\chapter{Quantitative differential absorbance} \label{cha:qta}

\clearpage

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{qta:eqn:ta_complete}
\end{eqnarray}

\autoref{qta:eqn: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{qta:eqn: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 into \autoref{qta:eqn: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}