4.4 Constitutive relations in the frequency domain and the Kronig-Kramers relations All materials are to some extent dispersive. If a field applied to a material undergoes sufficiently rapid change, there is a time lag in the response of the polarization or magnetization of the It has been found that such materials have constitutive ng pre in the frequency domain, and that the frequency-domain constitutive parameters are complex, frequency-dependent quantities. We shall restrict I case of anisotropic materials and refer the reader to and Lindell [113 for the more general case. For anisotropic materials we write P=60元eE, D=E·E=∈o+元E (4.13) B=乒·H=μ0[+元m]H (4.14) j=吞.E. (4.15) By the convolution theorem and the assumption of causality we immediately obtain the dyadic versions of(2. 29)-(2.31): D(r,)=∈0E(r,1)+/元2(r,t-1)·E(r,t)da B(r,1)=0(Hr,n)+/元m(r-1)Hr,t)dr J(r,t)=o(r, t-t).E(r, t,dr These describe the essential behavior of a dispersive material. The susceptance and conductivity, describing the response of the atomic structure to an applied field, depend not only on the present value of the applied field but on all past values as well. Now since D(r, 1), B(r, 1), and J(r, t)are all real, so are the entries in the dyadic matrices E(r, 1), A(r, t), and o(r, t). Thus, applying(4.3)to each entry we must have 元(r,-0)=无(r,),无m(r,-0)=元(r,), (4.16) and hence e(r,-)=e(r,o),乒(r,-0)='(r,a) (4.17) If we write the constitutive parameters in terms of real and imaginary parts as Aij+JRij these conditions become e(r,-)=e1(r,o),er(r,-m)=-e(r,) and so on. Therefore the real parts of the constitutive parameters are even functions of frequency, and the imaginary parts are odd functions of frequency 2001 by CRC Press LLC4.4 Constitutive relations in the frequency domain and the Kronig–Kramers relations All materials are to some extent dispersive. If a field applied to a material undergoes a sufficiently rapid change, there is a time lag in the response of the polarization or magnetization of the atoms. It has been found that such materials have constitutive relations involving products in the frequency domain, and that the frequency-domain constitutive parameters are complex, frequency-dependent quantities. We shall restrict ourselves to the special case of anisotropic materials and refer the reader to Kong [101] and Lindell [113] for the more general case. For anisotropic materials we write P˜ = 0χ˜¯ e · E˜ , (4.11) M˜ = χ˜¯ m · H˜ , (4.12) D˜ = ˜¯ · E˜ = 0[¯ I + χ˜¯ e] · E˜ , (4.13) B˜ = µ˜¯ · H˜ = µ0[¯ I + χ˜¯ m] · H˜ , (4.14) J˜ = σ˜¯ · E˜ . (4.15) By the convolution theorem and the assumption of causality we immediately obtain the dyadic versions of (2.29)–(2.31): D(r, t) = 0 E(r, t) + t −∞ χ¯ e(r, t − t  ) · E(r, t  ) dt  , B(r, t) = µ0 H(r, t) + t −∞ χ¯ m(r, t − t  ) · H(r, t  ) dt  , J(r, t) = t −∞ σ¯ (r, t − t  ) · E(r, t  ) dt . These describe the essential behavior of a dispersive material. The susceptances and conductivity, describing the response of the atomic structure to an applied field, depend not only on the present value of the applied field but on all past values as well. Now since D(r, t), B(r, t), and J(r, t) are all real, so are the entries in the dyadic matrices ¯(r, t), µ¯ (r, t), and σ¯ (r, t). Thus, applying (4.3) to each entry we must have χ˜¯ e(r, −ω) = χ˜¯ ∗ e (r, ω), χ˜¯ m(r, −ω) = χ˜¯ ∗ m(r, ω), σ˜¯ (r, −ω) = σ˜¯ ∗ (r, ω), (4.16) and hence ˜¯(r, −ω) = ˜¯ ∗ (r, ω), µ˜¯ (r, −ω) = µ˜¯ ∗ (r, ω). (4.17) If we write the constitutive parameters in terms of real and imaginary parts as ˜i j = ˜  i j + j˜  i j, µ˜ i j = µ˜  i j + jµ˜  i j, σ˜i j = σ˜ i j + jσ˜ i j, these conditions become ˜  i j(r, −ω) = ˜  i j(r, ω), ˜  i j(r, −ω) = −˜  i j(r, ω), and so on. Therefore the real parts of the constitutive parameters are even functions of frequency, and the imaginary parts are odd functions of frequency.