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\begin{document}
\title{Determining size and geometry of the particles from the
polarisation change of the light scattered from the particles}
\author{Aman Abhishek Tiwari}
\author{Dr. Pankaj Jain(Supervisor)}
\affiliation{Indian Institute of Technology Kanpur, Department of Physics}
\date{\today}
\begin{abstract}
An important problem in atmospheric physics is to characterize the
ambient aerosol distribution. While a majority of current laser-based
detectors can measure the size spectrum of the scattering particles,
they do not give information about the geometry of the scatterers. We
aim to compute the effect of the scatterers on the polarization of the
incoming radiation and to use the measured radiation to infer the size
as well as the geometry. In order to do so, we will write a code to
solve Maxwell's equations for arbitrary geometries using the
Discontinuous Galerkin method and then use this code to explore the
effect of scatterer geometry on the incoming radiation.
\end{abstract}
\maketitle
\section{Introduction}
Current optical aerosol counters used to characterize the particulate
matter suspended in air are only capable of measuring the size spectrum
of the particles, but they assume that the particles are spherical in
shape.
We aim to recover additional information about the particles by modeling
the particles as an ellipsoid and calculate the size spectrum $N(r)$ where
$r$ defines the size of the particles, eccentricity spectrum $N(e)$ and
where $e$ is the eccentricity of the particles.
\subsection{Experimental Setup}
A typical aerosol counter setup has a LASER light source, a sample
chamber in which air sample to be investigated is held and an array of
photosensitive detectors which detects the scattered light, it will
provide us our scattering data. Figure.\ref{fig:aerosol_counter} shows a
schematic of a typical aerosol counter.
\begin{figure}[h!]
\centering
\includegraphics[scale=1.0]{aerosol_counter.png}
\caption{\label{fig:aerosol_counter}Schematic of a typical aerosol
counter}
\end{figure}
\pagebreak
\section{Simulation}
\subsubsection{Simulation Domain}
We find the scattering solution for an arbitrarily shaped
particle inside a rectangular domain by solving the Maxwell's equations.
A schematic of the domain is shown in the figure
\ref{fig:domainOfInterest}.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.6]{domain_of_interest_2.png}
\caption{\label{fig:domainOfInterest}Figure showing an arbitrarily shaped
particle inside a rectangular domain in which we will solve Maxwell's equations.}
\end{figure}
\subsubsection{Maxwell's Equations}
The Maxwell's Equations are given by(ref \cite{book:electrodynamics_DJ}):
\begin{align} \label{eq:Maxwell_eq}
\nabla \cdot \vec{D} & = \rho \\
\nabla \cdot \vec{B} & = 0 \\
\nabla \times \vec{E} & = -\frac{\partial \vec{B}}{\partial t} \\
\nabla \times \vec{H} & = \vec{J_f}
+ \frac{\partial \vec{D}}{\partial t}
\end{align}
here,
$\vec{E}$ is the electric field.
$\vec{B}$ is the Magnetic field.
$\vec{H}$ is the magnetic field strength.
$\vec{D}$ is the electric displacement field.
$\vec{J_f}$ is the free current density.
\subsubsection{Tools and Methods to be used for simulation}
To solve Maxwell's equations for the domain shown in the figure
\ref{fig:domainOfInterest} we are divide the domain into second order
quadrangular elements. We will then use the Discontinuous Galerkin method
to solve Maxwell's equations over the domain.
Figure \ref{fig:meshingOverDomain} shows an example mesh made
over the domain shown in figure \ref{fig:domainOfInterest}.
Our mesh for solving Maxwell's equations will similar to this.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.6]{meshingOverDomain.png}
\caption{\label{fig:meshingOverDomain}$2^{nd}$ order Quadrangular meshing
done over the domain shown in the figure \ref{fig:domainOfInterest}.}
\end{figure}
\subsection{Code Verification}
To test our code for solving Maxwell's equations in the
domain shown in figure \ref{fig:domainOfInterest}, we plan to compare the
numerical scattering solution against analytic as well
as semi-analytic solutions.
We also plan to test the code by solving the scattering solution for a
homogeneous sphere. The analytic scattering solutions for scattering by
a homogeneous sphere is given by the Mie scattering solutions
\cite{wiki:mie_scattering}. We will compare our numerical solutions
against the Mie scatt ering solutions.
\subsection{Extracting particle characteristics}
The domain of our final version of our simulation will be a rectangular
domain similar to the one shown in figure \ref{fig:domainOfInterest}
but with much more number of particles inside it.
We aim to find the relationships $N(r)$ vs $r$ and $N(e)$ vs $e$ from the
simulation for a given input scattering data. We plan to do this by
iteratively varying the total number of particles $N_0$, and $r$, $e$, and
$\epsilon$(dielectric constant) for each of the particle inside the domain
until we get a scattering solution same as the input scattering
data.
\section{Conclusion}
Through this project we aim to devise an algorithm to extract more
information from the scattering data than is commonly available.
\medskip
\bibliographystyle{unsrt}
\bibliography{references.bib}
\end{document}
