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POYNTING'S THEOREM AND LUMINAL TOTAL ENERGY PHYSICAL REVIEW E 64 046610 Here P[1/(o-o) indicates the Cauchy Principal value 1(+a,Im[e(o)] distribution centered at o=w', and S(o-o)indicates the u(x, t)=Eo(x)+Em(x)2 Dirac 8 function (distribution) also centered at o=w.In- erting the fields(72)into(71), using the relevant versions of the distributional identities(73), and finally the Kramers- 5Eo(x)+Eo(x)Re[E(O)] Kronig relations(32)in reverse, we get(simplifying to the IsotropIc case JE(x, t)l E0(x)E0(x) pa2(x,1)=|Eo(x)Ree(9)]+ ∈(92) the second equality following from Kramers-Kronig for this EO(xEo(x) Isotropic case IIL GLOBAL ENERGY TRANSPORT VELOCITY We investigate the two extreme values of Eq. (74) In previous work, we investigated a certain"temporal ach point x) by limiting Eq (74)to the set of times at which center-of-mass"of an electromagnetic pulse [5]. We found the kinematic density is stationary. At those times(denoted among other things, that this formalism provided a frame- 1) we find that work wherein the classical notion of group velocity was meaningful even for broad-band pulses. The following rep- (x)E6(x) Eo(xEo(x) the weight of recent works on superluminal electromagnetic phenomena(for a"small"" sampling see [17), this issue of (75) the nature of global energy transport that we and others have addressed is clearly not the local one addressed by the (oth Note that both of these quantities are real at those times. The erwise very satisfying) classical Sommerfeld-Brillouin re- lues of the density(74)are then sult. Nevertheless, in the following one will see that, in con- trast to the"temporally oriented'" view of the properties of Appro (x, t )=Eo(x)Re[E(O) global total energy transport reported in [51, patially oriented' view is very much a global generalization of Som- +E(x)E0(x)e-2∈(0).(76) merfeld and Brillouin's local result We begin by defining the position of the total dynamical Using the fact that the second quantity in Eq.(76)is real at energy as the normalized, first spatial moment of the total dynamical energy density hese times. we realize that dxxu(x, t) e-2int+EO(x)E(x)2*(Q) E6(x)E0(x)∈(Q) dxu(x, t) in which case(76) becomes(after simplification) =E-l d'xxu(x,t) (82) approx(x,1)=|E0(x){Rete(9)]±|e(9)}.(78) The integrals are over all space and we have defined the total In Eq.(78)it is now clear that the approximate density does energy not have definite sign so long as Im[le(Q)] is not zero Note that in the limit of static fields, however, the dy E: dxu(x,t lamical and approximate results agree: Using real symmetry whereby Im[ E( )] goes to zero when n does, we see that Having defined the position of the total energy x, (t),we 74)becomes(after some simplification) then define the velocity of the total energy vu(r)in the natu- al way, i.e., by time differentiation of the position M npro (x 1)=2E(x)+E(xlE(0) RelE(o) E(x,)‖2 dx(t) as expected. Using identities(73)the dynamical energy den- Making use of the definition of the position(82)and by use sity(50)becomes, for the fields given in Eq. (72)at n2=0, of Poynting's conservation law(48)we find that 046610-9Here P@1/(v2v8)# indicates the Cauchy Principal value distribution centered at v5v8, and d(v2v8) indicates the Dirac d function ~distribution! also centered at v5v8. In￾serting the fields ~72! into ~71!, using the relevant versions of the distributional identities ~73!, and finally the Kramers￾Kronig relations ~32! in reverse, we get ~simplifying to the isotropic case! uapprox~x,t!5iE0~x!i 2Re@eˆ~V!#1 E0 T ~x!E0~x! 2 e22iVt eˆ~V! 1 E0 † ~x!E0 *~x! 2 e12iVt eˆ *~V!. ~74! We investigate the two extreme values of Eq. ~74! ~at each point x) by limiting Eq. ~74! to the set of times at which the kinematic density is stationary. At those times ~denoted ¯t) we find that E0 † ~x!E0 *~x! 2 e12iVt ¯ eˆ *~V!5 E0 T ~x!E0~x! 2 e22iVt ¯ eˆ~V!. ~75! Note that both of these quantities are real at those times. The extreme values of the density ~74! are then uapprox~x, ¯t!5iE0~x!i 2Re@eˆ~V!# 1E0 T ~x!E0~x!e22iVt ¯ eˆ~V!. ~76! Using the fact that the second quantity in Eq. ~76! is real at these times, we realize that e22iVt ¯ 56 E0 † ~x!E0 *~x!eˆ *~V! uE0 T ~x!E0~x!eˆ~V!u , ~77! in which case ~76! becomes ~after simplification! uapprox~x, ¯t!5iE0~x!i 2 $Re@eˆ~V!#6ueˆ~V!u%. ~78! In Eq. ~78! it is now clear that the approximate density does not have definite sign so long as Im@eˆ(V)# is not zero. Note that in the limit of static fields, however, the dy￾namical and approximate results agree: Using real symmetry, whereby Im@eˆ(V)# goes to zero when V does, we see that Eq. ~74! becomes ~after some simplification! uapprox~x,t!5 Re@eˆ~0!# 2 iE0~x!1E0 *~x!i 25eˆ~0! 2 iE~x,t!i 2, ~79! as expected. Using identities ~73! the dynamical energy den￾sity ~50! becomes, for the fields given in Eq. ~72! at V50, u~x,t!51 2 iE0~x!1E0 *~x!i 2 S 11 1 pPE 2` 1` dv Im@eˆ~v!# v20 D 5 1 2 iE0~x!1E0 *~x!i 2Re@eˆ~0!# 5 eˆ~0! 2 iE~x,t!i 2, ~80! the second equality following from Kramers-Kronig for this isotropic case. III. GLOBAL ENERGY TRANSPORT VELOCITY In previous work, we investigated a certain ‘‘temporal center-of-mass’’ of an electromagnetic pulse @5#. We found, among other things, that this formalism provided a frame￾work wherein the classical notion of group velocity was meaningful even for broad-band pulses. The following rep￾resents the spatial analog of that work. As is evidenced by the weight of recent works on superluminal electromagnetic phenomena ~for a ‘‘small’’ sampling see @17#!, this issue of the nature of global energy transport that we and others have addressed is clearly not the local one addressed by the ~oth￾erwise very satisfying! classical Sommerfeld-Brillouin re￾sult. Nevertheless, in the following one will see that, in con￾trast to the ‘‘temporally oriented’’ view of the properties of global total energy transport reported in @5#, the ‘‘spatially oriented’’ view is very much a global generalization of Som￾merfeld and Brillouin’s local result. We begin by defining the position of the total dynamical energy as the normalized, first spatial moment of the total dynamical energy density, xu~t!ª E d3x x u~x,t! E d3x u~x,t! ~81! 5E 21 E d3x x u~x,t!. ~82! The integrals are over all space and we have defined the total energy Eª E d3x u~x,t!. ~83! Having defined the position of the total energy xu(t), we then define the velocity of the total energy vu(t) in the natu￾ral way, i.e., by time differentiation of the position vu~t!ªd xu~t! dt . ~84! Making use of the definition of the position ~82! and by use of Poynting’s conservation law ~48! we find that POYNTING’S THEOREM AND LUMINAL TOTAL ENERGY . . . PHYSICAL REVIEW E 64 046610 046610-9
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