These are the Kronig-Kramers relations, named after R. de L Kronig and H.A. Kramers who derived them independently. The expressions show that causality requires the real and imaginary parts of the permittivity to depend upon each other through the Hilbert It is often more convenient to write the Kronig-Kramers relations in a form that employs only positive frequencies. This can be accomplished using the even-odd behavior of the real and imaginary parts of E. Breaking the integrals in(4. 35 )-(4.36)into the ranges(oo, 0)and(0, oo), and substituting from(4. 27), we can show that (4.37) E(r, o) E(r,2 ds- oo(r) (438) The symbol P.V. in this case indicates that values of the integrand around both &=0 and s2= o must be excluded from the integration. The details of the derivation of (4.37)-(4.38)are left as an exercise. We shall use(4.37)in 8 4.6 to demonstrate the Kronig-Kramers relationship for a model of complex permittivity of an actual material We cannot specify 2 arbitrarily; for a passive medium Ecmust be zero or negative at all values of a, and(4.36)will not necessarily return these required values. However, if we have a good measurement or physical model for 2, as might come from studies of the absorbing properties of the material, we can approximate the real part of the permittivity using(4.35). We shall demonstrate this using simple models for permittivity in 8 4.6 The Kronig-Kramers properties hold for u as well. We must for practical onsider the fact that magnetization becomes unimportant at a much lower frequency than does polarization, so that the infinite integrals in the Kronig-Kramers relations should be truncated at some upper frequency amax. If we use a model or measured values of A"to determine A', the form of the relation (4.37)should be [107] p'(r,o)-10=--PV cimax &u"(r, &2) where amax is the frequency at which magnetization ceases to be important, and above which A= uo. 4.5 Dissipated and stored energy in a dispersive medium Let us write down Poynting's power balance theorem for a dispersive medium. Writing J=J+ J we have(§2.9.5) ad J·E=JE+V·ExH+E· (4.39) We cannot express this in terms of the time rate of change of a stored energy density because of the difficulty in interpreting the term 2001 by CRC Press LLCThese are the Kronig–Kramers relations, named after R. de L. Kronig and H.A. Kramers who derived them independently. The expressions show that causality requires the real and imaginary parts of the permittivity to depend upon each other through the Hilbert transform pair [142]. It is often more convenient to write the Kronig–Kramers relations in a form that employs only positive frequencies. This can be accomplished using the even–odd behavior of the real and imaginary parts of ˜ c. Breaking the integrals in (4.35)–(4.36) into the ranges (−∞, 0) and (0,∞), and substituting from (4.27), we can show that ˜ c (r,ω) − 0 = − 2 π P.V. ∞ 0 ˜ c(r, ) 2 − ω2 d, (4.37) ˜ c(r,ω) = 2ω π P.V. ∞ 0 ˜ c (r, ) 2 − ω2 d − σ0(r) ω . (4.38) The symbol P.V. in this case indicates that values of the integrand around both  = 0 and  = ω must be excluded from the integration. The details of the derivation of (4.37)–(4.38) are left as an exercise. We shall use (4.37) in § 4.6 to demonstrate the Kronig–Kramers relationship for a model of complex permittivity of an actual material. We cannot specify ˜ c arbitrarily; for a passive medium ˜ c must be zero or negative at all values of ω, and (4.36) will not necessarily return these required values. However, if we have a good measurement or physical model for ˜ c, as might come from studies of the absorbing properties of the material, we can approximate the real part of the permittivity using (4.35). We shall demonstrate this using simple models for permittivity in § 4.6. The Kronig–Kramers properties hold for µ as well. We must for practical reasons consider the fact that magnetization becomes unimportant at a much lower frequency than does polarization, so that the infinite integrals in the Kronig–Kramers relations should be truncated at some upper frequency ωmax. If we use a model or measured values of µ˜  to determine µ˜  , the form of the relation (4.37) should be [107] µ˜  (r,ω) − µ0 = − 2 π P.V. ωmax 0 µ˜ (r, ) 2 − ω2 d, where ωmax is the frequency at which magnetization ceases to be important, and above which µ˜ = µ0. 4.5 Dissipated and stored energy in a dispersive medium Let us write down Poynting’s power balance theorem for a dispersive medium. Writing J = Ji + Jc we have (§ 2.9.5) − Ji · E = Jc · E +∇· [E × H] +  E · ∂D ∂t + H · ∂B ∂t . (4.39) We cannot express this in terms of the time rate of change of a stored energy density because of the difficulty in interpreting the term E · ∂D ∂t + H · ∂B ∂t (4.40)