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diff --git a/quantitative_ta/chapter.tex b/quantitative_ta/chapter.tex new file mode 100644 index 0000000..c8b594d --- /dev/null +++ b/quantitative_ta/chapter.tex @@ -0,0 +1,78 @@ +\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}
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