\documentstyle[12pt,epsf]{llncs}



\begin{document}


\title{Rough Surface Approximation for Ray Tracing Using a Simple Monte Carlo
Methods}



\author{K.Taft \inst{1}, V.N.Alexandrov \inst{1} and  E. Griese \inst{2} }

\institute{Department of  Computer Science,
University of Liverpool, Liverpool, UK \and
Siemens Nixdorf Informationssysteme AG / C-LAB,  Paderborn,
Germany}



\maketitle



%\newpage
\begin{abstract}


The initial problem is to attempt to characterise a planar wave-guide
 using ray tracing methods.
 The first problem, after of course implementing a system that generates
 rays that could be reflected between two planes subject to Snells law,
  $\sin(\Theta) > {n_1}/{n_2}$  ($n_2$ - cladding,  $n_1$ core
 refractive index), is how to treat rough surfaces at the core/cladding
 boundary which would be
 an unavoidable product of the manufacturing process.


The approach to this problem was in the first instance the production of a two dimensional
 model as planar wave-guides would support only the longitudinal wave, however the expansion
of the model to encompass bends in the wave-guides meant that a three dimensional model would
 be more practical. This of course lead to the interpretation of the model as a rectangular optical
 fibre with reflective sides.

We suggest a novel highly flexible model and approach where
there is no prior knowledge of the rough surface effects
nor  any statistical information
available on the wave-guides to be modeled  as
they were yet to be manufactured.

The first models for rough surface approximation looked at were
 the now classical approach of the far field analysis.
 There have been many papers published on the use of Greens function
to determine the scattering of a plane wave by rough surfaces.
 The reflection from
individual rough surfaces is very effective
 but when the action is repeated over a few thousand
 reflections then the mapping problems become
very time consuming.

The option to provide as simple a solution as possible
 is followed in this paper and as a result of this
the problem is approached  in terms
 of a weighted but stochastic exit angle of a ray
from an incident surface. The model could then be easily developed
in 2 and 3 dimensions.
The beam is, as in all rays tracing techniques composed of $n$ rays.

The algorithms  had obviously inherent problems in a sequential format,
e.g.  the generation and processing
 of individual rays.  Results on serial and  parallel implementations
 and relevant computational experiments will be presented.

\end{abstract}

\end{document}