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%TCIDATA{Created=Mon Sep 16 21:44:55 1996}
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\markright{{\sf Dielectric wave guides}}
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\section{8.10\quad Dielectric Wave Guides and Other Topics}

\subsection{8.10.1}

\quad \thinspace \textbf{a.} Dielectric waveguides and fiber optics are of
considerable (and growing) practical importance, but can only be discussed
briefly for lack of time.

\smallskip\ 

\textbf{b}. It is easy to understand how the fields can be confined to a
cylinder of index of refraction $n_1>1$ by recalling the phenomenon of total
internal reflection.

\smallskip\ 

\textbf{c. }The mathematical problem of matching the fields at the boundary
is hardly new. Unfortunately in this case there is no general way to reduce
the problem to a single scalar equation in a neat way, as in the separation
of TE and TM modes. Two coupled equations is the best one can do, according
to the discussion on 7.3.2; an exception is the case of azimuth-independent
modes in a circular cylinder, discussed by Jackson.

\smallskip\ 

\textbf{d.} If the dielectric constant inside is much larger than that
outside, $\varepsilon _1\gg \varepsilon _{0,}$ the dielectric wave guide has
(approximate) TM modes with the same frequencies (approximately) as the TE
modes in a guide with metallic walls. The reason is that in this case $%
E_{\perp }$ (nearly) vanishes at the walls, which corresponds to the (near)
vanishing of $H_{\perp }$ in a metallic wave guide, and that Maxwell's
equations are essentially symmetric in $\mathbf{E}$ and $\mathbf{H.}$

\smallskip\ 

\textbf{e. }The qualitative behavior of the relation between $\omega $ and $k
$ is similar to that of metallic wave guides, but for $c^{2}\gamma
^{2}>(\varepsilon _{0}-\varepsilon _{1})\omega ^{2}$ the fields are not
confined to the vicinity of the wave guide and radiate energy away.

\newpage 

\subsection{8.10.2 Other Topics in Ch. 8}

\quad \thinspace \textbf{a.} The topic of wave generation in a confined
geometry seems more logically to follow the topic of wave generation in
empty space (or space filled by a uniform medium) which is taken up in the
next chapter. Otherwise one risks to start thinking about the subject in the
way typical of electrical engineers, who are apt to regard free space as the
limiting case of a large cavity.

\smallskip\ 

\textbf{b}. The nice example of the earth and ionosphere as a resonant
cavity is also out of place at this point, since the theory of the Maxwell
equations in spherical geometries will be developed later.

\smallskip\ 

\textbf{c. } The problems of reflections by obstacles in a wave guide or of
leak from openings in the sides are virtuoso displays of mathematical
physics. The results are not very surprising: obstacles impede the flow and
openings leak.

\smallskip\ 

\textbf{d.} We will come back to some of these topics later on in the course
if time permits.

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