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diff --git a/irf/chapter.tex b/irf/chapter.tex new file mode 100644 index 0000000..07cd561 --- /dev/null +++ b/irf/chapter.tex @@ -0,0 +1,69 @@ +\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!
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