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

The complete fields follow from the rule of Section 7.8.2. Two compact formulas make the work short. For TM modes the transverse fields of a guide mode travelling along $\pm z$ are $E_{T} = \mp (j τ / h^{2}) \nabla_{T} E_{z}$ and $H_{T} = (j ω ε / h^{2}) \nabla_{T} E_{z} \times a_{z}$ . Since $\partial / \partial z$ acting on $e^{\mp j τ z}$ gives $\mp j τ$ , the first relation can be written for both directions at once as $E_{T} = (1/ h^{2}) \nabla_{T} (\partial E_{z} / \partial z)$ , and it then holds for any superposition of the two waves, in particular for the standing wave. The second relation contains no $τ$ and holds as it stands.

TMₘₙₚ modes

Take $E_{z}$ as the standing wave $cos (p π z / d)$ times the transverse pattern of the TM $_{mn}$ guide mode, with $h^{2} = h_{mn}^{2}$ :

E_{z} E_{x} E_{y} H_{x} H_{y} = E_{0} sin \frac{mπ x}{a} sin \frac{nπ y}{b} cos \frac{p π z}{d}, = - \frac{1}{h ^{2}} \frac{mπ}{a} \frac{p π}{d} E_{0} \times cos \frac{mπ x}{a} sin \frac{nπ y}{b} sin \frac{p π z}{d}, = - \frac{1}{h ^{2}} \frac{nπ}{b} \frac{p π}{d} E_{0} \times sin \frac{mπ x}{a} cos \frac{nπ y}{b} sin \frac{p π z}{d}, = \frac{j ω ε}{h ^{2}} \frac{nπ}{b} E_{0} \times sin \frac{mπ x}{a} cos \frac{nπ y}{b} cos \frac{p π z}{d}, = - \frac{j ω ε}{h ^{2}} \frac{mπ}{a} E_{0} \times cos \frac{mπ x}{a} sin \frac{nπ y}{b} cos \frac{p π z}{d}, (7.8.10)

with $H_{z} = 0$ , $m, n = 1, 2, \dots$ and $p = 0, 1, 2, \dots$ . To see where $E_{x}$ comes from, differentiate: $\partial E_{z} / \partial z = - (p π / d) E_{0} sin \frac{mπ x}{a} sin \frac{nπ y}{b} sin \frac{p π z}{d}$ , and the $x$ derivative of $sin (mπ x / a)$ is $(mπ / a) cos (mπ x / a)$ . For $H_{x}$ , note that $\nabla_{T} E_{z} \times a_{z} = (\partial E_{z} / \partial y) a_{x} - (\partial E_{z} / \partial x) a_{y}$ . Every component obeys the rule: $E_{x}$ and $E_{y}$ vanish on the end walls, $E_{z}$ and $H_{T}$ have their maxima there.

Unlike the guide, the cavity allows $p = 0$ for TM modes. In the TM $_{mn 0}$ mode $E_{z}$ is uniform along $z$ , $E_{T} = 0$ , and the end walls see only a normal $E$ , which they accept. Its frequency is $f_{mn 0} = f_{co, mn}$ : it is the guide mode at its own cutoff, where it does not travel at all, closed by two walls at any distance.

TEₘₙₚ modes

Here $H_{z}$ , the component normal to the end walls, must vanish there and goes as $sin (p π z / d)$ . With $H_{T} = (1/ h^{2}) \nabla_{T} (\partial H_{z} / \partial z)$ and $E_{T} = - (j ω μ / h^{2}) \nabla_{T} H_{z} \times a_{z}$ ,

H_{z} H_{x} H_{y} E_{x} E_{y} = H_{0} cos \frac{mπ x}{a} cos \frac{nπ y}{b} sin \frac{p π z}{d}, = - \frac{1}{h ^{2}} \frac{mπ}{a} \frac{p π}{d} H_{0} \times sin \frac{mπ x}{a} cos \frac{nπ y}{b} cos \frac{p π z}{d}, = - \frac{1}{h ^{2}} \frac{nπ}{b} \frac{p π}{d} H_{0} \times cos \frac{mπ x}{a} sin \frac{nπ y}{b} cos \frac{p π z}{d}, = \frac{j ω μ}{h ^{2}} \frac{nπ}{b} H_{0} \times cos \frac{mπ x}{a} sin \frac{nπ y}{b} sin \frac{p π z}{d}, = - \frac{j ω μ}{h ^{2}} \frac{mπ}{a} H_{0} \times sin \frac{mπ x}{a} cos \frac{nπ y}{b} sin \frac{p π z}{d}, (7.8.11)

with $E_{z} = 0$ , $m, n = 0, 1, 2, \dots$ (not both zero) and $p = 1, 2, \dots$ . The transverse electric field goes as $sin (p π z / d)$ and vanishes on both end walls, as it must. Here $p = 0$ is not allowed: it would make $H_{z}$ and $E_{T}$ vanish everywhere, and with them every other component. There would be no field at all.

Energy is stored, not carried

In both families $E$ and $H$ differ by the factor $j$ : they are $9 0^{\circ}$ out of phase in time. The time-average Poynting vector $\frac{1}{2} Re {E \times H^{*}}$ is then zero everywhere. Energy flows to and fro within each quarter period, but no power is carried on average. This is what distinguishes the standing wave of a cavity from the travelling wave of a guide.

Common mistake · TE and TM as absolute labels

Thinking that a mode is TE or TM in itself. The labels refer to the chosen reference axis $z$ , and a box has no preferred axis. The mode TE $_{101}$ with respect to $z$ has $E = E_{y} a_{y}$ and $H$ in the $x z$ plane, so it is TM with respect to $y$ . Relabel the axes and the same field pattern gets a different name. The resonant frequencies, which depend only on the three integers and the three sides, do not care.

7.8.4 TM and TE modes of the rectangular cavity

TMₘₙₚ modes

TEₘₙₚ modes