e(,-0)=e(r,m),(r,-a)=-(r,o) For an isotropic material it takes the particularly simple form +60+∈0 d we have (r,-①)=e(r,o),e"(r,-)=-e"(r,o) (4.27) 4.4.2 High and low frequency behavior of constitutive parameters At low frequencies the permittivity reduces to the electrostatic permittivity. Since 2 is even in o and e" is odd. we have for small o If the material has some dc conductivity oo, then for low frequencies the complex per mittivity behaves as If E or H changes very rapidly, there may be no polarization or magnetization effect at all. This occurs at frequencies so high that the atomic structure of the material cannot respond to the rapidly oscillating applied field. Above some frequency then, we can assume xe =0 and im =0 so that P=0.M=0, =μ In our simple models of dielectric materials($ 4.6) we find that as o becomes large Our assumption of a macroscopic model of matter provides a fairly strict upper frequency limit to the range of validity of the constitutive parameters. We must assume that the wavelength of the electromagnetic field is large compared to the size of the atomic struc ture. This limit suggests that permittivity and permeability might remain meaningful even at optical frequencies, and for dielectrics this is indeed the case since the values of P remain significant. However, M becomes insignificant at much lower frequencies, and B=10H[107] 4.4.3 The Kronig-Kramers relations The principle of causality is clearly implicit in(2. 29)-(2.31). We shall demonstrate that causality leads to explicit relationships between the real and imaginary parts of the frequency-domain constitutive parameters. For simplicity we concentrate on the isotropic case and merely note that the present analysis may be applied to all the dyadic com- ponents of an anisotropic constitutive parameter. We also concentrate on the complex permittivity and extend the results to permeability by induction. 2001 by CRC Press LLC
or ˜ c i j(r, −ω) = ˜ c i j(r, ω), ˜ c i j (r, −ω) = −˜ c i j (r, ω). For an isotropic material it takes the particularly simple form ˜ c = σ˜ jω + ˜ = σ˜ jω + 0 + 0χ˜e, (4.26) and we have ˜ c (r, −ω) = ˜ c (r, ω), ˜ c(r, −ω) = −˜ c(r, ω). (4.27) 4.4.2 High and low frequency behavior of constitutive parameters At low frequencies the permittivity reduces to the electrostatic permittivity. Since ˜ is even in ω and ˜ is odd, we have for small ω ˜ ∼ 0r, ˜ ∼ ω. If the material has some dc conductivity σ0, then for low frequencies the complex permittivity behaves as ˜ c ∼ 0r, ˜ c ∼ σ0/ω. (4.28) If E or H changes very rapidly, there may be no polarization or magnetization effect at all. This occurs at frequencies so high that the atomic structure of the material cannot respond to the rapidly oscillating applied field. Above some frequency then, we can assume χ˜¯ e = 0 and χ˜¯ m = 0 so that P˜ = 0, M˜ = 0, and D˜ = 0E˜ , B˜ = µ0H˜ . In our simple models of dielectric materials (§ 4.6) we find that as ω becomes large ˜ − 0 ∼ 1/ω2 , ˜ ∼ 1/ω3 . (4.29) Our assumption of a macroscopic model of matter provides a fairly strict upper frequency limit to the range of validity of the constitutive parameters. We must assume that the wavelength of the electromagnetic field is large compared to the size of the atomic structure. This limit suggests that permittivity and permeability might remain meaningful even at optical frequencies, and for dielectrics this is indeed the case since the values of P˜ remain significant. However, M˜ becomes insignificant at much lower frequencies, and at optical frequencies we may use B˜ = µ0H˜ [107]. 4.4.3 The Kronig–Kramers relations The principle of causality is clearly implicit in (2.29)–(2.31). We shall demonstrate that causality leads to explicit relationships between the real and imaginary parts of the frequency-domain constitutive parameters. For simplicity we concentrate on the isotropic case and merely note that the present analysis may be applied to all the dyadic components of an anisotropic constitutive parameter. We also concentrate on the complex permittivity and extend the results to permeability by induction
The implications of causality on the behavior of the constitutive parameters in the time domain can be easily identified. Writing(2.29)and(2. 31)after setting u=t-t and then u=t we have D(r, t)=EoE(r, t)+Eo/Xe(r, tE(r, t-t)dt J(r,t)=o(r,!E(r, t-I)dt We see that there is no contribution from values of xe(r, t) or o(r, t) for times t <0. So we can write Dr,)=∈0E(r,1)+60/x(r,t)E(r,t-)d' J(r, t)= o(r, t'E(r, t -)dr' th the additional assumption Xe(r, t)=0, (r,t)=0.t<0. (4.30) By(4.30) we can write the frequency-domain complex permittivity(4.26)as a(r, t')e-jonr'dr'+Eo/Xe(r,t' In order to derive the Kronig-Kramers relations we must understand the behavior of E(r, o)-Eo in the complex a-plane. Writing o= Or+ joi, we need to establish the following two properties Property 1: The function E(r, a)-Eo is analytic in the lower half-plane(o; 0) ept at o=0 We can establish the analyticity of o(r, o) by integrating over any closed contour in the lower half-plane. We have (r, o)do σ(r,t Note that an exchange in the order of integration in the above expression is only valid for o in the lower half-plane where limr'-oo e /or=0. Since the function f(o)=e- Jor is analytic in the lower half-plane, its closed contour integral is zero by the Cauchy-Goursat theorem. Thus, by(4.32)we have Then, since a may be assumed to be continuous medium, and since its closed path integral is zero for ible paths T, it is by Morera's theorem [110 analytic in the lower half-plane. By reasoning xe (r, o) is analytic in the lower half-plane. Since the function 1/o has a simple pole at o=0, the composite function E(r, a)-Eo given by(4.31)is analytic in the lower half-plane excluding o=0 here it has a simple pole 2001 by CRC Press LLC
The implications of causality on the behavior of the constitutive parameters in the time domain can be easily identified. Writing (2.29) and (2.31) after setting u = t − t and then u = t , we have D(r, t) = 0E(r, t) + 0 ∞ 0 χe(r, t )E(r, t − t ) dt , J(r, t) = ∞ 0 σ(r, t )E(r, t − t ) dt . We see that there is no contribution from values of χe(r, t) or σ(r, t) for times t < 0. So we can write D(r, t) = 0E(r, t) + 0 ∞ −∞ χe(r, t )E(r, t − t ) dt , J(r, t) = ∞ −∞ σ(r, t )E(r, t − t ) dt , with the additional assumption χe(r, t) = 0, t < 0, σ(r, t) = 0, t < 0. (4.30) By (4.30) we can write the frequency-domain complex permittivity (4.26) as ˜ c (r,ω) − 0 = 1 jω ∞ 0 σ(r, t )e− jωt dt + 0 ∞ 0 χe(r, t )e− jωt dt . (4.31) In order to derive the Kronig–Kramers relations we must understand the behavior of ˜ c(r,ω) − 0 in the complex ω-plane. Writing ω = ωr + jωi , we need to establish the following two properties. Property 1: The function ˜ c(r,ω) − 0 is analytic in the lower half-plane (ωi < 0) except at ω = 0 where it has a simple pole. We can establish the analyticity of σ(˜ r,ω) by integrating over any closed contour in the lower half-plane. We have σ(˜ r,ω) dω = ∞ 0 σ(r, t )e− jωt dt dω = ∞ 0 σ(r, t ) e− jωt dω dt . (4.32) Note that an exchange in the order of integration in the above expression is only valid for ω in the lower half-plane where limt →∞ e− jωt = 0. Since the function f (ω) = e− jωt is analytic in the lower half-plane, its closed contour integral is zero by the Cauchy–Goursat theorem. Thus, by (4.32) we have σ(˜ r,ω) dω = 0. Then, since σ˜ may be assumed to be continuous in the lower half-plane for a physical medium, and since its closed path integral is zero for all possible paths , it is by Morera’s theorem [110] analytic in the lower half-plane. By similar reasoning χe(r,ω) is analytic in the lower half-plane. Since the function 1/ω has a simple pole at ω = 0, the composite function ˜ c(r,ω) − 0 given by (4.31) is analytic in the lower half-plane excluding ω = 0 where it has a simple pole.
Figure 4.1: Complex integration contour used to establish the Kronig-Kramers relations ty 2: We h To establish this property we need the Riemann-Lebesgue lemma[142, which states that if f(r) is absolutely integrable on the interval (a, b) where a and b are finite or infinite constants, then lim f(t)e- o dt=0 From this we see that o(r, te ja dt=0 ime(r,o)-∈0=0 To establish the Kronig-Kramers relations we examine the integral where r is the contour shown in Figure 4.L. Since the points $2=0, o are excluded, the integrand is analytic everywhere within and on T, hence the integral vanishes by the Cauchy-Goursat theorem. By Property 2 we have e(r,92)一∈0 2001 by CRC Press LLC
Figure 4.1: Complex integration contour used to establish the Kronig–Kramers relations. Property 2: We have lim ω→±∞ ˜ c (r,ω) − 0 = 0. To establish this property we need the Riemann–Lebesgue lemma [142], which states that if f (t) is absolutely integrable on the interval (a, b) where a and b are finite or infinite constants, then lim ω→±∞ b a f (t)e− jωt dt = 0. From this we see that lim ω→±∞ σ(˜ r,ω) jω = lim ω→±∞ 1 jω ∞ 0 σ(r, t )e− jωt dt = 0, lim ω→±∞ 0χe(r,ω) = lim ω→±∞ 0 ∞ 0 χe(r, t )e− jωt dt = 0, and thus lim ω→±∞ ˜ c (r,ω) − 0 = 0. To establish the Kronig–Kramers relations we examine the integral ˜ c(r, ) − 0 − ω d where is the contour shown in Figure 4.l. Since the points = 0, ω are excluded, the integrand is analytic everywhere within and on , hence the integral vanishes by the Cauchy–Goursat theorem. By Property 2 we have lim R→∞ C∞ ˜ c(r, ) − 0 − ω d = 0,
dQ+Pv dg=0. (4.3) Co+C Here "P.V. " indicates that the integral is computed in the Cauchy principal value sense (see Appendix A). To evaluate the integrals over Co and Co, consider a function f(Z) analytic in the lower half of the Z-plane(Z= Z,+jZi). If the point z lies on the real axis as shown in Figure 4.1, we can calculate the integral F(x)= through the parameterization Z-z=8e/. since dz=jSeJe de we have F(z)=lim f(z+seJ [jseje]de=jf(2) de=jrf(2) Replacing Z by $2 and z by 0 we can compute e(r,)一∈0 to a(r,t We o(r, tdt'= go(r) as the dc conductivity and write mf(r2)-0 Too(r) If we replace Z by S and z by o we get e(r,2) -ds2= jE(r, o) Substituting these into(4.33)we have e(r,) P/(r,9)- If we write E(r, o)=E(r, o)+je(r, a) and equate real and imaginary parts in(4.34) we find that e(r,)-∈0=--PV ec(,sdS2, (4.35) e(r,3)-∈0 go(r) 36) 2001 by CRC Press LLC
hence C0+Cω ˜ c(r, ) − 0 − ω d + P.V. ∞ −∞ ˜ c(r, ) − 0 − ω d = 0. (4.33) Here “P.V.” indicates that the integral is computed in the Cauchy principal value sense (see Appendix A). To evaluate the integrals over C0 and Cω, consider a function f (Z) analytic in the lower half of the Z-plane (Z = Zr + j Zi). If the point z lies on the real axis as shown in Figure 4.1, we can calculate the integral F(z) = lim δ→0 f (Z) Z − z d Z through the parameterization Z − z = δe jθ . Since d Z = jδe jθ dθ we have F(z) = lim δ→0 0 −π f z + δe jθ δe jθ jδe jθ dθ = j f (z) 0 −π dθ = jπ f (z). Replacing Z by and z by 0 we can compute lim →0 C0 ˜ c(r, ) − 0 − ω d = lim →0 C0 1 j ∞ 0 σ(r, t )e− jt dt + 0 ∞ 0 χe(r, t )e− jt dt 1 −ω d = −π ∞ 0 σ(r, t ) dt ω . We recognize ∞ 0 σ(r, t ) dt = σ0(r) as the dc conductivity and write lim →0 C0 ˜ c(r, ) − 0 − ω d = −πσ0(r) ω . If we replace Z by and z by ω we get lim δ→0 Cω ˜ c(r, ) − 0 − ω d = jπ˜ c (r,ω) − jπ0. Substituting these into (4.33) we have ˜ c (r,ω) − 0 = − 1 jπ P.V. ∞ −∞ ˜ c(r, ) − 0 − ω d + σ0(r) jω . (4.34) If we write ˜ c(r,ω) = ˜ c (r,ω) + j˜ c(r,ω) and equate real and imaginary parts in (4.34) we find that ˜ c (r,ω) − 0 = − 1 π P.V. ∞ −∞ ˜ c(r, ) − ω d, (4.35) ˜ c(r,ω) = 1 π P.V. ∞ −∞ ˜ c (r, ) − 0 − ω d − σ0(r) ω . (4.36)
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 LLC
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 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)