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\section*{Overview of VASE: Basic Theory and Applications}
\begin{itemize}
	\item Reference: J.A Woolam\cite{woolam1}
\end{itemize}
\subsection*{Introduction}
\begin{itemize}
	\item Mathematical theory is based on Fresnel reflection/transmission equations
	\begin{itemize}
		\item These come from solutions to Maxwell's equations
		\item p polarised: $\vect{E}$ field parallel to plane of incidence
		\item s polarised: $\vect{E}$ field perpendicular to plane of incidence (parallel to surface)
	\end{itemize}
	\item Ellipsometric measurement:
	\begin{align*}
		\tan(\Psi) e^{i \Delta} = \rho = \frac{r_p}{r_s}
	\end{align*}
	\item Spectroscopic Ellipsometry (SE): Measure $\rho$ as a function of $\lambda$
	\item Variable Angle SE (VASE): Measure $\rho$ as a function of $\lambda$ and angle of incidence.
	\item Measure \emph{ratio} of 2 values $\implies$ accurate and reproducable
	\item Measure $\Delta$ (phase quantity), sensitive to presence of thin films
	\item VASE gathers many data points off a single sample, and is well suited to modelling and fitting
\end{itemize}
\subsection*{What can be Determined by VASE?}
\begin{itemize}
	\item Layer thickness
	\item Surface/interfacial roughness
	\item Optical constants $\implies$ any parameter that depends on these
	\item Gradients in properties vs depth in film
	\item Optical anisotropy
	\begin{align*}
		\langle\epsilon\rangle &= \langle\epsilon_1\rangle + i \langle\epsilon_2\rangle \\
				&= \langle\phasor{n}\rangle^2 = \left( \langle n\rangle + i \langle k \rangle \right)^2 \\
				&= \sin(\phi)^2 \left[ 1 + \tan(\phi)^2 \left(\frac{1 - \rho}{1 + \rho}\right)\right]
	\end{align*}
\end{itemize}
\pagebreak
\subsection*{Data Analysis}
\begin{itemize}
	\item Ellipsometry doesn't directly measure film parameters; measures $\Psi$ and $\Delta$
	\item Necessary to perform a model dependent analysis of $\Psi$ and $\Delta$ data
	\item VASE increases data points recorded $\implies$ good for fitting model to data
	\item Procedure:
	\begin{enumerate}
		\item Aquire Data
		\item Use assumed model to predict expected data
		\item Compare generated to experimental data, and adjust model parameters to fit
	\end{enumerate}
	\item Fitting algorithm - Marquardt-Levenberg
	\item Objective - minimise (root) Mean Squared Error (MSE)
	\begin{align*}
		\text{MSE} &= \sqrt{\frac{1}{2N - M} \displaystyle\sum_{i=1}^N \left[ \left(\frac{\Psi_i^{\text{Mod}} - \Psi_i^{\text{Exp}}}{\sigma_{\Psi_i}^{\text{Exp}}} \right)^2 + \left(\frac{\Delta_i^{\text{Mod}} - \Delta_i^{\text{Exp}}}{\sigma_{\Delta_i}^{\text{Exp}}} \right)^2\right]}
	\end{align*}
	\item Iterative process; start with simple model and refine
	\item ``...the danger in making the model more complex is that paramters become correlated, in which case multiple sets of paramters will give the same good MSE fit.''
\end{itemize}

\subsection*{Considerations for VASE Analysis}
\begin{itemize}
	\item Initial guesses must be close to the actual value
	\item This way the final MSE is the actual best fit, not a local minima as shown in \ref{MSE-woolam1}
\end{itemize}
	
\begin{center}
	\includegraphics[scale=0.80]{MSE-woolam1.png}
	\captionof{figure}{From \cite{woolam1}}
	\label{MSE-woolam1}
\end{center}

\subsection*{Building Optical Model}
\begin{itemize}
	\item Must have enough flexibility to accurately fit the experimental data
	\item Example: Surface roughness added to model increases accuracy of fit
\end{itemize}

\section*{Harris 1953}
\begin{itemize}
	\item Reference \cite{harris53}
	\item Measure conductivity of gold-blacks from reflection & transmission measurements in far IR
	\item Conductivity depends on wavelength
	\begin{enumerate}
		\item ``Condenser Effect''
		\begin{itemize}
			\item Structure of material; yarn like metal strands
			\item Gaps in metal strands act as condensers
			\item ``optical conductivity'' is actually an admittivity
			\item As $f$ of radiation increases, more strands capable of conducting current
			\item $\implies$ conductivity decreases as $\lambda$ increases
		\end{itemize}
		\item ``Relaxation Effect''
		\begin{itemize}
			\item Finite relaxation time of Electrons (Reference Drude and Zener)
			\item Electrons lag behind imposed EMF. Lag increases as $f$ increases.
			\item $\implies$ conductivity increases as $\lambda$ increases
		\end{itemize}
		\item At Resonance frequencies, optical absorbtivity (and hence conductivity) passes through a maximum.
	\end{enumerate}
	\item For metal blacks, the effects of the 3 factors dominate in different regions of $\lambda$
	\begin{itemize}
		\item Resonance $f$ lie in visible and near IR regions
		\item $\lambda > 100$microns
	\end{itemize}
	\item Relaxation effect is the dominant effect considered in this paper.
\end{itemize}


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