Lecture 19
More on Hollow Waveguides
19.1 Rectangular Waveguides, Contd.
19.1.1 TM Modes (E Modes or Ez 6 = 0 Modes)
The above exercise for TE modes can be repeated for the TM modes. The scalar wave function
(or eigenfunction/eigenmode) for the TM modes is
Ψes(x, y) = A sin
( mπ
a x
)
sin
( nπ
b y
)
(19.1.1)
Here, sine functions are chosen for the standing waves, and the chosen values of βx and βy
ensure that the homogeneous Dirichlet boundary condition is satisfied on the entire waveguide
wall. Neither of the m and n can be zero, lest the field is zero. In this case, both m > 0,
and n > 0 are needed. Thus, the lowest TM mode is the TM11 mode. Notice here that the
eigenvalue is
β2
s = β2
x + β2
y =
( mπ
a
)2
+
( nπ
b
)2
(19.1.2)
Therefore, the corresponding cutoff frequencies and cutoff wavelengths for the TMmn modes
are the same as the TEmn modes. These modes are degenerate in this case. For the lowest
modes, TE11 and TM11 modes have the same cutoff frequency. Figure 19.1 shows the disper-
sion curves for different modes of a rectangular waveguide. Notice that the group velocities of
all the modes are zero at cutoff, and then the group velocities approach that of the waveguide
medium as frequency becomes large. These observations can be explained physically.
179
180 Electromagnetic Field Theory
Figure 19.1: Dispersion curves for a rectangular waveguide. Notice that the lowest TM mode
is the TM11 mode, and k is equivalent to β in this course (courtesy of J.A. Kong [31]).
19.1.2 Bouncing Wave Picture
We have seen that the transverse variation of a mode in a rectangular waveguide can be
expanded in terms of sine and cosine functions which represent standing waves, or that they
are
[exp(−jβxx) ± exp(jβxx)] [exp(−jβy y) ± exp(jβy y)]
When the above is expanded and together with the exp(−jβz z) the mode propagating in the
z direction, we see four waves bouncing around in the xy directions and propagating in the z
direction. The picture of this bouncing wave can be depicted in Figure 19.2.
Figure 19.2: The waves in a rectangular waveguide can be thought of as bouncing waves off
the four walls as they propagate in the z direction.
More on Hollow Waveguides 181
19.1.3 Field Plots
Plots of the fields of different rectangular waveguide modes are shown in Figure 19.3. Higher
frequencies are needed to propagate the higher order modes or the high m and n modes.
Notice that for higher m’s and n’s, the transverse wavelengths are getting shorter, implying
that βx and βy are getting larger because of the higher frequencies involved.
Notice also how the electric field and magnetic field curl around each other. Since ∇×H =
jωεE and ∇ × E = −jωμH, they do not curl around each other “immediately” but with a
π/2 phase delay due to the jω factor. Therefore, the E and H fields do not curl around each
other at one location, but at a displaced location due to the π/2 phase difference. This is
shown in Figure 19.4.
Figure 19.3: Transverse field plots of different modes in a rectangular waveguide (courtesy
of Andy Greenwood. Original plots published in Lee, Lee, and Chuang, IEEE T-MTT, 33.3
(1985): pp. 271-274. [105]).
182 Electromagnetic Field Theory
Figure 19.4: Field plot of a mode propagating in the z direction of a rectangular waveguide.
Notice that the E and H fields do not exactly curl around each other.
19.2 Circular Waveguides
Another waveguide where closed-form solutions can be easily obtained is the circular hollow
waveguide as shown in Figure 19.5.
Figure 19.5: Schematic of a circular waveguide.
19.2.1 TE Case
For a circular waveguide, it is best first to express the Laplacian operator, ∇s2 = ∇s · ∇s, in
cylindrical coordinates. Such formulas are given in [31, 106]. Doing a table lookup, ∇sΨ =
More on Hollow Waveguides 183
ˆρ ∂
∂ρ Ψ + ˆφ 1
ρ
∂
∂φ , ∇s · A = 1
ρ
∂
∂ρ ρAρ + 1
ρ
∂
∂φ Aφ. Then
(∇s2 + βs2) Ψhs =
( 1
ρ
∂
∂ρ ρ ∂
∂ρ + 1
ρ2
∂2
∂φ2 + βs2
)
Ψhs(ρ, φ) = 0 (19.2.1)
The above is the partial differential equation for field in a circular waveguide. Using separation
of variables, we let
Ψhs(ρ, φ) = Bn(βsρ)e±jnφ (19.2.2)
Then ∂2
∂φ2 → −n2, and (19.2.1) becomes an ordinary differential equation which is
( 1
ρ
d
dρ ρ d
dρ − n2
ρ2 + βs2
)
Bn(βsρ) = 0 (19.2.3)
Here, we can let βsρ in (19.2.2) and (19.2.3) be x. Then the above can be rewritten as
( 1
x
d
dx x d
dx − n2
x2 + 1
)
Bn(x) = 0 (19.2.4)
The above is known as the Bessel equation whose solutions are special functions denoted as
Bn(x).
These special functions are Jn(x), Nn(x), H(1)
n (x), and H(2)
n (x) which are called Bessel,
Neumann, Hankel fuction of the first kind, and Hankel function of the second kind, respec-
tively, where n is the order, and x is the argument.1 Since this is a second order ordinary
differential equation, it has only two independent solutions. Therefore, two of the four com-
monly encountered solutions of Bessel equation are independent. Therefore, they can be
expressed then in term of each other. Their relationships are shown below:2
Bessel, Jn(x) = 1
2 [Hn(1)(x) + Hn(2)(x)] (19.2.5)
Neumann, Nn(x) = 1
2j [Hn(1)(x) − Hn(2)(x)] (19.2.6)
Hankel–First kind, Hn(1)(x) = Jn(x) + jNn(x) (19.2.7)
Hankel–Second kind, Hn(2)(x) = Jn(x) − jNn(x) (19.2.8)
It can be shown that
Hn(1)(x) ∼
√ 2
πx ejx−j(n+ 1
2 ) π
2 , x → ∞ (19.2.9)
Hn(2)(x) ∼
√ 2
πx e−jx+j(n+ 1
2 ) π
2 , x → ∞ (19.2.10)
They correspond to traveling wave solutions when x = βsρ → ∞. Since Jn(x) and Nn(x)
are linear superpositions of these traveling wave solutions, they correspond to standing wave
1Some textbooks use Yn(x) for Neumann functions.
2Their relations with each other are similar to those between exp(−jx), sin(x), and cos(x).
184 Electromagnetic Field Theory
solutions. Moreover, Nn(x), Hn(1)(x), and Hn(2)(x) → ∞ when x → 0. Since the field has to
be regular when ρ → 0 at the center of the waveguide shown in Figure 19.5, the only viable
solution for the waveguide is that Bn(βsρ) = AJn(βsρ). Thus for a circular hollow waveguide,
the eigenfunction or mode is of the form
Ψhs(ρ, φ) = AJn(βsρ)e±jnφ (19.2.11)
To ensure that the eigenfunction and the eigenvalue are unique, boundary condition for the
partial differential equation is needed. The homogeneous Neumann boundary condition on
the PEC waveguide wall then translates to
d
dρ Jn(βsρ) = 0, ρ = a (19.2.12)
Defining Jn′(x) = d
dx Jn(x), the above is the same as
Jn′(βsa) = 0 (19.2.13)
Plots of Bessel functions and their derivatives are shown in Figure 19.6. The above are the
zeros of the derivative of Bessel function and they are tabulated in many textbooks. The
m-th zero of Jn′(x) is denoted to be βnm in many books,3 and some of them are also shown
in Figure 19.7; and hence, the guidance condition for a waveguide mode is then
βs = βnm/a (19.2.14)
for the TEnm mode. From the above, β2
s can be obtained which is the eigenvalue of (19.2.1)
and (19.2.3). Using the fact that βz = √β2 − β2
s , then βz will become pure imaginary if β2
is small enough so that β2 < β2
s or β < βs. From this, the corresponding cutoff frequency of
the TEnm mode is
ωnm,c = 1
√με
βnm
a (19.2.15)
When ω < ωnm,c, the corresponding mode cannot propagate in the waveguide as βz becomes
pure imaginary. The corresponding cutoff wavelength is
λnm,c = 2π
βnm
a (19.2.16)
By the same token, when λ > λnm,c, the corresponding mode cannot be guided by the
waveguide. It is not exactly precise to say this, but this gives us the heuristic notion that if
wavelength or “size” of the wave or photon is too big, it cannot fit inside the waveguide.
3Notably, Abramowitz and Stegun, Handbook of Mathematical Functions [107]. An online version is
available at [108].
More on Hollow Waveguides 185
19.2.2 TM Case
The corresponding partial differential equation and boundary value problem for this case is
( 1
ρ
∂
∂ρ ρ ∂
∂ρ + 1
ρ2
∂2
∂φ2 + βs2
)
Ψes(ρ, φ) = 0 (19.2.17)
with the homogeneous Dirichlet boundary condition, Ψes(a, φ) = 0, on the waveguide wall.
The eigenfunction solution is
Ψes(ρ, φ) = AJn(βsρ)e±jnφ (19.2.18)
with the boundary condition that Jn(βsa) = 0. The zeros of Jn(x) are labeled as αnm is
many textbooks, as well as in Figure 19.7; and hence, the guidance condition is that for the
TMnm mode is that
βs = αnm
a (19.2.19)
where the eigenvalue for (19.2.17) is β2
s . With βz = √β2 − β2
s , the corresponding cutoff
frequency is
ωnm,c = 1
√με
αnm
a (19.2.20)
or when ω < ωnm,c, the mode cannot be guided. The cutoff wavelength is
λnm,c = 2π
αnm
a (19.2.21)
with the notion that when λ > λnm,c, the mode cannot be guided.
It turns out that the lowest mode in a circular waveguide is the TE11 mode. It is actually
a close cousin of the TE10 mode of a rectangular waveguide. Figure 19.6 shows the plot of
Bessel function Jn(x) and its derivative J′
n(x). Tables in Figure 19.7 show the roots of J′
n(x)
and Jn(x) which are important for determining the cutoff frequencies of the TE and TM
modes of circular waveguides.
186 Electromagnetic Field Theory
Figure 19.6: Plots of the Bessel function, Jn(x), and its derivatives J′
n(x).
Figure 19.7: Table 2.3.1 shows the zeros of J′
n(x), which are useful for determining the
guidance conditions of the TEmn mode of a circular waveguide. On the other hand, Table
2.3.2 shows the zeros of Jn(x), which are useful for determining the guidance conditions of
the TMmn mode of a circular waveguide.
More on Hollow Waveguides 187
Figure 19.8: Transverse field plots of different modes in a circular waveguide (courtesy of
Andy Greenwood. Original plots published in Lee, Lee, and Chuang [105]).
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