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 046610-2front 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