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\begin{document}




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\title{WAVE PROPAGATION ALONG NONUNIFORM  ANISOTROPIC OPEN WAVEGUIDES}
\author{
 I.G.Kondrat'ev, A.V.Kudrin,
             T.M.Zaboronkova}
{Radiophysical Research Institute (NIRFI) \\
25 B.Pecherskaya st., Nizhny Novgorod, Russia\\
Fax: +7 8312 369902; E-mail: zabr@nirfi.nnov.su}


      With the use of a full wave formulation, a study is made of propagation
  of electromagnetic waves along an axially nonuniform anisotropic
  waveguide immersed in a uniform anisotropic medium. To render the study
  more concrete, we specialize our treatment to nonuniform plasma channels
  embedded in a background magnetoplasma and oriented along an ambient
  magnetic field.

     We consider a cylindrical plasma waveguide whose parameters are
  allowed to be varying both across and along the ambient magnetic field
 $\vec B_0 = B_0\vec z_0 $. To describe the wave propagation  in such a channel, we
  apply Katsenelenbaum's method of normal modes ("method of cross-sections"),
  which is well-developed for closed waveguides [1] and open dielectric and
  acoustic waveguides [2], and extend this method to the case of open
  anisotropic guiding systems. Accordingly, we represent the field
  $\skew5\hat{\vec E}$, $\skew5\hat{\vec H}$ in a given cross-section
$z={\rm const}$ in terms of waves exiting
  in a certain auxiliary axially uniform waveguide whose parameters coincide
  with those that our nonuniform waveguide has in the cross-section considered.
  Assuming, for simplicity, the field to be independent of the azimuth about
  $\vec B_0$, [$\p/\p\f=0$], we can therefore write
    $$\skew5\hat{\vec E}(\rho,z)=\sum\limits_{\pm m}
d_m\,\vec E_m+\sum\limits_{s=(-)}^{(+)}\sum\limits_{\alpha =o}^e
\int\limits_0^\infty
d_{s,\alpha}(q)\,\vec E_{s,\alpha}(q)\,dq,$$
    $$\skew5\hat{\vec H}(\rho,z)=\sum\limits_{\pm m}
d_m\,\vec H_m+\sum\limits_{s=(-)}^{(+)}\sum\limits_{\alpha =o}^e
\int\limits_0^\infty
d_{s,\alpha}(q)\,\vec H_{s,\alpha}(q)\,dq$$
where $\vec E_m $, $\vec H_m $ and $\vec E_{s,\alpha}$,
$\vec H_{s,\alpha}$ are the waves of a discrete spectrum
and a continuous spectrum, respectively, for the corresponding auxiliary
waveguide [3] and $\rho $, $\phi $, $z$ are the cylindrical coordinates. The
signs ``$\pm$" mark the propagation directions and q is a normalized transverse
wavenumber: $q = \frac{k_{\perp}}{k_0} $, $k_0 = \frac{\omega}{c}$. The superscript $  $ label the waves whose
propagation constants are given by the relations appropriate to ordinary
( $\alpha = o$) and extraordinary ($\alpha = e$) principal modes of the surrounding
uniform medium.

   The amplitudes $d_m$ and $d_{s,\a}$ satisfy the system of equations
   $${\p d_m\over\p z}+ik_0p_md_m=\sum\limits_{\pm \tilde m}
S_{\tilde m}^m\,d_{\tilde m}+
\sum\limits_{\tl s=(-)}^{(+)}\,\sum\limits_{\tl \a=o}^e
\int\limits_0^\infty\,
S_{\tl s,\tl\a}^m(\tilde q)\,d_{\tl s,\tl\a}(\tilde  q)\,d\tilde q$$
$${\p d_{s,\a}\over\p z}+ik_0\,p_{s,\a}(q)\,d_{s,\a}(q)=
\sum\limits_{\pm \tl m}S_{\tl m}^{s,\a}(q)d_{\tl m}+
\sum\limits_{\tl s=(-)}^{(+)}\,\sum\limits_{\tl \a=o}^e
\int\limits_0^\infty\,
S_{\tl s,\tl\a}^{s,\a}(q,\tl q)\,d_{\tl s,\tl\a}(\tl q)\,d\tl q
$$
where  $p_{\pm m}$ and $p_{\pm,\a}$ are the normalized propagation constants of the waves
of the discrete spectrum and the continuous spectrum. We have found the
coupling coefficients $S_{\tilde m}^m$, $S_{\tl m}^{s,\a}$,
$S_{\tl s,\tl\a}^m$, $S_{\tilde s, \tilde\alpha}^{s,\alpha} $ and solved these equations
using WKB approximation  for some special cases.

   In applying the resultant formulation, we discuss some concrete problems
of wave channeling in nonuniform plasma structures.

  This work is supported by RFFI under grant N95-02-05001.
\begin{thebibliography}{5}
\bibitem{1}
V.Z. Katsenelenbaum, The Theory of Irregular Waveguides with Slowly Changing
Parameters, USSR Academy of Sciences Press, Moscow, 1961.
\bibitem{2}
V.V. Shevchenco, Smooth Transitions in Open Waveguides. An Introduction
to the Theory, Nauka, Moscow, 1969.
\bibitem{3}
I.G. Kondrat'ev, A.V. Kudrin, T.M. Zaboronkova, Radiophys. Quant. Electron.
1994, 37, pp.579-592.
\end{thebibliography}

\end{document}
