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

Подождите немного. Документ загружается.

1.8 Reciprocity 51

1.8 Reciprocity

The Lorentz reciprocity theorem is one of the most important theorems in

electromagnetic theory [24, 25, 37]. It is suitable for the ﬁelds and waves in

reciprocal media, in which the permittivity ² and permeability µ are symmet-

rical tensors of rank two for anisotropic media, and are scalars for isotropic

media.

Let E

, H

be the ﬁelds generated, in the volume V bounded by a closed

surface S, by a volume distribution of electric current J

and equivalent

magnetic current J

. Let E

, H

be the ﬁelds generated, in the same

volume, by a volume distribution of sources J

and J

. The two sets of

ﬁelds and sources both satisfy Maxwell’s equations

∇ × E

= −jωµ · H

− J

, ∇× H

= jω² · E

+ J

∇ × E

= −jωµ · H

− J

, ∇× H

= jω² · E

+ J

Expanding ∇·(E

×H

−E

×H

) and using the vector identity (B.38),

∇ · (A × B) = B · ∇ × A − A · ∇ × B,

we obtain

∇ · (E

× H

− E

× H

) = H

· ∇ × E

− E

· ∇ × H

· ∇ × E

− E

· ∇ × H

Substituting the curl of the ﬁeld vectors from Maxwell’s equations and noting

that

² · E = E · ², µ · H = H · µ,

because of the symmetry of the tensors ² and µ in reciprocal media, we have

∇·(E

×H

−E

×H

) = E

·J

−E

·J

−H

·J

. (1.286)

Integrating (1.286) over the volume V and using Gauss’s theorem (B.47) to

convert the volume integral of the divergence to a surface integral, give

×H

−E

×H

)·dS =

·J

−E

·J

−H

·J

)dV.

(1.287)

Equations (1.286) and (1.287) are the general form of the Lorentz reciprocity

theorem in derivative form and integral form, respectively. The follows are

some special cases

1. At a point without a source,

∇ · (E

× H

− E

× H

) = 0. (1.288)

In a source-free volume V ,

× H

− E

× H

) · dS = 0. (1.289)

52 1. Basic Electromagnetic Theory

2. In an adiabatic volume V , surrounded by a short-circuit surface, i.e.,

an electric wall or by an open-circuit surface, i.e., a magnetic wall,

× H

− E

× H

) · dS = 0.

If there are sources in the volume V , we have

· J

− H

· J

)dV =

· J

− H

· J

)dV. (1.290)

This means that in an adiabatic volume, the reaction of ﬁeld 1 on source

2 is equal to the reaction of ﬁeld 2 on source 1. The unbounded inﬁnite

space is also an adiabatic volume if all sources are of ﬁnite extent.

Using the reciprocity theorem, it can be proven that the receiving pattern

of any antenna constructed by reciprocal material is identical to its transmit-

ting pattern; and the characteristics of a probe in a reciprocal waveguide or

resonator as a receiving element is identical to that of an exciting element.

Note that, the reciprocity theorem is not suitable for the ﬁelds and waves

in non-reciprocal media, such as magnetized plasma and saturated magne-

tized ferrite. See Section 8.9 and 8.10.

The reciprocity in network theory may be derived from the reciprocity

theorem for ﬁelds, see Section 3.5.2.

Problems

1.1 Show that the equation of continuity may be derived from Maxwell’s

equations.

1.2 (1) Show that the volume charge density in a conducting medium with

conductivity σ and permittivity ² satisﬁes the following equation:

∂%

∂t

= 0.

(2) Show that any existing charge density within a conductor will be

damped exponentially and ﬁnd the relaxation time, i.e., the time re-

quired for it to be reduced to 1/e of its initial value.

(3) Find the relaxation times for copp er and glass. The conductivity

and relative permittivity of copper are 5.8 ×10

S/m and 1, and those

of a typical glass are 10

−12

S/m and 5, respectively.

Problems 53

1.3 For a coaxial cylindrical capacitor of radii a and b (b < a), calculate

the displacement current per unit length across a cylindrical surface of

radius r between the inner and outer conductors (b < r < a). The end

eﬀect can be neglected for the capacitor is long enough. Suppose the

voltage variation is sinusoidal in time and the frequency is low enough

so that the electric ﬁeld distribution is approximately the same as that

for static state. Show that the displacement current is independent of

r and equal to the conductive current for charging the capacitor.

1.4 Starting from Maxwell’s equation (1.26), prove that the total current is

continuous everywhere, i.e., for a closed surface

+ J

) · dS = 0,

where J

denotes the displacement current density and J

, the free

current density, including the conduction current density and the con-

vection current density.

1.5 Suppose we have the following expressions for the electric ﬁelds in a

source-free nonconducting region:

(a) E = ˆxE

cos(ωt − kz),

(b) E = ˆzE

cos(ωt − kz),

sinkzcos ωt,

(d) E = ˆxE

sink

y e

j(ωt−k

(e) E = (ˆx + jˆy)E

cos(ωt − kz),

(f) E = (ˆx + ˆz)E

cos(ωt − k|x − z|/

√

2).

(1) Which expressions satisfy the homogeneous wave equation? Which

ones do not?

(2) For the expressions that satisfy the homogeneous wave equation,

derive the expressions for the magnetic ﬁelds using Maxwell’s equa-

tions, and show the relations among the directions of the electric ﬁeld,

magnetic ﬁeld, and propagation.

1.6 The breakdown ﬁeld strength of air is approximately 3 ×10

V/m. Cal-

culate the maximum power ﬂow density in W/m

of a laser beam prop-

agating through air without breakdown.

1.7 Sunlight brings in average power ﬂow of 1376 W/m

approximately to

the earth. Assuming a plane polarized wave brings the same power ﬂow

density, ﬁnd the peak values of E and H in such a wave.

1.8 Given a cylindrical resistor carrying a current, ﬁnd the value of E and H

on the surface of the resistor, compute the Poynting vector, and show

that the amount of power ﬂowing into the resistor is just enough to

supply the Joule loss that appears as heat in the resistor.

54 1. Basic Electromagnetic Theory

1.9 Show that the instantaneous value of the Poynting vector for sinusoidal

ﬁelds may be found as follows:

S = <

S ±

E × He

j2ωt

Note:

<(A) =

(A + A

∗

), =(A) =

(A − A

∗

1.10 Assume an inﬁnitely long conducting wire along the

z axis. A current

I is suddenly applied to it at the time t = 0. Show that the vector

potential at a point perpendicular to the wire and at a distance r is

A(x, t) =

z(µ

I/2π) cosh

−1

(tc/r), t ≥ r/c,

A(x, t) = 0, t ≤ r/c.

Derive the electric and magnetic ﬁelds for t ≥ r/c , and show that the

Poynting vector is

S =

4π

r(c

− r

)

1.11 In a nonuniform medium the permittivity and permeability are func-

tions of coordinates ²(x) and µ(x). Derive the wave equations for E

and H in a source-free, nonconducting, nonuniform medium.

1.12 Derive the relationship between ϕ and A in the Coulomb gauge and

those in the Lorentz gauge.

1.13 Show that in a source-free region, ∇ · E = 0, the electric ﬁeld and

the magnetic ﬁeld may be expressed by means of the electric vector

potential A

as follows,

E = ∇ × A

H = jωA

−

∇∇ · A

j ωµ

and that A

satisﬁes the following Helmholtz’s equation

∇

+ ω

µ²A

= 0.

1.14 Prove that, in nonreciprocal anisotropic media, the Lorentz reciprocity

theorem is no longer satisﬁed.

Chapter 2

Introduction to Waves

J.C. Maxwell predicted that time-varying electromagnetic ﬁelds would exist

in the form of a wave propagating with the velocity of light. In Section 1.3.2,

the time-domain solution of the wave equation was given and the concept of

a uniform plane wave in unbounded space, the simplest form of waves, was

introduced.

In practice, a large variety of electromagnetic waves may exist. The form

of a wave in a speciﬁc system depends upon the nature of the media and the

boundary conditions. In the remainder of this book, waves in diﬀerent forms

will be discussed in detail.

In this chapter, the propagation, dissipation, polarization, reﬂection, and

refraction of sinusoidal uniform plane waves are presented.

2.1 Sinusoidal Uniform Plane Waves

The most fundamental and simplest electromagnetic waves in unbounded

homogeneous simple media are uniform plane waves. In a plane wave, the

ﬁelds propagate in a speciﬁc direction; the plane perpendicular to the direc-

tion of propagation is an equiphase plane or wave front. In a uniform plane

wave, the equiphase plane is also an equiamplitude plane. This means that

the ﬁeld strength is constant in the plane perpendicular to the direction of

propagation.

Strictly speaking, plane waves can be generated only by inﬁnite uniform

plane sources. In fact, waves at large distances from an arbitrary source have

negligible curvature, and are well represented by plane waves when observed

over a limited area. For example, solar light is a spherical wave, but when

we observe it on the earth, it can be identiﬁed as a plane wave.

56 2. Introduction to Waves

2.1.1 Uniform Plane Waves in Lossless Simple Media

Rewrite the Helmholtz’s equations in a lossless homogeneous simple medium

(1.156), (1.157),

∇

E + k

E = 0, (2.1)

∇

H + k

H = 0. (2.2)

and the corresponding source-free complex Maxwell equations (1.75)–(1.78),

∇ × E = −j ωµH, (2.3)

∇ × H = j ω²E, (2.4)

∇ · E = 0, (2.5)

∇ · H = 0. (2.6)

For Uniform plane waves propagating along z, E and H are functions of

z only and are independent of x and y, which determined the equiphase and

equiamplitude plane,

∂

∂x

= 0,

∂

∂y

= 0,

∂

∂z

6= 0.

Under the above conditions for uniform plane waves, the component equa-

tions of Maxwell’s equations (2.3) and (2.4) reduce to

= −jωµH

, (2.7)

= −jωµH

, (2.8)

0 = −jωµH

, (2.9)

= −jω²E

, (2.10)

= −jω²E

, (2.11)

0 = −jω²E

. (2.12)

We see from (2.9) and (2.12) that the longitudinal components of the

ﬁelds vanish,

= 0, H

= 0, (2.13)

and both the electric ﬁeld and the magnetic ﬁeld have only transverse compo-

nents perpendicular to the direction of propagation z. So the uniform plane

wave must be a transverse wave or so-called TEM wave.

The equations for transverse components become two independent sets,

the equations containing E

, H

, (2.8), (2.10) and the equations containing

, H

, (2.7), (2.11). The two sets of equations have the same form, so we

2.1 Sinusoidal Uniform Plane Waves 57

may deal with whichever set. Firstly, we deal with the equations containing

and H

For E

and H

, from maxwell’s equations (2.8) and (2.10) the one dimen-

sional scalar Helmholtz’s equations are derived,

= −ω

µ²E

, (2.14)

= −ω

µ²H

. (2.15)

Equations (2.8), (2.10), (2.14) and (2.15) are the complex forms of the corre-

sponding instantaneous equations (1.135), (1.137), (1.141) and (1.142) given

in Section 1.3.2.

In Section 1.3.2, with an arbitrary time dependence f(t), the time-domain

solution of the one-dimensional instantaneous wave equation in vacuum or

lossless media, (1.141) and (1.142), was given as follows, refer to (1.149) and

(1.150),

(z, t) = E

t −

+ E

−

t +

, (2.16)

(z, t) =

t −

−

t +

. (2.17)

This ia a linear polarized uniform plane wave with the electric ﬁeld in the

x direction, propagating along +z and −z, where v

= 1/

√

µ² denotes the

phase velocity, and η =

µ/² denotes the wave impedance.

For sinusoidal ﬁelds, the time dependence of the ﬁelds is explained by

f(t) = sin ωt, and the solutions (2.16) and (2.17) become

(z, t) = =E

(z, t) = E

sin ω (t − z/v

) + E

−

sin ω (t + z/v

)

= E

sin(ωt − kz) + E

−

sin(ωt + kz),

(z, t) = =H

(z, t) =

sin ω (t − z/v

) −

−

sin ω (t + z/v

)

sin(ωt − kz) −

−

sin(ωt + kz),

where k = ω/v

= ω

√

µ² (m

−1

), is the change in phase per unit length at

a particular frequency, which denotes the phase coeﬃcient or angular wave

number of uniform plane wave propagates in unbounded medium.

The complex amplitudes E

(z, t) and H

(z, t) are given by

(z, t) = E

j(ωt−kz)

+ E

−

j(ωt+kz)

, (2.18)

(z, t) =

j(ωt−kz)

−

j(ωt+kz)

. (2.19)

58 2. Introduction to Waves

Figure 2.1: Fields of a linear polarized uniform plane wave.

Solutions (2.18) and (2.19) can be explained as follows. Equations (2.14)

and (2.15) are second-order ordinary diﬀerential equations, so the solutions

of which must be the linear combination of two independent functions. The

functions that satisfy the diﬀerential equations (2.14) and (2.15) must have

the feature that their second-order derivative is the same function times a

constant −k

= −ω

µ². The only functions with this feature are exponential

functions and their linear combinations, sinusoidal and hyperbolic sinusoidal

functions. For positive k

, the solutions must be exponential functions with

imaginary arguments or sinusoidal functions. The solutions are obtained

precisely by solving equations (2.14) and (2.15) by means of the method of

inﬁnite series.

The two terms of the solutions (2.18) and (2.19) represent persistent

waves traveling along +z and −z directions, respectively. The x-y plane,

perpendicular to the direction of propagation, is the equiphase as well as the

equiamplitude plane.

The ﬁeld patterns of a linearly polarized uniform plane wave are shown

in Fig. 2.1 and a persistent traveling wave along +z is shown in Fig. 2.2.

Fields (2.18) and (2.19) are solutions of the equations (2.8) and (2.10)

containing E

, H

. This is a linear polarized uniform plane wave with the

electric ﬁeld in the x direction, called a x-polarized wave. Dealing with

equations (2.7) and (2.11), we can have another linear polarized uniform

plane wave containing E

, H

. This is a linear polarized uniform plane wave

with the electric ﬁeld in the y direction, called the y-polarized wave.

Note that, some authors, especially on optics, recognize the direction of

the magnetic ﬁeld instead of the electric ﬁeld as the direction of polarization.

2.1 Sinusoidal Uniform Plane Waves 59

Figure 2.2: Persistent traveling wave propagating along +z.

2.1.2 Uniform Plane Waves with an Arbitrary Direction

of Propagation

In the last section, we deal with the plane wave propagating along a speciﬁc

coordinate axis z. In isotropic media, the nature of a uniform plane wave is

independent of the orientation of coordinates. We would now like to reformu-

late the characteristics of a uniform plane wave with an arbitrary direction

of propagation.

Deﬁne k as the wave vector, the magnitude of which is the phase coeﬃ-

cient k, and the direction of which coincides with the direction of propagation

n perpendicular to the wave front:

k = kn =

, (2.20)

= k

+ k

. (2.21)

For an arbitrary point x, the equiphase plane must be the plane perpen-

dicular to k that has point x lying on it, see Fig. 2.3. The equation for the

equiphase plane is given by

ωt − k · x = ωt − k|x|cos θ = constant. (2.22)

The phase of the wave at point x can then be expressed as

k · x = k

x + k

y + k

z. (2.23)

The complex electric and magnetic ﬁelds of the plane wave propagating

in the +k direction can be written as

(x, t) = E

j(ωt−k·x)

, E

(x) = E

−jk·x

, (2.24)

(x, t) = H

j(ωt−k·x)

, H

(x) = H

−jk·x

, (2.25)

60 2. Introduction to Waves

Figure 2.3: Plane wave with an arbitrary direction of propagation.

where E

and H

are constant vectors in time and space, and

−jk·x

= e

−jk

. (2.26)

For a scalar functions ϕ = ϕ

−jk·x

, where ϕ

is a constant scalar, ∇ϕ

0, the operator ∇ takes the following form:

∇ϕ = ϕ

∇e

−jk·x

Applying (2.26) yields

∇e

−jk·x

∂

∂x

−jk·x

∂

∂y

−jk·x

∂

∂z

−jk·x

= −(

xjk

yjk

zjk

−jk·x

= −jke

−jk·x

So we have

∇ϕ = ϕ

∇e

−jk·x

= −jkϕ

−jk·x

= −jkϕ. (2.27)

For a vector functions A = A

−jk·x

, where A

is a constant vector,

∇ · A

= 0, and ∇ × A

= 0, using (B.37) and (B.40),

∇ · A = ∇ ·

−jk·x

= A

· ∇e

−jk·x

= −jk · A. (2.28)

∇ × A = ∇ ×

−jk·x

= ∇e

−jk·x

× A

= −jk × A. (2.29)

We have just seen that for a uniform plane wave, the complex form of a

sinusoidal function is e

−jk·x

, and the nabla operator becomes

∇ = −jk. (2.30)