Lecture 20

More on Waveguides and

Transmission Lines

20.1 Circular Waveguides, Contd.

As in the rectangular waveguide case, the guidance of the wave in a circular waveguide can

be viewed as bouncing waves in the radial direction. But these bouncing waves give rise

to standing waves expressible in terms of Bessel functions. The scalar potential (or pilot

potential) for the modes in the waveguide is expressible as

Ψαs(ρ, φ) = AJn(βsρ)e±jnφ (20.1.1)

where α = h for TE waves and α = e for TM waves. The Bessel function or wave is expressible

in terms of Hankel functions as in (19.2.5). Since Hankel functions are traveling waves, Bessel

functions represent standing waves. Therefore, the Bessel waves can be thought of as bouncing

traveling waves as in the rectangular waveguide case. In the azimuthal direction, one can

express e±jnφ as traveling waves in the φ direction, or they can be expressed as cos(nφ) and

sin(nφ) which are standing waves in the φ direction.

20.1.1 An Application of Circular Waveguide

When a real-world waveguide is made, the wall of the metal waveguide is not made of perfect

electric conductor, but with some metal of finite conductivity. Hence, tangential E field is

not zero on the wall, and energy can dissipate into the waveguide wall. It turns out that due

to symmetry, the TE01 mode of a circular waveguide has the lowest loss of all the waveguide

modes including rectangular waveguide modes. Hence, this waveguide mode is of interest to

astronomers who are interested in building low-loss and low-noise systems.1

The TE01 mode has electric field given by E = ˆφEφ. Furthermore, looking at the magnetic

field, the current is mainly circumferential flowing in the φ direction. Moreover, by looking

1Low-loss systems are also low-noise due to energy conservation and the fluctuation dissipation theorem

[103, 104, 109].

189

190 Electromagnetic Field Theory

at a bouncing wave picture of the guided waveguide mode, this mode has a small component

of tangential magnetic field on a waveguide wall: It becomes increasingly smaller as the

frequency increases (see Figure 20.1).

Figure 20.1: Bouncing wave picture of the Bessel wave inside a circular waveguide for the

TE01 mode.

The tangential magnetic field needs to be supported by a surface current on the waveguide

wall. This implies that the surface current on the waveguide wall becomes smaller as the

frequency increases. The wall loss (or copper loss or eddy current loss) of the waveguide,

hence, becomes smaller for higher frequencies. In fact, for high frequencies, the TE01 mode

has the smallest copper loss of the waveguide modes: It becomes the mode of choice (see

Figure 20.2). Waveguides supporting the TE01 modes are used to connect the antennas of

the very large array (VLA) for detecting extra-terrestrial signals in radio astronomy [110] as

shown in Figure 20.3.

More on Waveguides and Transmission Lines 191

Figure 20.2: Losses of different modes in a circular waveguide . It is seen that at high

frequencies, the TE01 mode has the lowest loss (courtesy of [111]).

192 Electromagnetic Field Theory

Figure 20.3: Picture of the Very Large Array (courtesy of [110]).

Figure 20.4 shows two ways of engineering a circular waveguide so that the TE01 mode

is enhanced: (i) by using a mode filter that discourages the guidance of other modes but

not the TE01 mode, and (ii), by designing ridged waveguide wall to discourage the flow of

axial current and hence, the propagation of the non-TE01 mode. More details of circular

waveguides can be found in [111]. Typical loss of a circular waveguide can be as low as 2

dB/km.

As shall be learnt later, an open circular waveguide can be made into an aperture antenna

quite easily, because the fields of the aperture are axially symmetric. Such antenna is called a

horn antenna. Because of this, the radiation pattern of such an antenna is axially symmetric,

which can be used to produce axially symmetric circularly polarized (CP) waves. Ways to

enhance the TE01 mode are also desirable [112] as shown in Figure 20.5.

More on Waveguides and Transmission Lines 193

Figure 20.4: Ways to enhance the TE01 mode in a circular waveguide. Such waveguide is

used in astronomy such as designing the communication between antennas in a very large

array (VLA [110]), or it is used in a circular horn antenna [112].

194 Electromagnetic Field Theory

Figure 20.5: Picture of a circular horn antenna where corrugated wall is used to enhance the

TE01 mode (courtesy of [113]).

20.2 Remarks on Quasi-TEM Modes, Hybrid Modes,

and Surface Plasmonic Modes

We have analyzed some simple structures where closed form solutions are available. These

solutions offer physical insight into how waves are guided, and how they are cutoff from

guidance. As has been shown, for some simple waveguides, the modes can be divided into

TEM, TE, and TM modes. However, most waveguides are not simple. We will remark on

various complexities that arise in real world applications.

20.2.1 Quasi-TEM Modes

Figure 20.6: Some examples of practical coaxial-like waveguides (left), and the optical fiber

(right). The environment of these waveguides is an inhomogeneous medium, and hence, a

pure TEM mode cannot propagate on these waveguides.

More on Waveguides and Transmission Lines 195

Many waveguides cannot support a pure TEM mode even when two conductors are present.

For example, two pieces of metal make a transmission line, and in the case of a circular coax,

a TEM mode can propagate in the waveguide. But most two-metal transmission lines do not

support a pure TEM mode: Instead, they support a quasi-TEM mode. In the optical fiber

case, when the index contrast of the fiber is very small, the mode is quasi-TEM as it has to

degenerate to the TEM case when the contrast is absent.

When a wave is TEM, it is necessary that the wave propagates with the phase velocity

of the medium. But when a uniform waveguide has inhomogeneity in between, as shown

in Figure 20.6, this is not possible anymore. We can prove this assertion by reductio ad

absurdum. From eq. (18.1.16) of the previous lecture, we have shown that for a TM mode,

Ez is given by

Ez = 1

jωεi

(β2

i − β2

z )Ψe (20.2.1)

The above derivation is valid in a piecewise homogeneous region. If this mode becomes TEM,

then Ez = 0 and this is possible only if βz = βi. In other words, the phase velocity of the

waveguide mode is the same as a plane TEM wave in the same medium.

Now assume that a TEM wave exists in both inhomogeneous regions of the microstrip line

or all three dielectric regions of the optical fiber in Figure 20.6. Then the phase velocities

in the z direction, determined by ω/βz of each region will be ω/βi of the respective region

where βi is the wavenumber of the i-th region. Hence, phase matching is not possible, and the

boundary condition cannot be satisfied at the dielectric interfaces. Nevertheless, the lumped

element circuit model of the transmission line is still a very good model for such a waveguide.

If the line capacitance and line inductances of such lines can be estimated, βz can still be

estimated. As shall be shown later, circuit theory is valid when the frequency is low, or the

wavelength is large compared to the size of the structures.

20.2.2 Hybrid Modes–Inhomogeneously-Filled Waveguides

For most inhomogeneously filled waveguides, the modes (eigenmodes or eigenfunctions) inside

are not cleanly classed into TE and TM modes, but with some modes that are the hybrid of

TE and TM modes. If the inhomogeneity is piecewise constant, some of the equations we have

derived before are still valid: In other words, in the homogeneous part (or constant part) of the

waveguide filled with piecewise constant inhomogeneity, the fields can still be decomposed into

TE and TM fields. But these fields are coupled to each other by the presence of inhomogeneity,

i.e., by the boundary conditions requisite at the interface between the piecewise homogeneous

regions. Or both TE and TM waves are coupled together and are present simultaneously, and

both Ez 6 = 0 and Hz 6 = 0. Some examples of inhomogeneously-filled waveguides where hybrid

modes exist are shown in Figure 20.7.

Sometimes, the hybrid modes are called EH or HE modes, as in an optical fiber. Never-

theless, the guidance is via a bouncing wave picture, where the bouncing waves are reflected

off the boundaries of the waveguides. In the case of an optical fiber or a dielectric waveguide,

the reflection is due to total internal reflection. But in the case of metalic waveguides, the

reflection is due to the metal walls.

196 Electromagnetic Field Theory

Figure 20.7: Some examples of inhomogeneously filled waveguides where hybrid modes exist:

(top-left) A general inhomogeneously filled waveguide, (top-right) slab-loaded rectangular

waveguides, and (bottom) an optical fiber with core and cladding.

20.2.3 Guidance of Modes

Propagation of a plane wave in free space is by the exchange of electric stored energy and

magnetic stored energy. So the same thing happens in a waveguide. For example. in the

transmission line, the guidance is by the exchange of electric and magnetic stored energy

via the coupling between the capacitance and the inductance of the line. In this case, the

waveguide size, like the cross-section of a coaxial cable, can be made much smaller than the

wavelength.

In the case of hollow waveguides, the E and H fields are coupled through their space and

time variations. Hence, the exchange of the energy stored is via the space that stores these

energies, like that of a plane wave. These waveguides work only when these plane waves can

“enter” the waveguide. Hence, the size of these waveguides has to be about half a wavelength.

The surface plasmonic waveguide is an exception in that the exchange is between the

electric field energy stored with the kinetic energy stored in the moving electrons in the

plasma instead of magnetic energy stored. This form of energy stored is sometimes referred

to as coming from kinetic inductance. Therefore, the dimension of the waveguide can be very

small compared to wavelength, and yet the surface plasmonic mode can be guided.

20.3 Homomorphism of Waveguides and Transmission

Lines

Previously, we have demonstrated mathematical homomorphism between plane waves in lay-

ered medium and transmission lines. Such homomorphism can be further extended to waveg-

More on Waveguides and Transmission Lines 197

uides and transmission lines. We can show this first for TE modes in a hallow waveguide,

and the case for TM modes can be established by invoking duality principle.2

20.3.1 TE Case

For this case, Ez = 0, and from Maxwell’s equations

∇ × H = jωεE (20.3.1)

By letting ∇ = ∇s + ∇z , H = Hs + Hz where ∇z = ˆz ∂

∂z , and Hz = ˆzHz , and the subscript

s implies transverse to z components, then

(∇s + ∇z ) × (Hs + Hz ) = ∇s × Hs + ∇z × Hs + ∇s × Hz (20.3.2)

where it is understood that ∇z × Hz = 0. Notice that the first term on the right-hand side

of the above is pointing in the z direction. Therefore, by letting E = Es + Ez , and equating

transverse components in (20.3.1), we have3

∇z × Hs + ∇s × Hz = jωεEs (20.3.3)

To simplify the above equation, we shall remove Hz from above. Next, from Faraday’s law,

we have

∇ × E = −jωμH (20.3.4)

Again, by letting E = Es + Ez , we can show that (20.3.4) can be written as

∇s × Es + ∇z × Es + ∇s × Ez = −jωμ(Hs + Hz ) (20.3.5)

Equating z components of the above, we have

∇s × Es = −jωμHz (20.3.6)

Using (20.3.6), Eq.(20.3.3) can be rewritten as

∇z × Hs + ∇s × 1

−jωμ ∇s × Es = +jωεEs (20.3.7)

The above can be further simplified by noting that

∇s × ∇s × Es = ∇s(∇s · Es) − ∇s · ∇sEs (20.3.8)

But since ∇ · E = 0, and Ez = 0 for TE modes, it implies that ∇s · Es = 0. Also, from

Maxwell’s equations, we have previously shown that for a homogeneous source-free medium,

(∇2 + β2)E = 0 (20.3.9)

2I have not seen exposition of such mathematical homomorphism elsewhere except in very simple cases [31].

3And from the above, it is obvious that ∇s × Hs = jωεEz , but this equation will not be used in the

subsequent derivation.

198 Electromagnetic Field Theory

or that

(∇2 + β2)Es = 0 (20.3.10)

Assuming that we have a guided mode, then

Es ∼ e∓jβz z , ∂2

∂z2 Es = −βz 2Es (20.3.11)

Therefore, (20.3.10) becomes

(∇s2 + β2 − βz 2)Es = 0 (20.3.12)

or that

(∇s2 + βs2)Es = 0 (20.3.13)

where β2

s = β2 − β2

z is the transverse wave number. Consequently, from (20.3.8)

∇s × ∇s × Es = −∇2

sEs = β2

s Es (20.3.14)

As such, (20.3.7) becomes

∇z × Hs = jωεEs + 1

jωμ ∇s × ∇s × Es

= jωεEs + 1

jωμ βs2Es

= jωε

(

1 − βs2

β2

)

= jωε βz 2

β2 Es (20.3.15)

Letting βz = β cos θ, then the above can be written as

∇z × Hs = jωε cos2 θEs (20.3.16)

The above now resembles one of the two telegrapher’s equations that we seek. Now looking

at (20.3.4) again, assuming Ez = 0, equating transverse components, we have

∇z × Es = −jωμHs (20.3.17)

More explicitly, we can rewrite (20.3.16) and (20.3.17) in the above as

∂

∂z ˆz × Hs = jωε cos2 θEs (20.3.18)

∂

∂z ˆz × Es = −jωμHs (20.3.19)

More on Waveguides and Transmission Lines 199

The above now resembles the telegrapher’s equations. We can multiply (20.3.19) by ˆz× to

get

∂

∂z Es = jωμˆz × Hs (20.3.20)

Now (20.3.18) and (20.3.20) look even more like the telegrapher’s equations. We can have

Es → V , ˆz × Hs → −I. μ → L, ε cos2 θ → C, and the above resembles the telegrapher’s

equations, or that the waveguide problem is homomorphic to the transmission line problem.

The characteristic impedance of this line is then

Z0 =

√ L

C =

√ μ

ε cos2 θ =

√ μ

ε

1

cos θ = ωμ

βz

(20.3.21)

Therefore, the TE modes of a waveguide can be mapped into a transmission problem. This

can be done, for instance, for the TEmn mode of a rectangular waveguide. Then, in the above

βz =

√

β2 −

( mπ

a

)2

−

( nπ

b

)2

(20.3.22)

Therefore, each TEmn mode will be represented by a different characteristic impedance Z0,

since βz is different for different TEmn modes.

20.3.2 TM Case

This case can be derived using duality principle. Invoking duality, and after some algebra,

then the equivalence of (20.3.18) and (20.3.20) become

∂

∂z Es = jωμ cos2 θˆz × Hs (20.3.23)

∂

∂z ˆz × Hs = jωεEs (20.3.24)

To keep the dimensions commensurate, we can let Es → V , ˆz × Hs → −I, μ cos2 θ → L,

ε → C, then the above resembles the telegrapher’s equations. We can thus let

Z0 =

√ L

C =

√ μ cos2 θ

ε =

√ μ

ε cos θ = βz

ωε (20.3.25)

Please note that (20.3.21) and (20.3.25) are very similar to that for the plane wave case, which

are the wave impedance for the TE and TM modes, respectively.

200 Electromagnetic Field Theory

Figure 20.8: A waveguide filled with layered medium is mathematically homomorphic to a

multi-section transmission line problem. Hence, transmission-line methods can be used to

solve this problem.

The above implies that if we have a waveguide of arbitrary cross section filled with layered

media, the problem can be mapped to a multi-section transmission line problem, and solved

with transmission line methods. When V and I are continuous at a transmission line junction,

Es and Hs will also be continuous. Hence, the transmission line solution would also imply

continuous E and H field solutions.

Figure 20.9: A multi-section waveguide is not exactly homormorphic to a multi-section trans-

mission line problem, circuit elements can be added at the junction to capture the physics at

the waveguide junctions as shown in the next figure.

20.3.3 Mode Conversion

In the waveguide shown in Figure 20.8, there is no mode conversion at the junction interface.

Assuming a rectangular waveguide as an example, what this means is that if we send at TE10

into the waveguide, this same mode will propagate throughout the length of the waveguide.

The reason is that only this mode alone is sufficient to satisfy the boundary condition at the

junction interface. To elaborate further, from our prior knowledge, the transverse fields of

the waveguide, e.g., for the TM mode, can be derived to be

Hs = ∇ × ˆzΨes(rs)e∓jβz z (20.3.26)

Es = ∓βz

ωε ∇sΨes(rs)e∓jβz z (20.3.27)

More on Waveguides and Transmission Lines 201

In the above, β2

s and Ψes(rs) are eigenvalue and eigenfunction, respectively, that depend

only on the geometrical shape of the waveguide, but not the materials filling the waveguide.

These eigenfunctions are the same throughout different sections of the waveguide. Therefore,

boundary conditions can be easily satisfied at the junctions.

However, for a multi-junction waveguide show in Figure 20.9, tangential E and H con-

tinuous condition cannot be satisfied by a single mode in each waveguide alone: V and I

continuous at a transmission line junction will not guarantee the continuity of tangential E

and tangential H fields at the waveguide junction. Multi-modes have to be assumed in each

section in order to match boundary conditions at the junction. Moreover, mode matching

method for multiple modes has to be used at each junction. Typically, a single mode inci-

dent at a junction will give rise to multiple modes reflected and multiple modes transmitted.

The multiple modes give rise to the phenomenon of mode conversion at a junction. Hence,

the waveguide may need to be modeled with multiple transmission lines where each mode is

modeled by a different transmission line with different characteristic impedances.

However, the operating frequency can be chosen so that only one mode is propagating at

each section of the waveguide, and the other modes are cutoff or evanescent. In this case, the

multiple modes at a junction give rise to localized energy storage at a junction. These energies

can be either inductive or capacitive. The junction effect may be modeled by a simple circuit

model as shown in Figure 20.10. These junction elements also account for the physics that

the currents and voltages are not continuous anymore across the junction. Moreover, these

junction lumped circuit elements account for the stored electric and magnetic energies at the

junction.

Figure 20.10: Junction circuit elements are used to account for stored electric and magnetic

energy at the junction. They also account for that the currents and voltages are not continuous

across the junctions anymore.

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