when the constitutive parameters have the form(2.29)-(2. 31). Physically, this term describes both the energy stored in the electromagnetic field and the energy dissipated by the material because of time lags between the application of e and h and the polarization or magnetization of the atoms(and thus the response fields D and B). In principle this term can also be used to describe active media that transfer mechanical or chemical energy of the material into field energy. nstead of attempting to interpret(4.40), we concentrate on the physical me V·S(r,1)=-V·[E(r,t)×H(r,t)] We shall postulate that this term describes the net flow of electromagnetic energy into the point r at time t. Then(4.39)shows that in the absence of impressed sources the energy How must act to(1) increase or decrease the stored energy density at r, (2) dissipate energy in ohmic losses through the term involving J, or 3)dissipate(or provide)energy through the term(40). Assuming linearity we may write S(r, t) (r, t)+=Wm(r, t) (4.41) re the terms on the right-hand side represent the time rates of change of, respectively, ed electric, stored magnetic, and dissipated energies 4.5.1 Dissipation in a dispersive material Although we may, in general, be unable to separate the individual terms (4.41),we can examine these terms under certain conditions. For example, consider a field that builds from zero starting from time t -oo and then decays back to zero at t v·S(dt=ucm(t=∞)-U-m(t=-∞)+u(=0)-u(=-0) where Wen We+Wm is the volume density of stored electromagnetic energy. This stored energy is zero at t= +oo since the fields are zero at those times. Thus, △uQ v·S(n)dt=o(t=∞)-uQ(t=-0) presents the volume density of the net energy dissipated by a lossy med um (or su by an active medium). We may thus classify materials according to the scheme active For an anisotropic material with the constitutive relations E·E,B=乒·H,J INote that in this section we suppress the r-dependence of most quantities for clarity of presentation 2001 by CRC Press LLC
when the constitutive parameters have the form (2.29)–(2.31). Physically, this term describes both the energy stored in the electromagnetic field and the energy dissipated by the material because of time lags between the application of E and H and the polarization or magnetization of the atoms (and thus the response fields D and B). In principle this term can also be used to describe active media that transfer mechanical or chemical energy of the material into field energy. Instead of attempting to interpret (4.40), we concentrate on the physical meaning of −∇ · S(r, t) = −∇ · [E(r, t) × H(r, t)]. We shall postulate that this term describes the net flow of electromagnetic energy into the point r at time t. Then (4.39) shows that in the absence of impressed sources the energy flow must act to (1) increase or decrease the stored energy density at r, (2) dissipate energy in ohmic losses through the term involving Jc, or (3) dissipate (or provide) energy through the term (40). Assuming linearity we may write −∇· S(r, t) = ∂ ∂t we(r, t) + ∂ ∂t wm(r, t) + ∂ ∂t wQ(r, t), (4.41) where the terms on the right-hand side represent the time rates of change of, respectively, stored electric, stored magnetic, and dissipated energies. 4.5.1 Dissipation in a dispersive material Although we may, in general, be unable to separate the individual terms in (4.41), we can examine these terms under certain conditions. For example, consider a field that builds from zero starting from time t = −∞ and then decays back to zero at t = ∞. Then by direct integration1 − ∞ −∞ ∇ · S(t) dt = wem(t = ∞) − wem(t = −∞) + wQ(t = ∞) − wQ(t = −∞) where wem = we +wm is the volume density of stored electromagnetic energy. This stored energy is zero at t = ±∞ since the fields are zero at those times. Thus, wQ = − ∞ −∞ ∇ · S(t) dt = wQ(t = ∞) − wQ(t = −∞) represents the volume density of the net energy dissipated by a lossy medium (or supplied by an active medium). We may thus classify materials according to the scheme wQ = 0, lossless, wQ > 0, lossy, wQ ≥ 0, passive, wQ < 0, active. For an anisotropic material with the constitutive relations D˜ = ˜¯ · E˜ , B˜ = µ˜¯ · H˜ , J˜ c = σ˜¯ · E˜ , 1Note that in this section we suppress the r-dependence of most quantities for clarity of presentation.
we find that dissipation is associated with negative imaginary parts of the constitutive parameters. To see this we write EGD=2/Ecom,D=/、Dra and thus find aD 1 J·E+E E(o)·E(u')·E(a')efa+aoja'dodo where 2 is the complex dyadic permittivity (4.24).Then △u=(2)2 E().2(a).E(o)+H(o). A(o').H(o) i(or+e) dt jodo dal (4.42) Using(A 4)and integrating over o we obtain △Q27[E(-0)(a)·E()+(-0)·k(a).BH(o) jo do.(443) Let us examine(4.43) more closely for the simple case of an isotropic material for 厂[-)-e"o-),Euo+ +ja(o)-A"(o)H(o).H(o)o'do' Using the frequency symmetry property for complex permittivity(4.17)(which also holds for permeability), we find that for isotropic materials e(r,ω)=e(r,-0),ec"(r,o)=-e"(r,-a) (4.44) p'(r,o)=n(r,-o),p"(r,o)=-p"(r,-0) (4.45) Thus, the products of a and the real parts of the constitutive parameters are odd functions, while for the imaginary parts these products are even. Since the dot products of the vector fields are even functions, we find that the integrals of the terms containing the real parts of the constitutive parameters vanish, leaving 46 Here we have used (4.3) in the form E(r,-0)=E*(r,o),H(r,-0)=r'(r,o) (4.47) Equation(4.46)leads us to associate the imaginary parts of the constitutive parameters with dissipation. Moreover, a lossy isotropic material for which Awg >0 must have at least one of E and u"less than zero over some range of positive frequencies, while an 2001 by CRC Press LLC
we find that dissipation is associated with negative imaginary parts of the constitutive parameters. To see this we write E(r, t) = 1 2π ∞ −∞ E˜(r,ω)e jωt dω, D(r, t) = 1 2π ∞ −∞ D˜ (r, ω )e jω t dω , and thus find Jc · E + E · ∂D ∂t = 1 (2π)2 ∞ −∞ ∞ −∞ E˜(ω) · ˜¯ c (ω ) · E˜(ω )e j(ω+ω )t jω dω dω where ˜¯ c is the complex dyadic permittivity (4.24). Then wQ = 1 (2π)2 ∞ −∞ ∞ −∞ E˜(ω) · ˜¯ c (ω ) · E˜(ω ) + H˜ (ω) · µ˜¯ (ω ) · H˜ (ω ) · · ∞ −∞ e j(ω+ω )t dt jω dω dω . (4.42) Using (A.4) and integrating over ω we obtain wQ = 1 2π ∞ −∞ E˜(−ω ) · ˜¯ c (ω ) · E˜(ω ) + H˜ (−ω ) · µ˜¯ (ω ) · H˜ (ω ) jω dω . (4.43) Let us examine (4.43) more closely for the simple case of an isotropic material for which wQ = 1 2π ∞ −∞ j˜ c (ω ) − ˜ c(ω ) E˜(−ω ) · E˜(ω )+ + jµ˜ (ω ) − µ˜ (ω ) H˜ (−ω ) · H˜ (ω ) ω dω . Using the frequency symmetry property for complex permittivity (4.17) (which also holds for permeability), we find that for isotropic materials ˜ c (r,ω) = ˜ c (r, −ω), ˜ c(r,ω) = −˜ c(r, −ω), (4.44) µ˜ (r,ω) = µ˜ (r, −ω), µ˜ (r,ω) = −µ˜ (r, −ω). (4.45) Thus, the products of ω and the real parts of the constitutive parameters are odd functions, while for the imaginary parts these products are even. Since the dot products of the vector fields are even functions, we find that the integrals of the terms containing the real parts of the constitutive parameters vanish, leaving wQ = 2 1 2π ∞ 0 −˜ c|E˜ | 2 − µ˜ |H˜ | 2 ω dω. (4.46) Here we have used (4.3) in the form E˜(r, −ω) = E˜ ∗(r, ω), H˜ (r, −ω) = H˜ ∗(r, ω). (4.47) Equation (4.46) leads us to associate the imaginary parts of the constitutive parameters with dissipation. Moreover, a lossy isotropic material for which wQ > 0 must have at least one of c and µ less than zero over some range of positive frequencies, while an
ctive isotropic medium must have at least one of these greater than zero. In general re speak of a lossy material as having negative imaginary constitutive parameters a lossless medium must have Things are not as simple in the more general anisotropic case. An integration of (4.42) over ay instead of o produces E(o).2(o).E()+H(o).A(-o.H(o)joda Adding half of this expression to half of (4.43)and using(4.25),(4. 17), and(4.47), we obtain △ E-EE·E”+m·.-H.,]jido Finally, using the dyadic identity(. 76), we have E(2-E")E+m(1-l)时jod where the dagger(t) denotes the hermitian(conjugate-transpose)operation. The condi- tion for a lossless anisotropic material is 乒=乒 These relationships imply that in the lossless case the diagonal entries of the constitutive dyadics are purely real Equations(4.50) show that complex entries in a permittivity or permeability matrix do not necessarily imply loss. For example, we will show in 8 4.6.2 that an electron plasma exposed to a z-directed dc magnetic field has a permittivity of the form j80 00∈ where E, Ez, and 8 are real functions of space and frequency. Since E is hermitian it describes a lossless plasma. Similarly, a gyrotropic medium such as a ferrite exposed to a z-directed magnetic field has a permeability dyadic 0 =|JKμ 000 hich also describes a lossless material 2001 by CRC Press LLC
active isotropic medium must have at least one of these greater than zero. In general, we speak of a lossy material as having negative imaginary constitutive parameters: ˜ c < 0, µ˜ < 0, ω> 0. (4.48) A lossless medium must have ˜ = µ˜ = σ˜ = 0 for all ω. Things are not as simple in the more general anisotropic case. An integration of (4.42) over ω instead of ω produces wQ = − 1 2π ∞ −∞ E˜(ω) · ˜¯ c (−ω) · E˜(−ω) + H˜ (ω) · µ˜¯ (−ω) · H˜ (−ω) jω dω. Adding half of this expression to half of (4.43) and using (4.25), (4.17), and (4.47), we obtain wQ = 1 4π ∞ −∞ E˜ ∗ · ˜¯ c · E˜ − E˜ · ˜¯ c∗ · E˜ ∗ + H˜ ∗ · µ˜¯ · H˜ − H˜ · µ˜¯ ∗ · H˜ ∗ jω dω. Finally, using the dyadic identity (A.76), we have wQ = 1 4π ∞ −∞ E˜ ∗ · ˜¯ c − ˜¯ c† · E˜ + H˜ ∗ · µ˜¯ − µ˜¯ † · H˜ jω dω where the dagger (†) denotes the hermitian (conjugate-transpose) operation. The condition for a lossless anisotropic material is ˜¯ c = ˜¯ c† , µ˜¯ = µ˜¯ † , (4.49) or ˜i j = ˜ ∗ ji, µ˜ i j = µ˜ ∗ ji, σ˜i j = σ˜ ∗ ji . (4.50) These relationships imply that in the lossless case the diagonal entries of the constitutive dyadics are purely real. Equations (4.50) show that complex entries in a permittivity or permeability matrix do not necessarily imply loss. For example, we will show in § 4.6.2 that an electron plasma exposed to a z-directed dc magnetic field has a permittivity of the form [˜¯] = − jδ 0 jδ 0 0 0 z where , z, and δ are real functions of space and frequency. Since ˜¯ is hermitian it describes a lossless plasma. Similarly, a gyrotropic medium such as a ferrite exposed to a z-directed magnetic field has a permeability dyadic [µ˜¯ ] = µ − jκ 0 jκ µ 0 0 0 µ0 , which also describes a lossless material.
4.5.2 Energy stored in a dis e material In the previous section we were able to isolate the dissipative effects for a dispersive material under special circumstances. It is not generally possible, however, to isolate a term describing the stored energy. The Kronig-Kramers relations imply that if the constitutive parameters of a material are frequency-dependent, they must have both real and imaginary parts; such a material, if isotropic, must be lossy. So dispersive materials are generally lossy and must have both dissipative and energy-storage characteristics However, many materials have frequency ranges called transparency ranges over which Ec and A"are small compared to 2c/ and A'. If we restrict our interest to these ranges, we may approximate the material as lossless and compute a stored energy. An important special case involves a monochromatic field oscillating at a frequency within this range To study the energy stored by a monochromatic field in a dispersive material we must consider the transient period during which energy accumulates in the fields. The assumption of a purely sinusoidal field variation would not include the effects described by the temporal constitutive relations(2.29)-(2.31), which show that as the field builds the energy must be added with a time lag. Instead we shall assume fields with the temporal variation E(r,t)=f(>ilE; (r)l cos(oot +5(r) where f(t) is an appropriate function describing the build-up of the sinusoidal field. To compute the stored energy of a sinusoidal wave we must parameterize f(t)so that we may drive it to unity as a limiting case of the parameter. A simple choice is r0=c-F()=、/es (4.52) Note that since f()approaches unity as a-0, we have the generalized Fourier trans- form relation limF(a)=2π8(a) 453) Substituting(4.51) into the Fourier transform formula(4.1) we find that E(r, o)= d,|Ei(r)le F(a-abxk \ilEi(r)le-jtE G)F(o+oo) We can simplify this by defining E(r)=∑E:(r) as the phasor vector field to obtain E(r,)2 E(r)F(o-o0)+E(r)F(o+oo) We shall discuss the phasor concept in detail in$4.7. The field E(r, t) is shown in Figure 4.2 as a function of t, while E(r, a) is shown in Figure 4.2 as a function of a. As a becomes small the spectrum of e(r, t) concentrates around o=+oo. We assume the material is transparent for all values a of interest so 2001 by CRC Press LLC
4.5.2 Energy stored in a dispersive material In the previous section we were able to isolate the dissipative effects for a dispersive material under special circumstances. It is not generally possible, however, to isolate a term describing the stored energy. The Kronig–Kramers relations imply that if the constitutive parameters of a material are frequency-dependent, they must have both real and imaginary parts; such a material, if isotropic, must be lossy. So dispersive materials are generally lossy and must have both dissipative and energy-storage characteristics. However, many materials have frequency ranges called transparency ranges over which ˜ c and µ˜ are small compared to ˜ c and µ˜ . If we restrict our interest to these ranges, we may approximate the material as lossless and compute a stored energy. An important special case involves a monochromatic field oscillating at a frequency within this range. To study the energy stored by a monochromatic field in a dispersive material we must consider the transient period during which energy accumulates in the fields. The assumption of a purely sinusoidal field variation would not include the effects described by the temporal constitutive relations (2.29)–(2.31), which show that as the field builds the energy must be added with a time lag. Instead we shall assume fields with the temporal variation E(r, t) = f (t) 3 i=1 ˆii|Ei(r)| cos[ω0t + ξ E i (r)] (4.51) where f (t) is an appropriate function describing the build-up of the sinusoidal field. To compute the stored energy of a sinusoidal wave we must parameterize f (t) so that we may drive it to unity as a limiting case of the parameter. A simple choice is f (t) = e−α2t 2 ↔ F˜ (ω) = π α2 e − ω2 4α2 . (4.52) Note that since f (t) approaches unity as α → 0, we have the generalized Fourier transform relation lim α→0 F˜ (ω) = 2πδ(ω). (4.53) Substituting (4.51) into the Fourier transform formula (4.1) we find that E˜(r,ω) = 1 2 3 i=1 ˆii|Ei(r)|e jξ E i (r) F˜ (ω − ω0) + 1 2 3 i=1 ˆii|Ei(r)|e− jξ E i (r) F˜ (ω + ω0). We can simplify this by defining Eˇ(r) = 3 i=1 ˆii|Ei(r)|e jξ E i (r) (4.54) as the phasor vector field to obtain E˜(r,ω) = 1 2 Eˇ(r)F˜ (ω − ω0) + Eˇ ∗(r)F˜ (ω + ω0) . (4.55) We shall discuss the phasor concept in detail in § 4.7. The field E(r, t) is shown in Figure 4.2 as a function of t, while E˜(r, ω) is shown in Figure 4.2 as a function of ω. As α becomes small the spectrum of E(r, t) concentrates around ω = ±ω0. We assume the material is transparent for all values α of interest so
△ Oot 2.0-1.5-1.00.50.00 01.52 Figure 4.2: Temporal (top) and spectral magnitude(bottom) dependences of E used to compute energy stored in a dispersive material that we may treat e as real. Then, since there is no dissipation, we conclude that the effects of field build-up. Hence the interpretation? ored energy at time t, including the term(4.40)represents the time rate of change of ad dw ab a H We shall concentrate on the electric field term and later obtain the magnetic field term by induction. Since for periodic signals it is more convenient to deal with the time-averaged stored energy than with the instantaneous stored energy, we compute the time average of we(r, t) over the period of the sinusoid centered at the time origin. That is, we compute )= e(t)di 4.5 where T= 2/o0. With a-0, this time-average value is accurate for all periods of the sinusoidal wave Because the most expedient approach to the computation of (4.56) is to employ the Fourier spectrum of e, we us E(r d aD(r, t) (oD(r, o)e/do (-jO)D(r, e- jo'rdw' Note that in this section we suppress the r-dependence of most quantities for clarity of presentation 2001 by CRC Press LLC
-40 -20 0 20 40 ω t -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 ω/ω 0 0 Figure 4.2: Temporal (top) and spectral magnitude (bottom) dependences of E used to compute energy stored in a dispersive material. that we may treat as real. Then, since there is no dissipation, we conclude that the term (4.40) represents the time rate of change of stored energy at time t, including the effects of field build-up. Hence the interpretation2 E · ∂D ∂t = ∂we ∂t , H · ∂B ∂t = ∂wm ∂t . We shall concentrate on the electric field term and later obtain the magnetic field term by induction. Since for periodic signals it is more convenient to deal with the time-averaged stored energy than with the instantaneous stored energy, we compute the time average of we(r, t) over the period of the sinusoid centered at the time origin. That is, we compute we = 1 T T/2 −T/2 we(t) dt (4.56) where T = 2π/ω0. With α → 0, this time-average value is accurate for all periods of the sinusoidal wave. Because the most expedient approach to the computation of (4.56) is to employ the Fourier spectrum of E, we use E(r, t) = 1 2π ∞ −∞ E˜(r,ω)e jωt dω = 1 2π ∞ −∞ E˜ ∗(r, ω )e− jω t dω , ∂D(r, t) ∂t = 1 2π ∞ −∞ (jω)D˜ (r,ω)e jωt dω = 1 2π ∞ −∞ (− jω )D˜ ∗(r, ω )e− jω t dω . 2Note that in this section we suppress the r-dependence of most quantities for clarity of presentation.