\documentclass[letterpaper,12pt]{article}
\usepackage{epsfig}

% expand the size a little so we fill to within 1.0 inch of
% 8.5 x 11.0 inch paper.  Let the page numbers into the margin.

\vsize=6truein
\hoffset=-0.5truein
\voffset=-0.0truein
\setlength\paperheight{12in}
\setlength\textheight{8.9in}

\begin{document}
%\setcounter{page}{}
\columnwidth=6.5truein
\hsize=6.5truein
\title{Resonant $n>0$ Modes Trapped by a Dielectric in a Coaxial Line}
\author{Lawrence R. Doolittle\\
Tunnel Dust, Inc.\\
Walnut Creek, California, USA
}

\maketitle

\section{Introduction}
The simplest possible coaxial window geometry has a dielectric material
inserted between coaxial metal cylinders, which extend undisturbed
beyond the ends of the window, as shown in figure~\ref{coax-window}.
The dielectric reduces the impedance of the line, so some form of
impedance match is required.

\begin{figure}[htb]
\centering
\epsfig{file=coax_window.eps, width=0.75\columnwidth}
\caption{Coaxial window geometry}
\label{coax-window}
\end{figure}

One approach is to use a two-step quarter-wave match, from a high
($\sim 50\thinspace \Omega$) impedance line to
low ($\sim 5\thinspace \Omega$) impedance window.
This is simple, compact, and leaves the region in the vicinity
of the window undisturbed from the form shown in figure~\ref{coax-window}.

Scaling laws provide an incentive to make a high power window as large
as possible, since (at constant $l$, at least) losses scale inversely
with radius, and both heat flux and temperature rise scale inversely
with the radius squared.  One phenomenon that limits how large a window
can be made is the presence of higher order
modes in the structure.  Conventional wisdom is to limit the average
circumference to one wavelength, so that these modes cannot propagate.

This rule makes sense, and the derivation below will confirm it for
the vacuum section of the line.  Applying that rule to the dielectric
loaded section of the line is too conservative.  While the dielectric
can trap modes below the cutoff frequency of the vacuum line, those
narrow band resonances can be computed (analytically) and placed
away from frequencies of operation.

\section{HOM Math}

As discussed in Langmuir\cite{langmuir},
modes of a coaxial line have a radial variation given by
$$ R(r) = A_nJ_n(kr)+B_nN(kr)~~.$$
The TM boundary conditions at $r=r_i$ and $r=r_o$ give
$$ {N_n(k_rr_i) \over J_n(k_rr_i)} = {N_n(k_rr_o) \over J_n(k_rr_o)} = -{A_n \over B_n}$$
and the corresponding conditions for TE waves are
$$ {N_n'(k_rr_i) \over J_n'(k_rr_i)} = {N_n'(k_rr_o) \over J_n'(k_rr_o)} = -{A_n \over B_n}~~.$$

\begin{figure}[htb]
\centering
\epsfig{file=show1.eps, width=0.75\columnwidth}
\caption{Bessel function ratios}
\label{show1}
\end{figure}


When $r_i$ is within a factor of 2 of $r_o$, the family of modes with the lowest
cutoff frequencies are TE modes.  These have $k_r=n/r_a$, where $n$ is the
azimuthal quantum number, 
and $r_a\approx (r_i+r_o)/2$, but $r_a$ really has a weak dependence on
both $n$ and $r_o/r_i$.

\begin{figure}[htb]
\centering
\epsfig{file=show2.eps, width=0.75\columnwidth}
\caption{Normalized transverse separation constants}
\label{show2}
\end{figure}

The fact that small values of $k$ can apply to even closely spaced
conductors ($r_i\approx r_o$) is possibly surprising.  Inspection
of a graph of the above Bessel function ratios (figure~\ref{show1},
in which the TE$_n$ curves show $N_n'(x)/J_n'(x)$,
and the TM$_n$ curves show $N_n(x)/J_n(x)$
) makes it clear how this can work.
The peaks in the TE style curves represent
cases where two closely spaced $x$ values can have the same $f(x)$.
This arrangement does not happen with the TM curves.
Representing such solutions ($f(kr_i)=f(kr_o)$) of these graphs
in terms of $\alpha=r_o/r_i$
gives us the curves in figure~\ref{show2}.

Treat a section of dielectric of length $l$ in a coaxial line with inner
radius $r_i$ and outer radius $r_o$, as depicted in figure~\ref{coax-window}.
Consider behavior at angular frequency $\omega$,
for which the free-space propagation constant is $k_0=\omega/c$,
and the corresponding unguided propagation constant in a medium of
dielectric constant is $k_\epsilon=\sqrt{\epsilon}k_0$.
Assume $k_0 < k_r < k_\epsilon$, so HOMs don't propagate in the $\epsilon=1$
section, but do propagate inside the dielectric.
Then the wave equation's solution has
$z$ dependence $\sin k_1 z$ and/or $\cos k_1 z$ in the dielectric, and
$e^{\pm az}$ in the adjoining vacuum.
The separation constants satisfy $k_1^2=k_\epsilon^2-k_r^2$
and $a^2=k_r^2-k_0^2$.

If we match $H_z$ and $dH_z/dz$ on the dielectric-vacuum boundaries $z=\pm l/2$,
and take the $\cos z$ style solution in the dielectric, then we get relations
$$\cos k_1 {l\over 2} = A$$
$$-k_1 \sin k_1 {l\over 2} = -aA.$$
Eliminating $A$ from these equations, we get the condition for resonance
$$l = {2\over k_1} \tan^{-1}{a\over k_1}.$$
Note that the multi-valued nature of $\tan^{-1}$ gives a whole class of
solutions, separated by $\pi$ in the arctangent.
Changing from $\cos$ (even) solutions to $\sin$ (odd) solutions
produces solutions that are spaced $\pi/2$ from the first set.
Combine this with the relations
$$ k_1 = {1\over r_a} \sqrt{\epsilon t^2-n^2}$$
$$ a   = {1\over r_a} \sqrt{n^2 - t^2}$$
where $t=\omega r_a/c$.  A suitably normalized form of the result is
$${l\over r_a} = {2\over \sqrt{\epsilon t^2-n^2}} \left( \tan^{-1}
   \sqrt{ n^2-t^2\over \epsilon t^2-n^2 } + m{\pi\over 2} \right)~~, $$
where $m$ is the longitudinal quantum number.
This relation is displayed in figure~\ref{coax-mode-plot}.
It is clear that trapped (resonant) modes can only exist if
the normalized frequency $t$ satisfies $n/\sqrt{\epsilon} < t < n$.

\begin{figure}[htb]
\centering
\epsfig{file=coax_mode_plot.eps, width=0.8\columnwidth}
\caption{Normalized resonant frequencies for $\epsilon=10$}
\label{coax-mode-plot}
\end{figure}

\section{Example}

The preceding math may seem a little abstract and opaque, so here is an
example that may clarify the implications.

If we work with Al$_2$O$_3$ ceramic that has $\epsilon=10$, and the
window assembly fits in a traditional $50\thinspace \Omega$ line, a magic value
of $\alpha=r_o/r_i$ for the window region is 1.30, because the
$60\ln \alpha = 15.8\thinspace \Omega$ vacuum line impedance can be used as
a quarter-wave matching section to the
$60/\sqrt{\epsilon} \cdot \ln \alpha = 5\thinspace \Omega$
dielectric loaded section.

Suppose a window is needed for 700\thinspace MHz.
To get a reasonable extinction length for HOMs, we should choose $r_a$
somewhat less than the critical value of $c/2\pi f = 6.82$\thinspace cm.
Capriciously choosing a value of 20\thinspace cm for the extinction length, we
derive a value of 6.45\thinspace cm for $r_a$.  Thus the normalized frequency
$t$ in the section above is 0.947.

Solving for $r_i$ and $r_o$ in terms of $r_a=(r_i+r_o)/2$ and $\alpha=r_o/r_i$,
we get $r_i=2r_a/(1+\alpha)=5.61$\thinspace cm and $r_o=7.30$\thinspace cm.

Inserting all these parameters to find $l$ for various modes gives as
set of lengths to avoid for the window.  For $n=1$ modes, the first two lengths
to avoid are 0.52~and 7.71\thinspace cm.
The first $n=2$ mode will line up with the
operation frequency if the window length is 3.88\thinspace cm.

Suppose the window is made 3.0\thinspace cm long, which normalizes
to $l/r_a=0.456$.
Although the equation for $l/r_a$ is not analytically invertible,
it is easy to solve for the resonant frequency numerically,
and get $t=0.641$, or 474\thinspace MHz.
The field pattern for this mode, and and the next two lowest frequency
modes, are shown in figure~\ref{example-field-pattern}.
With a loop probe some distance away from the window,
these modes could presumably be excited on purpose.  The $Q_0$ of such
trapped modes would be a useful low power indicator of window losses.

\begin{figure}[htb]
\centering
\epsfig{file=pattern.eps, width=0.7\columnwidth}
\caption{Longitudinal $H_z$ dependence of resonant modes discussed
in the example}
\label{example-field-pattern}
\end{figure}

\begin{thebibliography}{1}
\bibitem{langmuir}
Robert V. Langmuir, {\bf Electromagnetic Fields and Waves},
McGraw-Hill, New York, 1961, section 12.5.

\end{thebibliography}
\end{document}