Chapter 4 Temporal and spatial frequency domain representation 4.1 Interpretation of the temporal transform When a field is represented by a continuous superposition of elemental components, the esulting decomposition can simplify computation and provide physical insight. Such rep- esentation is usually accomplished through the use of an integral transform. Although everal different transforms are used in electromagnetics. we shall concentrate on the powerful and efficient Fourier transform Let us consider the Fourier transform of the electromagnetic field. The field depends on x, y, z, t, and we can transform with respect to any or all of these variables. However, a consideration of units leads us to consider a transform over t separately. Let y(r, t) represent any rectangular component of the electric or magnetic field. Then the temporal transform will be designated by y(r, a) ψ(r,1)←v(r,o) Here o is the transform variable. The transform field y is calculated using(A. The inverse transform is, by(A. 2) ψ(r,o) ejon d e Since y is complex it may be written in amplitude-phase form y(r, o)=ly(r, o). Since y(r, t) must be real, (4. 1)shows that ψ(r,-)=v’(r,ω) Furthermore, the transform of the derivative of y may be found by differentiating(4.2) r(r, =2/ 2001 by CRC Press LLC
Chapter 4 Temporal and spatial frequency domain representation 4.1 Interpretation of the temporal transform When a field is represented by a continuous superposition of elemental components, the resulting decomposition can simplify computation and provide physical insight. Such representation is usually accomplished through the use of an integral transform. Although several different transforms are used in electromagnetics, we shall concentrate on the powerful and efficient Fourier transform. Let us consider the Fourier transform of the electromagnetic field. The field depends on x, y,z, t, and we can transform with respect to any or all of these variables. However, a consideration of units leads us to consider a transform over t separately. Let ψ(r, t) represent any rectangular component of the electric or magnetic field. Then the temporal transform will be designated by ψ( ˜ r,ω): ψ(r, t) ↔ ψ( ˜ r, ω). Here ω is the transform variable. The transform field ψ˜ is calculated using (A.1): ψ( ˜ r,ω) = ∞ −∞ ψ(r, t) e− jωt dt. (4.1) The inverse transform is, by (A.2), ψ(r, t) = 1 2π ∞ −∞ ψ( ˜ r,ω) e jωt dω. (4.2) Since ψ˜ is complex it may be written in amplitude–phase form: ψ( ˜ r,ω) = |ψ( ˜ r,ω)|e jξ ψ (r,ω), where we take −π<ξ ψ (r,ω) ≤ π. Since ψ(r, t) must be real, (4.1) shows that ψ( ˜ r, −ω) = ψ˜ ∗(r, ω). (4.3) Furthermore, the transform of the derivative of ψ may be found by differentiating (4.2). We have ∂ ∂t ψ(r, t) = 1 2π ∞ −∞ jωψ( ˜ r,ω) e jωt dω,
(r,t)分joy(r,o) By virtue of(4.2), any electromagnetic field component can be decomposed into a contin- uous, weighted superposition of elemental temporal terms eJor. Note that the weighting factor y(r, a), often called the frequency spectrum of y(r, t), is not arbitrary because y(r, t)must obey a scalar wave equation such as(2. 327). For a source-free region of pace we have for do =o Differentiating under the integral sign we have 2T/[(v2-joHua +a ue)v(r, a)Jelo do =0 hence by the Fourier integral theorem he is the wavenumber. Equation(4.5)is called the scalar Helmholtz equation, and represents the wave equation in the temporal frequency domain 4.2 The frequency-domain Maxwell equations If the region of interest contains sources, we can return to Maxwells equations and represent all quantities using the temporal inverse Fourier transform. We have, for ex- (r,t) E(r, o)eor do he r, o) ∑E:(r (4.6) All other field quantities will be written similarly with an appropriate superscript on the phase. Substitution into Ampere's law gives r D(r, o)ejr do+ H(r,)-joD(r,o)一 0 2001 by CRC Press LLC
hence ∂ ∂t ψ(r, t) ↔ jωψ( ˜ r, ω). (4.4) By virtue of (4.2), any electromagnetic field component can be decomposed into a continuous, weighted superposition of elemental temporal terms e jωt . Note that the weighting factor ψ( ˜ r,ω), often called the frequency spectrum of ψ(r, t), is not arbitrary because ψ(r, t) must obey a scalar wave equation such as (2.327). For a source-free region of space we have ∇2 − µσ ∂ ∂t − µ ∂2 ∂t 2 1 2π ∞ −∞ ψ( ˜ r,ω) e jωt dω = 0. Differentiating under the integral sign we have 1 2π ∞ −∞ ∇2 − jωµσ + ω2 µ ψ( ˜ r,ω) e jωt dω = 0, hence by the Fourier integral theorem ∇2 + k2 ψ( ˜ r,ω) = 0 (4.5) where k = ω √µ 1 − j σ ω is the wavenumber . Equation (4.5) is called the scalar Helmholtz equation, and represents the wave equation in the temporal frequency domain. 4.2 The frequency-domain Maxwell equations If the region of interest contains sources, we can return to Maxwell’s equations and represent all quantities using the temporal inverse Fourier transform. We have, for example, E(r, t) = 1 2π ∞ −∞ E˜(r,ω) e jωt dω where E˜(r,ω) = 3 i=1 ˆii E˜i(r,ω) = 3 i=1 ˆii|E˜i(r,ω)|e jξ E i (r,ω). (4.6) All other field quantities will be written similarly with an appropriate superscript on the phase. Substitution into Ampere’s law gives ∇ × 1 2π ∞ −∞ H˜ (r,ω) e jωt dω = ∂ ∂t 1 2π ∞ −∞ D˜ (r,ω) e jωt dω + 1 2π ∞ −∞ J˜(r,ω) e jωt dω, hence 1 2π ∞ −∞ [∇ × H˜ (r,ω) − jωD˜ (r,ω) − J˜(r,ω)]e jωt dω = 0
after we differentiate under the integ ne terms V×H D+J by the Fourier integral theorem. This version of Ampere's law involves only the frequency domain fields. By similar reasoning we have V×E=-joB, P (4.9) 0, (4.10) J Equations(4.7)-(4.10) govern the temporal spectra of the electromagnetic fields. We may manipulate them to obtain wave equations, and apply the boundary conditions from the following section. After finding the frequency-domain fields we may find the temporal fields by Fourier inversion. The frequency-domain equations involve one fewer derivative (the time derivative has been replaced by multiplication by jo), hence may be easier solve. However. the inverse transform may be difficult to compute 4.3 Boundary conditions on the frequency-domain fields Several boundary conditions on the source and mediating fields were derived in$ 2.8.2 or example, we found that the tangential electric field must obey n12×E1(r,1)-n12×E2(r,t)=-Jms(r,t) The technique of the previous section gives us 12×[E1(r,o)-E2(r,o)=-Jm(r,o) as the condition satisfied by the frequency-domain electric field. The remaining boundary conditie e treated similarly. Let us summarize the results, including the effects of fictitious magnetic sources H2)=J, n12×(E1-E2) n2·OD1-D2)=5 12·(1-J2)=-V4J,-jop, Here f12 points into region 1 from region 2. 2001 by CRC Press LLC
after we differentiate under the integral signs and combine terms. So ∇ × H˜ = jωD˜ + J˜ (4.7) by the Fourier integral theorem. This version of Ampere’s law involves only the frequencydomain fields. By similar reasoning we have ∇ × E˜ = − jωB˜ , (4.8) ∇ · D˜ = ρ,˜ (4.9) ∇ · B˜(r,ω) = 0, (4.10) and ∇ · J˜ + jωρ˜ = 0. Equations (4.7)–(4.10) govern the temporal spectra of the electromagnetic fields. We may manipulate them to obtain wave equations, and apply the boundary conditions from the following section. After finding the frequency-domain fields we may find the temporal fields by Fourier inversion. The frequency-domain equations involve one fewer derivative (the time derivative has been replaced by multiplication by jω), hence may be easier to solve. However, the inverse transform may be difficult to compute. 4.3 Boundary conditions on the frequency-domain fields Several boundary conditions on the source and mediating fields were derived in § 2.8.2. For example, we found that the tangential electric field must obey nˆ 12 × E1(r, t) − nˆ 12 × E2(r, t) = −Jms(r, t). The technique of the previous section gives us nˆ 12 × [E˜ 1(r,ω) − E˜ 2(r,ω)] = −J˜ms(r,ω) as the condition satisfied by the frequency-domain electric field. The remaining boundary conditions are treated similarly. Let us summarize the results, including the effects of fictitious magnetic sources: nˆ 12 × (H˜ 1 − H˜ 2) = J˜s, nˆ 12 × (E˜ 1 − E˜ 2) = −J˜ms, nˆ 12 · (D˜ 1 − D˜ 2) = ρ˜s, nˆ 12 · (B˜ 1 − B˜ 2) = ρ˜ms, and nˆ 12 · (J˜1 − J˜2) = −∇s · J˜s − jωρ˜s, nˆ 12 · (J˜m1 − J˜m2) = −∇s · J˜ms − jωρ˜ms. Here nˆ 12 points into region 1 from region 2
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 LLC
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 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.
In most instances, the presence of an imaginary part in the constitutive parameters implies that the material is either dissipative(lossy ), transforming some of the electro- magnetic energy in the fields into thermal energy, or active, transforming the chemical or mechanical energy of the material into energy in the fields. We investigate this further We can also write the constitutive equations in amplitude-phase form. Letting 石=同;e周, l周,=同Gle, and using the field notation(4.6), we can write(4.13)-(4. 15)as D=D=∑Ee+, (4.18) B1=1B=∑la1e 方=1=∑GEA Here we remember that the amplitudes and phases may be functions of both r and a For isotropic materials these reduce te D1=|D|e"=|ele+), (4.21) Bi= biles=lall hilejt (4.22) 1=1 4.4.1 The complex permittivity As mentioned above, dissipative effects may be associated with complex entries in the permittivity matrix. Since conduction effects can also lead to dissipation, the permittivit and conductivity matrices are often combined to form a compler permittivity. Writing the current as a sum of impressed and secondary conduction terms(J=Ji+Jc)and substituting(4.13) and(4.15)into Ampere's law, we find V×H=J+aE+jo是.E. Defining the complex permittivity e(r.o-o(r, o (4.24) V×H=+joeE. Using the complex permittivity we can include the effects of conduction current by merely replacing the total current with the impressed current. Since Faraday s law is unaffected any equation(such as the wave equation) derived previously using total current retains its form with the same substitutio By(4.16)and(4.17) the complex permittivity obeys E(r,-a)=e*(r,o) (4.25) 2001 by CRC Press LLC
In most instances, the presence of an imaginary part in the constitutive parameters implies that the material is either dissipative (lossy), transforming some of the electromagnetic energy in the fields into thermal energy, or active, transforming the chemical or mechanical energy of the material into energy in the fields. We investigate this further in § 4.5 and § 4.8.3. We can also write the constitutive equations in amplitude–phase form. Letting ˜i j = |˜i j|e jξ i j, µ˜ i j = |µ˜ i j|e jξµ i j, σ˜i j = |σ˜i j|e jξ σ i j, and using the field notation (4.6), we can write (4.13)–(4.15) as D˜ i = |D˜ i|e jξ D i = 3 j=1 |˜i j||E˜ j|e j[ξ E j +ξ i j] , (4.18) B˜i = |B˜i|e jξ B i = 3 j=1 |µ˜ i j||H˜ j|e j[ξ H j +ξµ i j] , (4.19) J˜ i = |J˜ i|e jξ J i = 3 j=1 |σ˜i j||E˜ j|e j[ξ E j +ξ σ i j] . (4.20) Here we remember that the amplitudes and phases may be functions of both r and ω. For isotropic materials these reduce to D˜ i = |D˜ i|e jξ D i = |˜||E˜i|e j(ξ E i +ξ ) , (4.21) B˜i = |B˜i|e jξ B i = |µ˜ ||H˜i|e j(ξ H i +ξµ) , (4.22) J˜ i = |J˜ i|e jξ J i = |σ˜||E˜i|e j(ξ E i +ξ σ ) . (4.23) 4.4.1 The complex permittivity As mentioned above, dissipative effects may be associated with complex entries in the permittivity matrix. Since conduction effects can also lead to dissipation, the permittivity and conductivity matrices are often combined to form a complex permittivity. Writing the current as a sum of impressed and secondary conduction terms (J˜ = J˜i + J˜ c) and substituting (4.13) and (4.15) into Ampere’s law, we find ∇ × H˜ = J˜i + σ˜¯ · E˜ + jω˜¯ · E˜ . Defining the complex permittivity ˜¯ c (r,ω) = σ˜¯ (r,ω) jω + ˜¯(r, ω), (4.24) we have ∇ × H˜ = J˜i + jω˜¯ c · E˜ . Using the complex permittivity we can include the effects of conduction current by merely replacing the total current with the impressed current. Since Faraday’s law is unaffected, any equation (such as the wave equation) derived previously using total current retains its form with the same substitution. By (4.16) and (4.17) the complex permittivity obeys ˜¯ c (r, −ω) = ˜¯ c∗ (r,ω) (4.25)
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)
