Lecture 18 Hollow Waveguides and Rectangular Modes

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Lecture 18

Hollow Waveguides

Hollow waveguides are useful for high-power microwaves. Air has a higher breakdown voltage

compared to most materials, and hence, could be a good medium for propagating high power

microwave. Also, hollow waveguides are sufficiently shielded from the rest of the world so

that interference from other sources is minimized. Furthermore, for radio astronomy, they

can provide a low-noise system immune to interference. Air generally has less loss than

materials, and loss is often the source of thermal noise. A low loss waveguide is also a low

noise waveguide.1

Many waveguide problems can be solved in closed form. An example is the coaxial waveg-

uide previously discussed. But there are many other waveguide problems that have closed

form solutions. Closed form solutions to Laplace and Helmholtz equations are obtained by the

separation of variables method. The separation of variables method works only for separable

coordinate systems. (There are 11 separable coordinates for Helmholtz equations, but 13 for

Laplace equation.) Some examples of separable coordinate systems are cartesian, cylindrical,

and spherical coordinates. But these three coordinates are about all we need to know for

solving many engineering problems. More complicated cases are now handled with numerical

methods using computers.

When a waveguide has a center conductor or two conductors like a coaxial cable, it can

support a TEM wave where both the electric field and the magnetic field are orthogonal to

the direction of propagation. The uniform plane wave is an example of a TEM wave, for

instance.

However, when the waveguide is hollow or is filled completely with a homogeneous medium,

without a center conductor, it cannot support a TEM mode as we shall prove next. Much of

the materials of this lecture can be found in [31, 75, 84].

1There is a fluctuation dissipation theorem [103, 104] that says that when a system loses energy to the

environment, it also receives the same amount of energy from the environment in order to conserve energy.

Hence, a lossy system loses energy to its environment, but it receives energy back from the environment in

terms of thermal noise.

171

172 Electromagnetic Field Theory

18.1 Hollow Waveguides

18.1.1 Absence of TEM Mode in a Hollow Waveguide

Figure 18.1: Absence of TEM mode in a hollow waveguide enclosed by a PEC wall. The

magnetic field lines form a closed loop due to the absence of magnetic charges.

We would like to prove by contradiction (reductio ad absurdum) that a hollow waveguide

as shown in Figure 18.1 (i.e. without a center conductor) cannot support a TEM mode as

follows. If we assume that TEM mode does exist, then the magnetic field has to end on itself

due to the absence of magnetic charges. It is clear that ¸

C Hs · dl 6 = 0 about any closed

contour following the magnetic field lines. But Ampere’s law states that the above is equal

to ˛

Hs · dl = jω

D · dS +

J · dS (18.1.1)

Hence, this equation cannot be satisfied unless there are Ez 6 = 0 component, or that Jz 6 = 0

inside the waveguide. The right-hand side of the above cannot be entirely zero, or this implies

that Ez 6 = 0 unless a center conductor carrying a current J is there. This implies that a TEM

mode in a hollow waveguide without a center conductor cannot exist.

Therefore, in a hollow waveguide filled with homogeneous medium, only TEz or TMz

modes can exist like the case of a layered medium. For a TEz wave (or TE wave), Ez = 0,

Hz 6 = 0 while for a TMz wave (or TM wave), Hz = 0, Ez 6 = 0. These classes of problems

can be decomposed into two scalar problems like the layerd medium case, by using the pilot

potential method. However, when the hollow waveguide is filled with a center conductor, the

TEM mode can exist in addition to TE and TM modes.

We will also study some closed form solutions to hollow waveguides, such as the rectan-

gular waveguides. These closed form solutions offer us physical insight into the propagation

of waves in a hollow waveguide. Another waveguide where closed form solutions can be ob-

tained is the circular hollow waveguide. The solutions need to be sought in terms of Bessel

functions. Another waveguide with closed form solutions is the elliptical waveguide. However,

the solutions are too complicated to be considered.

Hollow Waveguides 173

18.1.2 TE Case (Ez = 0, Hz 6 = 0)

In this case, the field inside the waveguide is TE to z or TEz . To ensure a TE field, we can

write the E field as

E(r) = ∇ × ˆzΨh(r) (18.1.2)

Equation (18.1.2) will guarantee that Ez = 0 due to its construction. Here, Ψh(r) is a scalar

potential and ˆz is called the pilot vector.2

The waveguide is assumed source free and filled with a lossless, homogeneous material.

Eq. (18.1.2) also satisfies the source-free condition since ∇·E = 0. And hence, from Maxwell’s

equations, it can be shown that the electric field E(r) satisfies the following Helmholtz wave

equation, or partial differential equation that

(∇2 + β2)E(r) = 0 (18.1.3)

where β2 = ω2με. Substituting (18.1.2) into (18.1.3), we get

(∇2 + β2)∇ × ˆzΨh(r) = 0 (18.1.4)

In the above, we assume that ∇2∇ × ˆzΨ = ∇ × ˆz(∇2Ψ), or that these operators commute.3

Then it follows that

∇ × ˆz(∇2 + β2)Ψh(r) = 0 (18.1.5)

Thus, if Ψh(r) satisfies the following Helmholtz wave equation of partial differential equa-

tion

(∇2 + β2)Ψh(r) = 0 (18.1.6)

then (18.1.5) is satisfied, and so is (18.1.3). Hence, the E field constructed with (18.1.2)

satisfies Maxwell’s equations, if Ψh(r) satisfies (18.1.6).

2It “pilots” the field so that it is transverse to z.

3This is a mathematical parlance, and a commutator is defined to be [A, B] = AB − BA for two operators

A and B. If these two operators commute, then [A, B] = 0.

174 Electromagnetic Field Theory

Figure 18.2: A hollow metallic waveguide with a center conductor (left), and without a center

conductor (right).

Next, we look at the boundary condition for Ψh(r) which is derivable from the boundary

condition for E. The boundary condition for E is that ˆn × E = 0 on C, the PEC wall of the

waveguide. But from (18.1.2), using the back-of-the-cab (BOTC) formula,

ˆn × E = ˆn × (∇ × ˆzΨh) = −ˆn · ∇Ψh = 0 (18.1.7)

In applying the BOTC formula, one has to be mindful that ∇ operates on a function to its

right, and the function Ψh should be placed to the right of the ∇ operator.

In the above ˆn · ∇ = ˆn · ∇s where ∇s = ˆx ∂

∂x + ˆy ∂

∂y since ˆn has no z component. The

boundary condition (18.1.7) then becomes

ˆn · ∇sΨh = ∂nΨh = 0 on C (18.1.8)

which is also known as the homogeneous Neumann boundary condition.

Furthermore, in a waveguide, just as in a transmission line case, we are looking for traveling

solutions of the form exp(∓jβz z) for (18.1.6), or that

Ψh(r) = Ψhs(rs)e∓jβz z

(18.1.9)

where rs = ˆxx+ ˆyy, or in short, Ψhs(rs) = Ψhs(x, y). Thus, ∂nΨh = 0 implies that ∂nΨhs = 0.

With this assumption, ∂2

∂z2 → −βz 2, and (18.1.6) becomes even simpler, namely,

(∇s2 + β2 − βz 2)Ψhs(rs) = (∇s2 + β2

s )Ψhs(rs) = 0 , ∂nΨhs(rs) = 0, on C (18.1.10)

where β2

s = β2 − β2

z . The above is a boundary value problem for a 2D waveguide problem.

The above 2D wave equation is also called the reduced wave equation.

Hollow Waveguides 175

18.1.3 TM Case (Ez 6 = 0, Hz = 0)

Repeating similar treatment for TM waves, the TM magnetic field is then

H = ∇ × ˆzΨe(r) (18.1.11)

where

(∇2 + β2)Ψe(r) = 0 (18.1.12)

We need to derive the boundary condition for Ψe(r) when we know that ˆn × E = 0 on the

waveguide wall. To this end, we find the corresponding E field by taking the curl of the

magnetic field in (18.1.11), and thus the E field is proportional to

E ∼ ∇ × ∇ × ˆzΨe(r) = ∇∇ · (ˆzΨe) − ∇2 ˆzΨe = ∇ ∂

∂z Ψe + ˆzβ2Ψe (18.1.13)

where we have used the BOTC formula to simplify the above. Taking the z component of

the above, we get

Ez ∼ ∂2

∂z2 Ψe + β2Ψe (18.1.14)

Assuming that we have a propagating mode inside the waveguide so that

Ψe ∼ e∓jβz z

(18.1.15)

then in (18.1.14), ∂2/∂z2 → −β2

z , and

Ez ∼ (β2 − β2

z )Ψe (18.1.16)

Therefore, if

Ψe(r) = 0 on C, (18.1.17)

then,

Ez (r) = 0 on C (18.1.18)

Equation (18.1.16) is also called the homogeneous Dirichlet boundary condition. One can

further show from (18.1.13) that the homogeneous Dirichlet boundary condition also implies

that the other components of tangential E are zero, namely ˆn × E = 0 on the waveguide wall

Thus, with some manipulation, the boundary value problem related to equation (18.1.12)

reduces to a simpler 2D problem, i.e.,

(∇s2 + β2

s )Ψes(rs) = 0 (18.1.19)

with the homogeneous Dirichlet boundary condition that

Ψes(rs) = 0, rs on C (18.1.20)

In the above, we have assumed that

Ψe(r) = Ψes(rs)e∓jβz z

(18.1.21)

To illustrate the above theory, we can solve some simple waveguides problems.

176 Electromagnetic Field Theory

18.2 Rectangular Waveguides

Rectangular waveguides are among the simplest waveguides to analyze because closed form

solutions exist in cartesian coordinates. One can imagine traveling waves in the xy directions

bouncing off the walls of the waveguide causing standing waves to exist inside the waveguide.

As shall be shown, it turns out that not all electromagnetic waves can be guided by

a hollow waveguide. Only when the wavelength is short enough, or the frequency is high

enough that an electromagnetic wave can be guided by a waveguide. When a waveguide

mode cannot propagate in a waveguide, that mode is known to be cut-off. The concept of

cut-off for hollow waveguide is quite different from that of a dielectric waveguide we have

learned previously.

18.2.1 TE Modes (H Mode or Hz 6 = 0 Mode)

For this mode, the scalar potential Ψhs(rs) satisfies

(∇s2 + βs2)Ψhs(rs) = 0, ∂

∂n Ψhs(rs) = 0 on C (18.2.1)

where βs2 = β2 − βz 2. A viable solution using separation of variables4 for Ψhs(x, y) is then

Ψhs(x, y) = A cos(βxx) cos(βy y) (18.2.2)

where βx2 + β2

y = β2

s . One can see that the above is the representation of standing waves

in the xy directions. It is quite clear that Ψhs(x, y) satisfies equation (18.2.1). Furthermore,

cosine functions, rather than sine functions are chosen with the hindsight that the above

satisfies the homogenous Neumann boundary condition at x = 0, and y = 0 surfaces.

Figure 18.3: The schematic of a rectangular waveguide. By convention, the length of the

longer side is usually named a.

4For those who are not familiar with this topic, please consult p. 385 of Kong [31].

Hollow Waveguides 177

To further satisfy the boundary condition at x = a, and y = b surfaces, it is necessary

that the boundary condition for eq. (18.1.8) is satisfied or that

∂xΨhs(x, y)|x=a ∼ sin(βxa) cos(βy y) = 0, (18.2.3)

∂y Ψhs(x, y)|y=b ∼ cos(βxx) sin(βy b) = 0, (18.2.4)

The above puts constraints on βx and βy , implying that βxa = mπ, βy b = nπ where m and

n are integers. Hence (18.2.2) becomes

Ψhs(x, y) = A cos

( mπ

a x

)

cos

( nπ

b y

)

(18.2.5)

where

β2

x + β2

y =

( mπ

( nπ

= β2

s = β2 − βz 2 (18.2.6)

Clearly, (18.2.5) satisfies the requisite homogeneous Neumann boundary condition at the

entire waveguide wall.

At this point, it is prudent to stop and ponder on what we have done. Equation (18.2.1)

is homomorphic to a matrix eigenvalue problem

A · xi = λixi (18.2.7)

where xi is the eigenvector and λi is the eigenvalue. Therefore, β2

s is actually an eigenvalue,

and Ψhs(rs) is an eigenfunction (or an eigenmode), which is analogous to an eigenvector. Here,

the eigenvalue β2

s is indexed by m, n, so is the eigenfunction in (18.2.5). The corresponding

eigenmode is also called the TEmn mode.

The above condition on β2

s is also known as the guidance condition for the modes in the

waveguide. Furthermore, from (18.2.6),

βz = √β2 − β2

s =

√

β2 −

( mπ

−

( nπ

(18.2.8)

And from (18.2.8), when the frequency is low enough, then

β2

s =

( mπ

( nπ

> β2 = ω2με (18.2.9)

and βz becomes pure imaginary and the mode cannot propagate or become evanescent in the

z direction.5 For fixed m and n, the frequency at which the above happens is called the cutoff

frequency of the TEmn mode of the waveguide. It is given by

ωmn,c = 1

√με

√( mπ

( nπ

(18.2.10)

5We have seen this happening in a plasma medium earlier and also in total internal reflection.

178 Electromagnetic Field Theory

When ω < ωmn,c, the TEmn mode is evanescent and cannot propagate inside the waveguide.

A corresponding cutoff wavelength is then

λmn,c = 2

[( m

)2 + ( n

)2]1/2 (18.2.11)

So when λ > λmn,c, the mode cannot propagate inside the waveguide.

When m = n = 0, then Ψh(r) = Ψhs(x, y) exp(∓jβz z) is a function independent of x and

y. Then E(r) = ∇ × ˆzΨh(r) = ∇s × ˆzΨh(r) = 0. It turns out the only way for Hz 6 = 0 is for

H(r) = ˆzH0 which is a static field in the waveguide. This is not a very interesting mode, and

thus TE00 propagating mode is assumed not to exist and not useful. So the TEmn modes

cannot have both m = n = 0. As such, the TE10 mode, when a > b, is the mode with the

lowest cutoff frequency or longest cutoff wavelength.

For the TE10 mode, for the mode to propagate, from (18.2.11), it is needed that

λ < λ10,c = 2a (18.2.12)

The above has the nice physical meaning that the wavelength has to be smaller than 2a in

order for the mode to fit into the waveguide. As a mnemonic, we can think that photons have

“sizes”, corresponding to its wavelength. Only when its wavelength is small enough can the

photons go into (or be guided by) the waveguide. The TE10 mode, when a > b, is also the

mode with the lowest cutoff frequency or longest cutoff wavelength.

It is seen with the above analysis, when the wavelength is short enough, or frequency is

high enough, many modes can be guided. Each of these modes has a different group and

phase velocity. But for most applications, a single guided mode only is desirable. Hence,

the knowledge of the cutoff frequencies of the fundamental mode (the mode with the lowest

cutoff frequency) and the next higher mode is important. This allows one to pick a frequency

window within which only a single mode can propagate in the waveguide.

It is to be noted that when a mode is cutoff, the field is evanescent, and there is no real

power flow down the waveguide: Only reactive power is conveyed by such a mode.

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