Lecture 18 Hollow Waveguides - Purdue University

Lecture 18

Hollow Waveguides

Hollow waveguides are useful for high-power microwaves. Air has a higher breakdown voltagecompared to most materials, and hence, could be a good medium for propagating high powermicrowave. Also, hollow waveguides are sufficiently shielded from the rest of the world sothat interference from other sources is minimized. Furthermore, for radio astronomy, theycan provide a low-noise system immune to interference. Air generally has less loss thanmaterials, and loss is often the source of thermal noise. A low loss waveguide is also a lownoise 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 closedform solutions. Closed form solutions to Laplace and Helmholtz equations are obtained by theseparation of variables method. The separation of variables method works only for separablecoordinate systems. (There are 11 separable coordinates for Helmholtz equations, but 13 forLaplace equation.) Some examples of separable coordinate systems are cartesian, cylindrical,and spherical coordinates. But these three coordinates are about all we need to know forsolving many engineering problems. More complicated cases are now handled with numericalmethods using computers.

When a waveguide has a center conductor or two conductors like a coaxial cable, it cansupport a TEM wave where both the electric field and the magnetic field are orthogonal tothe direction of propagation. The uniform plane wave is an example of a TEM wave, forinstance.

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 ofthe 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 theenvironment, 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 interms 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. Themagnetic 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 waveguideas shown in Figure 18.1 (i.e. without a center conductor) cannot support a TEM mode asfollows. If we assume that TEM mode does exist, then the magnetic field has to end on itselfdue to the absence of magnetic charges. It is clear that

¸C

Hs · dl 6= 0 about any closedcontour following the magnetic field lines. But Ampere’s law states that the above is equalto ˛

C

Hs · dl = jω

ˆS

D · dS +

ˆS

J · dS (18.1.1)

Hence, this equation cannot be satisfied unless there are Ez 6= 0 component, or that Jz 6= 0inside the waveguide. The right-hand side of the above cannot be entirely zero, or this impliesthat Ez 6= 0 unless a center conductor carrying a current J is there. This implies that a TEMmode 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 problemscan be decomposed into two scalar problems like the layerd medium case, by using the pilotpotential method. However, when the hollow waveguide is filled with a center conductor, theTEM 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 propagationof 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 Besselfunctions. 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 canwrite the E field as

E(r) = ∇× zΨh(r) (18.1.2)

Equation (18.1.2) will guarantee that Ez = 0 due to its construction. Here, Ψh(r) is a scalarpotential and z 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’sequations, it can be shown that the electric field E(r) satisfies the following Helmholtz waveequation, 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)∇× zΨh(r) = 0 (18.1.4)

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

Then it follows that

∇× z(∇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.


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

Next, we look at the boundary condition for Ψh(r) which is derivable from the boundarycondition for E. The boundary condition for E is that n×E = 0 on C, the PEC wall of thewaveguide. But from (18.1.2), using the back-of-the-cab (BOTC) formula,

n×E = n× (∇× zΨh) = −n · ∇Ψh = 0 (18.1.7)

In applying the BOTC formula, one has to be mindful that ∇ operates on a function to itsright, and the function Ψh should be placed to the right of the ∇ operator.

In the above n · ∇ = n · ∇s where ∇s = x ∂∂x + y ∂

∂y since n has no z component. The

boundary condition (18.1.7) then becomes

n · ∇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 travelingsolutions of the form exp(∓jβzz) for (18.1.6), or that

Ψh(r) = Ψhs(rs)e∓jβzz (18.1.9)

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

With this assumption, ∂2

∂z2 → −βz2, and (18.1.6) becomes even simpler, namely,

(∇s2 + β2 − βz2)Ψhs(rs) = (∇s2 + β2s )Ψhs(rs) = 0 , ∂nΨhs(rs) = 0, on C (18.1.10)

where β2s = 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.


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

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

H = ∇× zΨe(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 n × E = 0 on thewaveguide wall. To this end, we find the corresponding E field by taking the curl of themagnetic field in (18.1.11), and thus the E field is proportional to

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

∂zΨe + zβ2Ψe (18.1.13)

where we have used the BOTC formula to simplify the above. Taking the z component ofthe above, we get

Ez ∼∂2

∂z2Ψe + β2Ψe (18.1.14)

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

Ψe ∼ e∓jβzz

(18.1.15)

then in (18.1.14), ∂2/∂z2 → −β2z , and

Ez ∼ (β2 − β2z )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 canfurther show from (18.1.13) that the homogeneous Dirichlet boundary condition also impliesthat the other components of tangential E are zero, namely n×E = 0 on the waveguide wallC.

Thus, with some manipulation, the boundary value problem related to equation (18.1.12)reduces to a simpler 2D problem, i.e.,

(∇s2 + β2s )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∓jβzz (18.1.21)

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


18.2 Rectangular Waveguides

Rectangular waveguides are among the simplest waveguides to analyze because closed formsolutions exist in cartesian coordinates. One can imagine traveling waves in the xy directionsbouncing 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 bya hollow waveguide. Only when the wavelength is short enough, or the frequency is highenough that an electromagnetic wave can be guided by a waveguide. When a waveguidemode cannot propagate in a waveguide, that mode is known to be cut-off. The concept ofcut-off for hollow waveguide is quite different from that of a dielectric waveguide we havelearned 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,

∂

∂nΨhs(rs) = 0 on C (18.2.1)

where βs2 = β2 − βz2. A viable solution using separation of variables4 for Ψhs(x, y) is then

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

where βx2 + β2

y = β2s . 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 abovesatisfies 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 thelonger side is usually named a.

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


To further satisfy the boundary condition at x = a, and y = b surfaces, it is necessarythat the boundary condition for eq. (18.1.8) is satisfied or that

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

∂yΨhs(x, y)|y=b ∼ cos(βxx) sin(βyb) = 0, (18.2.4)

The above puts constraints on βx and βy, implying that βxa = mπ, βyb = nπ where m andn are integers. Hence (18.2.2) becomes

Ψhs(x, y) = A cos(mπax)

cos(nπby)

(18.2.5)

where

β2x + β2

y =(mπa

)2

+(nπb

)2

= β2s = β2 − βz2 (18.2.6)

Clearly, (18.2.5) satisfies the requisite homogeneous Neumann boundary condition at theentire 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, β2s 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 correspondingeigenmode is also called the TEmn mode.

The above condition on β2s is also known as the guidance condition for the modes in the

waveguide. Furthermore, from (18.2.6),

βz =√β2 − β2

s =

√β2 −

(mπa

)2

−(nπb

)2

(18.2.8)

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

β2s =

(mπa

)2

+(nπb

)2

> β2 = ω2µε (18.2.9)

and βz becomes pure imaginary and the mode cannot propagate or become evanescent in thez direction.5 For fixed m and n, the frequency at which the above happens is called the cutofffrequency of the TEmn mode of the waveguide. It is given by

ωmn,c =1√µε

√(mπa

)2

+(nπb

)2

(18.2.10)

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


When ω < ωmn,c, the TEmn mode is evanescent and cannot propagate inside the waveguide.A corresponding cutoff wavelength is then

λmn,c =2

[(ma

)2+(nb

)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(∓jβzz) is a function independent of x and

y. Then E(r) = ∇× zΨh(r) = ∇s × zΨh(r) = 0. It turns out the only way for Hz 6= 0 is forH(r) = zH0 which is a static field in the waveguide. This is not a very interesting mode, andthus TE00 propagating mode is assumed not to exist and not useful. So the TEmn modescannot have both m = n = 0. As such, the TE10 mode, when a > b, is the mode with thelowest 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 inorder 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 thephotons go into (or be guided by) the waveguide. The TE10 mode, when a > b, is also themode with the lowest cutoff frequency or longest cutoff wavelength.

It is seen with the above analysis, when the wavelength is short enough, or frequency ishigh enough, many modes can be guided. Each of these modes has a different group andphase 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 lowestcutoff frequency) and the next higher mode is important. This allows one to pick a frequencywindow 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 realpower flow down the waveguide: Only reactive power is conveyed by such a mode.

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Lecture 18 Hollow Waveguides - Purdue University

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