From ff2e66aeef80259d60df46261305d387ad604baa Mon Sep 17 00:00:00 2001
From: Blaise Thompson <blaise@untzag.com>
Date: Thu, 5 Apr 2018 17:29:20 -0500
Subject: 2018-04-05 17:29

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+\chapter{Instrumental response function} \label{cha:irf}
+
+The instrumental response function (IRF) is a classic concept in analytical science.  %
+Defining IRF becomes complex with instruments as complex as these, but it is still useful to
+attempt.  %
+
+It is particularly useful to define bandwidth.
+
+\subsubsection{Time Domain}
+
+I will use four wave mixing to extract the time-domain pulse-width.   %
+I use a driven signal \textit{e.g.} near infrared carbon tetrachloride response.  %
+I'll homodyne-detect the output.  %
+In my experiment I'm moving pulse 1 against pulses 2 and 3 (which are coincident).  %
+
+The driven polarization, $P$, goes as the product of my input pulse \textit{intensities}:
+
+\begin{equation}
+P(T) = I_1(t-T) \times I_2(t) \times I_3(t)
+\end{equation}
+
+In our experiment we are convolving $I_1$ with $I_2 \times I_3$.  %
+Each pulse has an \textit{intensity-level} width, $\sigma_1$, $\sigma_2$, and $\sigma_3$. $I_2
+\times I_3$ is itself a Gaussian, and  
+\begin{eqnarray}
+\sigma_{I_2I_3} &=& \dots \\
+&=& \sqrt{\frac{\sigma_2^2\sigma_3^2}{\sigma_2^2 + \sigma_3^2}}.
+\end{eqnarray}
+
+The width of the polarization (across $T$) is therefore
+
+\begin{eqnarray}
+\sigma_P &=& \sqrt{\sigma_1^2 + \sigma_{I_2I_3}^2} \\
+&=& \dots \\ 
+&=& \sqrt{\frac{\sigma_1^2 + \sigma_2^2\sigma_3^2}{\sigma_1^2 + \sigma_2^2}}. \label{eq:generic}
+\end{eqnarray}
+
+% TODO: determine effect of intensity-level measurement here
+
+I assume that all of the pulses have the same width.   %
+$I_1$, $I_2$, and $I_3$ are identical Gaussian functions with FWHM $\sigma$. In this case,
+\autoref{eq:generic} simplifies to  
+
+\begin{eqnarray}
+\sigma_P &=& \sqrt{\frac{\sigma^2 + \sigma^2\sigma^2}{\sigma^2 + \sigma^2}} \\
+&=& \dots \\
+&=& \sigma \sqrt{\frac{3}{2}}
+\end{eqnarray}
+
+Finally, since we measure $\sigma_P$ and wish to extract $\sigma$:
+
+\begin{equation}
+\sigma = \sigma_P \sqrt{\frac{2}{3}}
+\end{equation}
+
+Again, all of these widths are on the \textit{intensity} level.
+
+\subsubsection{Frequency Domain}
+
+We can directly measure $\sigma$ (the width on the intensity-level) in the frequency domain using a
+spectrometer.  %
+A tune test contains this information.  %
+
+\subsubsection{Time-Bandwidth Product}
+
+For a Gaussian, approximately 0.441
+
+% TODO: find reference
+% TODO: number defined on INTENSITY level!
\ No newline at end of file
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