\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