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Authors: Mani Chandra, Balavarun P, Aman Abhishek Tiwari
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\begin{document}
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\title{Determining size and geometry of the particles from the
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polarisation change of the light scattered from the particles}
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\author{Aman Abhishek Tiwari}
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\author{Dr. Pankaj Jain(Supervisor)}
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\affiliation{Indian Institute of Technology Kanpur, Department of Physics}
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\date{\today}
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\begin{abstract}
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An important problem in atmospheric physics is to characterize the
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ambient aerosol distribution. While a majority of current laser-based
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detectors can measure the size spectrum of the scattering particles,
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they do not give information about the geometry of the scatterers. We
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aim to compute the effect of the scatterers on the polarization of the
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incoming radiation and to use the measured radiation to infer the size
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as well as the geometry. In order to do so, we will write a code to
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solve Maxwell's equations for arbitrary geometries using the
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Discontinuous Galerkin method and then use this code to explore the
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effect of scatterer geometry on the incoming radiation.
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\end{abstract}
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\maketitle
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\section{Introduction}
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Current optical aerosol counters used to characterize the particulate
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matter suspended in air are only capable of measuring the size spectrum
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of the particles, but they assume that the particles are spherical in
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shape.
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We aim to recover additional information about the particles by modeling
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the particles as an ellipsoid and calculate the size spectrum $N(r)$ where
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$r$ defines the size of the particles, eccentricity spectrum $N(e)$ and
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where $e$ is the eccentricity of the particles.
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\subsection{Experimental Setup}
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A typical aerosol counter setup has a LASER light source, a sample
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chamber in which air sample to be investigated is held and an array of
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photosensitive detectors which detects the scattered light, it will
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provide us our scattering data. Figure.\ref{fig:aerosol_counter} shows a
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schematic of a typical aerosol counter.
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\begin{figure}[h!]
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\centering
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\includegraphics[scale=1.0]{aerosol_counter.png}
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\caption{\label{fig:aerosol_counter}Schematic of a typical aerosol
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counter}
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\end{figure}
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\pagebreak
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\section{Simulation}
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\subsubsection{Simulation Domain}
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We find the scattering solution for an arbitrarily shaped
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particle inside a rectangular domain by solving the Maxwell's equations.
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A schematic of the domain is shown in the figure
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\ref{fig:domainOfInterest}.
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\begin{figure}[h!]
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\centering
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\includegraphics[scale=0.6]{domain_of_interest_2.png}
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\caption{\label{fig:domainOfInterest}Figure showing an arbitrarily shaped
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particle inside a rectangular domain in which we will solve Maxwell's equations.}
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\end{figure}
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\subsubsection{Maxwell's Equations}
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The Maxwell's Equations are given by(ref \cite{book:electrodynamics_DJ}):
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\begin{align} \label{eq:Maxwell_eq}
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\nabla \cdot \vec{D} & = \rho \\
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\nabla \cdot \vec{B} & = 0 \\
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\nabla \times \vec{E} & = -\frac{\partial \vec{B}}{\partial t} \\
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\nabla \times \vec{H} & = \vec{J_f}
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+ \frac{\partial \vec{D}}{\partial t}
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\end{align}
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here,
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$\vec{E}$ is the electric field.
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$\vec{B}$ is the Magnetic field.
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$\vec{H}$ is the magnetic field strength.
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$\vec{D}$ is the electric displacement field.
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$\vec{J_f}$ is the free current density.
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\subsubsection{Tools and Methods to be used for simulation}
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To solve Maxwell's equations for the domain shown in the figure
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\ref{fig:domainOfInterest} we are divide the domain into second order
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quadrangular elements. We will then use the Discontinuous Galerkin method
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to solve Maxwell's equations over the domain.
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Figure \ref{fig:meshingOverDomain} shows an example mesh made
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over the domain shown in figure \ref{fig:domainOfInterest}.
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Our mesh for solving Maxwell's equations will similar to this.
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\begin{figure}[h!]
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\centering
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\includegraphics[scale=0.6]{meshingOverDomain.png}
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\caption{\label{fig:meshingOverDomain}$2^{nd}$ order Quadrangular meshing
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done over the domain shown in the figure \ref{fig:domainOfInterest}.}
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\end{figure}
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\subsection{Code Verification}
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To test our code for solving Maxwell's equations in the
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domain shown in figure \ref{fig:domainOfInterest}, we plan to compare the
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numerical scattering solution against analytic as well
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as semi-analytic solutions.
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We also plan to test the code by solving the scattering solution for a
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homogeneous sphere. The analytic scattering solutions for scattering by
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a homogeneous sphere is given by the Mie scattering solutions
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\cite{wiki:mie_scattering}. We will compare our numerical solutions
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against the Mie scatt ering solutions.
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\subsection{Extracting particle characteristics}
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The domain of our final version of our simulation will be a rectangular
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domain similar to the one shown in figure \ref{fig:domainOfInterest}
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but with much more number of particles inside it.
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We aim to find the relationships $N(r)$ vs $r$ and $N(e)$ vs $e$ from the
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simulation for a given input scattering data. We plan to do this by
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iteratively varying the total number of particles $N_0$, and $r$, $e$, and
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$\epsilon$(dielectric constant) for each of the particle inside the domain
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until we get a scattering solution same as the input scattering
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data.
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\section{Conclusion}
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Through this project we aim to devise an algorithm to extract more
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information from the scattering data than is commonly available.
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\medskip
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\bibliographystyle{unsrt}
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\bibliography{references.bib}
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% \begin{thebibliography}{5}
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% \bibitem{WHO_aq_guide}Ambient (outdoor) air quality and health, WHO
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% \url{<http://www.who.int/mediacentre/factsheets/fs313/en/>}
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% \bibitem{mie_scattering}Mie scattering, Wikipedia
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% \url{<https://en.wikipedia.org/wiki/Mie_scattering>}
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% \end{thebibliography}
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\end{document}
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