From 9c6afe7dc748bf26e4ab6f6df0c70d57e74b610f Mon Sep 17 00:00:00 2001
From: Blaise Thompson <blaise@untzag.com>
Date: Sat, 7 Apr 2018 14:56:52 -0500
Subject: 2018-04-07 14:56

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 quantitative_ta/chapter.tex | 78 +++++++++++++++++++++++++++++++++++++++++++++
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+\chapter{Quantitative transient absorbance} \label{cha:qta}
+
+\subsubsection{Quantitative TA}
+
+Transient absorbance (TA) spectroscopy is a self-heterodyned technique.  %
+Through chopping you can measure nonlinearities quantitatively much easier than with homodyne
+detected (or explicitly heterodyned) experiments.
+
+\begin{figure}
+	\includegraphics[width=\textwidth]{"spectroscopy/TA setup"}
+	\label{fig:ta_and_tr_setup}
+	\caption{CAPTION TODO}
+\end{figure}
+
+\autoref{fig:ta_and_tr_setup} diagrams the TA measurement for a generic sample.  %
+Here I show measurement of both the reflected and transmitted probe beam \dots not important in
+opaque (pyrite) or non-reflective (quantum dot) samples \dots  %
+
+Typically one attempts to calculate the change in absorbance $\Delta A$ \dots  %
+
+\begin{eqnarray}
+\Delta A &=& A_{\mathrm{on}} - A_{\mathrm{off}} \\
+&=& -\log_{10}\left(\frac{I_\mathrm{T}+I_\mathrm{R}+I_{\Delta\mathrm{T}} + I_{\Delta\mathrm{R}}}{I_0}\right) + \log\left(\frac{I_\mathrm{T}+I_\mathrm{R}}{I_0}\right) \\
+&=& -\left(\log_{10}(I_\mathrm{T}+I_\mathrm{R}+I_{\Delta\mathrm{T}}+ I_{\Delta\mathrm{R}})-\log_{10}(I_0)\right)+\left(\log_{10}(I_\mathrm{T}+I_\mathrm{R})-\log_{10}(I_0)\right) \\
+&=& -\left(\log_{10}(I_\mathrm{T}+I_\mathrm{R}+I_{\Delta\mathrm{T}}+ I_{\Delta\mathrm{R}})-\log_{10}(I_\mathrm{T}+I_\mathrm{R})\right) \\
+&=& -\log_{10}\left(\frac{I_\mathrm{T}+I_\mathrm{R}+I_{\Delta\mathrm{T}}+ I_{\Delta\mathrm{R}}}{I_\mathrm{T}+I_\mathrm{R}}\right) \label{eq:ta_complete}
+\end{eqnarray}
+
+\autoref{eq:ta_complete} simplifies beautifully  if reflectivity is negligible \dots
+
+Now I define a variable for each experimental measurable:
+\begin{center}
+	\begin{tabular}{c | l}
+		$V_\mathrm{T}$ & voltage recorded from transmitted beam, without pump \\
+		$V_\mathrm{R}$ & voltage recorded from reflected beam, without pump \\
+		$V_{\Delta\mathrm{T}}$ & change in voltage recorded from transmitted beam due to pump \\
+		$V_{\Delta\mathrm{R}}$ & change in voltage recorded from reflected beam due to pump
+	\end{tabular}
+\end{center}
+
+We will need to calibrate using a sample with a known transmisivity and reflectivity constant:
+\begin{center}
+	\begin{tabular}{c | l}
+		$V_{\mathrm{T},\,\mathrm{ref}}$ & voltage recorded from transmitted beam, without pump \\
+		$V_{\mathrm{R},\,\mathrm{ref}}$ & voltage recorded from reflected beam, without pump \\
+		$\mathcal{T}_\mathrm{ref}$ & transmissivity \\
+		$\mathcal{R}_\mathrm{ref}$ & reflectivity
+	\end{tabular}
+\end{center}
+
+Define two new proportionality constants...
+\begin{eqnarray}
+C_\mathrm{T} &\equiv& \frac{\mathcal{T}}{V_\mathrm{T}} \\
+C_\mathrm{R} &\equiv& \frac{\mathcal{R}}{V_\mathrm{R}}
+\end{eqnarray}
+These are explicitly calibrated (as a function of probe color) prior to the experiment using the
+calibration sample.  %
+
+Given the eight experimental measurables ($V_\mathrm{T}$, $V_\mathrm{R}$, $V_{\Delta\mathrm{T}}$,
+$V_{\Delta\mathrm{R}}$, $V_{\mathrm{T},\,\mathrm{ref}}$, $V_{\mathrm{R},\,\mathrm{ref}}$,
+$\mathcal{T}_\mathrm{ref}$, $\mathcal{R}_\mathrm{ref}$) I can express all of the intensities in
+\autoref{eq:ta_complete} in terms of $I_0$.  %
+
+\begin{eqnarray}
+C_\mathrm{T} &=& \frac{\mathcal{T}_\mathrm{ref}}{V_{\mathrm{T},\,\mathrm{ref}}} \\
+C_\mathrm{R} &=& \frac{\mathcal{R}_\mathrm{ref}}{V_{\mathrm{R},\,\mathrm{ref}}} \\
+I_\mathrm{T} &=& I_0 C_\mathrm{T} V_\mathrm{T} \\
+I_\mathrm{R} &=& I_0 C_\mathrm{R} V_\mathrm{R} \\
+I_{\Delta\mathrm{T}} &=& I_0 C_\mathrm{T} V_{\Delta\mathrm{T}} \\
+I_{\Delta\mathrm{R}} &=& I_0 C_\mathrm{R} V_{\Delta\mathrm{R}}
+\end{eqnarray} 
+
+Wonderfully, the $I_0$ cancels when plugged back in to \autoref{eq:ta_complete}, leaving a final
+expression for $\Delta A$ that only depends on my eight measurables.  %
+
+\begin{equation}
+\Delta A = - \log_{10} \left(\frac{C_\mathrm{T}(V_\mathrm{T} + V_{\Delta\mathrm{T}}) + C_\mathrm{R}(V_\mathrm{R} + V_{\Delta\mathrm{R}})}{C_\mathrm{T} V_\mathrm{T} + C_\mathrm{R} V_\mathrm{R}}\right)
+\end{equation}
\ No newline at end of file
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