Zhang K., Li D. Electromagnetic Theory for Microwaves and Optoelectronics

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1.4 Poynting’s Theorem 31

In conclusion, for linear, non-dispersive reciprocal media, including recip-

rocal anisotropic media and isotropic media (isotropic media are certainly

reciprocal),

E ·

∂D

∂t

∂

∂t

E · D

, (1.165)

H ·

∂B

∂t

∂

∂t

H · B

. (1.166)

Equations (1.161) and (1.162) then become

−∇ · (E × H) =

∂

∂t

E · D

∂

∂t

H · B

+ σE

+ E · J + H · J

, (1.167)

−

(E×H)·dS =

∂

∂t

E ·D

∂

∂t

H·B

+ σE

+ E ·J +H·J

dV. (1.168)

In the above two equations, at ﬁrst, we are familiar with the third term

in the right-hand side, which represents the volume density of Joule’s power

loss or conducting dissipation, caused by the conduction electric current,

= σE

. (1.169)

This is just the joule’s Law in derivative form, which can be derived from the

circuit theory.

The fourth and the ﬁfth terms

= E · J , p

= H · J

(1.170)

are the volume densities of power dissipation caused by the electric current

other than conduction current and the equivalent magnetic current, respec-

tively, for positive values. Their negative values represent the volume densi-

ties of energy produced by the currents, which means the J and E or J

and H have the components in the opposite direction, i.e., the angle between

vectors J and E or J

and H is larger than π/2. The eﬀect of such a

current represents a power source.

For isotropic, non-dispersive media, D = ²E, B = µH, the factors in

expressions (1.165) and (1.166) become

E · D

²E

, w

H · B

µH

. (1.171)

They represent the instantaneous volume densities of energy stored in the

electric and the magnetic ﬁelds, respectively, which are the same as the ex-

pressions for the density of energy stored in a static electric ﬁeld and in a

stationary magnetic ﬁeld, respectively.

For reciprocal anisotropic media, the instantaneous volume densities of

stored energy remain in the earlier form:

32 1. Basic Electromagnetic Theory

E · D

, w

H · B

. (1.172)

Investigating Poynting’s equation (1.168), we see that its right-hand side

represents the power dissipation and the rate of increase of the energy stored

in the volume. According to the law of conservation of energy, there must be

net energy ﬂowing into the volume through the closed surface S. The left-

hand side of (1.168) must then represent this power ﬂow. We interpret the

vector E ×H as the surface density of the power ﬂow of the electromagnetic

ﬁelds,

S = E × H. (1.173)

Vector S gives the magnitude and direction of the power ﬂow per unit area

at any point in space and is called the Poynting vector. Nevertheless, this

interpretation is a matter of convenience and does not follow directly from

Poynting’s theorem, which gives only the total power ﬂow throughout the

whole closed surface. However, we never get results that disagree with ex-

periments if we assume that the Poynting vector S is the surface density of

the power ﬂow at a point.

Then equations (1.167) and (1.168), i.e., the law of energy conservation

for electromagnetic ﬁelds, may be rewritten as follows:

−∇ · S =

∂w

∂t

∂w

∂t

+ p

, (1.174)

−

S · dS =

∂w

∂t

∂w

∂t

+ p

dV. (1.175)

1.4.2 Frequency-Domain Poynting Theorem

For steady-state sinusoidal time-varying ﬁelds, Poynting’s theorem in com-

plex form, or the so-called frequency-domain Poynting theorem, may be de-

rived from Maxwell’s equations in complex form, (1.111)–(1.114).

In circuit theory, for sinusoidal time-varying state, the time average power

delivered to a circuit element with a voltage v(t) = =(

√

2V e

jωt

) across its

terminals and a current i(t) = =(

√

2Ie

jωt

) into the terminals is

P = v(t)i(t)=

v(t)i(t)dt =

√

2V e

jωt

)=(

√

2Ie

jωt

)dt.

Using the formulas for complex functions

<(A) =

(A + A

∗

), =(A) =

(A − A

∗

the time average power becomes

P =

(V e

jωt

− V

∗

−jωt

)

(Ie

jωt

− I

∗

−jωt

))dt

1.4 Poynting’s Theorem 33

(V I

∗

+ V

∗

I)dt −

(V Ie

j2ωt

+ V

∗

−j2ωt

)dt

(V I

∗

+ V

∗

I) − 0 =

[V I

∗

+ (V I

∗

)

∗

] = <[V I

∗

where V and I are the complex eﬀective value of the voltage and the current,

respectively, and I

∗

is the complex conjugate of I.

If we use complex amplitude value V

√

2V and I

√

2I instead of

eﬀective value V and I, the time average p ower becomes

P = v(t)i(t) = <

∗

, where

P = V I

∗

= P + jQ

denotes the complex p ower,

P is the time average power or active power and

Q denotes the reactive power.

Similar to the deﬁnition of complex power in circuit theory, deﬁne a com-

plex power ﬂow density, i.e., complex Poynting vector:

S =

E × H

∗

= S + j q, (1.176)

where E and H are complex vectors in amplitude value (but not eﬀect value

as usually used in circuit theory), H

∗

is the complex conjugate of H. The

real part of the complex Poynting vector

S denotes the time-average power

ﬂow density or the active component of the complex power ﬂow density, and

the imaginary part

q denotes the reactive component of the complex power

ﬂow density.

Rewrite the complex Maxwell equation (1.111) and write the complex

conjugate of (1.112):

∇ × E = −j ωB − J

, (1.177)

∇ × H

∗

= −j ωD

∗

+ σE

∗

+ J

∗

. (1.178)

Substituting them into the vector identity (B.38), we obtain

−∇·

S = −∇ ·

E × H

∗

· (∇ × E) −

E · (∇ × H

∗

)

= j2ω

∗

·B

−

E·D

∗

+ σ

E·E

∗

E·J

∗

·J

. (1.179)

Integrating the above expression over the volume V and applying the diver-

gence theorem gives

−

S·dS = −

E × H

∗

· dS

j2ω

∗

·B

−

E·D

∗

+ σ

E·E

∗

E·J

∗

·J

dV. (1.180)

34 1. Basic Electromagnetic Theory

This is the frequency-domain or complex Poynting theorem for non-

dispersive isotropic and anisotropic media. In this equation, 2ω is the angular

frequency of the alternative energy. The complex energy densities stored in

the electric and magnetic ﬁelds are identiﬁed as

˙w

E · D

∗

E ·

∗

· E

∗

, ˙w

∗

· B

∗

µ · H

(1.181)

where

² and

µ are complex tensors independent of frequency. For lossless

media, ² and µ become real tensors and (1.181) become average energy den-

sities:

E · ² · E

∗

· µ · H

(1.182)

For isotropic, non-dispersive medium, the permittivity and permeability

become complex scalars:

D = ˙²E, D

∗

= ˙²

∗

, B = ˙µH,

˙² = ²

− j ²

, ˙²

∗

= ²

+ j ²

, ˙µ = µ

− j µ

For non-dispersive media, ²

, ²

, µ

, and µ

are constants with respect to the

frequency ω. Then (1.179) and (1.180) become

−∇ ·

S = −∇ ·

E × H

∗

= j 2 ω

˙µH

−

˙²E

σE

E · J

∗

· J

, (1.183)

−

S · dS = −

E × H

∗

· dS

j2ω

˙µH

−

˙²E

σE

E · J

∗

· J

dV. (1.184)

Equations (1.183) and (1.184) may be separated into real and imaginary

parts:

−∇ ·

S = −∇ · <

E × H

∗

ω²

ωµ

σE

+ <

E · J

∗

· J

, (1.185)

−∇ ·

q = −∇ · =

E × H

∗

= 2 ω

−

+ =

E · J

∗

· J

, (1.186)

1.4 Poynting’s Theorem 35

−

S · dS = −

E × H

∗

· dS

ω²

ωµ

dV+

σE

dV+<

E·J

∗

·J

dV,(1.187)

−

q · dS = −

E × H

∗

· dS

= 2ω

−

dV + =

E·J

∗

·J

dV, (1.188)

where σE

/2 is the Joule power loss density, ω²

/2 is the polarization

damping loss density, and ωµ

/2 is the magnetization damping loss den-

sity. All of them are time-average values.

The time-average energy densities stored in the electric and magnetic

ﬁelds are

. (1.189)

For lossless media, ²

= 0, µ

= 0, and ² = ²

, µ = µ

, then we have

²E

µH

. (1.190)

Equations (1.187) and (1.188) describe the power equilibrium conditions

in sinusoidal time-varying electromagnetic ﬁelds. We investigate the exam-

ple of an existing electromagnetic ﬁeld in a source-free volume ﬁlled with a

lossless medium and bounded by a perfectly conducting wall. We know that

J = 0, J

= 0, σ = 0, ²

= 0 and µ

= 0, and on the boundary, E

= 0,

so that

E × H

∗

= 0. Equations (1.187) and (1.188) become

²E

dV =

µH

dV.

The time-average energy stored in the electric ﬁeld is equal to that stored

in the magnetic ﬁeld within a closed adiabatic volume. This is the natural

oscillating condition of a resonator. So any closed adiabatic volume forms an

electromagnetic cavity resonator.

1.4.3 Poynting’s Theorem for Dispersive Media

Poynting’s theorem as derived in the previous subsection is suitable for non-

dispersive media only, where constitutional parameters are scalars or tensors,

independent of frequency. In this subsection, expressions for the energy den-

sities for dispersive, isotropic media and for dispersive, reciprocal, anisotropic

media will be given.

36 1. Basic Electromagnetic Theory

(1) Energy Densities in Lossless, Dispersive, Isotropic Media

The average energy density stored in the ﬁelds in dispersive media is obtained

as follows [78]. It has been shown in Sects. 1.1.2 that for dispersive media the

constitutive parameters depend upon not only the instantaneous values but

also the derivatives of ﬁelds with respect to time. For time-harmonic ﬁelds,

the constitutive parameters are functions of frequency . In dispersive media,

relations (1.165) and (1.166) and the expressions for the energy densities

(1.171), (1.181), (1.189) and( 1.190) are no longer valid. We must go back to

(1.161) and (1.162), in which

∂w

∂t

= E ·

∂D

∂t

∂w

∂t

= H ·

∂B

∂t

. (1.191)

We see that, in general, the rate of change of the electric energy density is

equal to the scalar product of the electric ﬁeld and the displacement current

density and the rate of change of the magnetic energy density is equal to the

scalar product of the magnetic ﬁeld and the displacement magnetic current

density. The energy densities become

(t) =

E ·

∂D

∂t

dt + C

, w

(t) =

H ·

∂B

∂t

dt + C

, (1.192)

where C

and C

are integration constants whose values depend on how the

ﬁelds are established. This means that the instantaneous energy density in

a dispersive medium are not fully determined by means of the instantaneous

value of the ﬁeld. We assume that the wave is quasi-monochromatic, then for

t → −∞ we have E(−∞) = 0, w

(−∞) = 0, and hence C

= 0. In a similar

way the integration constant for magnetic energy is also shown to be zero.

That is, for a quasi-monochromatic wave that starts with a value of zero in

the remote past and builds up gradually, the integration constants are zero

and w

(t) and w

(t) are fully determined.

First we deal with the electric energy in a lossless, electric-dispersive

medium, and assume that the time dependence of the electric ﬁeld has the

form of a slowly modulated high-frequency function of time:

E(t) = E sin ∆ωt sin ωt =

E[cos(ω − ∆ω)t − cos(ω + ∆ω)t], (1.193)

where E is a constant vector that denotes the peak value of the modulated

wave, and the modulation frequency ∆ω is suﬃciently small compared to the

carrier frequency ω. Thus E(t) has the form of a high-frequency carrier sin ωt

whose modulation envelope sin ∆ωt varies slowly with time. The electric

induction or electric displacement vector is then given by

D(t) = ²(ω)E(t)

E[²

(ω −∆ω)

cos(ω − ∆ω)t − ²

(ω +∆ω)

cos(ω + ∆ω)t], (1.194)

1.4 Poynting’s Theorem 37

and the resulting displacement current density is

∂D(t)

∂t

= −

E[(ω − ∆ω)²

(ω −∆ω)

sin(ω − ∆ω)t

− (ω + ∆ω)²

(ω +∆ω)

sin(ω + ∆ω)t]. (1.195)

In the above expression, we suppose that the permittivity is a constant or

slow-varying function with respect to time, so that the derivative of the per-

mittivity with respect to time can be neglected.

The Taylor series expansion of function f(ω ± ∆ω) at ω is given by

f(ω ± ∆ω) = f(ω) ±

df(ω)

dω

∆ω ∓ ···.

Then we get the approximate expressions

(ω + ∆ω) ²

(ω +∆ω)

≈ ω² +

∂ω²

∂ω

∆ω, (1.196)

(ω − ∆ω) ²

(ω −∆ω)

≈ ω² −

∂ω²

∂ω

∆ω. (1.197)

Substituting them into (1.195), we have

∂D(t)

∂t

= E

ω² sin(∆ωt) cos(ωt) +

∂ω²

∂ω

∆ω cos(∆ωt) sin(ωt)

. (1.198)

According to (1.192), the energy density gained during the time interval

from t

to t is given by

(t) − w

) =

E(t) ·

∂D(t)

∂t

dt.

From (1.193) it is evident that E(0) = 0, and the time required for E(t) to

build up from zero to its maximum value is ∆ωt = π/2 or t = π/2∆ω. The

energy density gained during the time interval from t

= 0 to t = π/2∆ω is

given by

π/ 2∆ ω

E(t) ·

∂D(t)

∂t

dt. (1.199)

Substituting (1.193) and (1.198) into (1.199), we have

= E ·E ω²

π/ 2∆ ω

sin

(∆ωt) sin(ωt) cos(ωt)dt

+ E · E ∆ω

∂ω²

∂ω

π/ 2∆ ω

sin

(ωt) sin(∆ωt) cos(∆ωt)dt. (1.200)

38 1. Basic Electromagnetic Theory

Under the condition that ∆ω ¿ ω, the ﬁrst integral on the right-hand side

is negligibly small,

sin

(∆ωt) sin(ωt) cos(ωt)dt =

sin(2ωt)(1 − cos 2∆ωt) dt

sin 2 (ωt) dt−

[sin 2 (ω+∆ω)t+sin 2 (ω−∆ω)t] dt

≈ 0,

and the second integral may approximately be

π/ 2∆ ω

sin

(ωt) sin(∆ωt) cos(∆ωt) dt =

π/ 2∆ ω

sin(2∆ωt) (1−cos 2ωt) dt

(

π/ 2∆ ω

sin(2∆ωt) dt−

π/ 2∆ ω

[sin 2 (ω+∆ω)t−sin 2 (ω−∆ω)t] dt

)

≈

π/ 2∆ ω

sin(2∆ωt) dt =

4∆ω

Substituting them into (1.200) we have the time-average electric energy den-

sity

∂ω²

∂ω

E · E =

∂ω²

∂ω

. (1.201)

Similarly, the time-average magnetic energy density in lossless, magnetic-

dispersive medium is given by

∂ωµ

∂ω

H · H =

∂ωµ

∂ω

. (1.202)

If we take the complex forms of the ﬁelds E(t) = =[

E sin ∆ωt e

jωt

], H(t) =

H sin ∆ωt e

jωt

] instead of (1.193), then the time-average electric energy

density and magnetic energy density in lossless, dispersive media becomes

∂ω²

∂ω

E ·

∗

∂ω²

∂ω

, (1.203)

∂ωµ

∂ω

H ·

∗

∂ωµ

∂ω

. (1.204)

(2) Energy Densities in Lossless, Dispersive, Reciprocal,

Anisotropic Media

In dispersive, anisotropic media the permittivity or permeability becomes a

frequency-dependent tensor. We investigate the relation between the energy

and ﬁeld with a small perturbation in frequency and try to ﬁnd the pertur-

bation formulation of Poynting’s theorem [38, 78].

We deal with a nonconducting, dispersive, reciprocal, anisotropic medium

with negligibly small p olarization and magnetization loss, which means ²(ω)

1.4 Poynting’s Theorem 39

and µ(ω) are real symmetrical tensors. Suppose there is a small current drive

δJ introduced into an originally source-free ﬁeld E, H. The current drive

corresponds to perturbations in the ﬁeld, δE, δH, with complex frequency

deviation

δω. The perturbation formulation of Poynting’s theorem is given

− ∇ ·

∗

× δH + δE × H

∗

)

∗

·(∇×δH)−δH·(∇×E

∗

)−H

∗

·(∇×δE)+δE·(∇×H

∗

)]. (1.205)

The original ﬁelds E and H satisfy the source-free Maxwell equations

because J = 0:

∇ × E = −j ωµ · H, ∇ × H = j ω² · E, (1.206)

∇ × E

∗

= j ωµ · H

∗

, ∇×H

∗

= −j ω² · E

∗

, (1.207)

where ² and µ are functions of frequency, i.e., ²(ω) and µ(ω).

The above equations are perturbed with δJ at a frequency ω +

δω, where

δω is complex, so that the time dependence of the ﬁeld becomes

exp[j (ω +

δω)t] = exp[−(=δω)t] exp[j (ω + <δω)t], (1.208)

corresponding to a rate of exponential build up of the ﬁeld. The equations

(1.206) perturbed to the ﬁrst order (E → E + δE, H → H + δH) are

∇ × δE = δ(∇×E) = −jδ(ωµ · H) = −jωµ · δH − jδ(ωµ) · H, (1.209)

∇×δH = δ(∇×H) = jδ(ω² ·E) + δJ = jω² ·δE + jδ(ω²) ·E + δJ. (1.210)

Substituting (1.207), (1.209), and (1.210) into (1.205), yields

−∇ ·

∗

× δH + δE ×H

∗

·jω² · δE + E

∗

·jδ(ω²) · E + E

∗

·δJ

−δH·j ωµ · H

∗

·jωµ · δH +H

∗

·jδ(ωµ) · H − δE ·j ω² · E

∗

Using the tensor identities (E.44)

A · a · B = B · a

· A,

and noting that for lossless reciprocal anisotropic media the constitutional

tensors are real symmetrical matrices and the transposed matrices are equal

to the original matrices, we have

−∇ ·

∗

× δH + δE × H

∗

)

= j

∗

· δ(ω²) · E + H

∗

· δ(ωµ) · H] +

∗

· δJ

. (1.211)

40 1. Basic Electromagnetic Theory

The deviations δ(ω²) and δ(ωµ) are

δ(ω²) = δω

∂(ω²)

∂ω

, δ(ωµ) = δω

∂(ωµ)

∂ω

Equation (1.211) becomes

−∇ ·

∗

× δH + δE × H

∗

)

= j (2δω)

∗

∂(ω²)

∂ω

· E +

∗

∂(ωµ)

∂ω

· H

∗

· δJ

. (1.212)

In the above expression

∗

· δJ

is the power per unit volume dissipated or supplied by the current perturba-

tion δJ, and

∗

× δH + δE × H

∗

)

is identiﬁed as the perturbation of the Poynting vector caused by δJ .

The energy is quadratic in the ﬁeld amplitudes, so, for the time-

dependence of ﬁeld, exp[ −=(δω)t ], the growth of the energy must be with

the time dep endence exp[ −2=(δω)t ]. The growth rate of the energy is equiv-

alent to multiplication by =(2δω). Hence, the remaining term in (1.212) is

identiﬁed as the time-average energy density stored in the ﬁeld,

w =

∗

∂(ω²)

∂ω

· E +

∗

∂(ωµ)

∂ω

· H

and we have the average electric and magnetic energy densities in lossless

dispersive reciprocal anisotropic media:

∗

∂(ω²)

∂ω

· E, w

∗

∂(ωµ)

∂ω

· H, (1.213)

where ²(ω) and µ(ω) are real symmetrical tensor functions of ω.

For dispersive isotropic media, tensor functions ²(ω) and µ(ω) reduce to

scalar functions ²(ω) and µ(ω), and (1.213) reduces to (1.203) and (1.204),

∂ω²

∂ω

, w

∂ωµ

∂ω

For non-dispersive anisotropic media, tensors ² and µ are independent of

frequency, so that (1.213) reduces to (1.182),

E · ² · E

∗

· µ · H

Finally, for non-dispersive isotropic media, tensors ² and µ become scalars ²

and µ, then (1.213) and (1.203), (1.204) reduces to (1.190),

²E

µH