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

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4.11 Approximate Solution of Helmholtz’s Equations 231

But the trial ﬁelds do not satisfy the ﬁeld-matching conditions (4.5) on the

inner boundaries S

. Now, the question is: for the trial functions in the

ith and jth subregions E

, H

and E

, H

, what conditions must be

satisﬁed on the boundary S

so that the errors in the approximate eigenvalue

and the approximate eigenfunction become minimum.

The true eigenvalue and the true eigenfunction satisfy (4.258), but the

trial function E

does not satisfy it. We try to make the trial function E

satisfy the following equation as the basis of the approximate solution,

i=1

|∇ × E

i=1

, (4.291)

and to ﬁnd the proper boundary conditions that satisfy the above equation.

Taking the scalar product of E

∗

with the equation for E

(4.288), and

integrating it over the volume V

gives

∗

· E

dV =

∗

· ∇ × ∇ × E

dV. (4.292)

Using the vector identity (B.38) yields

∗

· ∇ × ∇ × E

= |∇ × E

− ∇ · (E

∗

× ∇ × E

and (4.292) becomes

dV =

|∇×E

dV −

∗

×∇× E

) ·n

dS. (4.293)

Taking the sum of (4.293) over all V

yields

i=1

dV =

i=1

|∇ × E

dV −

i=1

∗

× ∇ × E

) · n

dS,

(4.294)

where n

denotes the outward normal unit vector of S

, the boundary of V

By using the formula of the triple scalar product (B.29), we have the second

term of the right-hand side in (4.293):

· (E

∗

× ∇ × E

) = ∇ × E

· (n

× E

∗

)

· (E

∗

× ∇ × E

) = −E

∗

· (n

× ∇ × E

) = jωµE

∗

· (n

× H

Since the trial ﬁelds E

(x) or H

(x) satisfy the zero boundary conditions

on the outer boundaries S

(4.290), the surface integral of the second term

on the right-hand side of (4.293) and (4.294) on S

is zero and the rest is

the surface integral on the boundaries S

∗

× ∇ × E

) · n

dS =

∗

× ∇ × E

) · n

dS, (4.295)

232 4. Time-Varying Boundary Value Problems

where n

denotes the normal unit vector of S

in the direction from V

The boundary S

is shared by both V

and V

and is included in both

and S

. The sum for all S

must include two surfaces over S

, one for V

and the other for V

, and the directions of n

and n

are opposite to each

other. As a result, we have

i=1

∗

× ∇ × E

) · n

i=1

j=1

[(E

∗

× ∇ × E

) − (E

∗

× ∇ × E

)] · n

dS.

Substituting it into (4.294) yields

i=1

|∇×E

i=1

−

i=1

j=1

[(E

∗

×∇×E

) − (E

∗

×∇×E

)] · n

i=1

. (4.296)

If k

and E

satisfy the basis of the approximate solution (4.291), the second

term on the right-hand side of the above expression must be zero, so that

i=1

j=1

[(E

∗

× ∇ × E

) − (E

∗

× ∇ × E

)] · n

dS = 0.

The trial ﬁelds E

, E

satisfy Helmholtz’s and Maxwell’s equations, so we

have

i=1

j=1

[(E

∗

× H

) − (E

∗

× H

)] · n

dS = 0. (4.297)

This condition means that the total power ﬂowing through all the inner

boundaries must be continuous.

The above condition must be satisﬁed if the power ﬂows through each

inner boundary are continuous:

∗

× H

) · n

dS =

∗

× H

) · n

dS. (4.298)

This is known as the power-ﬂow matching condition and can be used as the

matching condition for the optimum approximate solution. By using the

triple scaler product formula (B.29), the condition (4.298) becomes

· (n × E

∗

)dS =

· (n × E

∗

)dS,

4.11 Approximate Solution of Helmholtz’s Equations 233

i.e.,

dS =

dS,

where E

and H

denote the tangential components of the ﬁelds.

In most cases, one of the tangential components of electric and magnetic

trial ﬁelds can be continuous everywhere on the boundary, for example, the

tangential components of electric trial ﬁeld is continuous everywhere on the

boundary,

n × E

= n × E

, i.e., E

= E

. (4.299)

If the electric ﬁeld is uniform on the boundary, then the matching condi-

tion of the magnetic ﬁeld becomes

dS =

dS. (4.300)

This means that the surface integral of the tangential component of the mag-

netic trial ﬁeld on S

or the average value of H

on S

must be continuous.

This is known as the integral matching or the average matching condition.

If we take H

be matched everywhere on S

, then E

must satisfy the

average matching condition.

The other approximate matching condition is the speciﬁc-point matching

condition in which trial ﬁeld at a speciﬁc point on S

, instead of the average

value, is continuous. It gives satisfactory approximate solutions too.

234 4. Time-Varying Boundary Value Problems

Problems

4.1 Give the boundary conditions of the Borgnis’ potentials U

(x)

, V

(x)

, and

(y )

, V

(y )

on perfect conducting boundaries in transverse and longitu-

dinal directions.

4.2 Plot the dispersion curves (ω–β diagram) of the uniform plane-wave and

the TEM wave in a transmission line and the fast wave in a metallic

waveguide. Point out the phase velocity and the group velocity of the

wave corresponding to a speciﬁc point on the curves.

4.3 Derive the expressions for transverse ﬁeld components (4.100) to (4.103)

by using the component form of complex Maxwell’s equations.

4.3 In spherical coordinates, neglect the condition ϕ = −∇ · Π

given in

Section 1.6 and re-deﬁne the Hertzian vectors as follows

= ˆrΠ

, φ

= −

∂Π

∂r

E = k

− ∇φ

, H = jω² ∇ × Π

and

= ˆrΠ

, φ

= −

∂Π

∂r

E = −jω²∇ × Π

, H = k

− ∇φ

Prove that the above Π

and Π

are equivalent to the Borgnis’ poten-

tials U and V . Refer to [106].

4.4 Prove the variational principle (stationary formula) for the trial ﬁeld

that satisﬁes the boundary conditions but does not satisfy Helmholtz’s

equation.

4.5 Prove the variational principle for the two-dimensional boundary-value

problems of Helmholtz’s equation.

Chapter 5

Metallic Waveguides and

Resonant Cavities

There are two sorts of electromagnetic waves, the wave in unbounded media

and the wave conﬁned by material boundaries, which is known as a guided-

wave. In Chapter 2, the uniform plane wave in unbounded simple medium

was given as the simplest example of the electromagnetic waves. In this

and the next two chapters, guided waves conﬁned in metallic boundaries,

dielectric boundaries, and periodic boundaries will be introduced.

During the early years, the open two-wire line was the only guided-wave

system, which is adequate for low frequencies when the wire spacing is very

much less than the wavelength. At higher frequencies, when the wavelength

approaches the cross-sectional dimensions of the open line, the phase dif-

ference between the currents ﬂowing through the two wires are no longer

negligibly small, the open line becomes a radiator and give rise to radiation

loss. A coaxial line or coaxial waveguide is suitable for high-frequency ap-

plications, since it is a completely enclosed system and the radiation loss is

avoided. It has been mentioned in Chapter 4 that the dominant mode of all

transmission lines with two or more insulated conductors is the TEM mode,

which has zero cut-oﬀ frequency and can be analyzed by means of either

circuit theory, given in Chapter 2 or ﬁeld theory, to be given in this chapter.

Guided-wave system widely used at high frequencies, especially in the

microwave band, is a hollow metallic tube and is known as the metallic

waveguide. The metallic waveguide is also a completely enclosed system,

so radiation loss is prevented. Furthermore, the resistive loss of the inner

conductor and the dielectric loss of the insulator supporting the inner con-

ductor for the coaxial line are eliminated. In hollow metallic waveguides with

only one conductor, however, the TEM mode with zero cutoﬀ frequency does

not exist and only TE and TM modes with nonzero cutoﬀ frequency exist.

The cutoﬀ frequency for TE and TM modes are decided by the transverse

236 5. Metallic Waveguides and Resonant Cavities

Figure 5.1: Examples of metallic waveguides.

dimensions of the waveguide, and single-mode transmission can be realized

only for frequencies higher than the cutoﬀ frequency of the dominant mode

and lower than that of the next higher mode. This means that, in practice,

metallic waveguides are suitable for centimeter and millimeter wave bands,

i.e., the microwave band. The two-conductor transmission lines including

coaxial lines are also considered as waveguides in which the lowest mode, i.e.,

TEM mode with zero cutoﬀ frequency can be supported as well as TE and

TM modes.

For high frequencies, the dimensions of ordinary lumped-circuit elements

are comparable to wavelengths, and energy will be lost by radiation. In

addition, the resistance of ordinary wire circuits may become high because

of the skin eﬀect. Hence L-C resonant circuits with lumped elements are

no longer suitable for the microwave band. It is suggested that the circuit

should be completely enclosed by a good conductor to prevent radiation, and

the current paths should be made with as large an area as possible to reduce

the resistive loss. The resulting resonant elements for the microwave band is

known as the resonant cavity, which is simply a hollow metallic box with the

electromagnetic energy conﬁned within the box.

5.1 General Characteristics of Metallic

Waveguides

The metallic waveguide discussed in this chapter includes all inﬁnitely-long

cylindrical systems bounded by one or several insulated good conductors.

Some examples of metallic waveguides are given in Fig. 5.1.

5.1 General Characteristics of Metallic Waveguides 237

5.1.1 Ideal-Waveguide Model

Waveguides constructed with good conductor boundaries and ﬁlled with low-

loss medium can be approximately analyzed as an ideal waveguide in which

the waveguide walls are considered to be perfect conductors or short-circuit

surfaces and the medium inside the waveguide is considered to be uniform

lossless perfect dielectric material.

According to the general principle given in Section 4.6, the electromag-

netic problem of an inﬁnitely long cylindrical system enclosed by short-circuit

boundaries is a two-dimensional eigenvalue problem. The transverse eigen-

value T

of such a problem must be zero or p ositive and the possible modes

in the system are TEM modes and fast-wave modes. In a hollow metallic

waveguide, the only possible modes are fast-wave modes. It will be seen that

the boundary conditions of uniform metallic waveguides can be satisﬁed by

any one of the TE or TM modes, which means that the TE or TM modes

can exist in the metallic waveguide independently.

TE mode or H mode: U = 0, V 6= 0, or E

= 0, H

6= 0.

TM mode or E mode: U 6= 0, V = 0, or E

6= 0, H

= 0.

In the waveguide, some modes are degenerate. The combination of two or

more degenerate modes forms a hybrid mode denoted by HEM mode. Some

hybrid modes have their electric or magnetic ﬁelds laid on a longitudinal

section called Longitudinal-section modes denoted by LSE or LSM modes.

5.1.2 Propagation Characteristics

According to the general description of the guided waves in cylindrical sys-

tems given in Section 4.6, the cutoﬀ angular wave number k

of a speciﬁc

mode is equal to the transverse angular wave number T , which is the eigen-

value of the two-dimensional boundary-value problem and is determined by

the shape and the dimension of the waveguide cross section. The correspond-

ing cutoﬀ angular frequency is ω

and the cutoﬀ frequency is f

, so we have

= T, ω

√

µ²

√

, f

2π

√

µ²

2π

√

. (5.1)

When the frequency is higher than the cutoﬀ frequency of a metallic

waveguide mode, there are standing waves along the transverse coordinates

and traveling waves along the longitudinal coordinate, which is the transmis-

sion state of the mode or guided mode. When the frequency is lower than the

cutoﬀ frequency, there are standing waves along the transverse coordinates

and decaying ﬁelds along the longitudinal coordinate, which is the cutoﬀ state

of the mode or cutoﬀ mode. The transverse wavelength in the waveguide is

equal to the wavelength in the dielectric at the cutoﬀ frequency:

2π

√

µ²

√

, (5.2)

238 5. Metallic Waveguides and Resonant Cavities

and the corresponding wavelength in vacuum is known as the cutoﬀ wave-

length or critical wavelength,

2π

√

. (5.3)

The waveguide mode is in the propagation state when ω > ω

, k > T ,

and the longitudinal propagation constant β = k

and the longitudinal phase

velocity v

are determined by (4.112) and (4.121), respectively,

β = k

= k

1 −

, v

√

1 − T

. (5.4)

The longitudinal wavelength λ

or guided wavelength λ

is given by

= λ

2π

= λ

1 − T

√

1 − T

, (5.5)

where λ is the wavelength of a plane wave in the medium with ² and µ and

is that in vacuum. The group velocity in the metallic waveguide in a

propagation state is given by (4.122):

dω

dβ

√

µ²

1 −

, (5.6)

and we have

µ²

; in vacuum v

= c

. (5.7)

The waveguide mode is in the cutoﬀ state when ω < ω

, k < T , and the

longitudinal propagation constant β becomes imaginary. Let jβ = α, then

we have

α = k

− 1, (5.8)

and the ﬁeld in the waveguide becomes a decaying ﬁeld along z. These modes

are known as cutoﬀ modes or evanescent modes.

We have mentioned in Section 4.10.4 that the solutions of Helmholtz’s

equations that satisfy speciﬁc boundary conditions in a cylindrical coordinate

system is a complete set of inﬁnite number of normal modes. When the

electromagnetic ﬁeld with certain frequency exists in the uniform waveguide,

A ﬁnite number of modes are guided modes and the rests of them are cutoﬀ

modes or evanescent modes. When a waveguide is excited by a source, or

a imperfection or discontinuity is located in the guide, all guided modes

and evanescent modes are excited to satisfy the boundary conditions of the

waveguide and the source or discontinuity. Guided modes propagate along

the guide to the remote distance while evanescent modes damp out in a short

distance and are localized ﬁelds that store reactive energy. These give the

5.1 General Characteristics of Metallic Waveguides 239

0 1 2 3 4

-4 -3 -2 -1 0 1 2 3 4

= c



= k

(a) v

, v

(b) k -

E

k -



= k

E

diagram

Figure 5.2: Dispersion characteristics of a metallic waveguide.

discontinuity its reactive properties, because the characteristic impedance of

cutoﬀ mode is reactive, see Section 5.1.4. If the waveguide is uniform and

inﬁnitely long and the source is located far enough, then only the ﬁelds of

the guided modes exist in the guide.

5.1.3 Dispersion Relations

The relation of the phase velocity v

versus the frequency f is known as

the dispersion characteristics or dispersion curves of the modes in transmis-

sion system. The normalized dispersion curve of a speciﬁc mode in metallic

waveguide is shown in Fig. 5.2(a), where the curve of group velocity v

is also

given.

The other useful expression for the dispersion characteristics is the ω–β

diagram or the k–β diagram. The normalized k–β diagram for a metallic

waveguides is shown in Fig. 5.2(b), in which, the curve of α versus k below

cutoﬀ is also given.

Phase velocity and group velocity are shown in the ω–β curve simulta-

240 5. Metallic Waveguides and Resonant Cavities

neously. According to the deﬁnition of the phase velocity and the group

velocity, the slope of the straight line connected from the origin to a p oint

on the ω–β curve represents the phase velocity v

and the slope of the tan-

gential line at that point represents the group velocity v

. The slope of the

asymptote of the dispersion curve is the velocity of light in unbounded space

c. The corresponding slopes on the k–β diagram represent v

/c and v

/c,

respectively, see Fig. 5.2(c). It can be seen from Fig. 5.2(c) that in metallic

waveguides, the phase velocity is always larger than, and group velocity is

always less than the velocity of light c, whereas the directions of the phase

velocity and the group velocity are always the same.

5.1.4 Wave Impedance

The ﬁeld components of a wave propagating along the +z direction in a

cylindrical waveguide are given by (4.63)–(4.68).

For a uniform plane wave or TEM wave, the ratio of the electric ﬁeld to the

magnetic ﬁeld, which are both transverse, is deﬁned as the wave impedance or

characteristic impedance of the transmission system. Similarly, for non-TEM

waves, the wave impedance or characteristic impedance is deﬁned by the ratio

of the transverse component of the electric ﬁeld to the transverse component

of the magnetic ﬁeld. The sign of the wave impedance is determined by the

right-hand screw rule in the sequence of E–H–k.

In accordance with (4.65) and (4.68), for a TE mode, U = 0, V 6= 0,

yields

ωµ

= η

1 − T

, (5.9)

and for a TM mode, U 6= 0, V = 0, yields

ω²

= η

1 −

. (5.10)

According to the right-hand screw rule, from (4.66) and (4.67), we see

that the ratio of E

to H

becomes a negative wave impedance.

For a guided mode, T

< k

, β is real and the characteristic impedance

is real or resistive; whereas for a cutoﬀ mode, the characteristic impedance

is imaginary or reactive, since T

> k

and β becomes imaginary.

5.1.5 Power Flow

The power ﬂow along a metallic waveguide is the surface integral of the

Poynting vector over the cross section of the waveguide:

P =

(E × H

∗

) · dS =

∗

− E

∗

) dS.