkowksi debate in this view is not a defi- nition of “the” momentum of light in dielectric media but rather a prescription of when to use which type of momen￾tum. An attempt is made in this paper to find out for which physical situation the total field-plus-matter momentum of the auxiliary field model is a useful quantity. The main shortcoming of the auxiliary field model is that it does not take into account deformation or displacement of the material medium. It is assumed that the position of each material point is kept fixed throughout the interaction with the electromagnetic field. This implies that the distinction between the space-fixed coordinate frame and the coordinate frame fixed to the material points is lost so that a clear iden￾tification as to which quantity is momentum and which quan￾tity is pseudomomentum cannot be made. This indistinguish￾ability of uniformity of space and homogeneity of matter has the consequence that only one meaningful momentumlike￾conserved quantity exists within the model. This total system momentum corresponds to what is called pseudomomentum by Gordon 2, wave momentum
the sum of momentum and pseudomomentum by Nelson 3 and canonical momentum by Garrison and Chiao 4. The paper is organized as follows. In Sec. II the equations of motion are derived and shown to be equivalent to Max￾well’s equations in general linear dielectrics. The conserva￾tion laws are treated in Sec. III, and Sec. IV focuses on the energy and momentum of a one-dimensional wave packet. The paper is concluded in Sec. V with a discussion of the obtained results and an outlook on possibilities for future explorations. Concerning the notation, it is mentioned that in the fol￾lowing the dependence of E and B on position r and time t and the dependence of F on
angular eigenfrequency
, position r, and time t is suppressed, except when this com￾pact notation can give rise to ambiguity. Vector notation is used if convenient and the tensor notation in all other cases. The partial derivative with respect to time is denoted by
t , the partial derivative with respect to the spatial coordinates by
, where =x, y ,z, and the Einstein summation conven￾tion is used. Partial derivatives only apply to the quantity directly following the derivative unless brackets indicate oth￾erwise. The tensor  is the Kronecker tensor
= 1 if  = and 0 otherwise, and the tensor  is the Levi-Civita tensor
 = 1 for  even permutations of xyz, −1 for odd permutations, and 0 otherwise. II. EQUATIONS OF MOTION The action is the integral over time and space of the La￾grangian density I =  dt  d3 rL,
3 where the Lagrangian density is the sum of an electromag￾netic contribution, a contribution from the material field, and an interaction contribution L = 0 2 E2 − 1 2 0 B2 + 0  0  d
ˆ



t F 2 −
2 F2 + 2F · E.
4 The function ˆ

is positive for all nonzero
and defined for negative angular frequencies by ˆ

=ˆ
−
. The ab￾sence of free charges and currents implies that ˆ

→0 if
→0. It may be defined for complex
by analytical con￾tinuation and is assumed to have no poles in the upper half complex plane
in view of causality. The electromagnetic part of the Lagrangian density is just the vacuum electromag￾netic Lagrangian density, the material part describes a con￾tinuous set of harmonic oscillators, and the interaction term describes the interaction of the electric field with a continu￾ous set of electric dipoles. The polarization P is thus entirely defined in terms of the material field F P = 20   0  d
ˆ

F.
5 The dielectric displacement D and magnetic field H are then defined by D =
L
E = 0E + 20   0  d
ˆ

F,
6 H = −
L
B = 1 0 B.
7 The scalar potential and vector potential A are introduced via E = −
−
t A,
8 SJOERD STALLINGA PHYSICAL REVIEW E 73, 026606
2006 026606-2

ENERGY AND MOMENTUM OF LIGHT IN DIELECTRIC PHYSICAL REVIEW E 73. 026606(2006) B=V×A e(1)=at)+-|do0(a)(t) which solves the two homogeneous Maxwell equations v×E+aB=0 (10) d(o) TJo 02-(o'+iy? exp(io't) v.B=0 (19) The Euler-Lagrange equations for the potentials are the two It follows that the Fourier transform of the dielectric function inhomogeneous" Maxwell equations where is given by because in the present context there are no free charges and (a) currents so that these equations are in fact homogeneous as e(o)=1+= do-m2-(o+iy)2 v.D=0 (12) 1+ ,(a2 v×H-aD=0 1+ (20) which can be demonstrated with textbook manipulations a d,G(w) [12,13] where it has been used that d(o)=G(o). Using that The Euler-Lagrange equation for the material field F is the equation of a driven harmonic oscillator =P一+ima), (21) 2F+ (14) The inhomogeneous solution of this equation is (with depen. where the capital"P"indicates the principal value, it follows dence on o and t explicit) G(o' Gw) id(o) (22) where g(o, t)is a Greens function of the harmonic oscillator equation. The homogeneous solution is not present in this classical theory. However, in the quantum theory it must be By construction, this function satisfies the Kramers-Kronig taken into account. There it describes a noise polarization, a relations as well as the symmetry relation E(o)=E(-o).As quantity which can even be interpreted as the basic ingredi- a consequence, the dielectric function in the time domain is ent of the quantum theory on which all other fields depend real [e(t=a(t)'1 and causal [e()=0 if tsoJ. This proves [14-17]. In the classical theory it turns out that the solution that the constitutive relation for media with arbitrary disper- is causal provided the Green's function is chosen to be the sion and absorption is properly described by the present retarded Greens function model. As a consequence, the equations of motion for the electromagnetic field in such media are formally equivalent G(a,t)=6(r) sin(ot) (16) to the Euler-Lagrange equations for the proposed Lagrangian dens where A(r) is the step function [a(r=l if t>0, a(0)=1/2 if IIL CONSERVATION LAWS 1=0, 0(0)=0 if t<o]. The retarded Greens function has Fourier representation The transport and dissipation of electromagnetic energy is 2Tw'-('+iy)2 p(-io'D,(17 described by the energy balance equation a I M+VS=-w (23) where y is a positive infinitesimal quantity. The resulting where the electromagnetic field energy density u M, energy expression for the material field F leads to a dielectric dis- flux density S( Poynting vector), and the density of the rate of work on the material subsystem W are defined by D)=Eodr’e(t-t)E() n=E2+-B2, with the dielectric function S=E×H 026606-3

B =
 A,
9 which solves the two homogeneous Maxwell equations
 E +
t B = 0,
10
· B = 0.
11 The Euler-Lagrange equations for the potentials are the two “inhomogeneous” Maxwell equations where we use quotes because in the present context there are no free charges and currents so that these equations are in fact homogeneous as well
· D = 0,
12
 H −
t D = 0,
13 which can be demonstrated with textbook manipulations 12,13. The Euler-Lagrange equation for the material field F is the equation of a driven harmonic oscillator
t 2 F +
2 F = E.
14 The inhomogeneous solution of this equation is
with depen￾dence on
and t explicit F

,t =  −  dt
G

,t − t
E
t
,
15 where G

,t is a Green’s function of the harmonic oscillator equation. The homogeneous solution is not present in this classical theory. However, in the quantum theory it must be taken into account. There it describes a noise polarization, a quantity which can even be interpreted as the basic ingredi￾ent of the quantum theory on which all other fields depend 14–17. In the classical theory it turns out that the solution is causal provided the Green’s function is chosen to be the retarded Green’s function G

,t = 
t sin

t
,
16 where 
t is the step function 
t= 1 if t0, 
t= 1/ 2 if t=0, 
t= 0 if t0. The retarded Green’s function has a Fourier representation G

,t =  −  d

2 1
2 −


+ i 2 exp
− i

t,
17 where is a positive infinitesimal quantity. The resulting expression for the material field F leads to a dielectric dis￾placement D
t = 0 −  dt

t − t
E
t
,
18 with the dielectric function
t = 
t + 2  0  d
ˆ


t sin

t
=  −  d

2 1 + 2  0  d
ˆ


2 −


+ i 2 exp
− i

t.
19 It follows that the Fourier transform of the dielectric function is given by ˆ

=1+ 2  0  d

ˆ




2 −

+ i 2 =1+ 1  0  d

ˆ




 1

−
− i − 1

+
+ i =1+ 1  −  d

ˆ




1

−
− i ,
20 where it has been used that ˆ

=ˆ
−
. Using that 1
− i = P 1
+ i

,
21 where the capital “P” indicates the principal value, it follows that ˆ

=1+ 1  P −  d

ˆ







−
+ iˆ


=1+ 2  P 0  d

ˆ




2 −
2 + iˆ


.
22 By construction, this function satisfies the Kramers-Kronig relations as well as the symmetry relation ˆ

=ˆ
−
* . As a consequence, the dielectric function in the time domain is real
t=
t *  and causal
t= 0 if t0. This proves that the constitutive relation for media with arbitrary disper￾sion and absorption is properly described by the present model. As a consequence, the equations of motion for the electromagnetic field in such media are formally equivalent to the Euler-Lagrange equations for the proposed Lagrangian density. III. CONSERVATION LAWS A. Energy The transport and dissipation of electromagnetic energy is described by the energy balance equation
t uEM +
· S = − W,
23 where the electromagnetic field energy density uEM, energy flux density S
Poynting vector, and the density of the rate of work on the material subsystem W are defined by uEM = 0 2 E2 + 1 2 0 B2,
24 S = E  H,
25 ENERGY AND MOMENTUM OF LIGHT IN DIELECTRIC... PHYSICAL REVIEW E 73, 026606
2006 026606-3

SJOERD STALLINGA PHYSICAL REVIEW E 73. 026606(2006) W=dP.E dod(o[(OF)2+0F2-FE] This energy balance equation follows directly from Max- wells equations [12] It appears that the rate of work can be written as the tin =中,E-3P=DE-3D,E,(35 derivative of a quantity that may be interpreted as the energy of the material subsystem been used, in a way similar to the derivation of Eq (27. b where the equation of motion naterial field f ha ap. E=Eo dod(a)oFE nondispersive to dispersive dissipation W is approximately zero in case dispersion is small. The total energy may then be approximated by the nondispersive contribution u. This doi(o)aF·(aF+a2F) leads to the default expression for the electromagnetic energy in a dielectric appearing in many textbooks [12, 29] doi(o)(aF)2+o3F].(27) B Momentum he transport and dissipation of electromagnetic (linear) Here, the equation of motion of the material field F, Eq (14), entum is described by the momentum balance equation is used to eliminate e in favor of F. The energy of the ma- terial subsystem thus follows as (36) uM=Eo dod(ol(a,F)+o'F].(28) where the momentum density Em, the momentum flux den- (28) sity (stress tensor)TEM and the density of the force on the material subsystem fa are given by Conservation of energy of the total system is expressed by ga(=ε0∈ eaByEBB (37) ,+v.S=0 where the total energy density is given by E0E2EB-=BaBB+。E2+ (38) E eo dod(o)l(a,F)2+0F] f (39) (30) These expressions correspond to the Abraham momentum density, the Maxwell stress tensor, and the Lorentz force den The total energy is the sum of squares, and therefore always sity. The momentum balance equation for the electromag- positive, which guarantees thermodynamic stability. The netic field can be derived from Maxwell,'s equations in a same energy conservation law has been found previously by straightforward manner [12] Tip [10], and by Glasgow, Ware, and Peatross by deduction The Lorentz force density can be written as the sum of from Maxwell's equations and the constitutive relation [18. temporal and spatial derivatives. This implies the existence The total energy may be split into parts in a variety of f a momentum balance equation without a source term, i.e., ways. A division between nondispersive and dispersive con- an equation that expresses the conservation of the total mo- mentum of the combined field-matter system. This rewriting is done in a number of steps. First, using Faraday's law it E.D+一HB follows that fa=fa+a(EaB,PpB,dp(-EaPB+EPSa p=50|do(o(a2+aF2-FE],(32) which gives rise to energy balance equations for the two where du D+VS=-w, (33) fa=3PBdEB-EBo PB=DB0EB-EBdoDE A second of the equation of motion for the where the dissipation from the nondispersive to the disper- material field F, Eq (14), in order to eliminate E in favor of sive energy density is given by F in the expression for f, 026606-4

W =
t P · E.
26 This energy balance equation follows directly from Max￾well’s equations 12. It appears that the rate of work can be written as the time derivative of a quantity that may be interpreted as the energy of the material subsystem
t P · E = 0  0  d
ˆ


t F · E = 0  0  d
ˆ


t F ·

t 2 F +
2 F =
t 0  0  d
ˆ



t F 2 +
2 F2  .
27 Here, the equation of motion of the material field F, Eq.
14, is used to eliminate E in favor of F. The energy of the ma￾terial subsystem thus follows as uMT = 0  0  d
ˆ



t F 2 +
2 F2 .
28 Conservation of energy of the total system is expressed by
t u +
· S = 0,
29 where the total energy density is given by u = uEM + uMT = 0 2 E2 + 1 2 0 B2 + 0  0  d
ˆ



t F 2 +
2 F2 .
30 The total energy is the sum of squares, and therefore always positive, which guarantees thermodynamic stability. The same energy conservation law has been found previously by Tip 10, and by Glasgow, Ware, and Peatross by deduction from Maxwell’s equations and the constitutive relation 18. The total energy may be split into parts in a variety of ways. A division between nondispersive and dispersive con￾tributions uND and uDS may be defined by uND = 1 2 E · D + 1 2 H · B,
31 uDS = 0  0  d
ˆ



t F 2 +
2 F2 − F · E,
32 which gives rise to energy balance equations for the two parts
t uND +
· S = − W
,
33
t uDS = W
,
34 where the dissipation from the nondispersive to the disper￾sive energy density is given by W
=
t 0  0  d
ˆ



t F 2 +
2 F2 − F · E = 1 2
t P · E − 1 2 P ·
t E = 1 2
t D · E − 1 2 D ·
t E,
35 where the equation of motion for the material field F has been used, in a way similar to the derivation of Eq.
27. The nondispersive to dispersive dissipation W
is approximately zero in case dispersion is small. The total energy may then be approximated by the nondispersive contribution uND. This leads to the default expression for the electromagnetic energy in a dielectric appearing in many textbooks 12,29. B. Momentum The transport and dissipation of electromagnetic
linear momentum is described by the momentum balance equation
t g EM +
T EM = − f,
36 where the momentum density g EM, the momentum flux den￾sity
stress tensor T EM and the density of the force on the material subsystem f are given by g EM = 0 EB ,
37 T EM = − 0EE − 1 0 BB +  0 2 E2 + 1 2 0 B2 ,
38 f = −
PE + 
tPB .
39 These expressions correspond to the Abraham momentum density, the Maxwell stress tensor, and the Lorentz force den￾sity. The momentum balance equation for the electromag￾netic field can be derived from Maxwell’s equations in a straightforward manner 12. The Lorentz force density can be written as the sum of temporal and spatial derivatives. This implies the existence of a momentum balance equation without a source term, i.e., an equation that expresses the conservation of the total mo￾mentum of the combined field-matter system. This rewriting is done in a number of steps. First, using Faraday’s law it follows that f = f
+
t
 PB +
− EP + 1 2 E · P ,
40 where f
= 1 2 P
E − 1 2 E
P = 1 2 D
E − 1 2 E
D.
41 A second step is the use of the equation of motion for the material field F, Eq.
14, in order to eliminate E in favor of F in the expression for f
SJOERD STALLINGA PHYSICAL REVIEW E 73, 026606
2006 026606-4

ENERGY AND MOMENTUM OF LIGHT IN DIELECTRIC PHYSICAL REVIEW E 73. 026606(2006) fa= dod w)( Fp) TaB=TEn+TMT -,BB+ED+HB saB Eo dod(o)(Fp0 ,FB-2 FBdFp) doGo)[(O,F)2-0'F2+FE]sag.(50) The total system momentum proposed here corresponds to o, =0 dod()(Fp0.d FB-d Fgd Fg .(42) the pseudomomentum of Gordon [2]. the wave momentum of Nelson [3] and the canonical momentum of Garrison and A third step is rewriting this expression using the following Chiao [4] According to nelson the wave momentum is the sum of momentum and pseudomomentum. The momentum contri bution from the material subsystem in the present theory corresponds to Nelsons pseudomomentum contribution to the wave momentum. a difference with Nelson is in the gen- Eo dwd(o)[awl eral form of the momentum density and stress tensor. These quantities are not unique in the sense that terms can be shifted from the density to the fux density and vice versa. In +FB0 FB+a FBd FBl particular, any multiple of the identity Eq. (43)can be added or subtracted from the total momentum conservation law Eq =o- dod(o)(FB0ad, FB+0. FB0FB) (48). An example of such a redefinition of the momentum density and stress tensor using the identity Eq.(43)is 1时 ub,、2e0doao)lFBF This identity follows from the equation of motion of the material field Eq (14). This gives that TaB=-EODB-HBB+ED+HB SaB f=al doG(o)F80,, F Eo dod(o)[(F)2-0 F+F E]8og. (52) dodo[(aF)2-0F+F These forms correspond quite closely to the density and flux The Lorentz force density can now be expressed as (44) density of wave momentum of Nelson [3].Apparently, an independent requirement is needed to justify the form of these quantities. The point of view taken here is motivated fo=gm+aRTo by an an of wave packets, and will be discussed in the next section. It here the material momentum density and momentum flux turns out that the present choice, Eqs. (49)and(50), results nsity are given by in transport of energy and momentum with the same velocity, s opposed to the alternative choice, Eqs. (51)and(52) saT=EapxPBBy ed dod o)Fpoa FB.(46)which leads to transport of energy and momentum at differ mentum travel at the same speed, which implies that Eqs Ta=-EPp-Eo dd( o)[(O, F)2-w0F210ng 49)and(50) are the correct forms of the density and flux density of the total momentum. Similar to the energy case the total momentum can be (47) divided into nondispersive and dispersive parts, with densi- Conservation of momentum of the total system is expressed tes B aga+dBlaB=0 where the momentum density and stress tensor of the total dodo)FBdoo, F system are and flux densities gaM+ga=∈DpB, dod(o)Fga. aF 026606-5

f
= 0  0  d
ˆ


F
E − E
F = 0  0  d
ˆ


F

t 2 F −
t 2 F
F =
t 0  0  d
ˆ


F

t F −
t F
F .
42 A third step is rewriting this expression using the following identity: 0 =
 0  0  d
ˆ



t 2 F +
2 F − EF = 0  0  d
ˆ




2 FF − EF + F

t 2 F +
F
t 2 F =
t 0  0  d
ˆ


F

t F +
t F
F −
 0  0  d
ˆ



t F 2 −
2 F2 + F · E .
43 This identity follows from the equation of motion of the material field Eq.
14. This gives that f
=
t 20   0  d
ˆ

F

t F −
 0  0  d
ˆ



t F 2 −
2 F2 + F · E .
44 The Lorentz force density can now be expressed as f =
t g MT +
T MT,
45 where the material momentum density and momentum flux density are given by g MT =  PB + 20   0  d
ˆ

F

t F,
46 T MT = − EP − 0  0  d
ˆ



t F 2 −
2 F2 .
47 Conservation of momentum of the total system is expressed by
t g +
T = 0,
48 where the momentum density and stress tensor of the total system are g = g EM + g MT =  DB + 20   0  d
ˆ

F
t
F,
49 T = T EM + T MT = − ED − HB +  1 2 E · D + 1 2 H · B  − 0  0  d
ˆ



t F 2 −
2 F2 + F · E.
50 The total system momentum proposed here corresponds to the pseudomomentum of Gordon 2, the wave momentum of Nelson 3, and the canonical momentum of Garrison and Chiao 4. According to Nelson, the wave momentum is the sum of momentum and pseudomomentum. The momentum contri￾bution from the material subsystem in the present theory corresponds to Nelson’s pseudomomentum contribution to the wave momentum. A difference with Nelson is in the gen￾eral form of the momentum density and stress tensor. These quantities are not unique in the sense that terms can be shifted from the density to the flux density and vice versa. In particular, any multiple of the identity Eq.
43 can be added or subtracted from the total momentum conservation law Eq.
48. An example of such a redefinition of the momentum density and stress tensor using the identity Eq.
43 is g
=  DB − 20   0  d
ˆ


t F
F,
51 T
= − ED − HB +  1 2 E · D + 1 2 H · B  + 0  0  d
ˆ



t F 2 −
2 F2 + F · E.
52 These forms correspond quite closely to the density and flux density of wave momentum of Nelson 3. Apparently, an independent requirement is needed to justify the form of these quantities. The point of view taken here is motivated by an analysis of the relation between energy and momentum of wave packets, and will be discussed in the next section. It turns out that the present choice, Eqs.
49 and
50, results in transport of energy and momentum with the same velocity, as opposed to the alternative choice, Eqs.
51 and
52, which leads to transport of energy and momentum at differ￾ent velocities 19. It seems natural to have energy and mo￾mentum travel at the same speed, which implies that Eqs.
49 and
50 are the correct forms of the density and flux density of the total momentum. Similar to the energy case the total momentum can be divided into nondispersive and dispersive parts, with densi￾ties g ND =  DB ,
53 g DS = 20   0  d
ˆ

F

t F,
54 and flux densities T ND = − ED − HB +  1 2 E · D + 1 2 H · B ,
55 ENERGY AND MOMENTUM OF LIGHT IN DIELECTRIC... PHYSICAL REVIEW E 73, 026606
2006 026606-5