S. GLASGOW. M. WARE AND J PEATROSS PHYSICAL REVIEW E 64 046610 We next use Eq (25)to eliminate explicit reference to the polarization vector in Eq. (24). To do this, we inverse Fou Relxeo)l (32) rier transform(25)to obtain We can use these relationships between the real and dTGE(t-TE(T) (27) Imaginary parts of the susceptibilities to show that the in- phase and out-of-phase components of the electric and mag- netic convolution kernels are not independent. These two where the convolution kernel GE(t) is defined in terms of the components of the convolution kernels are defined in terms susceptibilityvia d (28) doe Relf(o) (33) We need the time derivative of the polarization. Via Eq(27) G"()=2]- dwe ImL XF(o)] we see that this is obtained through the formula Note that G(o=GF(o+GE(o) aP(=J。 drat gee(-nE( (29) We now show that the in- and out-of-phase components of the convolution kernels are identical for positive argu- ment,i.e, GF(1=GF(0), 1>0. To that end we rewrite (Note: The rapid vanishing of the susceptibilities at large Eq(33)via Eq(32)and obtain frequencies renders the kernels differentiable everywhere but at a single time where they are, fortunately, continuous. Thus exchange of orders of the operations of integration and ImLxF(o)] differentiation is justified. We now use the various properties of the susceptibilities (35) to reduce Eq. (29)to an equivalent expression that can be used to directly demonstrate the conserved energy. The first Exchanging the orders of the integrations(and simplifying), (and usual)simplification is to note that the integral(28)can we obtain be evaluated explicitly for (<0. We use Cauchys integral theorem with contours constructed from great semicircles in GF(o 2m2/ dop/d e-ior Im[X(o’)] the upper-half o plane, closed along the real axis. Since the susceptibilities are analytic and rapidly vanish with increas ing radius in the region enclosed by these contours, it is readily shown that for (<o the integration over the real in- The inner integral can be evaluated via Cauchy s theorem by terval defining the convolution kernel gives zero Ise of a large semicircular contour that extends into the lower-half plane(for (>0)and that, for example, contains a Gg(1)=0;t<0. (30Small semicircular dimple excluding the pole at w=@.Al- ternatively, one can recognize the integral as a Hilbert trans- form and consult a table. Either way the result is that 0 indicates the zero matrix. )The formula expressing the time derivative of the polarization vector in terms of the electric field, Eq. (29), then reduces to integration up to time d -a't.t>0. (37) Using this result in Eq.(36) gives (31) The previous formula involves the convolution kernel GE, which is constructed from the susceptibility by Eq(28 ). according to definition (34) In particular, it appears from that construction that both the Our formula allowing us to eliminate the polarization(31) real and imaginary parts of the susceptibility are important. can now be expressed We now show that the convolution kernel can be constructed entirely from the imaginary part of the susceptibility which, in turn, will allow us to use passivity (e) to deduce certain 'A rigorous exchange can be made by writing the Cauchy princi important properties of this kernel. To that end, we note that pal value operation as a limit and by restricting the fields to certain in terms of a susceptibility, the Kramers-Kronig relations physically reasonable function spaces. Similar statements apply to [causality (a)] can be expressed much of what follows 046610-4We next use Eq. ~25! to eliminate explicit reference to the polarization vector in Eq. ~24!. To do this, we inverse Fou￾rier transform ~25! to obtain P~t!5 E 2` 1` dt Gˆ E~t2t!E~t!, ~27! where the convolution kernel Gˆ E(t) is defined in terms of the susceptibility via Gˆ E~t!ª 1 2pE 2` 1` dv e2ivt xˆ E~v!. ~28! We need the time derivative of the polarization. Via Eq. ~27!, we see that this is obtained through the formula ] ]t P~t!5 E 2` 1` dt ] ]t Gˆ E~t2t!E~t!. ~29! ~Note: The rapid vanishing of the susceptibilities at large frequencies renders the kernels differentiable everywhere but at a single time where they are, fortunately, continuous. Thus the exchange of orders of the operations of integration and differentiation is justified.! We now use the various properties of the susceptibilities to reduce Eq. ~29! to an equivalent expression that can be used to directly demonstrate the conserved energy. The first ~and usual! simplification is to note that the integral ~28! can be evaluated explicitly for t,0. We use Cauchy’s integral theorem with contours constructed from great semicircles in the upper-half v plane, closed along the real axis. Since the susceptibilities are analytic and rapidly vanish with increas￾ing radius in the region enclosed by these contours, it is readily shown that for t,0 the integration over the real in￾terval defining the convolution kernel gives zero: Gˆ E~t!50ˆ; t,0. ~30! (0ˆ indicates the zero matrix.! The formula expressing the time derivative of the polarization vector in terms of the electric field, Eq. ~29!, then reduces to integration up to time t5t: ] ]t P~t!5 E 2` t dt ] ]t Gˆ E~t2t!E~t!. ~31! The previous formula involves the convolution kernel Gˆ E , which is constructed from the susceptibility by Eq. ~28!. In particular, it appears from that construction that both the real and imaginary parts of the susceptibility are important. We now show that the convolution kernel can be constructed entirely from the imaginary part of the susceptibility which, in turn, will allow us to use passivity ~c! to deduce certain important properties of this kernel. To that end, we note that in terms of a susceptibility, the Kramers-Kronig relations @causality ~a!# can be expressed as Re@xˆ F~v!#5 1 p PE 2` 1` dv8 Im@xˆ F~v8!# v82v . ~32! We can use these relationships between the real and imaginary parts of the susceptibilities to show that the in￾phase and out-of-phase components of the electric and mag￾netic convolution kernels are not independent. These two components of the convolution kernels are defined in terms of the real and imaginary parts of the susceptibilities via Gˆ F in~t!ª 1 2pE 2` 1` dv e2ivt Re@xˆ F~v!#, ~33! Gˆ F out~t!ª i 2pE 2` 1` dv e2ivt Im@xˆ F~v!#. ~34! Note that Gˆ F(t)5Gˆ F in(t)1Gˆ F out(t). We now show that the in- and out-of-phase components of the convolution kernels are identical for positive argu￾ment, i.e., Gˆ F in(t)5Gˆ F out(t), t.0. To that end we rewrite Eq. ~33! via Eq. ~32! and obtain Gˆ F in~t!ª 1 2pE 2` 1` dv e2ivt 1 p PE 2` 1` dv8 Im@xˆ F~v8!# v82v . ~35! Exchanging the orders of the integrations1 ~and simplifying!, we obtain Gˆ F in~t!5 1 2p2 E 2` 1` dv8S PE 2` 1` dv e2ivt v82vD Im@xˆ F~v8!#. ~36! The inner integral can be evaluated via Cauchy’s theorem by use of a large semicircular contour that extends into the lower-half plane ~for t.0) and that, for example, contains a small semicircular dimple excluding the pole at v5v8. Al￾ternatively, one can recognize the integral as a Hilbert trans￾form and consult a table. Either way the result is that PE 2` 1` dv e2ivt v82v 5ipe2iv8t ; t.0. ~37! Using this result in Eq. ~36! gives Gˆ F in~t!5 i 2pE 2` 1` dv8e2iv8t Im@xˆ F~v8!#5:Gˆ F out~t!; t.0, ~38! according to definition ~34!. Our formula allowing us to eliminate the polarization ~31! can now be expressed as 1 A rigorous exchange can be made by writing the Cauchy princi￾pal value operation as a limit and by restricting the fields to certain physically reasonable function spaces. Similar statements apply to much of what follows. S. GLASGOW, M. WARE, AND J. PEATROSS PHYSICAL REVIEW E 64 046610 046610-4