S. GLASGOW. M. WARE AND J PEATROSS PHYSICAL REVIEW E 64 046610 tant to establish if u is to be interpreted as a meaningful n(x1)=|E(x,l)P2+|H(x,2 dynamical energy density that not only has the units of en- 2 ergy but can also prescribe the qualitative features of system dynamics. Such features include the boundedness(as well as dol ae(x, o) dEe(x, 7) existence and uniqueness) of solutions for all time, the asymptotic state of the solutions, and, since our dynamical equations(1)and(2)constitute a system of wave equations au(x,o) dreH(x, T) (50) the"domain of dependence"of solutions, i.e., the classical side the light cone of compactly supported initial data [4] Here we remind the reader that over real frequencies the a In another publication [3] we discuss in greater detail how tensors are related to the susceptibilities and hence permit- the structure of dynamical energy density(50) suggests a mechanism for the Garrett and Mc Cumber [6 and Chiao effects [7]. For now we limit our discussion to demonstrating E(x, ) ae(x, o)=oImLxE(x, o)]=wIm[E(x, o)-1 that a causal medium responds to virtual frequencies and to giving a very geometric proof of the property of luminal sImlE(x, o)] (51) front velocity. In addition, we discuss the connection be- tween the dynamical energy density(50)and an approximate aj(x, o)an(x,)=Im[Xn(x, o)]=w[u(x, o)- expression often employed (52) 1. The medium responds to a virtual, instantaneous spectrum These last two formulas should also remind the reader that The form of Eq. (50)can be used to explain the phenom- what is required in Eq. (50)is the imaginary parts of the ena by which the leading portion of an electromagnetic pulse spatially varying permittivity and permeability. Thus if, as in the trailing portion [3]. To see that this is possible, rewrite 14], composite media are considered, long range"effec tive''constitutive relations cannot be used to obtain Eq. (50 Eq (50)as but rather recourse to the original, spatially resolved relations must be made. It is only the latter that are guaranteed to n(x,1)=i|E(x,)|2+;|H(x,l)2 satisfy all the requirements of causality. In particular the ef- fective constitutive parameters mentioned in Ref. [14] do not satisfy the high frequency asymptotics of causality (a)ensur- doo[E(x, w; t)Im[E(x, o)JE(x, o; t) ing luminal front velocity. This does not mean that the com- posite media in such constructions are not causal(physically impossible), but only that the formulas for the effective con- +H(x,o t)ImLu(x, JH(x, w t) (54) stitutive relations are approximate, applying only for the low where the instantaneous spectrum at time t, F(x, o, t) Frequencies associated with the long range spatial averaging (F=E or H) is defined by We note that the expression for the current total dynam cal energy density u(x, t) Eq. (50) contains the classical ex- X U dTef(x, T) pression for the(heat)energy eventually dissipated to the medium. Due to propagation, we expect the fields to eventu- ally vanish at any given position x as time t-++oo. Thus via The instantaneous spectrum F(x, o; t)is just the spectrum of Eq. (50)we expect the density of energy ""left behind"(as a modified version of the"signal"F(x, r)truncated or I-+oo) at any given position to be obtained only via the "turned off" at time T=t, third, temporally nonlocal term. (x,7) u(x, +oo)= do o[E (x, o)Im[E(x, o )JE(x, o ;t<了<+∞ (56) This formula is the well known classical expression(53) Note that in the limit t-,00, the instantaneous spectrum Is +H(x, Imu(x,o)H(x,o) imply the Fourier transform of F(x, t) as per Eq.(7). That the energy density in a physical system must depend energy eventually dissipated to the medium [14] on the fields this way is made clear by causality: the energy at a given time t cannot depend on future values of the fields C. Discussion of the total dynamical energy density producing it. It is also clear that the instantaneous spectra can be much broader at certain finite times than at its asymptotic Definition(50) demonstrates that the density represented (t-00)value. In particular it can be shown to be broadest at by u in the conservation law Eq(48)is a positive definite a given position x when the signal achieves its peak value quadratic form in the fields. The positivity property is impor- there-i.e, when truncation produces the greatest disconti- 046610-6u~x,t!ª1 2 iE~x,t!i 21 1 2 iH~x,t!i 2 1 E 2` 1` dvFI aˆ E~x,v! 1 A2p E 2` t dt eivtE~x,t!I 2 1I aˆ H~x,v! 1 A2p E 2` t dt eivt H~x,t!I 2 G . ~50! Here we remind the reader that over real frequencies the aˆ tensors are related to the susceptibilities and hence permittivity and permeability as follows: aˆ E † ~x,v!aˆ E~x,v!5v Im@xˆ E~x,v!#5v Im@eˆ~x,v!2I ˆ# 5v Im@eˆ~x,v!#, ~51! aˆ H † ~x,v!aˆ H~x,v!5v Im@xˆ H~x,v!#5v Im@mˆ ~x,v!2I ˆ# 5v Im@mˆ ~x,v!#. ~52! These last two formulas should also remind the reader that what is required in Eq. ~50! is the imaginary parts of the spatially varying permittivity and permeability. Thus if, as in @14#, composite media are considered, long range ‘‘effective’’ constitutive relations cannot be used to obtain Eq. ~50!, but rather recourse to the original, spatially resolved relations must be made. It is only the latter that are guaranteed to satisfy all the requirements of causality. In particular the effective constitutive parameters mentioned in Ref. @14# do not satisfy the high frequency asymptotics of causality ~a! ensuring luminal front velocity. This does not mean that the composite media in such constructions are not causal ~physically impossible!, but only that the formulas for the effective constitutive relations are approximate, applying only for the low frequencies associated with the long range spatial averaging that give rise to such formulas ~see also @15#!. We note that the expression for the current total dynamical energy density u(x,t) Eq. ~50! contains the classical expression for the ~heat! energy eventually dissipated to the medium. Due to propagation, we expect the fields to eventually vanish at any given position x as time t→6`. Thus via Eq. ~50! we expect the density of energy ‘‘left behind’’ ~as t→1`) at any given position to be obtained only via the third, temporally nonlocal term, u~x,1`!5 E 2` 1` dv v@E† ~x,v!Im@eˆ~x,v!#E~x,v! 1H† ~x,v!Im@mˆ ~x,v!#H~x,v!#. ~53! This formula is the well known classical expression for the energy eventually dissipated to the medium @14#. C. Discussion of the total dynamical energy density Definition ~50! demonstrates that the density represented by u in the conservation law Eq. ~48! is a positive definite quadratic form in the fields. The positivity property is important to establish if u is to be interpreted as a meaningful dynamical energy density that not only has the units of energy but can also prescribe the qualitative features of system dynamics. Such features include the boundedness ~as well as existence and uniqueness! of solutions for all time, the asymptotic state of the solutions, and, since our dynamical equations ~1! and ~2! constitute a system of wave equations, the ‘‘domain of dependence’’ of solutions, i.e., the classical Sommerfeld-Brillouin result of vanishing of the fields outside the light cone of compactly supported initial data @4#. In another publication @3# we discuss in greater detail how the structure of dynamical energy density ~50! suggests a mechanism for the Garrett and McCumber @6# and Chiao effects @7#. For now we limit our discussion to demonstrating that a causal medium responds to virtual frequencies and to giving a very geometric proof of the property of luminal front velocity. In addition, we discuss the connection between the dynamical energy density ~50! and an approximate expression often employed. 1. The medium responds to a virtual, instantaneous spectrum The form of Eq. ~50! can be used to explain the phenomena by which the leading portion of an electromagnetic pulse exchanges energy with the causal medium differently than the trailing portion @3#. To see that this is possible, rewrite Eq. ~50! as u~x,t!ª1 2 iE~x,t!i 21 1 2 iH~x,t!i 2 1 E 2` 1` dv v@E† ~x,v;t!Im@eˆ~x,v!#E~x,v;t! 1H† ~x,v;t!Im@mˆ ~x,v!#H~x,v;t!#, ~54! where the instantaneous spectrum at time t, F(x,v;t), (F5E or H) is defined by F~x,v;t!ª 1 A2p E 2` t dt eivt F~x,t!. ~55! The instantaneous spectrum F(x,v;t) is just the spectrum of a modified version of the ‘‘signal’’ F(x,t) truncated or ‘‘turned off’’ at time t5t, F~x,t! ; 2`,t,t 0 ; t,t,1`. ~56! @Note that in the limit t→`, the instantaneous spectrum is simply the Fourier transform of F(x,t) as per Eq. ~7!.# That the energy density in a physical system must depend on the fields this way is made clear by causality: the energy at a given time t cannot depend on future values of the fields producing it. It is also clear that the instantaneous spectra can be much broader at certain finite times than at its asymptotic (t→`) value. In particular it can be shown to be broadest at a given position x when the signal achieves its peak value there—i.e., when truncation produces the greatest discontiS. GLASGOW, M. WARE, AND J. PEATROSS PHYSICAL REVIEW E 64 046610 046610-6