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