ECE 604, Lecture 20

Mon, Feb 25, 2019

Contents
1 Circular Waveguides, Contd. 2
1.1 An Application of Circular Waveguide . . . . . . . . . . . . . . . 2

2 Remarks on Quasi-TEM Modes, Hybrid Modes, and Surface

Plasmonic Modes 3
2.1 Quasi-TEM Modes . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Hybrid Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.3 Guidance of Modes . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3 Homomorphism of Waveguides and Transmission Lines 4

3.1 TE Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.2 TM Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Printed on March 24, 2019 at 16 : 47: W.C. Chew and D. Jiao.

1
ECE 604, Lecture 20 Mon, Feb 25, 2019

1 Circular Waveguides, Contd.

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 with 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 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. The TE01 mode
has electric field given by E = φ̂Eφ . Furthermore, looking at the magnetic field,
the current is mainly circumferential flowing in the φ direction.
Figure 3 shows two ways of engineering a circular waveguide so that the
TE01 mode is enhanced: by using a mode filter that discourages the guidance of
other modes but not the TE01 mode, and second, by designing ridged waveguide
wall to discourage the flow of axial current and hence, the propagation of the
non-TE01 mode.

Figure 1:

2
ECE 604, Lecture 20 Mon, Feb 25, 2019

2 Remarks on Quasi-TEM Modes, Hybrid Modes,

and Surface Plasmonic Modes
We have analyzed some simple structures where closed form solutions are avail-
able. These solutions offer physical insight into how waves are guided, and
how they are cutoff from guidance. 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.

2.1 Quasi-TEM Modes

Figure 2:

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.
However, most two-metal transmission lines do not support a pure TEM mode
but a quasi-TEM mode. When a wave is TEM, it is necessary that the wave
propagates with the phase velocity of the medium. But when a uniform waveg-
uide has inhomogeneity in between, as shown in Figure 2, this is not possible
anymore, and only a quasi-TEM mode can propagate. We can prove this asser-
tion by reductio ad absurdum as before. From eq. (1.16) of Lect. 18, we have
shown that for a TM mode, Ez is given by

Ez ∼ (β 2 − βz2 )Ψe (2.1)

If this mode becomes TEM, then Ez = 0 and this is possible only if βz = β. 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 regions of the microstrip line
or all three dielectric regions of the optical fiber in Figure 2, 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 model of the
transmission line is still a very good model for such a waveguide. If the line

3
ECE 604, Lecture 20 Mon, Feb 25, 2019

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

2.2 Hybrid Modes

For most inhomogeneously filled waveguides, the modes inside are not cleanly
classed into TE and TM modes, but with some modes that are the hybrid of
TE and TM modes. In this case, both TE and TM waves are coupled together
and are present simultaneously. In other words, both Ez 6= 0 and Hz 6= 0.
Sometimes, the hybrid modes are called EH or HE modes, as in an optical fiber.
Nevertheless, 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.

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 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 exchange of 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 stored energy with the kinetic energy stored in the
moving electrons in the plasma instead of magnetic energy stored. Therefore,
the dimension of the waveguide can be very small compared to wavelength, and
yet the surface plasmonic mode can be guided.

3 Homomorphism of Waveguides and Transmis-

sion Lines
Previously, we have demonstrated mathematical homomorphism between plane
waves in layered medium and transmission lines. Such homomorphism can be
further extended to waveguides 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.

4
ECE 604, Lecture 20 Mon, Feb 25, 2019

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

∇ × H = jωεE (3.1)
∂
By letting ∇ = ∇s + ∇z , H = Hs + Hz where ∇z = ẑ ∂z , and Hz = ẑHz , and
the subscript s implies transverse to z components, then

(∇s + ∇z ) × (Hs + Hz ) = ∇s × Hs + ∇z × Hs + ∇s × Hz (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 (3.1), we have

∇z × Hs + ∇s × Hz = jωεEs (3.3)

Next, from Faraday’s law, we have

∇ × E = −jωµH (3.4)

Again, by letting E = Es + Ez , we can show that (3.4) can be written as

∇s × Es + ∇z × Es + ∇s × Ez = −jωµ(Hs + Hz ) (3.5)

Equating z components of the above, we have

∇s × Es = −jωµHz (3.6)

Using (3.6), Eq.(3.3) can be rewritten as

1
∇z × H s + ∇ s × ∇s × Es = +jωεEs (3.7)
−jωµ
But

∇s × ∇s × Es = ∇s (∇s · Es ) − ∇s · ∇s Es (3.8)

and 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 + β 2 )E = 0 (3.9)

or that

(∇2 + β 2 )Es = 0 (3.10)

Assuming that we have a guided mode, then

∂2
Es ∼ e∓jβz z , Es = −βz 2 Es (3.11)
∂z 2

5
ECE 604, Lecture 20 Mon, Feb 25, 2019

Therefore, (3.10) becomes

(∇s 2 + β 2 − βz 2 )Es = 0 (3.12)

or that

(∇s 2 + βs 2 )Es = 0 (3.13)

where βs2 = β 2 − βz is the transverse wave number. Consequently, from (3.8)

∇s × ∇s × Es = −∇2s Es = βs2 Es (3.14)

As such, (3.7) becomes

1
∇z × Hs = jωεEs + ∇s × ∇s × Es
jωµ
1
= jωεEs + βs 2 Es
jωµ
βs 2 βz 2
 
= jωε 1 − 2 = jωε 2 Es (3.15)
β β
Letting βz = β cos θ, then the above can be written as

∇z × Hs = jωε cos2 θEs (3.16)

Now looking at (3.4) again, assuming Ez = 0, equating transverse components,

we have

∇z × Es = −jωµHs (3.17)

More explicitly, we can rewrite (3.16) and (3.17) the above as

∂
ẑ × Hs = jωε cos2 θEs (3.18)
∂z

∂
ẑ × Es = −jωµHs (3.19)
∂z
The above resembles the telegrapher’s equation. We can multiply (3.19) by ẑ×
to get
∂
Es = jωµẑ × Hs (3.20)
∂z
Now (3.18) and (3.20) look even more like the telegrapher’s equation. We can
have Es →, ẑ × 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
r r r
L µ µ 1 ωµ
Z0 = = 2
= = (3.21)
C ε cos θ ε cos θ βz

6
ECE 604, Lecture 20 Mon, Feb 25, 2019

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
r  mπ 2  nπ 2
βz = β 2 − − (3.22)
a b
Therefore, each TEmn mode will be represented by a different characteristic
impedance Z0 , since βz are different for different TEmn modes.

3.2 TM Case
This case can be derived using duality principle. Invoking duality, and after
some algebra, then the equivalence of (3.18) and (3.20) become

∂
Es = jωµ cos2 θẑ × Hs (3.23)
∂z

∂
ẑ × Hs = jωεEs (3.24)
∂z
To keep the dimension commensurate, we can let Es → V , ẑ × Hs → −I,
µ cos2 θ → L, ε → C, then the above resembles the telegrapher’s equations. We
can thus let
r r r
L µ cos2 θ µ βz
Z0 = = = cos θ = (3.25)
C ε ε ωε

Please note that (3.21) and (3.25) are very similar to that for the plane wave
case, which are the wave impedance for the TE and TM modes, respectively.

Figure 3:

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

7
ECE 604, Lecture 20 Mon, Feb 25, 2019

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 4:

However, for a multi-junction waveguide show in Figure 4, tangential E and

H continuous 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 con-
ditions at the junction. Moreover, mode matching method for multiple modes
has to be used at each junction. Typically, a single mode incident 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 line.
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 capac-
itive. The junction effect may be modeled by a simple circuit model as shown
in Figure 5.

Figure 5: