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 collection 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 passivity. ~E.g., in a crystal the eigenvalues of eˆ give the permittivity in the direction prescribed by the corresponding eigenvectors. 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 medium, ~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 relationship 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 relevant to the general assumptions made in the preceding section. 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 identify this quantity as the total dynamical energy density, comprising recoverable and irrecoverable mechanical energies as well as the energy stored solely in the electromagnetic field. To achieve this goal we temporarily introduce the polarization 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 properties 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 properties ~b! and ~c!. They also demonstrate real symmetry: xˆ F *(v)5xˆ F(2v*). ~To avoid repetition, here and in the following 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 electric 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