《电动力学》课程参考文献：Poynting’s theorem and luminal total energy transport in passive dielectric media

PHYSICAL REVIEW E, VOLUME 64, 046610 Poynting's theorem and luminal total energy transport in passive dielectric media Glasgow I M. Ware 2 and J. Peatross Department of Mathematics, Brigham Young University, Provo, Utah 84601 2Department of Physics, Brigham Young University, Provo, Utah 84601 (Received 26 June 2000; revised manuscript received 17 May 2001; published 25 September 2001) Without approximation the energy density in Poynting,s theorem for the generally dispersive and passive dielectric medium is demonstrated to be a system total dynamical energy density. Thus the density in Poynting's theorem is a conserved form that by virtue of its positive definiteness prescribes important quali tative and quantitative features of the medium-field dynamics by rendering the system dynamically closed. This fully three-dimensional result, applicable to anisotropic and inhomogeneous media, is model independent, elying solely on the complex-analytic consequences of causality and passivity. As direct applications of this result, we show ()that a causal medium responds to a virtual, ""instantaneous" field spectrum, (2) that a causal, passive medium supports only a luminal front velocity, (3)that the spatial"center-of-mass"motion of the total dynamical energy is also always luminal and (4) that contrary to (3)the spatial center-of-mass speed of subsets of the total dynamical energy can be arbitrarily large. Thus we show that in passive media super- luminal estimations of energy transport velocity for spatially extended pulses is inextricably associated with incomplete energy accounting DOI:10.1103/ hysRevE64.046610 PACS number(s): 42.25.B . INTRODUCTION superluminal' nature of energy transport in dielectrics do not have series representations that converge in large enough Recently several groups have published the outcomes of intervals to capture the cause of the anomalous behavior experiments in which superluminal electromagnetic pulse This is because these effects are associated with medium propagation has been observed in various senses. These field resonances that are given mathematically by singulari overages have varied from the moderate [l], to the extreme ties in the relevant constitutive relations. Thus, in order to [2]. In most(but not all) of these recent works the authors establish an unambiguous notion of the global properties of have freely expressed the conservative sentiment that noth- energy transport for finite energy medium-field excitations, ing particularly disturbing has occurred with respect to rela- we introduce the moments of various components of the total tivity. Indeed it is well known that all of thethe predictions energy(analogous to a center of mass). Energy naturally of the current classes of superluminal phenomenology have lends itself to this method since expectations are most in- been inspired by classical theory, which is heavily circum- structive when the analog of a probability distribution(i.e,a scribed by the limitations of relativity. One of the purposes positive definite form) is used. With regard to superluminal of the theoretical work presented here is to point out ways in phenomena the evolutions of these various moments are not which these conservative sentiments can be made precise only enlightening and subject to concrete analyst of group is. but also To accomplish this, we address and clarify the central give the relevant and unambiguous generalization issue of energy transport in dissipative/dispersive dielectrics. velocity for arbitrarily complicated pulses [5] (Here we limit to the passive case and address the active case The main results of this paper are given in a theorem and elsewhere [3].)We make these clarifications by introducing a a corollary. The first is given by Eqs.(48)-(50), and the new theorem and an immediate corollary that address the second by Eqs. (88)through(92). Most of this paper is de phenomena of global energy flow in causal media. It is only voted to their development, with only a limited amount of in this global sense that the various authors have ventured to space given to their application. In another publication [3] predict and, recently, to verify superluminal electromagnetic we show how the theorem can be used to precisely (i.e pulse propagation, the local sense having been authorit- quantitatively) explain both the Garrett and McCumber [6] tively proscribed by the theorems of Sommerfeld and Bril- and Chiao [7] effects(as demonstrated through experiment louin [4] almost 85 years ago. (The global theory presented by Chu and Wong [8] and Wang et al. [2], respectively). We here also contains the main implication of the local also discuss elsewhere [9] how the traditional, local concept Sommerfeld-Brillouin theory as an important corollary of energy transport velocity and the global concept of the In order to produce a notion of global energy transport velocity of the energy's spatialcenter-of-mass" both pre hat is unambiguous, we employ the method of moments or scribe upper bounds on the signal velocity expectations(more often seen and used in quantum mechan This paper is organized as follows: in Sec. II we develop ics and kinetic theory than in electromagnetic theory). These Poynting's theorem for a passive dielectric. In Sec. II A we techniques allow one to pass beyond the(often severe)ana- present Maxwells equations and the assumptions that apply lytic limitations of the local analyses usually employed in most generally to a passive linear dielectric. In Sec. II B this area of research. For example, a commonly employed then show how this structure produces a positive definite local tool is the Taylor series. Importantly, many of the ob- form for the total dynamical system energy density. Section jects to which this local tool is applied when analyzing the llc discusses this form and shows how it implies luminal 1063-651X200164(4)/04661014)S20.00 64046610-1 C2001 The American Physical Society

Poynting’s theorem and luminal total energy transport in passive dielectric media S. Glasgow,1 M. Ware,2 and J. Peatross2 1 Department of Mathematics, Brigham Young University, Provo, Utah 84601 2 Department of Physics, Brigham Young University, Provo, Utah 84601 ~Received 26 June 2000; revised manuscript received 17 May 2001; published 25 September 2001! Without approximation the energy density in Poynting’s theorem for the generally dispersive and passive dielectric medium is demonstrated to be a system total dynamical energy density. Thus the density in Poynting’s theorem is a conserved form that by virtue of its positive definiteness prescribes important quali￾tative and quantitative features of the medium-field dynamics by rendering the system dynamically closed. This fully three-dimensional result, applicable to anisotropic and inhomogeneous media, is model independent, relying solely on the complex-analytic consequences of causality and passivity. As direct applications of this result, we show ~1! that a causal medium responds to a virtual, ‘‘instantaneous’’ field spectrum, ~2! that a causal, passive medium supports only a luminal front velocity, ~3! that the spatial ‘‘center-of-mass’’ motion of the total dynamical energy is also always luminal and ~4! that contrary to ~3! the spatial center-of-mass speed of subsets of the total dynamical energy can be arbitrarily large. Thus we show that in passive media super￾luminal estimations of energy transport velocity for spatially extended pulses is inextricably associated with incomplete energy accounting. DOI: 10.1103/PhysRevE.64.046610 PACS number~s!: 42.25.Bs I. INTRODUCTION Recently several groups have published the outcomes of experiments in which superluminal electromagnetic pulse propagation has been observed in various senses. These overages have varied from the moderate @1#, to the extreme @2#. In most ~but not all! of these recent works the authors have freely expressed the conservative sentiment that noth￾ing particularly disturbing has occurred with respect to rela￾tivity. Indeed it is well known that all of the the predictions of the current classes of superluminal phenomenology have been inspired by classical theory, which is heavily circum￾scribed by the limitations of relativity. One of the purposes of the theoretical work presented here is to point out ways in which these conservative sentiments can be made precise. To accomplish this, we address and clarify the central issue of energy transport in dissipative/dispersive dielectrics. ~Here we limit to the passive case and address the active case elsewhere @3#.! We make these clarifications by introducing a new theorem and an immediate corollary that address the phenomena of global energy flow in causal media. It is only in this global sense that the various authors have ventured to predict and, recently, to verify superluminal electromagnetic pulse propagation, the local sense having been authorita￾tively proscribed by the theorems of Sommerfeld and Bril￾louin @4# almost 85 years ago. ~The global theory presented here also contains the main implication of the local Sommerfeld-Brillouin theory as an important corollary.! In order to produce a notion of global energy transport that is unambiguous, we employ the method of moments or expectations ~more often seen and used in quantum mechan￾ics and kinetic theory than in electromagnetic theory!. These techniques allow one to pass beyond the ~often severe! ana￾lytic limitations of the local analyses usually employed in this area of research. For example, a commonly employed local tool is the Taylor series. Importantly, many of the ob￾jects to which this local tool is applied when analyzing the ‘‘superluminal’’ nature of energy transport in dielectrics do not have series representations that converge in large enough intervals to capture the cause of the anomalous behavior. This is because these effects are associated with medium- field resonances that are given mathematically by singulari￾ties in the relevant constitutive relations. Thus, in order to establish an unambiguous notion of the global properties of energy transport for finite energy medium-field excitations, we introduce the moments of various components of the total energy ~analogous to a center of mass!. Energy naturally lends itself to this method since expectations are most in￾structive when the analog of a probability distribution ~i.e., a positive definite form! is used. With regard to superluminal phenomena the evolutions of these various moments are not only enlightening and subject to concrete analysis, but also give the relevant and unambiguous generalization of group velocity for arbitrarily complicated pulses @5#. The main results of this paper are given in a theorem and a corollary. The first is given by Eqs. ~48!–~50!, and the second by Eqs. ~88! through ~92!. Most of this paper is de￾voted to their development, with only a limited amount of space given to their application. In another publication @3# we show how the theorem can be used to precisely ~i.e., quantitatively! explain both the Garrett and McCumber @6# and Chiao @7# effects ~as demonstrated through experiment by Chu and Wong @8# and Wang et al. @2#, respectively!. We also discuss elsewhere @9# how the traditional, local concept of energy transport velocity and the global concept of the velocity of the energy’s spatial ‘‘center-of-mass’’ both pre￾scribe upper bounds on the signal velocity. This paper is organized as follows: in Sec. II we develop Poynting’s theorem for a passive dielectric. In Sec. II A we present Maxwell’s equations and the assumptions that apply most generally to a passive linear dielectric. In Sec. II B we then show how this structure produces a positive definite form for the total dynamical system energy density. Section II C discusses this form and shows how it implies luminal PHYSICAL REVIEW E, VOLUME 64, 046610 1063-651X/2001/64~4!/046610~14!/$20.00 ©2001 The American Physical Society 64 046610-1

S. GLASGOW. M. WARE AND J PEATROSS PHYSICAL REVIEW E 64 046610 front speed without the usual recourse to path integrals, as With both the permittivity and permeability tensors non well as pointing out the crucial distinctions between the dy- trivial (i.e, not proportional to the identity) and depending namic total energy density and the quantity FE D+!. H,(locally) on the spatial coordinate(as well as nonlocally on which is sometimes referred to [10]. In Sec. Ill we present time), we are prepared to analyze inhomogeneous and aniso- he simple corollary to the theorem of Sec. Il B that aug- tropic media with both electric and magnetic effects. The ments the local Sommerfeld-Brillouin theorems by showing development of the total energy density in the following sec. that total energy transport is also globally luminal. Finally, in tion can be greatly simplified leaving out anisotropy, but we Sec IV, and in contrast to the unsurprising results of Sec. Ill, include the more general derivation since interest has re- we show that a certain subset of the total energy can have emerged recently in considering these effects [11, 12] superluminal global transport propertie as is obvious in these constitutive relations. we have adopted the common practice of using the same symbols to IL. POYNTING'S THEOREM AND CONSERVATION denote the fields as well as their temporal Fourier transforms OF TOTAL DYNAMICAL ENERGY distinguishing the two sets only by explicit reference to ther time t or frequency a: for F(r)any one of the original A. Assumptions four fields, we define F(o)via We start with Maxwells equations for the four real mac oscopic fields. These fields are the electric field e(x, n), the electric displacement D(x, 1), the magnetic induction B(x, t) F(o dt e F(r) and the magnetic field H(x, n). x and t denote, respectively, the spatial and temporal coordinates. We currently exclude nd then note the inversion formula he possibility of macroscopic currents so that we are dealing with a true dielectric. The dynamical equations are then(in the Heaviside-Lorentz system of units) F(1) doe f(o D(t)-cV×H(1)=0, (1) Since the original fields are real, the transforms manifest the symmetry F*(o)=F(-O). Via Eqs. (3) and(4), we then see that the permittivity and permeability tensors possess the 一B(1)+cV×E(1)=0 same symmetry:e.g,E(O)=E(-O) In the following, we refer to this symmetry as real symmetry Here and in much of the following we explicitly denote only In addition to assuming the validity of the macroscopic the time coordinate since we assume only temporally nono- Maxwell's equations, we limit the constitutive relations (3) cal constitutive relations -i.e,, we assume temporal but not and(4)to physically reasonable ones via the following three patial dispersion. We assume these relations are, neverthe- assumptions less, local in the frequency domain(stationary in time)and (a) Causality. E(o)-I and u(o)-I are rapidly vanishing and analytic(termwise) in the upper-half complex o plane(I D()=e(u)E(), is the identity tensor). This implies the Kramers-Kronig re- lations. Among these we will need that, for real o, B(o=u(o)H(o) Rele(ol=l E and u are, respectively, the (electric) permittivity and mP/do, mle(a") (magnetic) permeability tensors. Since we currently exclude nonlinear effects, e and u are tensors of rank 2, and since we +o, Im[u(o)] can think of the fields as three-component column vectors, we can interpret these tensors as 3 X3 matrices. The right REi(2。do-0 hand sides of Eqs.(3)and(4)are then interpreted in the Here the symbol P re Note that the permittivity and permeability tensors can Cauchy principal ale ers to the operation of taking the sense of matrix multiplication so depend locally on the space coordinate x (b)Kinetic symmetry. In the absence of a strong, external static magnetic field, we have from near-equilibrium thermo- dynamic considerations [13] that We will suppress this dependence for the time being as it (a)=k(o) (12) does not enter the calculations immediately, but we empha size that this spatial dependence is important in the end to Here and in the following superscript T indicates the trans- achieve finite and, hence, physical total energy ose 046610-2

front speed without the usual recourse to path integrals, as well as pointing out the crucial distinctions between the dy￾namic total energy density and the quantity 1 2 E•D1 1 2 B•H, which is sometimes referred to @10#. In Sec. III we present the simple corollary to the theorem of Sec. II B that aug￾ments the local Sommerfeld-Brillouin theorems by showing that total energy transport is also globally luminal. Finally, in Sec. IV, and in contrast to the unsurprising results of Sec. III, we show that a certain subset of the total energy can have superluminal global transport properties. II. POYNTING’S THEOREM AND CONSERVATION OF TOTAL DYNAMICAL ENERGY A. Assumptions We start with Maxwell’s equations for the four real mac￾roscopic fields. These fields are the electric field E(x,t), the electric displacement D(x,t), the magnetic induction B(x,t), and the magnetic field H(x,t). x and t denote, respectively, the spatial and temporal coordinates. We currently exclude the possibility of macroscopic currents so that we are dealing with a true dielectric. The dynamical equations are then ~in the Heaviside-Lorentz system of units! ] ]t D~t!2c“3H~t!50, ~1! ] ]t B~t!1c“3E~t!50. ~2! Here and in much of the following we explicitly denote only the time coordinate since we assume only temporally nonlo￾cal constitutive relations – i.e., we assume temporal but not spatial dispersion. We assume these relations are, neverthe￾less, local in the frequency domain ~stationary in time! and also linear: D~v!5eˆ~v!E~v!, ~3! B~v!5mˆ ~v!H~v!. ~4! eˆ and mˆ are, respectively, the ~electric! permittivity and ~magnetic! permeability tensors. Since we currently exclude nonlinear effects, eˆ and mˆ are tensors of rank 2, and since we can think of the fields as three-component column vectors, we can interpret these tensors as 333 matrices. The right hand sides of Eqs. ~3! and ~4! are then interpreted in the sense of matrix multiplication. Note that the permittivity and permeability tensors can also depend locally on the space coordinate x, eˆ5eˆ~x,v!, ~5! mˆ 5mˆ ~x,v!. ~6! We will suppress this dependence for the time being as it does not enter the calculations immediately, but we empha￾size that this spatial dependence is important in the end to achieve finite and, hence, physical total energy. With both the permittivity and permeability tensors non￾trivial ~i.e., not proportional to the identity! and depending ~locally! on the spatial coordinate ~as well as nonlocally on time!, we are prepared to analyze inhomogeneous and aniso￾tropic media with both electric and magnetic effects. The development of the total energy density in the following sec￾tion can be greatly simplified leaving out anisotropy, but we include the more general derivation since interest has re￾emerged recently in considering these effects @11,12#. As is obvious in these constitutive relations, we have adopted the common practice of using the same symbols to denote the fields as well as their temporal Fourier transforms, distinguishing the two sets only by explicit reference to ei￾ther time t or frequency v: for F(t) any one of the original four fields, we define F(v) via F~v!ª 1 A2p E 2` 1` dt eivt F~t!, ~7! and then note the inversion formula F~t!5 1 A2p E 2` 1` dv e2ivt F~v!. ~8! Since the original fields are real, the transforms manifest the symmetry F*(v)5F(2v*). Via Eqs. ~3! and ~4!, we then see that the permittivity and permeability tensors possess the same symmetry: e.g., eˆ *(v)5eˆ(2v*). In the following, we refer to this symmetry as real symmetry. In addition to assuming the validity of the macroscopic Maxwell’s equations, we limit the constitutive relations ~3! and ~4! to physically reasonable ones via the following three assumptions. ~a! Causality. eˆ(v)2I ˆ and mˆ (v)2I ˆ are rapidly vanishing and analytic ~termwise! in the upper-half complex v plane (I ˆ is the identity tensor!. This implies the Kramers-Kronig re￾lations. Among these we will need that, for real v, Re@eˆ~v!#5I ˆ1 1 p PE 2` 1` dv8 Im@eˆ~v8!# v82v , ~9! Re@mˆ ~v!#5I ˆ1 1 pPE 2` 1` dv8 Im@mˆ ~v8!# v82v . ~10! Here the symbol P refers to the operation of taking the Cauchy principal value. ~b! Kinetic symmetry. In the absence of a strong, external, static magnetic field, we have from near-equilibrium thermo￾dynamic considerations @13# that eˆ T~v!5eˆ~v!, ~11! mˆ T~v!5mˆ ~v!. ~12! Here and in the following superscript T indicates the trans￾pose. S. GLASGOW, M. WARE, AND J. PEATROSS PHYSICAL REVIEW E 64 046610 046610-2

POYNTING'S THEOREM AND LUMINAL TOTAL ENERGY PHYSICAL REVIEW E 64 046610 (c)Passivity. We assume that the spectra p (i. e, the col- and dot the first of our Maxwells equations(1) into the lection of eigenvalues)of the imaginary parts of E and u are electric field E(t), and add to this the result of dotting the positive for positive frequencies o second Eq.(2)into the magnetic field H(t), p{Im[e(o)]}>0 (13) E(t)-D(t)+H()·B()+cV[E(1)×H(t)=0 p{Im[A(o)]}>0 (14) Note that this assumption of passivity combined with the Here we have used the usual identity from vector kinetic symmetry assumption(b)shows that the imaginary calculus, namely that H(D VXE(D-E(0. VXH(D parts of the spectra of e and u are also positive for positive =V.E(OXH()I frequencies(which property we call dissipation) The goal of this section is to express the first two terms in (c)Dissipation. Eq.(21)as the time derivative of a positive definite quantity [quadratic in the electric and magnetic fields, E(n) and H(o)I Im[P{∈(o)}>0 (15) under the assumptions made in the last section. We will iden tify this quantity as the total dynamical energy density, com- Im[p{A(o)}>0. (16) prising recoverable and irrecoverable mechanical energies as well as the energy stored solely in the electromagnetic field At first(c')might seem a more natural definition of passiv- To achieve this goal we temporarily introduce the polariza ity. (E g, in a cry stal the eigenvalues of e give the permit- Heaviside-Lorentz system of units [10) via vectors. The imaginary parts of the eigenvalues then describe absorption. However, we will eventually see that(c) is the P(t):=D(1)-E(t) (22) more useful assumption from the complex-analytic point of view. At any rate, in the case that these tensors encode the (23) electromagnetic properties of a crystal or an isotropic me- Using these to eliminate D(o and B(O) from Eq.(21),we dium, (c) and(c)are equivalent since the eigenvectors of ese tensors can be taken to be real (e.g, the directions of he crystals principle axes). For a discussion of the relation- a ship between what we have called dissipation and what we arI2E(2+2H(O +E() a: P()+H(o ar M(O Using real symmetry, we see that the imaginary parts of e +cV[E(1)×H(t)]=0. (24) and u are odd functions of real frequency o. Consequently according to the passivity property()[Eqs. (13)and(14) As the first term of this expression is manifestly the time we have that for all real frequencies derivative of a positive definite quadratic form in E(t)and H(n), we now need only to recognize the second and third p{oIm[e(a)]}≥0 (17 terms in Eq.(24)as such. To that end we introduce and define the electric and magnetic susceptibility tensors p{ulmA(o)]}≥0, (18) XE(O)=E(o)-I and XH(a):=A(o)-1. The transforms of the polarization and magnetization vectors, P(o) and M(o) with equality possibly holding only at o=0. We use the fact can be expressed locally in terms of the transforms of the that these two tensors are non-negative in order to factor electric and magnetic fields via them and thereby make their spectral properties obvious ere are tensor-valued functions ae o) and aH(o) such P(O)=XEOE(o) hat wImlE(O)]=ado)aEo), (19) 1()=XH(oH(o) Note that from their definitions, and from the relevant prop- o ImU(o)]=ah(o)aHo) (20) erties of the permittivity and permeability tensors [properties for all real frequencies o (a)-(c)] the susceptibility tensors are analytic and rapidly vanishing in the upper half o plane, and also possess prop- erties(b)and(c). They also demonstrate real symmetry B. Derivation of the total dynamical energy density xF(o)=xF-O*). To avoid repetition, here and in the fol lowing F will stand for either e or H. Also. owing to the Here we derive the version of Poynting's theorem rel- symmetry between the two pairs(, E)and(M, H), in the evant to the general assumptions made in the preceding sec- following we abbreviate by only presenting the derivation of tion. To our knowledge, this is the first time that this general the quadratic form associated with the polarization and elec case has been handled correctly. We begin in the usual way tric field. In the end we present the results for both pairs 046610-3

~c! Passivity. We assume that the spectra r ~i.e., the col￾lection of eigenvalues! of the imaginary parts of eˆ and mˆ are positive for positive frequencies v: r$Im@eˆ~v!#%.0, ~13! r$Im@mˆ ~v!#%.0. ~14! Note that this assumption of passivity combined with the kinetic symmetry assumption ~b! shows that the imaginary parts of the spectra of eˆ and mˆ are also positive for positive frequencies ~which property we call dissipation!. (c8) Dissipation. Im@r$eˆ~v!%#.0, ~15! Im@r$mˆ ~v!%#.0. ~16! At first (c8) might seem a more natural definition of passiv￾ity. ~E.g., in a crystal the eigenvalues of eˆ give the permit￾tivity in the direction prescribed by the corresponding eigen￾vectors. The imaginary parts of the eigenvalues then describe absorption.! However, we will eventually see that ~c! is the more useful assumption from the complex-analytic point of view. At any rate, in the case that these tensors encode the electromagnetic properties of a crystal or an isotropic me￾dium, ~c! and (c8) are equivalent since the eigenvectors of these tensors can be taken to be real ~e.g., the directions of the crystal’s principle axes!. For a discussion of the relation￾ship between what we have called dissipation and what we have called passivity see the Appendix. Using real symmetry, we see that the imaginary parts of eˆ and mˆ are odd functions of real frequency v. Consequently, according to the passivity property ~c! @Eqs. ~13! and ~14!#, we have that for all real frequencies r$v Im@eˆ~v!#%>0, ~17! r$v Im@mˆ ~v!#%>0, ~18! with equality possibly holding only at v50. We use the fact that these two tensors are non-negative in order to factor them and thereby make their spectral properties obvious: there are tensor-valued functions aˆ E(v) and aˆ H(v) such that v Im@eˆ~v!#5aˆ E † ~v!aˆ E~v!, ~19! v Im@mˆ ~v!#5aˆ H † ~v!aˆ H~v! ~20! for all real frequencies v. B. Derivation of the total dynamical energy density in Poynting’s theorem Here we derive the version of Poynting’s theorem rel￾evant to the general assumptions made in the preceding sec￾tion. To our knowledge, this is the first time that this general case has been handled correctly. We begin in the usual way and dot the first of our Maxwell’s equations ~1! into the electric field E(t), and add to this the result of dotting the second Eq. ~2! into the magnetic field H(t), E~t!• ] ]t D~t!1H~t!• ] ]t B~t!1c“•@E~t!3H~t!#50. ~21! Here we have used the usual identity from vector calculus, namely that H(t)•“3E(t)2E(t)•“3H(t) 5“•@E(t)3H(t)#. The goal of this section is to express the first two terms in Eq. ~21! as the time derivative of a positive definite quantity @quadratic in the electric and magnetic fields, E(t) and H(t)# under the assumptions made in the last section. We will iden￾tify this quantity as the total dynamical energy density, com￾prising recoverable and irrecoverable mechanical energies as well as the energy stored solely in the electromagnetic field. To achieve this goal we temporarily introduce the polariza￾tion P(t) and magnetization M(t). They are defined ~in the Heaviside-Lorentz system of units @10#! via P~t!ªD~t!2E~t!, ~22! M~t!ªB~t!2H~t!. ~23! Using these to eliminate D(t) and B(t) from Eq. ~21!, we obtain ] ]t S 1 2 iE~t!i 21 1 2 iH~t!i 2 D 1E~t!• ] ]t P~t!1H~t!• ] ]t M~t! 1c“•@E~t!3H~t!#50. ~24! As the first term of this expression is manifestly the time derivative of a positive definite quadratic form in E(t) and H(t), we now need only to recognize the second and third terms in Eq. ~24! as such. To that end we introduce and define the electric and magnetic susceptibility tensors xˆ E(v)ªeˆ(v)2I ˆ and xˆ H(v)ªmˆ (v)2I ˆ. The transforms of the polarization and magnetization vectors, P(v) and M(v), can be expressed locally in terms of the transforms of the electric and magnetic fields via P~v!5xˆ E~v!E~v!, ~25! M~v!5xˆ H~v!H~v!. ~26! Note that from their definitions, and from the relevant prop￾erties of the permittivity and permeability tensors @properties ~a!–~c!#, the susceptibility tensors are analytic and rapidly vanishing in the upper half v plane, and also possess prop￾erties ~b! and ~c!. They also demonstrate real symmetry: xˆ F *(v)5xˆ F(2v*). ~To avoid repetition, here and in the fol￾lowing F will stand for either E or H. Also, owing to the symmetry between the two pairs (P,E) and (M,H), in the following we abbreviate by only presenting the derivation of the quadratic form associated with the polarization and elec￾tric field. In the end we present the results for both pairs.! POYNTING’S THEOREM AND LUMINAL TOTAL ENERGY . . . PHYSICAL REVIEW E 64 046610 046610-3

S. GLASGOW. M. WARE AND J PEATROSS PHYSICAL REVIEW E 64 046610 We next use Eq (25)to eliminate explicit reference to the polarization vector in Eq. (24). To do this, we inverse Fou Relxeo)l (32) rier transform(25)to obtain We can use these relationships between the real and dTGE(t-TE(T) (27) Imaginary parts of the susceptibilities to show that the in- phase and out-of-phase components of the electric and mag- netic convolution kernels are not independent. These two where the convolution kernel GE(t) is defined in terms of the components of the convolution kernels are defined in terms susceptibilityvia d (28) doe Relf(o) (33) We need the time derivative of the polarization. Via Eq(27) G"()=2]- dwe ImL XF(o)] we see that this is obtained through the formula Note that G(o=GF(o+GE(o) aP(=J。 drat gee(-nE( (29) We now show that the in- and out-of-phase components of the convolution kernels are identical for positive argu- ment,i.e, GF(1=GF(0), 1>0. To that end we rewrite (Note: The rapid vanishing of the susceptibilities at large Eq(33)via Eq(32)and obtain frequencies renders the kernels differentiable everywhere but at a single time where they are, fortunately, continuous. Thus exchange of orders of the operations of integration and ImLxF(o)] differentiation is justified. We now use the various properties of the susceptibilities (35) to reduce Eq. (29)to an equivalent expression that can be used to directly demonstrate the conserved energy. The first Exchanging the orders of the integrations(and simplifying), (and usual)simplification is to note that the integral(28)can we obtain be evaluated explicitly for (<0. We use Cauchys integral theorem with contours constructed from great semicircles in GF(o 2m2/ dop/d e-ior Im[X(o’)] the upper-half o plane, closed along the real axis. Since the susceptibilities are analytic and rapidly vanish with increas ing radius in the region enclosed by these contours, it is readily shown that for (<o the integration over the real in- The inner integral can be evaluated via Cauchy s theorem by terval defining the convolution kernel gives zero Ise of a large semicircular contour that extends into the lower-half plane(for (>0)and that, for example, contains a Gg(1)=0;t<0. (30Small semicircular dimple excluding the pole at w=@.Al- ternatively, one can recognize the integral as a Hilbert trans- form and consult a table. Either way the result is that 0 indicates the zero matrix. )The formula expressing the time derivative of the polarization vector in terms of the electric field, Eq. (29), then reduces to integration up to time d -a't.t>0. (37) Using this result in Eq.(36) gives (31) The previous formula involves the convolution kernel GE, which is constructed from the susceptibility by Eq(28 ). according to definition (34) In particular, it appears from that construction that both the Our formula allowing us to eliminate the polarization(31) real and imaginary parts of the susceptibility are important. can now be expressed We now show that the convolution kernel can be constructed entirely from the imaginary part of the susceptibility which, in turn, will allow us to use passivity (e) to deduce certain 'A rigorous exchange can be made by writing the Cauchy princi important properties of this kernel. To that end, we note that pal value operation as a limit and by restricting the fields to certain in terms of a susceptibility, the Kramers-Kronig relations physically reasonable function spaces. Similar statements apply to [causality (a)] can be expressed much of what follows 046610-4

We next use Eq. ~25! to eliminate explicit reference to the polarization vector in Eq. ~24!. To do this, we inverse Fou￾rier transform ~25! to obtain P~t!5 E 2` 1` dt Gˆ E~t2t!E~t!, ~27! where the convolution kernel Gˆ E(t) is defined in terms of the susceptibility via Gˆ E~t!ª 1 2pE 2` 1` dv e2ivt xˆ E~v!. ~28! We need the time derivative of the polarization. Via Eq. ~27!, we see that this is obtained through the formula ] ]t P~t!5 E 2` 1` dt ] ]t Gˆ E~t2t!E~t!. ~29! ~Note: The rapid vanishing of the susceptibilities at large frequencies renders the kernels differentiable everywhere but at a single time where they are, fortunately, continuous. Thus the exchange of orders of the operations of integration and differentiation is justified.! We now use the various properties of the susceptibilities to reduce Eq. ~29! to an equivalent expression that can be used to directly demonstrate the conserved energy. The first ~and usual! simplification is to note that the integral ~28! can be evaluated explicitly for t,0. We use Cauchy’s integral theorem with contours constructed from great semicircles in the upper-half v plane, closed along the real axis. Since the susceptibilities are analytic and rapidly vanish with increas￾ing radius in the region enclosed by these contours, it is readily shown that for t,0 the integration over the real in￾terval defining the convolution kernel gives zero: Gˆ E~t!50ˆ; t,0. ~30! (0ˆ indicates the zero matrix.! The formula expressing the time derivative of the polarization vector in terms of the electric field, Eq. ~29!, then reduces to integration up to time t5t: ] ]t P~t!5 E 2` t dt ] ]t Gˆ E~t2t!E~t!. ~31! The previous formula involves the convolution kernel Gˆ E , which is constructed from the susceptibility by Eq. ~28!. In particular, it appears from that construction that both the real and imaginary parts of the susceptibility are important. We now show that the convolution kernel can be constructed entirely from the imaginary part of the susceptibility which, in turn, will allow us to use passivity ~c! to deduce certain important properties of this kernel. To that end, we note that in terms of a susceptibility, the Kramers-Kronig relations @causality ~a!# can be expressed as Re@xˆ F~v!#5 1 p PE 2` 1` dv8 Im@xˆ F~v8!# v82v . ~32! We can use these relationships between the real and imaginary parts of the susceptibilities to show that the in￾phase and out-of-phase components of the electric and mag￾netic convolution kernels are not independent. These two components of the convolution kernels are defined in terms of the real and imaginary parts of the susceptibilities via Gˆ F in~t!ª 1 2pE 2` 1` dv e2ivt Re@xˆ F~v!#, ~33! Gˆ F out~t!ª i 2pE 2` 1` dv e2ivt Im@xˆ F~v!#. ~34! Note that Gˆ F(t)5Gˆ F in(t)1Gˆ F out(t). We now show that the in- and out-of-phase components of the convolution kernels are identical for positive argu￾ment, i.e., Gˆ F in(t)5Gˆ F out(t), t.0. To that end we rewrite Eq. ~33! via Eq. ~32! and obtain Gˆ F in~t!ª 1 2pE 2` 1` dv e2ivt 1 p PE 2` 1` dv8 Im@xˆ F~v8!# v82v . ~35! Exchanging the orders of the integrations1 ~and simplifying!, we obtain Gˆ F in~t!5 1 2p2 E 2` 1` dv8S PE 2` 1` dv e2ivt v82vD Im@xˆ F~v8!#. ~36! The inner integral can be evaluated via Cauchy’s theorem by use of a large semicircular contour that extends into the lower-half plane ~for t.0) and that, for example, contains a small semicircular dimple excluding the pole at v5v8. Al￾ternatively, one can recognize the integral as a Hilbert trans￾form and consult a table. Either way the result is that PE 2` 1` dv e2ivt v82v 5ipe2iv8t ; t.0. ~37! Using this result in Eq. ~36! gives Gˆ F in~t!5 i 2pE 2` 1` dv8e2iv8t Im@xˆ F~v8!#5:Gˆ F out~t!; t.0, ~38! according to definition ~34!. Our formula allowing us to eliminate the polarization ~31! can now be expressed as 1 A rigorous exchange can be made by writing the Cauchy princi￾pal value operation as a limit and by restricting the fields to certain physically reasonable function spaces. Similar statements apply to much of what follows. S. GLASGOW, M. WARE, AND J. PEATROSS PHYSICAL REVIEW E 64 046610 046610-4

POYNTING'S THEOREM AND LUMINAL TOTAL ENERGY PHYSICAL REVIEW E 64 046610 P(1=2 dT-GE (T-TE(T) P(OE(n dol aeo dEe(T) The advantage of this expression over Eq(31)is that the auxiliary field is now related to the electric field only through aas(o)」dreE()(4 the imaginary part of the susceptibility, about which we have the restrictions of passivity().(Recall that we have no di This expression would be an obvious perfect derivative if the ect restriction on the real part of this tensor. vectors that are multiplied were not complex conjugates We are trying to re-express the term E()(alan)P() in However, while the individual terms in the frequency inte (24)so as to recognize it as the derivative of a positive defi- grand are complex, the integration clearly gives a real result. nite quadratic form in the electric field E(n). For uniformity Thus the integrand can be re-expressed in terms of only its of notation between dot products and matrix/tensor products, real part. We write this as we will denote this scalar product by juxtaposition of ad 2 dol ae(o) dreE(T) E(,P(=E(P(=xPOE()(40) X= aeo) dreE(T)+cc In passing from the second to the third expression we have (45) used that the fields are real Using the third form of the expression in Eq (40)and Eq. Here c c. denotes the complex conjugate (39)to eliminate the auxiliary field P, as well as definition This object is now clearly a perfect time derivative to (34)to eliminate the out-of-phase component of the conyo. which the product rule has been applied, and so can be re- lution kernel, we find that the dot product can be expressed written as in terms of only the electric field and the imaginary part of the susceptibility. The formula is 1 P()E(t)= dt 2 P(DE( ×dreE(r) Xe-iof(I-o ImE(JE(TE(t) Here the norm symbol *l indicates that one takes the length of its argument as a complex 3 vector. )This expres- (41) sion is manifestly the time derivative of a positive definite quadratic form in the electric field, albeit nonlocal in time We now remember that, from passivity (c)and real symme- Repeating the above steps for the pair (M, H) we get an try, o ImLxE(o) is a non-negative tensor for all real fre- analogous formula, LEq (19)]and so can be fac M(O H(O do an() P(D)E(1) de dOeTH(T) GE()aEOE(TE(O) We can now express the dispersive, dissipative version of (42) Poynting's theorem(in the absence of macroscopic currents) Emphasizing the spatial dependencies heretofore suppressed Interchanging the orders of integration and rearranging terms this conservation law is in a more symmetric fashion, we get the suggestive form (x,) v·S(x,)=0, P(D)E(1) de dTeTaEOE(T where the energy flux S(x, t) is the usual Poynting vector. XeaEOE(r S(x,)=E(X,D1)×H(x,D) (49) which is immediately recognized as a sum of the Hermitian The total energy density u(x, t)is now somewhat more com- products of various vectors with their derivatives plicated than in the usual case 046610-5

] ]t P~t!52 E 2` t dt ] ]t Gˆ E out~t2t!E~t!. ~39! The advantage of this expression over Eq. ~31! is that the auxiliary field is now related to the electric field only through the imaginary part of the susceptibility, about which we have the restrictions of passivity ~c!. ~Recall that we have no di￾rect restriction on the real part of this tensor.! We are trying to re-express the term E(t)•(]/]t)P(t) in ~24! so as to recognize it as the derivative of a positive defi- nite quadratic form in the electric field E(t). For uniformity of notation between dot products and matrix/tensor products, we will denote this scalar product by juxtaposition of ad￾joints, E~t!• ] ]t P~t!5E† ~t! ] ]t P~t!5F ] ]t P~t!G † E~t!. ~40! In passing from the second to the third expression we have used that the fields are real. Using the third form of the expression in Eq. ~40! and Eq. ~39! to eliminate the auxiliary field P, as well as definition ~34! to eliminate the out-of-phase component of the convo￾lution kernel, we find that the dot product can be expressed in terms of only the electric field and the imaginary part of the susceptibility. The formula is F ] ]t P~t!G † E~t!5 1 pFE 2` t dt E 2` 1` dv 3e2iv(t2t) v Im@xˆ E~v!#E~t!G † E~t!. ~41! We now remember that, from passivity ~c! and real symme￾try, v Im@xˆ E(v)# is a non-negative tensor for all real fre￾quencies @Eq. ~19!# and so can be factored, F ] ]t P~t!G † E~t!5 1 pFE 2` t dt E 2` 1` dv 3e2iv(t2t) aˆ E † ~v!aˆ E~v!E~t!G † E~t!. ~42! Interchanging the orders of integration and rearranging terms in a more symmetric fashion, we get the suggestive form F ] ]t P~t!G † E~t!5 1 pE 2` 1` dvFE 2` t dt eivt aˆ E~v!E~t!G † 3eivt aˆ E~v!E~t!, ~43! which is immediately recognized as a sum of the Hermitian products of various vectors with their derivatives: F ] ]t P~t!G † E~t!5 1 pE 2` 1` dvF aˆ E~v!E 2` t dt eivtE~t!G † 3 ] ]t F aˆ E~v!E 2` t dt eivtE~t!G . ~44! This expression would be an obvious perfect derivative if the vectors that are multiplied were not complex conjugates. However, while the individual terms in the frequency inte￾grand are complex, the integration clearly gives a real result. Thus the integrand can be re-expressed in terms of only its real part. We write this as F ] ]t P~t!G † E~t!5 1 2pE 2` 1` dvHF aˆ E~v!E 2` t dt eivtE~t!G † 3 ] ]t F aˆ E~v!E 2` t dt eivtE~t!G 1c.c.J . ~45! Here c.c. denotes the complex conjugate. This object is now clearly a perfect time derivative to which the product rule has been applied, and so can be re￾written as F ] ]t P~t!G † E~t!5 ] ]t H 1 2pE 2` 1` dvI aˆ E~v! 3 E 2` t dt eivtE~t!I 2 J . ~46! ~Here the norm symbol i*i indicates that one takes the length of its argument as a complex 3 vector.! This expres￾sion is manifestly the time derivative of a positive definite quadratic form in the electric field, albeit nonlocal in time. Repeating the above steps for the pair (M,H) we get an analogous formula, S ] ]t M~t!D † H~t!5 ] ]t H 1 2pE 2` 1` dvI aˆ H~v! 3 E 2` t dt eivt H~t!I 2 J . ~47! We can now express the dispersive, dissipative version of Poynting’s theorem ~in the absence of macroscopic currents!. Emphasizing the spatial dependencies heretofore suppressed, this conservation law is ]u~x,t! ]t 1c“•S~x,t!50, ~48! where the energy flux S(x,t) is the usual Poynting vector, S~x,t!5E~x,t!3H~x,t!. ~49! The total energy density u(x,t) is now somewhat more com￾plicated than in the usual case, POYNTING’S THEOREM AND LUMINAL TOTAL ENERGY . . . PHYSICAL REVIEW E 64 046610 046610-5