Electromagnetic Fields and Waves: Notes, Solved Problems and Interactive Lab

The equation and the boundary conditions

For the TM case $H_{z} = 0$ but $E_{z} \neq = 0$ , and $E_{z}$ satisfies the Helmholtz equation

\nabla^{2} E_{z} (x, y, z) + k^{2} E_{z} (x, y, z) = 0. (7.3.7)

Putting $H_{z} = 0$ in Eq. (7.3.3), the transverse fields follow from $E_{z}$ alone:

E_{x} H_{x} = - \frac{j τ}{h ^{2}} \frac{\partial E _{z}}{\partial x}, = \frac{j ω ε}{h ^{2}} \frac{\partial E _{z}}{\partial y}, E_{y} H_{y} = - \frac{j τ}{h ^{2}} \frac{\partial E _{z}}{\partial y}, = - \frac{j ω ε}{h ^{2}} \frac{\partial E _{z}}{\partial x} . (7.3.8)

On a perfect conductor the tangential electric field vanishes. $E_{z}$ is tangential to all four walls, so

E_{z} (x, 0, z) E_{z} (0, y, z) = 0, E_{z} (x, b, z) = 0, = 0, E_{z} (a, y, z) = 0. (7.3.9)

These four conditions are enough. On the walls $y = 0$ and $y = b$ the other tangential component is $E_{x}$ , which by Eq. (7.3.8) is proportional to $\partial E_{z} / \partial x$ . If $E_{z}$ vanishes all along the wall, so does its derivative along the wall, and $E_{x} = 0$ there automatically. In the same way $E_{y} \propto \partial E_{z} / \partial y$ vanishes on $x = 0$ and $x = a$ .

Separation of variables

With $\partial^{2} / \partial z^{2} \to - τ^{2}$ and $h^{2} = k^{2} - τ^{2}$ , equation (7.3.7) becomes a two-dimensional equation in the cross-section,

\frac{\partial ^{2} E _{z}}{\partial x ^{2}} + \frac{\partial ^{2} E _{z}}{\partial y ^{2}} + h^{2} E_{z} = 0. (7.3.10)

The cross-section is a rectangle and the boundary conditions are stated on lines $x =$ const and $y =$ const, so we try a product of a function of $x$ alone and a function of $y$ alone:

E_{z} (x, y, z) = X (x) Y (y) e^{- j τ z} . (7.3.11)

Substituting and dividing by $X Y$ ,

\frac{1}{X} \frac{d ^{2} X}{d x ^{2}} + \frac{1}{Y} \frac{d ^{2} Y}{d y ^{2}} + h^{2} = 0. (7.3.12)

The first term depends on $x$ only, the second on $y$ only, and their sum is the constant $- h^{2}$ for every $x$ and every $y$ . Keep $y$ fixed and vary $x$ : the first term cannot change. Each term must therefore be a constant by itself. Calling the constants $- k_{x}^{2}$ and $- k_{y}^{2}$ ,

\frac{1}{X} \frac{d ^{2} X}{d x ^{2}} = - k_{x}^{2}, \frac{1}{Y} \frac{d ^{2} Y}{d y ^{2}} = - k_{y}^{2}, h^{2} = k_{x}^{2} + k_{y}^{2}, (7.3.13)

two ordinary equations of the harmonic oscillator, with the general solutions

X (x) Y (y) = A_{1} cos (k_{x} x) + A_{2} sin (k_{x} x), = B_{1} cos (k_{y} y) + B_{2} sin (k_{y} y) . (7.3.14)

Applying the walls

The conditions (7.3.9) become conditions on $X$ and $Y$ :

X (0) X (a) Y (0) Y (b) = A_{1} = 0 \Rightarrow X (x) = A_{2} sin (k_{x} x), = A_{2} sin (k_{x} a) = 0 \Rightarrow k_{x m} = \frac{mπ}{a}, = B_{1} = 0 \Rightarrow Y (y) = B_{2} sin (k_{y} y), = B_{2} sin (k_{y} b) = 0 \Rightarrow k_{y n} = \frac{nπ}{b},

with $m, n = 1, 2, 3, \dots$ The constants $A_{2}$ and $B_{2}$ cannot vanish, otherwise there is no field at all, so $sin (k_{x} a) = 0$ and $sin (k_{y} b) = 0$ , and only a discrete set of $k_{x}$ and $k_{y}$ is allowed. Then

h_{mn}^{2} τ_{mn} = k_{x m}^{2} + k_{y n}^{2} = (\frac{mπ}{a})^{2} + (\frac{nπ}{b})^{2}, = k^{2} - (\frac{mπ}{a})^{2} - (\frac{nπ}{b})^{2} . (7.3.15)

$E_{z}$ is a standing wave across the guide (Figure 7.3.2a): $m$ half-periods of a sine fit across the width $a$ and $n$ across the height $b$ , with nodes on the walls. Along $z$ it is a travelling wave. Writing $C_{mn} = A_{2 m} B_{2 n}$ for the product of the two constants, the TM $_{mn}$ mode is

E_{z, mn} (x, y, z) = C_{mn} sin (\frac{mπ}{a} x) sin (\frac{nπ}{b} y) \times e^{- j τ_{mn} z} . (7.3.16)

Neither $m$ nor $n$ can be zero: $m = 0$ or $n = 0$ makes $sin (0) = 0$ , so $E_{z} \equiv 0$ , and with it every transverse component, since they are all derivatives of $E_{z}$ . The lowest TM mode is TM $_{11}$ .

(a) TM: X(x)=\sin(m\pi x/a). — (a) TM: $X (x) = sin (mπ x / a)$ .

(b) TE: X(x)=\cos(m\pi x/a). — (a) TM: $X (x) = sin (mπ x / a)$ .

The complete TM field

The general TM solution is the sum of all the modes, $E_{z} = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} E_{z, mn}$ , and the transverse components follow from Eq. (7.3.8) by differentiating each term:

E_{z} (x, y, z) = m = 1 \sum \infty n = 1 \sum \infty C_{mn} sin (\frac{mπ}{a} x) \times sin (\frac{nπ}{b} y) e^{- j τ_{mn} z}, E_{x} (x, y, z) = - j m = 1 \sum \infty n = 1 \sum \infty \frac{mπ τ _{mn}}{a h _{mn}^{2}} C_{mn} \times cos (\frac{mπ}{a} x) sin (\frac{nπ}{b} y) e^{- j τ_{mn} z}, E_{y} (x, y, z) = - j m = 1 \sum \infty n = 1 \sum \infty \frac{nπ τ _{mn}}{b h _{mn}^{2}} C_{mn} \times sin (\frac{mπ}{a} x) cos (\frac{nπ}{b} y) e^{- j τ_{mn} z}, H_{x} (x, y, z) = j ω ε m = 1 \sum \infty n = 1 \sum \infty \frac{nπ}{b h _{mn}^{2}} C_{mn} \times sin (\frac{mπ}{a} x) cos (\frac{nπ}{b} y) e^{- j τ_{mn} z}, H_{y} (x, y, z) = - j ω ε m = 1 \sum \infty n = 1 \sum \infty \frac{mπ}{a h _{mn}^{2}} C_{mn} \times cos (\frac{mπ}{a} x) sin (\frac{nπ}{b} y) e^{- j τ_{mn} z} . (7.3.17)

Each term is one TM $_{mn}$ mode, which travels with its own $τ_{mn}$ . Two checks are worth making. First, the ratios of the transverse components of one mode are

\frac{E _{x}}{H _{y}} = - \frac{E _{y}}{H _{x}} = \frac{τ _{mn}}{ω ε} = Z_{T M},

the TM wave impedance of Eq. (7.3.4). Second, $E_{x} \propto sin (nπ y / b)$ vanishes on $y = 0$ and $y = b$ , and $E_{y} \propto sin (mπ x / a)$ vanishes on $x = 0$ and $x = a$ : the tangential electric field is zero on all four walls, as it must be.

Cutoff frequencies of the TM modes

$τ_{mn}$ in Eq. (7.3.15) is real, and the mode propagates, only if $k = ω μ ε > h_{mn}$ . Setting $ω μ ε = h_{mn}$ gives the cutoff frequency of the TM $_{mn}$ mode,

f_{co, mn} = \frac{1}{2 μ ε} (\frac{m}{a})^{2} + (\frac{n}{b})^{2}, (7.3.18)

and the lowest TM mode is TM $_{11}$ , with $f_{co, 11} = \frac{v}{2} 1/ a^{2} + 1/ b^{2}$ .

TM$_{mn}$ modes of the rectangular guide

E_{z} = C_{mn} sin \frac{mπ x}{a} sin \frac{nπ y}{b} e^{- j τ_{mn} z}, h_{mn}^{2} = (\frac{mπ}{a})^{2} + (\frac{nπ}{b})^{2}, m, n = 1, 2, 3, \dots (7.3.19)

The transverse fields follow from Eq. (7.3.8), $τ_{mn} = k^{2} - h_{mn}^{2}$ , the cutoff is Eq. (7.3.18) and the wave impedance is $Z_{T M} = τ_{mn} / ω ε$ . The lowest TM mode is TM $_{11}$ .

In plain words

The walls force $E_{z}$ to be zero all round the edge of the cross-section, so it can only be a pattern of half sine waves, $m$ across and $n$ up. Each pattern is one mode, and the finer the pattern, the higher the frequency at which it starts to travel.

Common mistake · Calling TM

_{11}

the dominant mode

TM $_{11}$ is the lowest TM mode, not the lowest mode of the guide. The TE $_{10}$ mode of the next subsection has the cutoff $v /2 a$ , which is lower than $f_{co, 11} = \frac{v}{2} 1/ a^{2} + 1/ b^{2}$ for any $b$ . The dominant mode of the rectangular guide is TE $_{10}$ .

7.3.2 TM modes

The equation and the boundary conditions

Separation of variables

Applying the walls

The complete TM field

Cutoff frequencies of the TM modes