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\title{FAST ELECTROMAGNETIC HEATING OF A THIN CONDUCTING CYLINDER BY
A DIFFRACTING GAUSSIAN BEAM}
\author{A. M. Akhmeteli}{
Russian Federation, 121609 Moscow, Rublevskoe 38-2-139, e-mail
akhm@sol.msk.ru }

In many applications, in particular, to create active media of
short-wavelength lasers, it is necessary to achieve high specific input of
power of an electromagnetic beam. Diffraction constraints set the lower
limit to the transverse dimensions of the beam and represent a fundamental
obstacle for heating of small or distant regions. Some standard ways to
circumvent this difficulty are discussed in this work, but it deals mostly
with the possibility of efficient heating of a thin conducting cylinder by a
broad electromagnetic beam through proper matching of geometry of heating
and of electric properties of the cylinder. The conditions of efficient
absorption of an electromagnetic beam in a cylinder were studied in several
works (see, e.g., \cite{zharov} and references therein), but typically only
transverse geometry (a cylindrical wave converging on a cylinder) and the
condition of 100\% absorption were considered. The latter limitation
resulted in resonant conditions of efficient heating. As in practice fast
heating is accompanied by the change of electric properties of the target,
nonresonant conditions are of greater interest. In this work such
conditions were found for the longitudinal geometry (a free Gaussian beam
propagates along the axis of a cylinder with transverse dimensions much less
than those of the beam) and for the transverse geometry (an axially
symmetric beam converges on a cylinder with transverse dimensions much less
than the wavelength).\ The Maxwell equations were solved rigorously as
follows. The field of the incident beam was expanded into cylindrical waves,
for every cylindrical wave the exact solution of the problem of diffraction
on a cylinder \cite{wait} was calculated, and the refracted and reflected
fields were calculated by the inverse Hankel transform. This procedure was
used to control the semi-quantitative one-wave approximation \cite{akhm2}
and confirmed the validity of the latter.

Let the complex permittivity and the radius of the cylinder be $\varepsilon
=\varepsilon ^{\prime }+i\varepsilon ^{\prime \prime }$ and $a$
correspondingly, the system of units is such that the wave vector in vacuum
is equal to unity modulo. In the longitudinal geometry let the field in the
beam be circularly polarized, the beam's waist's radius is $r\gg a$, the
azimuthal number $n=\pm 1$ (otherwise there is no efficient absorption), the
beam is long ($l_w\gg r$, where $l_w$ is the length of the beam's waist
region), $\kappa _1=2/r\ll 1$ (the beam's waist is much greater than the
wavelength), $p_1=\kappa _1a\ll 1$ (the transverse dimensions of the beam
are much greater than those of the cylinder), $\kappa _2=\sqrt{\varepsilon
-1+\kappa _1^2}$, $\left| \kappa _2\right| \gg \kappa _1$, $p_2=p_2^{\prime
}+p_2^{\prime \prime }=\kappa _2a$. Then, assuming that $l=\left| \ln
p_1\right| \gg 1$ and $\frac 1{p_1^2}\gg \left| \frac{J_1^{\prime }(p_2)}{%
p_2J_1(p_2)}\right| $ ($p_1$ is much less than any other parameter), we
obtain (in the one-wave approximation) an asymptotic formula for the part of
the incident beam power absorbed in the cylinder: $\eta \approx 4$Im$((p_2%
\frac{J_0(p_2)}{J_1(p_2)}(\frac 1{p_2^2}+\frac 1{2a^2})+\ln p_1)^{-1})$ (to
take into account the attenuation of the beam due to the absorption it would
be more accurate to use the expression $1-\exp (-\eta )$ ). Using this
formula, we can find the parameter domains that provide efficient absorption
(typically, $\eta =\frac{2\div 5}l$, that means in practice that tens
percent of the beam's power are absorbed in the cylinder). The following
domains seem to be of most practical interest: 1.1. $\left| p_2\right| \ll 1$%
, $x=\frac 1{a^2l}<1$, $\left| x-1\right| \gg \frac 1l$, $\varepsilon
^{\prime \prime }\sim \max (\left| y+\frac{2x}{x-1}\right| ,\frac{2x}{%
(x-1)^2l})$, $\left| y\right| 
\lesim 
\frac{2x}{(1-x)^2}$, where $y=\varepsilon ^{\prime }-1$. This parameter
domain may be defined roughly as follows: the skin-depth $\delta _s$, the
cylinder diameter, and the wavelength are of the same order of magnitude
(cf. \cite{akhm2}). 2.1. $p_2^{\prime \prime }
\gesim 
1$, $4l
\lesim 
\varepsilon ^{\prime \prime }
\lesim 
8a^2l^2$, $y\sim 4a^2l^2$. 2.3. $p_2^{\prime \prime }
\gesim 
1$, $\left| y\right|
\lesim 
4a^2l^2$, $\varepsilon ^{\prime \prime }\sim 8a^2l^2$, $a^2
\gesim 
\frac 1{2l}$. In the domain 2.3 the cylinder diameter is of the same or
greater order of magnitude as the skin-depth and the wavelength, and $\frac{%
2a}{\delta _s}\sim \frac \sigma \omega $, where $\sigma $ and $\omega $ are
the conductivity and the frequency, correspondingly ($\varepsilon ^{\prime
\prime }=\frac{4\pi \sigma }\omega )$. The parameter domains 1.1, 2.1, and
2.3 are nonresonant, so inside these domains the conductivity may typically
vary by an order of magnitude (and in some cases, e.g., in the domain 2.1, -
by several orders) without significant loss of efficiency of absorption.

The analysis shows that these domains do not allow efficient heating of
cylinders that are much thinner than the wavelength. Such possibility was
found in the transverse geometry. If the electric field in the incident wave
is directed along the axis of the cylinder, $a\ll 1$ (the cylinder radius is
much less than the wavelength), $\left| \ln a\right| \gg 1$, $\varepsilon
^{\prime \prime }
\gesim 
\left| \varepsilon ^{\prime }\right| $, $\varepsilon ^{\prime \prime }\sim
\frac 2{a^2\left| \ln a\right| }$ (roughly speaking, the skin-depth is of
the same order of magnitude as the cylinder diameter), then the part of the
power absorbed in the cylinder $\eta \sim \frac \pi {\left| \ln a\right| }$
is quite significant. The length of the region under heating may be of the
order of the wavelength.

Applications of the above results were discussed in 
\cite{akhm2},\cite{akhm1}, and \cite{akhm3} (see also references therein).
\vspace*{-0.5cm}

\begin{thebibliography}{9}
\vspace*{-0.1cm}
\bibitem{zharov} Zharov A.A, Zaboronkova T.M. {\it Fizika Plazmy}, v.9, no.5,
p.995 (1983) (in Russian) 
\vspace*{-0.1cm}
\bibitem{wait}  Wait J.R. {\it Electromagnetic radiation from cylindrical
structures}, London, Pergamon Press, 1959
\vspace*{-0.1cm}
\bibitem{akhm2}  Akhmeteli A.M. {\it Pis'ma Zhurn.Tekhn.Fiz.}, v.17, no.11,
p.21 (1991) (in Russian)
\vspace*{-0.1cm}
\bibitem{akhm1}  Akhmeteli A.M. {\it Int-l Conf.Adv.Laser Technol. ALT'92},
Moscow, 1992, Book of Summaries, part 6, p.103
\vspace*{-0.5cm}
\bibitem{akhm3}  Akhmeteli A.M. {\it Int-l Conf. on Short Wavelength Radiation},
Zvenigorod, 1994, p.Tu19.
\vspace*{-0.1cm}
\end{thebibliography}


\end{document}
