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%TCIDATA{Created=Wed Sep 11 15:08:24 1996}
%TCIDATA{LastRevised=Wed Sep 11 16:37:56 1996}
%TCIDATA{Language=American English}

\input tcilatex
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\markright{{\sf Cylindrical cavities and wave guides}}
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\begin{document}


\section{8.3\quad Cylindrical Cavities and Wave Guides.}

\quad \quad \quad (Also includes 8.2, 8.4 and 8.7)

\subsection{8.3.1}

\textbf{\quad \thinspace a. }These are very important in practice, both in
engineering and in experimental physics. The walls will be assumed to be
perfect conductors for the purpose of computing the normal modes.
(travelling waves in a guide and standing waves in a cavity). The losses are
then computed by finding the dissipated power in the skin depth, either
following the method of previous section, or by treating the finite
penetration as a perturbation (see 8.6).

\smallskip\ 

\textbf{b. }We solve first for a wave guide and then go on to a cavity by
imposing the appropriate boundary conditions on the end walls.

\smallskip\ 

\textbf{c.} In an infinitely long guide of constant section, with axis $z$,
the fundamental solutions for the fields depend on $z$ and $t$ as $\exp
(i(k_zz-\omega t)).$ Further, the two modes with $k_z$ $=\pm k$ have the
same frequency $\omega .$ All this follows simply from translational
invariance in the $z$ direction and would be true if we had never heard of
Maxwell. In this section we always take $\omega \geq 0,k\geq 0$ (if real)
and the meaning of $k$ is $\left| k_z\right| $ (not $\sqrt{\varepsilon \mu }$
$\omega /c).$

\smallskip\ 

\textbf{d. }Every component of the fields is then of the from 
\[
\func{Re}\left[ u(x,y)e^{\pm ikz-i\omega t}\right] 
\]
and $u$ obeys 
\begin{equation}
\frac{\partial ^2u}{\partial x^2}+\frac{\partial ^2u}{\partial y^2}+\left(
\mu \varepsilon \frac{\omega ^2}{c^2}-k^2\right) u=0.  \label{1}
\end{equation}

\newpage

\subsection{8.3.2}

\quad \thinspace \textbf{a.} In particular, $E_z$ obeys equation (\ref{1})
and the boundary conditions $E_z=0$ on the side walls. Thus $E_z(x,$ $y)$ is
obtained by solving a familiar boundary-value problem for the Helmholtz
equation with Dirichlet boundary conditions in two dimensions.

\smallskip\ 

\textbf{b. }Similarly, $B_{z}(x,y)$ obeys equation (\ref{1}) with the
Neumann boundary conditions $\partial B_{z}/\partial n=0$ on the side walls.
These conditions follow from the fact that $\partial B_{z}/\partial
n-\partial B_{n}/\partial z$ gives the component of $\mathbf{D}$ parallel to
the walls in the $xy$ plane, which must vanish. Further, $B_{n}$ vanishes at
the walls for all $z,$ thus $\partial B_{n}/\partial z=0.$

\smallskip\ 

\textbf{c. }Since $E_z$ and $B_z$ each obey boundary conditions
independently of all other field components, it is possible to find
solutions of Maxwell's equations in a wave guide where either

- $B_z=0$ everywhere and $E_z=0$ at the walls (TM waves)

- $E_{z}=0$ everywhere and $\partial B_{z}/\partial n=0$ at the walls (TE
waves).

\smallskip\ 

\textbf{d. }The \textsl{Transverse Magnetic} (TM) waves are also sometimes
called Electric (or ``E'') waves.

The \textsl{Transverse Electric} (TE) waves are sometimes called Magnetic
(or ``M'') waves.

This serves mostly to generate confusion.

\smallskip\ 

\textbf{e. }It may also be possible to have transverse electromagnetic (TEM)
waves, with $E_{z}=B_{z}=0$ everywhere (see 8.3.7 below).

\smallskip\ 

\textbf{f }The word ``transverse'' here does not mean simply
``divergenceless'' (all Maxwell waves are divergenceless). It means, quite
properly, ``not in the direction of propagation''.

\newpage

\subsection{8.3.3}

\quad \thinspace \textbf{a.} The importance of the components $E_z,B_z$ lies
further in the fact that they can be used as ``super potentials'' (in the
sense of Sect. 7.33e) for TM and TE waves respectively.

\smallskip\ 

\textbf{b.} To see this for TM waves, for instance, take the $x$ component
of $\mathbf{\nabla }\times \mathbf{E}=i\omega \mathbf{B}/c$ and the $y$
component of $\mathbf{\nabla }\times \mathbf{H=-}i\omega \mathbf{D}/c.$
Obtain (see Appendix on Cavities) 
\[
\frac{\partial E_{z}}{\partial y}-\frac{\partial E_{y}}{\partial z}=i\frac{%
\omega }{c}\mu H_{x}
\]
\[
\frac{\partial H_{x}}{\partial z}-\frac{\partial H_{z}}{\partial x}=i\frac{%
\omega }{c}\varepsilon E_{y}.
\]
With $H_{z}=0$ and $\partial /\partial z\rightarrow ik_{z},$ the second
equation gives 
\begin{equation}
k_{z}H_{x}=-\frac{\varepsilon \omega }{c}E_{y}  \label{2}
\end{equation}
and then the first equation gives 
\begin{equation}
\frac{\partial E_{z}}{\partial y}=\left( ik_{z}-i\frac{\omega }{c}\mu \frac{%
\omega \varepsilon }{ck_{z}}\right) E_{y}.  \label{3}
\end{equation}
Thus $E_{y}$ and $H_{x}$ can be obtained from $\partial E_{z}/\partial y,$
and a similar procedure hods for $E_{x}$ and $H_{y}.$

\smallskip\ 

\textbf{c.} All formulae of type (\ref{2}) and (\ref{3}) are written more
compactly on the next page, using Jackson's notation that $\mathbf{E=E}_t+E_z%
\mathbf{e}_3$ and $\psi $ obeys 
\begin{equation}
\left( \nabla _t^2+\gamma ^2\right) \psi =0  \label{4}
\end{equation}
with 
\begin{equation}
\gamma ^2=\mu \varepsilon \frac{\omega ^2}{c^2}-k^2.  \label{4'}
\end{equation}
The subscript $t$ means ``transverse to the guide'' and not ``tangential''.

\newpage

\subsection{8.3.4}

\quad \thinspace \textbf{a. }\textsl{For TM Waves:} 
\begin{eqnarray*}
\mathbf{E}_t &=&\frac{ik_z}{\gamma ^2}\mathbf{\nabla }_t\psi e^{ik_zz}\quad
\quad \quad \quad \quad \quad \quad \quad \quad \left( \psi \equiv E_z\right)
\\
\frac{ck_z}{\varepsilon \omega }\mathbf{H}_t &=&\mathbf{e}_3\times \mathbf{E}%
_t\quad \quad \quad \text{or \quad \quad \quad }\mathbf{H}_t=\frac{%
i\varepsilon \omega }{c\gamma ^2}\mathbf{e}_3\times \mathbf{\nabla }_t\psi
e^{ik_zz}\,.
\end{eqnarray*}

\smallskip\ 

\textbf{b. }\textsl{For TE waves:} 
\begin{eqnarray*}
\mathbf{H}_t &=&\frac{ik_z}{\gamma ^2}\mathbf{\nabla }_t\psi e^{ik_zz}\quad
\quad \quad \quad \quad \quad \quad \quad \quad \quad \left( \psi \equiv
H_z\right) \\
\frac{ck_z}{\varepsilon \omega }\mathbf{E}_t &=&-\mathbf{e}_3\times \mathbf{H%
}_t\quad \quad \quad \text{or \quad \quad \quad }\mathbf{E}_t=-\frac{i\mu
\omega }{c\gamma ^2}\mathbf{e}_3\times \mathbf{\nabla }_t\psi e^{ik_zz}\,.
\end{eqnarray*}

\smallskip\ 

\textbf{c.} Especially in the engineering literature, formulae are often
written in terms of the wave impedance $Z,$ defined as the ratio of the
normal field at the surface $E_{\perp }$ to the surface current $K_{z}$
(i.e. $Z=E_{\perp }/K_{z})$ for the forward-going wave ( $k_{z}=k).$ By the
argument in 8.1.5a, (4$\pi /c)K_{z}=H_{t/\!/}$ and for TM waves $%
H_{t/\!/}=(\varepsilon \omega /ck$)$E_{\perp }.$ Hence 
\[
Z=\frac{c}{4\pi }\frac{ck}{\varepsilon \omega }\quad \quad \quad \quad \quad
\quad \text{(TM\ waves)}
\]
[For the wave going in the $-z$ direction, $k_{z}=-k$ and $K_{z}$ changes
sign, but $E_{\perp }$ remains the same; hence $E_{\perp }/K_{z}=-Z\,].$
Similarly 
\[
Z=\frac{c}{4\pi }\frac{\mu \omega }{ck}\quad \quad \quad \quad \quad \text{%
(TE}\;\text{waves)}
\]
Jackson inconsistently omits the factor $(c/4\pi )$ in these formulae,
perhaps because it does not appear in MKS units (see Appendix).

\newpage

\subsection{8.3.5}

\quad \thinspace \thinspace \textbf{a. }The whole problem is reduced to
solving 
\begin{equation}
\nabla ^2\psi +\gamma ^2\psi =0\quad \quad \quad \quad \quad \quad \text{in
2-d}  \label{4}
\end{equation}
with the boundary condition $\psi =0$ or $\partial \psi /\partial n=0.$ In
either case from (\ref{4}) \textsl{and} the boundary conditions 
\[
\int \left| \mathbf{\nabla }\psi \right| ^2d^2r=\gamma ^2\int \left| \psi
\right| ^2d^2r 
\]
which shows that $\gamma ^2\geq 0.$ Physically, this occurs because $\psi $
must be oscillatory to have nodes (or crests) on opposite boundaries.

\smallskip\ 

\textbf{b. }If the boundary is closed, (\ref{4}) has a discrete spectrum of
eigenvalues $\gamma _{\lambda }^{2}.$ For each $\lambda $, the wave vector
is $k_{\lambda }^{2}=\mu \varepsilon (\omega ^{2}/c^{2})-\gamma _{\lambda
}^{2}.$ Propagation occurs only for $\omega \geq $ $\omega _{\lambda },$
with 
\[
\omega _{\lambda }=\frac{c\gamma _{\lambda }}{\sqrt{\varepsilon \mu }}\quad
\quad \text{the \textsl{cutoff frequency} for the }\lambda \text{ mode.}
\]
The spectrum of $\omega $ vs. $k$ (or $k$ vs. $\omega )$ has many branches,
\[
\omega ^{2}=\omega _{\lambda }^{\,2}+\frac{c^{2}k^{2}}{\varepsilon \mu }
\]
all extending to infinity along the ``full space'' line 
\[
\omega =\frac{ck}{\sqrt{\varepsilon \mu }}\quad \quad \quad \quad \quad (%
\text{see figure).}
\]
One often plots $k$ vs. $\omega $, as Jackson does, because $\omega $ is the
variable under direct control in practice (by tuning the resonator at the
input end of the waveguide).

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\newpage 

\subsection{8.3.6}

\quad \thinspace \thinspace \textbf{a. }The simplest example is the \textsl{%
rectangular wave guide} (which of course can be treated directly without
recourse to all the previous theory). For TE waves 
\[
H_{z}=\psi _{mn}e^{ik_{z}z}=H_{o}\cos \left( \frac{m\pi x}{a}\right) \cos
\left( \frac{n\pi y}{b}\right) e^{ik_{z}z}
\]
and 
\[
\gamma _{mn}^{2}=\pi ^{2}\left( \frac{m^{2}}{a^{2}}+\frac{n^{2}}{b^{2}}%
\right) \quad \quad \quad \quad \quad m,n\text{ integers}\geq 0.
\]
The cutoff frequencies are 
\[
\omega _{mn}=\frac{c\pi }{\sqrt{\varepsilon \mu }}\left( \frac{m^{2}}{a^{2}}+%
\frac{n^{2}}{b^{2}}\right) ^{1/2}.
\]
The solution with $m=n=0,$ corresponding to $B_{z}=$ const. is not
acceptable, because it implies that all the fields vanish (by 8.3.4b this
implies $\mathbf{E}_{t}=0,$ hence $\omega B_{z}=\mathbf{\nabla }_{t}\times 
\mathbf{E}_{t}=0$

\smallskip\ 

\textbf{b. }For $a>b$ the lowest cutoff frequency is $\omega _{1,0}=\pi c/%
\sqrt{\varepsilon \mu a}.$ $\;$ The next is either $\omega _{1,1}$ or $%
\omega _{2,0}\,,$ depending on $(b/a)$.

\smallskip\ 

\textbf{c.} For TM waves 
\[
\psi _{mn}=E_{0}\sin \frac{m\pi x}{a}\sin \frac{n\pi y}{b},\quad \quad
E_{z}=\psi _{mn}e^{ik_{z}z}.
\]
The smallest acceptable values of $m$ \textsl{and} $n$ is now $1$ and the
lowest cutoff frequency is $\omega _{1,1\,.}$ Note that for $n\neq 0$ there
is degeneracy between TE and TM modes. In any event, in a suitable range of
frequencies above $\omega _{1,0}$ only the $(1,0)T$E mode will be excited.
The wave guide is usually operated under these conditions.

\newpage

\subsection{8.3.7}

\quad \thinspace \thinspace \textbf{a. }There remains to examine the
possibility of solutions with $E_z=0$ \textsl{and} $H_z=0.$ In this case
Maxwell's equations (see Appendix) give $\partial E_x/\partial y=\partial
E_y/\partial x,$ which implies $\mathbf{E}_t=\mathbf{\nabla }_t\varphi (x,y)$
and $\partial E_x/\partial x+\partial E_y/\partial y=0,$ which implies $%
\nabla _t^2\varphi =0.$ In other words, $\mathbf{E}_t$ is a two-dimensional
electrostatic field. A similar conclusion holds for $\mathbf{H}_t.$

\smallskip\ 

\textbf{b. }From electrostatics and potential theory, there are non-trivial
solutions only if the cross section of the wave guide is multiply connected,
as in a coaxial cable or a transmission line.

\smallskip\ 

\textbf{c. }Solutions of this type are called \textsl{Transverse
Electro-Magnetic} (TEM) waves.

\smallskip\ 

\textbf{d. }The remaining Maxwell equations (see Appendix) (such as the
pair: $ik_zE_y=i(\omega /c)\mu H_x$ and $-ik_zH_x=-i(\omega /c)\varepsilon
E_y)$ imply that $k_z^2(=k^2)=\varepsilon \mu (\omega /c)^2$ as in full
space. Thus there are no frequency cutoffs.

\smallskip\ 

\textbf{e. }The relations between the components of $\mathbf{B}$ and $%
\mathbf{E}$ for TEM waves can be written compactly as 
\[
\mathbf{B}_{\text{TEM}}=\pm \sqrt{\mu \varepsilon }\mathbf{e}_{3}\times 
\mathbf{E}_{\text{TEM}}
\]
(with $\pm $ for $k_{z}=\pm k$) again as in full space. [``Full space'' here
means an unbounded medium with constants $\varepsilon ,\mu \,].$ 

\newpage

\subsection{8.3.8}

\quad \thinspace \thinspace \textbf{a.} The fields in a \textsl{cylindrical
cavity} are obtained by superimposing the travelling waves in the
corresponding guide, so as to satisfy the boundary conditions at the end
walls. We consider cavities closed at both ends).

\smallskip\ 

\textbf{b.} For TM modes, the boundary conditions are $\mathbf{E}_{t}=0$ at $%
z=0$ and $z=d.$ With $\mathbf{E}_{t}$ given by 8.3.4a, the correct
combination of $k_{z}=k$ and $k_{z}=-k$ is clearly 
\[
\mathbf{E}_{t}=\frac{\mathbf{\nabla }_{t}\psi }{2\gamma ^{2}}\left(
ike^{ikz}+i(-k)e^{-ikz}\right) ,\quad \quad \quad k=\frac{p\pi }{d}\quad
\quad p=0,1,2,...
\]
(Here the $2$ in the denominator is arbitrary). Thus the transverse
component are proportional to corresponds to $\sin (p\pi z/d).$On the other
hand, $E_{z}=\psi \cos (p\pi z/d).$

\smallskip\ 

\textbf{c. }For TE modes the boundary conditions are simply $H_z=0.$ Thus $%
H_z=\psi \sin (p\pi z/d),$ $p=1,2,3,...$

\smallskip\ 

\textbf{d.} The other field components are derived as on 8.3.4 and are
written out in Jackson, (8.76) and (8.77).

\smallskip\ 

\textbf{e.} The important new feature is that the possible values of $k$ are
now discrete and that correspondingly one has a discrete spectrum of
eigenfrequencies

\[
\omega _{\lambda p}^2=\frac{c^2}{\mu \varepsilon }\left( \gamma _\lambda
^2+\left( \frac{p\pi }d\right) ^2\right) =\omega _\lambda ^2+\frac{c^2}{\mu
\varepsilon }\left( \frac{p\pi }d\right) ^2\,. 
\]
One must notice that $\lambda $ denotes in fact two discrete indices, such
as $m,n$ for a rectangular waveguide (page 6) or the radial and azimuthal
quantum numbers in the case of a circular cylinder (next page).

\newpage

\subsection{8.3.9}

\quad \thinspace \thinspace \textbf{a.} The superpotentials for a circular
wave guide, and for a \textsl{right circular cylindrical cavity}, are
expressed in terms of cylindrical Bessel functions: 
\[
\psi (\rho ,\varphi )=J_{m}(\gamma _{mn}\rho )e^{im\varphi }\,,
\]
where $m$ is an integer and $n$ is a positive integer that numbers the roots
of $J_{m}(\gamma R)=0$ for TM modes and $J_{m}^{\,\prime }(\gamma R)=0$ for
TE modes. $R$ is the radius. All one needs to find the eigenfrequencies of a
cavity are tables of zeroes of the Bessel functions and their first
derivatives.

\smallskip\ 

\textbf{b.} The lowest mode in a cavity of length $d$ has frequency 
\[
\omega _{1,1,1}=\frac{1.841}{\sqrt{\mu \varepsilon }}\frac{c}{R}\left(
1+2.912\frac{R^{2}}{d^{2}}\right) ^{1/2}\,,
\]
and is TE in character, provided that $d>2.03R;$ otherwise the lowest mode
is TM with frequency 
\[
\omega _{0,1,0}=\frac{2.405}{\sqrt{\mu \varepsilon }}\frac{c}{R}\,,
\]
independent of $d.$

\smallskip\ 

\textbf{c. }This is true in general: in a short cavity the lowest mode is TM
and length-independent, but beyond a certain length the lowest mode is TE$.$

\end{document}
