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 numbers: 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 dissipative systems. The second aspect concerns the difference between 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 displacements of the dielectric gives rise to conservation of pseudomomentum. Depending on the experimental circumstances one or the other, or even a combination of both types of momenta seems useful. The difficulty in describing dissipative 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 attendant conservation laws for the enlarged system, and interpret 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 dielectric media with arbitrary dispersion and absorption introduced 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 freedom interacting with the electromagnetic field. The material field F effectively describes the harmonic motions of the charges inside the dielectric. A difference between the electromagnetic 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 eigenfrequency of the harmonic material motions. The electromagnetic 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 consisting 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 density 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 consisting 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 1539-3755/2006/732/02660612/$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 vectorial 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 irreversibility 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 arbitrary, 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 interpretation is thus how to relate these quantities to the absorbed heat, force, and torque on the medium that are actually 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 momentum. 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 identification as to which quantity is momentum and which quantity is pseudomomentum cannot be made. This indistinguishability of uniformity of space and homogeneity of matter has the consequence that only one meaningful momentumlikeconserved 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 Maxwell’s equations in general linear dielectrics. The conservation 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 following 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 compact 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 convention is used. Partial derivatives only apply to the quantity directly following the derivative unless brackets indicate otherwise. 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 Lagrangian density I = dt d3 rL, 3 where the Lagrangian density is the sum of an electromagnetic contribution, a contribution from the material field, and an interaction contribution L = 0 2 E2 − 1 2 0 B2 + 0 0 dˆt 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 absence of free charges and currents implies that ˆ→0 if →0. It may be defined for complex by analytical continuation 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 electromagnetic Lagrangian density, the material part describes a continuous set of harmonic oscillators, and the interaction term describes the interaction of the electric field with a continuous set of electric dipoles. The polarization P is thus entirely defined in terms of the material field F P = 20 0 dˆF. 5 The dielectric displacement D and magnetic field H are then defined by D = L E = 0E + 20 0 dˆF, 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 dependence on and t explicit F,t = − dtG,t − tEt, 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 ingredient 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 sint , 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− it, 17 where is a positive infinitesimal quantity. The resulting expression for the material field F leads to a dielectric displacement Dt = 0 − dtt − tEt, 18 with the dielectric function t = t + 2 0 dˆt sint = − d 2 1 + 2 0 d ˆ 2 − + i 2 exp− it. 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 dispersion 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 Maxwell’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 dˆt F · E = 0 0 dˆt F · t 2 F + 2 F = t 0 0 dˆt 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 material subsystem thus follows as uMT = 0 0 dˆt 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 dˆt 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 contributions uND and uDS may be defined by uND = 1 2 E · D + 1 2 H · B, 31 uDS = 0 0 dˆt 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 dispersive energy density is given by W = t 0 0 dˆt 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 density 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 density. The momentum balance equation for the electromagnetic 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 momentum 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 PE − 1 2 EP = 1 2 DE − 1 2 ED. 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 dˆFE − EF = 0 0 dˆFt 2 F − t 2 FF = t 0 0 dˆFt F − t FF . 42 A third step is rewriting this expression using the following identity: 0 = 0 0 dˆt 2 F + 2 F − EF = 0 0 dˆ2 FF − EF + Ft 2 F + Ft 2 F = t 0 0 dˆFt F + t FF − 0 0 dˆt 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 dˆFt F − 0 0 dˆt 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 dˆFt F, 46 T MT = − EP − 0 0 dˆt 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 dˆFt F, 49 T = T EM + T MT = − ED − HB + 1 2 E · D + 1 2 H · B − 0 0 dˆt 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 contribution 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 general 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 dˆt FF, 51 T = − ED − HB + 1 2 E · D + 1 2 H · B + 0 0 dˆt 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 different velocities 19. It seems natural to have energy and momentum 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 densities g ND = DB , 53 g DS = 20 0 dˆFt 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
SJOERD STALLINGA PHYSICAL REVIEW E 73. 026606(2006) dod(o[(O,F)-oF+F E]8aB(56) a ja+agaB=0 (60) where the total angular momentum density and angular mo- The nondispersive contributions to the momentum density mentum flux density are given by and momentum flux density are recognized as the Minkow ski momentum density and the Minkowski stress tensor, re spectively. The momentum balance equations for the two dodt(o)EaB FBd, Fr (62) Division of the total angular momentum into field and matter (58) contributions, and into dispersive and nondispersive contri- where the Minkowski force density I' is given by (41). This butions are completely analogous to the linear momentum force density is approximately zero when dispersion may be case. A division into spin and orbital parts can be developed neglected. In that case the total momentum may be approxi- along the lines of Refs. [20,21] but will not be pursued here ated by the nondispersive (Minkowski) momentum. In general, however, the dispersive terms need to be taken into IV WAVE PACKETS account. The importance of including dispersive contribu tions has also been stressed by Nelson [3] and Garrison and Certain interesting features of the auxiliary field model Chiao [4] become apparent when studying wave packets In this sec- tion, one-dimensional propagation along the z axis of a lin- early polarized wave packet is considered. Then the electric C Angular momentum field only has a nonzero x component given by Our treatment of angular momentum will be brief, as it is quite similar to the case of linear momentum treated previ E(z,) ously. The angular momentum quantities are simply found 2r (2, @exp(-ior) from the linear momentum quantities by taking the cross product with the position vector. where An issue frequently popping up in discussions about an- lar momentum conservation is the symmetry, or lack of it, E(z, 0)=E(o)explik(o)z (64) of the stress tensor. It appears that dispersion can result in an asymmetric stress tensor, although the medium is isotropic. Here E(o)=E(-o), because E(z, t) is real, and where the This can be seen as follows. The dielectric displacement D at (magnitude of) the wave vector is given by a specific time t depends on the electric field E at all previ ous times. The electric field at these times t'<t is not nec- essarily oriented in the same direction as the electric field at k()=[n(o)+ik(o) (65) time t. It follows that D and e at time t are not necessarily parallel, implying that the stress tensor defined by Eq (50)is with n(a) the refractive index and K(o) the absorption coef- generally asymmetric. This seemingly points to nonconser- ficient. These are related to the dielectric function by vation of angular momentum. however. it turns out that antisymmetric part of the stress tensor can be expressed as a E(o)=[n()+ik(o)]2 time derivative of a quantity, which may be interpreted as contributing to the internal angular momentum so that the real and imaginary parts can be written as EaByPBE dwd(o)EaB FBa, Fy+oFy) E(a)=n(a)2-k(a)2 (67) o) =2n(o)k(o) (68) dodt(o)EaB FBd, F The other field components of interest are The internal angular momentum contribution depends on the cross product of the material field and the time derivative of he material field. It follows that this contribution is only e(o)E(z, )exp(-iat), (69) nonzero if the orientation of the fields changes with time Dz)=80」27 which corroborates the qualitative argument given previ ously. An alternative, equally valid, way of dealing with this asymmetry is to absorb it into a redefinition of the linear B(,1)=oH,(2) momentum density Conservation of angular momentum is thus expressed by (70) 026606-6
T DS = − 0 0 dˆt F 2 − 2 F2 + F · E. 56 The nondispersive contributions to the momentum density and momentum flux density are recognized as the Minkowski momentum density and the Minkowski stress tensor, respectively. The momentum balance equations for the two parts are t g ND + T ND = − f , 57 t g DS + T DS = f , 58 where the Minkowski force density f is given by 41. This force density is approximately zero when dispersion may be neglected. In that case the total momentum may be approximated by the nondispersive Minkowski momentum. In general, however, the dispersive terms need to be taken into account. The importance of including dispersive contributions has also been stressed by Nelson 3 and Garrison and Chiao 4. C. Angular momentum Our treatment of angular momentum will be brief, as it is quite similar to the case of linear momentum treated previously. The angular momentum quantities are simply found from the linear momentum quantities by taking the cross product with the position vector. An issue frequently popping up in discussions about angular momentum conservation is the symmetry, or lack of it, of the stress tensor. It appears that dispersion can result in an asymmetric stress tensor, although the medium is isotropic. This can be seen as follows. The dielectric displacement D at a specific time t depends on the electric field E at all previous times. The electric field at these times tt is not necessarily oriented in the same direction as the electric field at time t. It follows that D and E at time t are not necessarily parallel, implying that the stress tensor defined by Eq. 50 is generally asymmetric. This seemingly points to nonconservation of angular momentum. However, it turns out that the antisymmetric part of the stress tensor can be expressed as a time derivative of a quantity, which may be interpreted as contributing to the internal angular momentum T = PE = 20 0 dˆ Ft 2 F + 2 F = t 20 0 dˆ Ft F . 59 The internal angular momentum contribution depends on the cross product of the material field and the time derivative of the material field. It follows that this contribution is only nonzero if the orientation of the fields changes with time, which corroborates the qualitative argument given previously. An alternative, equally valid, way of dealing with this asymmetry is to absorb it into a redefinition of the linear momentum density. Conservation of angular momentum is thus expressed by tj + M = 0, 60 where the total angular momentum density and angular momentum flux density are given by j = rg + 20 0 dˆ Ft F , 61 M = r T. 62 Division of the total angular momentum into field and matter contributions, and into dispersive and nondispersive contributions are completely analogous to the linear momentum case. A division into spin and orbital parts can be developed along the lines of Refs. 20,21, but will not be pursued here. IV. WAVE PACKETS Certain interesting features of the auxiliary field model become apparent when studying wave packets. In this section, one-dimensional propagation along the z axis of a linearly polarized wave packet is considered. Then the electric field only has a nonzero x component given by Exz,t = − d 2 E ˆz,exp− it, 63 where E ˆz, = E ˆexpikz. 64 Here E ˆ=E ˆ− * , because Exz,t is real, and where the magnitude of the wave vector is given by k = n + i c , 65 with n the refractive index and the absorption coef- ficient. These are related to the dielectric function by ˆ = n + i2, 66 so that the real and imaginary parts can be written as ˆr = n 2 − 2, 67 ˆi = ˆ = 2n. 68 The other field components of interest are Dxz,t = 0 − d 2 ˆE ˆz,exp− it, 69 Byz,t = 0Hyz,t = − d 2 k E ˆz,exp− it, 70 SJOERD STALLINGA PHYSICAL REVIEW E 73, 026606 2006 026606-6
ENERGY AND MOMENTUM OF LIGHT IN DIELECTRIC PHYSICAL REVIEW E 73. 026606(2006) F(o, z,= (2, o)exp(- iot) (71) Using these expressions, the density and flow of energy and 0o-(o+iy)- w0-(o'-iy linear me nsity of the rate of work and force can be calculated. The attention is restricted to energy [E(o)-1o-[e(o)-1]ja and linear momentum, as angular momentum does not play a a’+iy role for the wave packets under discussion E(o)o-E()o G(ω)+G(o In the following, the shorthand D(o,)')f(o,o')≡ (22f(o,)E(, a) where it has been used that y is infinitesimally small and where the expression(20)for the dielectric function is used The total energy density may now be divided into two parts XE(, w)exp[-i(o-w)r] (72) gating part, the nonpropagating part being due to dissipation alone will be used, which is convenient as most relevant quantities are bilinear in field components. The integrand may be split u=uPr +I P into parts f(o,')=fs(o, )+fA(o,w), where fs(o, o') =fs(o, o) and fA(o, o')=-fA(o,o). As D( D(o', o)it follows that only the part fs(o, o')contributes to the integral. This often helps to simplify equations. L cont o o) where the propagating and nonpropagating parts are 2E0 D(o, w EAo)o-E(o")/ +vE(oE(o') (77) A Energy For the field part of the energy density it is found that D(o, w ') (78) 0)+iy The nonpropagating contribution may be further rewritten 2p0 using the Fourier representation of the step function D(au)[1+v(o)E(a’)].(73) I dre(r)exp[i(o-o)I'].(79) The material part is more involved to the time integral of the rate of work T drw"(r) (80) Eo D(o, o')X(o, a'). (74) where the rate of work is given by W=Eo D(o,o'[G(o)+G(G) (81) X(o, c') This may be written as the product of the dissipative part of he current density and the electric field [an2-(a+iy)2∞2-( W"=j(x,)E2(z,1), (82) where the dissipative current density is given by (a+iy)2-(a-iy)2 woo(oo) j=8J27 GoE(z, ) exp(- iot).(83) This dissipative current density is the time derivative of the dissipative part of the dielectric polarization, i.e., the part of the dielectric polarization that involves only the imaginary u′+2iy)丌Jo part of the susceptibility. The remaining, conservative part of (a+iy)2+oo′(a-iy)2+oo′ the dielectric polarization contributes to the energy of the propagating wave. It follows that the nonpropagating part of the total energy may be identified as the local energy of the 026606-7
Fx,z,t = − d 2 1 2 − + i 2E ˆz,exp− it. 71 Using these expressions, the density and flow of energy and linear momentum, and the density of the rate of work and force can be calculated. The attention is restricted to energy and linear momentum, as angular momentum does not play a role for the wave packets under discussion. In the following, the shorthand D,f, − − dd 2 2 f,E ˆz, E ˆz, * exp− i − t, 72 will be used, which is convenient as most relevant quantities are bilinear in field components. The integrand may be split into parts f,= fS,+ fA,, where fS, = fS, * and fA,=−fA, * . As D, =D, * it follows that only the part fS, contributes to the integral. This often helps to simplify equations. A. Energy For the field part of the energy density it is found that uEM = 1 2 0Ex 2 + 1 2 0 By 2 = 1 2 0 D,1 + ˆˆ * . 73 The material part is more involved uMT = 0 0 d0ˆ0t Fx 2 + 0 2 Fx 2 = 1 2 0 D,X,, 74 with X, = 2 0 d0ˆ0 0 2 + 0 2 − + i 2 0 2 − − i 2 = 1 + i 2 − − i 2 2 0 d0ˆ0 0 2 + 0 2 − + i 2 − 0 2 + 0 2 − − i 2 = 1 + − + 2i 2 0 d0ˆ0 + i 2 + 0 2 − + i 2 − − i 2 + 0 2 − − i 2 = 1 − + i 2 − d0ˆ0 0 2 − + i 2 − 0 2 − − i 2 = ˆ − 1 − ˆ * − 1 − + i = ˆr − ˆr − − 1 + iˆ + ˆ − + i , 75 where it has been used that is infinitesimally small and where the expression 20 for the dielectric function is used. The total energy density may now be divided into two parts in yet a third way, namely into a propagating and nonpropagating part, the nonpropagating part being due to dissipation alone u = uPR + uNP, 76 where the propagating and nonpropagating parts are uPR = 1 2 0 D, ˆr − ˆr − + ˆˆ * , 77 uNP = 1 2 0 D, iˆ + ˆ − + i . 78 The nonpropagating contribution may be further rewritten using the Fourier representation of the step function i − + i = − dttexpi − t, 79 to the time integral of the rate of work uNP = − t dtWt, 80 where the rate of work is given by W = 1 2 0 D,ˆ + ˆ. 81 This may be written as the product of the dissipative part of the current density and the electric field W = jxz,tExz,t, 82 where the dissipative current density is given by jxz,t = 0 − d 2 ˆE ˆz,exp− it. 83 This dissipative current density is the time derivative of the dissipative part of the dielectric polarization, i.e., the part of the dielectric polarization that involves only the imaginary part of the susceptibility. The remaining, conservative part of the dielectric polarization contributes to the energy of the propagating wave. It follows that the nonpropagating part of the total energy may be identified as the local energy of the ENERGY AND MOMENTUM OF LIGHT IN DIELECTRIC... PHYSICAL REVIEW E 73, 026606 2006 026606-7
SJOERD STALLINGA PHYSICAL REVIEW E 73. 026606(2006) continuous reservoir of oscillators into which the wave dis- d[n(oo)ool sipates energy. The reservoir gains energy by dissipation and, because of causality, depends only on the electric fields dab Eo(c,m/ d[n(oo)wo]S doo previous times. The energy flux is directed along the z axis and has a ma proving that the energy of the wave packet travels at the S,=eOc D(o, c)[VE(o)+Ve(')].(84 (84) group velocity c/n,o)with the group refractive index g(o)dn(o)o (94) It turns out that the nonpropagating energy density satisfies do lim I P=0 (85) It appears that Eo(z, t)is a slowly varying function if Eo(O, t) is a slowly varying function, provided that z is sufficiently small, irrespective of whether oo is close to a resonance or lim P= dtW=E0_ 2T O)E(, o).(86) not [22]. In this transparency regime the group velocity can exceed the speed of light in vacuum and can even be nega- ive, as demonstrated experimentally in Ref [23]. The arrival It follows that the loss in the propagating part of the time of a pulse can be given a well-defined meaning, even in density over the total duration of the pulse is always these exotic regimes [24]. In turn, if z is sufficiently large as d(o)is positive for all o. This irreversibility is Eo(z, t)cannot be a slowly varying function, even if eo(o, 1) ment with expectations. The integral of the propagating en- is, and if wo is far away from a resonance. In this latter ergy density and energy flux density over the duration of the regime, the asymptotic regime, a different treatment is pulse follow as needed [25, 26] de,oo +|(o)E(o)P B Momentum The nondispersive (Minkowski) part of the momentum (87) density is given by dts,=eoc (8)820=DB≈0|Dao)aoa(0)+a(o)va(o) Consider now a pulse The dispersive part of the momentum density E(z, t)=Eo(z, t)exp(-ioot)+Eo(z, t)exp(iwor) dwod(oo)Fr d,0, Fr (89) where wo is a carrier frequency and where Eo(z, t)is a slowly |Daa)aVl(oa2+l(a)a arying envelope function for all z concerned. It then follows that the spectrum E(z, a)has narrow peaks at too so that the (96) dielectric function can be taken constant across the integra ion range. In this approximation the propagating energy, rate with of work, and energy flux density are given by Y(o, w) d[e(oo)wol J。d0(a-(+1)厘m-(-1y W"=2EoG oleO(z D) (91) (a+o')(a-a’+iy) S=Eo(oocEo(z, 12 +a)( d(o) d(w) If the absorption is small the nonpropagating contribution may be neglected and the total energy (which is equal to the propagating energy in this limit)may be approximated as The total momentum can be rewritten using 026606-8
continuous reservoir of oscillators into which the wave dissipates energy. The reservoir gains energy by dissipation and, because of causality, depends only on the electric fields at previous times. The energy flux is directed along the z axis and has a magnitude Sz = 1 2 0c D, ˆ + ˆ * . 84 It turns out that the nonpropagating energy density satisfies lim t→− uNP = 0, 85 lim t→+ uNP = − dtW = 0 − d 2 ˆ E ˆz, 2. 86 It follows that the loss in the propagating part of the energy density over the total duration of the pulse is always positive, as ˆ is positive for all . This irreversibility is in agreement with expectations. The integral of the propagating energy density and energy flux density over the duration of the pulse follow as − dtuPR = 1 2 0 − d 2 dˆr d + ˆ E ˆz, 2, 87 − dtSz = 0c − d 2 n E ˆ 2. 88 Consider now a pulse Exz,t = 1 2 E0z,texp− i0t + 1 2 E0z,t * expi0t, 89 where 0 is a carrier frequency and where E0z,t is a slowly varying envelope function for all z concerned. It then follows that the spectrum E ˆz, has narrow peaks at ±0 so that the dielectric function can be taken constant across the integration range. In this approximation the propagating energy, rate of work, and energy flux density are given by uPR = 1 4 0 dˆr00 d0 + ˆ0 E0z,t 2, 90 W = 1 2 0ˆ0 E0z,t 2, 91 Sz = 1 2 0n0c E0z,t 2. 92 If the absorption is small the nonpropagating contribution may be neglected and the total energy which is equal to the propagating energy in this limit may be approximated as u = 1 2 0n0 dn00 d0 E0z,t 2 = dn00 d0 Sz c , 93 proving that the energy of the wave packet travels at the group velocity c/ng with the group refractive index ng = dn d . 94 It appears that E0z,t is a slowly varying function if E00,t is a slowly varying function, provided that z is sufficiently small, irrespective of whether 0 is close to a resonance or not 22. In this transparency regime the group velocity can exceed the speed of light in vacuum and can even be negative, as demonstrated experimentally in Ref. 23. The arrival time of a pulse can be given a well-defined meaning, even in these exotic regimes 24. In turn, if z is sufficiently large E0z,t cannot be a slowly varying function, even if E00,t is, and if 0 is far away from a resonance. In this latter regime, the asymptotic regime, a different treatment is needed 25,26. B. Momentum The nondispersive Minkowski part of the momentum density is given by gz ND = DxBy = 0 2c D,ˆ ˆ * + ˆ * ˆ. 95 The dispersive part of the momentum density is gz DS = 20 0 d0ˆ0Fxt zFx = 0 2c D,Y, ˆ2 + ˆ * 2 , 96 with Y, = 2 0 d0ˆ0 1 0 2 − + i 2 0 2 − − i 2 = ˆ − ˆ * + − + i = 1 + − + i ˆr − ˆr + iˆ + ˆ . 97 The total momentum can be rewritten using SJOERD STALLINGA PHYSICAL REVIEW E 73, 026606 2006 026606-8
ENERGY AND MOMENTUM OF LIGHT IN DIELECTRIC PHYSICAL REVIEW E 73. 026606(2006) E(a)a2+VE(a)°o2 (a+a3)(a-a'+iy) r2=--|do0(ao(aF)2-02F2+FE VE()o+VE(a)o' vE(o)-VE(o')w' D(o,oQ(o, a'). (105) (98) so that a division of total momentum into propagating and with nonpropagating parts 8 =8 tg can be made such that BPR=E D(o, c E(o)E(aIVE()+et() a2-(+iy)2∞2-(o+iy2 E(o)-e, ( o"l(a o Ve(),'1 (a+iy)2][n2-(a'-iy)2 e(u)-8(u) E(o-E(o (ol-V(a1)o].(100 The zz component of the total stress tensor then follows as “py XIvE(o)o+vE(o)o'] Eo ilo(o)+G(o) (106) e(ω)+V(o) (( (101) The nonpropagating momentum density satisfies The nonpropagating momentum density may be written as lim NP=0 (107) the integral of the force density 82P=dr"(e). (108) I=4c D(wu, w )(G( o)+d()IVE()+ve()] from which it may be concluded that the dissipation of propagating momentum integrated over the entire pulse is ach Fourier component of th (103) momentum dissipation is a factor n(o)/c times the Fourier The flow of z momentum in the z direction is given by the omponent of the integrated energy dissipation. The sum of the nondispersive(Minkowski) stress tensor compo. tegral of the propagating momentum density and flux density nent dE (o) TWD=E, D,+3H, By 广_=广 EEo D(o, a')E(o)+E(a')+2ve(o)e(') Xn(o)E(z, o)2 do d (eo)w (104) +|(a)n(o)E(z,o)2 and the dispersive stress tensor component (109)
2 ˆ2 + ˆ * 2 + − + i = ˆ + ˆ * − + i + ˆ − ˆ * + , 98 so that a division of total momentum into propagating and nonpropagating parts gz = gz PR + gz NP, 99 can be made, such that gz PR = 0 4c D,2 ˆˆ * ˆ + ˆ * + ˆr − ˆr − ˆ + ˆ * + ˆ − ˆ * + ˆ − ˆ * , 100 gz NP = 0 4c D, iˆ/ + ˆ/ − + i ˆ + ˆ * = 0 4c D, iˆ + ˆ − + i ˆ + ˆ * . 101 The nonpropagating momentum density may be written as the integral of the force density gz NP = − t dtfz t, 102 with fz = 0 4c D,ˆ + ˆ ˆ + ˆ * . 103 The flow of z momentum in the z direction is given by the sum of the nondispersive Minkowski stress tensor component Tzz ND = 1 2 ExDx + 1 2 HyBy = 1 4 0 D,ˆ + ˆ * + 2 ˆˆ * , 104 and the dispersive stress tensor component Tzz DS = − 0 0 d0ˆ0t Fx 2 − 0 2 Fx 2 + FxEx = 0 2 D,Q,, 105 with Q, = 2 0 d0ˆ02 + 0 2 0 2 − + i 2 0 2 − − i 2 − 1 0 2 − + i 2 − 1 0 2 − + i 2 =− 2 0 d0ˆ0 − 2 0 2 − + i 2 0 2 − − i 2 = − − ˆ − ˆ * + . The zz component of the total stress tensor then follows as Tzz = 1 4 0 D,ˆ + ˆ * + 2 ˆˆ * − 2 − ˆ − ˆ * + . 106 The nonpropagating momentum density satisfies lim t→− gz NP = 0, 107 lim t→+ gz NP = − dtfz = 0 c − d 2 ˆn E ˆz, 2, 108 from which it may be concluded that the dissipation of propagating momentum integrated over the entire pulse is always positive. Each Fourier component of the integrated momentum dissipation is a factor n/c times the Fourier component of the integrated energy dissipation. The time integral of the propagating momentum density and flux density are − dtgz PR = 0 c − d 2 ˆ − 2 + 1 2 dˆr d n E ˆz, 2 = 0 2c − d 2 dˆr d + ˆ n E ˆz, 2, 109 ENERGY AND MOMENTUM OF LIGHT IN DIELECTRIC... PHYSICAL REVIEW E 73, 026606 2006 026606-9
SJOERD STALLINGA PHYSICAL REVIEW E 73. 026606(2006) function of the electric field) need to be taken into account diT 1-2[e, ()+ E()IE(z, )12 [27-29]. This would also give rise to separate conservation laws for momentum and pseudomomentum as then there are 广c two independent continuous translation symmetries, one re- O)2E(z,) (110) flecting uniformity of space and one reflecting homogeneity of matter in the undeformed reference state Similar to the dissipation of energy it follows that each Fou Instead of such a first principles approach we may also rier component of the integrated density and flux density of introduce inhomogeneity in an ad hoc manner by making the the propagating momentum is a factor n(a)/c times the Fou- conduction function space dependent, i.e., by replacing Gf() rier component of the integrated density and flux density of by G(r, o)everywhere. This does not alter the equations of the propagating energy. It is this relation between energy and motion of the model, nor the expression of the conservation momentum that has motivated the choice of the density and of energy. The total momentum is no longer conserved be flux density of momentum given by Eqs.(49)and(50)over cause of the broken translational symmetry(the Lagrangian the forms given in Eqs. (51)and(52). It is mentioned that the density depends explicitly on the spatial coordinates). It turns dispersive momentum flux density does not contribute to the out that now time integral of the total momentum flux density, the only nonzero contribution comes from the nondispersive 0,ga+aBab= (Minkowski) stress tensor. where the dissipation of momentum due to the inhomogene For the narrow-band pulse it is found that ity is given by 2=14+1E fint=-Eol dud d(r, o)[(G,F)2-0]2+2F.E] (111) The implication is that inhomogeneities are accompanied by f"=-doln(wo)Eo(z, 1)P (112) forces on the system. As a consequence, the total field-plus matter system considered so far must be an open system, as external forces are needed to maintain the static inhomoge =2=(0akx (113) neity of the system when an electromagnetic field is applied These external forces can be identified as the mechanical forces that have been excluded from the description in the If the absorption is small the nonpropagating momentum beginning. The open character of the system has also been density may be neglected, and the total momentum may be noticed by Garrison and Chiao as important for the applica approximated as bility of the total momentum [4] 8:=;n(an)2n(o dn(oo)]. An explicit expression for the force density may be found dao Eo(z, in the Fourier domain(similar to the expressions derived for the wave packets studied in Sec. IV) (114) proving that the momentum of the wave packet travels at a --2eo D(r, o, o)a el(r, so) o'+e(r, o"y'o o+a speed equal to the group velocity, just as the energy. The ratios of the density, Alux density, and dissipation of energy (118) and the density, flux density, and dissipation of momentum where the shorthand D(r, a, a')is defined by are all equal to dodo R INP S D(r,o,o')f(a,3)= DE f(o, w) (115 (2丌) 8 rk 8 tx f n(oo) i.e., the ratio between total energy and total momentum of XE(r, o) exp (a-t he wave packet is equal to the phase velocity 119) This gives the time integral V DISCUSSION The momentum conservation law that has been derived dtf= E(r,o)P2ae,(r,o),(120) this paper applies to homogeneous dielectrics only, as defor- mations of the medium are excluded from the start. A more so that the dissipation of momentum integrated over the du- general theory should address the deformability of the mate- ration of the interaction between the medium and the elec rial medium. Then the kinetic energy, kinetic momentum, tromagnetic field is proportional to the spectral average of hydrostatic forces(for fluids), and elastic forces(for solids), the product of the square of the electric field and the gradient and effects such as electrostriction (change in density as a of the real part of the dielectric function. This agrees with the 026606-10
− dtTzz = 1 2 0 − d 2 ˆr + ˆ E ˆz, 2 = 0 − d 2 n 2 E ˆz, 2. 110 Similar to the dissipation of energy it follows that each Fourier component of the integrated density and flux density of the propagating momentum is a factor n/c times the Fourier component of the integrated density and flux density of the propagating energy. It is this relation between energy and momentum that has motivated the choice of the density and flux density of momentum given by Eqs. 49 and 50 over the forms given in Eqs. 51 and 52. It is mentioned that the dispersive momentum flux density does not contribute to the time integral of the total momentum flux density, the only nonzero contribution comes from the nondispersive Minkowski stress tensor. For the narrow-band pulse it is found that gz PR = 1 2 0 c dˆr00 d0 + ˆ0 n0 E0z,t 2, 111 fz = 1 2 0 c ˆ0n0 E0z,t 2, 112 Tzz = 1 2 0n0 2 E0z,t 2. 113 If the absorption is small the nonpropagating momentum density may be neglected, and the total momentum may be approximated as gz = 1 2 0n0 2 dn00 d0 E0z,t 2 = dn00 d0 Tzz c , 114 proving that the momentum of the wave packet travels at a speed equal to the group velocity, just as the energy. The ratios of the density, flux density, and dissipation of energy and the density, flux density, and dissipation of momentum are all equal to uPR gz PR = uNP gz NP = Sz Tzz = W fz = c n0 , 115 i.e., the ratio between total energy and total momentum of the wave packet is equal to the phase velocity. V. DISCUSSION The momentum conservation law that has been derived in this paper applies to homogeneous dielectrics only, as deformations of the medium are excluded from the start. A more general theory should address the deformability of the material medium. Then the kinetic energy, kinetic momentum, hydrostatic forces for fluids, and elastic forces for solids, and effects such as electrostriction change in density as a function of the electric field need to be taken into account 27–29. This would also give rise to separate conservation laws for momentum and pseudomomentum as then there are two independent continuous translation symmetries, one re- flecting uniformity of space and one reflecting homogeneity of matter in the undeformed reference state. Instead of such a first principles approach we may also introduce inhomogeneity in an ad hoc manner by making the conduction function space dependent, i.e., by replacing ˆ by ˆr, everywhere. This does not alter the equations of motion of the model, nor the expression of the conservation of energy. The total momentum is no longer conserved because of the broken translational symmetry the Lagrangian density depends explicitly on the spatial coordinates. It turns out that now t g + T = − f inh, 116 where the dissipation of momentum due to the inhomogeneity is given by f inh = − 0 0 dˆr,t F 2 − 2 F2 + 2F · E. 117 The implication is that inhomogeneities are accompanied by forces on the system. As a consequence, the total field-plusmatter system considered so far must be an open system, as external forces are needed to maintain the static inhomogeneity of the system when an electromagnetic field is applied. These external forces can be identified as the mechanical forces that have been excluded from the description in the beginning. The open character of the system has also been noticed by Garrison and Chiao as important for the applicability of the total momentum 4. An explicit expression for the force density may be found in the Fourier domain similar to the expressions derived for the wave packets studied in Sec. IV f inh = − 1 2 0 Dr,, ˆr, + ˆr, * + , 118 where the shorthand Dr,, is defined by Dr,,f, − − dd 2 2 f,E ˆ r, E ˆ r, * exp− i − t. 119 This gives the time integral − dtf inh = − 1 2 0 − d 2 Er, 2 ˆrr,, 120 so that the dissipation of momentum integrated over the duration of the interaction between the medium and the electromagnetic field is proportional to the spectral average of the product of the square of the electric field and the gradient of the real part of the dielectric function. This agrees with the SJOERD STALLINGA PHYSICAL REVIEW E 73, 026606 2006 026606-10
