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)