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 \chapter{Acquistion}

 

-Details of acquisition software.  %
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+% TODO: cool quote, if I can think of one

+

+\clearpage

+

+In the Wright Group, \gls{PyCMDS} replaces the old acquisition softwares `ps control', written by

+Kent Meyer and `Control for Lots of Research in Spectroscopy' written by Schuyler Kain.

+

+PyCMDS directly addresses the hardware during experiments.

+

+\section{Overview}  % =============================================================================

+

+PyCMDS has, through software improvements alone, dramatically lessened scan times...

+

+\begin{itemize}[topsep=-1.5ex, itemsep=0ex, partopsep=0ex, parsep=0ex, label=$\rightarrow$]

+	\item simultaneous motor motion

+	\item digital signal processing  % TODO: reference section when it exists

+	\item ideal axis positions \ref{sec:ideal_axis_positions}

+\end{itemize}

+

+\section{Future directions}  % ====================================================================

+

+\subsection{Ideal Axis Positions}\label{sec:ideal_axis_positions}  % ------------------------------

+

+Frequency domain multidimensional spectroscopy is a time-intensive process.  %

+A typical \gls{pixel} takes between one-half second and three seconds to acquire.  %

+Depending on the exact hardware being scanned and signal being detected, this time may be mostly

+due to hardware motion or signal collection.  %

+Due to the \gls{curse of dimensionality}, a typical three-dimensional CMDS experiment contains

+roughly 100,000 pixels.  %

+CMDS hardware is transiently-reliable, so speeding up experiments is a crucial component of

+unlocking ever larger dimensionalities and higher resolutions.  %

+

+One obvious way to decrease the scan-time is to take fewer pixels.  %

+Traditionally, multidimensional scans are done with linearly arranged points in each axis---this is

+the simplest configuration to program into the acquisition software.  %

+Because signal features are often sparse or slowly varying (especially so in high-dimensional

+scans) linear stepping means that \emph{most of the collected pixels} are duplicates or simply

+noise.  %

+A more intelligent choice of axis points can capture the same nonlinear spectrum in a fraction of

+the total pixel count.  %

+

+An ideal distribution of pixels is linearized in \emph{signal}, not coordinate.  %

+This means that every signal level (think of a contour in the N-dimensional case) has roughly the

+same number of pixels defining it.  %

+If some generic multidimensional signal goes between 0 and 1, one would want roughly 10\% of the

+pixels to be between 0.9 and 1.0, 10\% between 0.8 and 0.9 and so on.  %

+If the signal is sparse in the space explored (imagine a narrow two-dimensional Lorentzian in the

+center of a large 2D-Frequency scan) this would place the majority of the pixels near the narrow

+peak feature(s), with only a few of them defining the large (in axis space) low-signal floor.  %

+In contrast linear stepping would allocate the vast majority of the pixels in the low-signal 0.0 to

+0.1 region, with only a few being used to capture the narrow peak feature.  %

+Of course, linearizing pixels in signal requires prior expectations about the shape of the

+multidimensional signal---linear stepping is still an appropriate choice for low-resolution

+``survey'' scans.  %

+

+CMDS scans often posses correlated features in the multidimensional space.  %

+In order to capture such features as cheaply as possible, one would want to define regions of

+increased pixel density along the correlated (diagonal) lineshape.  %

+As a concession to reasonable simplicity, our acquisition software (PyCMDS) assumes that all scans

+constitute a regular array with-respect-to the scanned axes.  %

+We can acquire arbitrary points along each axis, but not for the multidimensional scan.  %

+This means that we cannot achieve strictly ideal pixel distributions for arbitrary datasets.  %

+Still, we can do much better than linear spacing.

+% TODO: refer to PyCMDS/WrightTools 'regularity' requirement when that section exists

+

+Almost all CMDS lineshapes (in frequency and delay) can be described using just a few lineshape

+functions:

+\begin{ditemize}

+	\item exponential

+	\item Gaussian

+	\item Lorentzian

+	\item bimolecular

+\end{ditemize}

+

+Exponential and bimolecular dynamics fall out of simple first and second-order kinetics (I will

+ignore higher-order kinetics here).  %

+Gaussians come from our Gaussian pulse envelopes or from normally-distributed inhomogeneous

+broadening.  %

+The measured line-shapes are actually convolutions of the above.  %

+I will ignore the convolution except for a few illustrative special cases.  %

+More exotic lineshapes are possible in CMDS---quantum beating and breathing modes, for example---I

+will also ignore these.  %

+Derivations of the ideal pixel positions for each of these lineshapes appear below.

+% TODO: cite Wright Group quantum beating paper, Kambempati breathing paper

+

+\subsection{Exponential}

+

+Simple exponential decays are typically used to describe population and coherence-level dynamics in

+CMDS.  %

+For some generic exponential signal $S$ with time constant $\tau$,

+\begin{equation} \label{eq:simple_exponential_decay}

+S(t) = \me^{-\frac{t}{\tau}}.

+\end{equation}

+We can write the conjugate equation to \ref{eq:simple_exponential_decay}, asking ``what $t$ do I

+need to get a cerain signal level?'':

+\begin{eqnarray}

+\log{(S)} &=& -\frac{t}{\tau} \\

+t &=& -\taulog{(S)}.

+\end{eqnarray}

+So to step linearly in $t$, my step size has to go as $-\tau\log{(S)}$.

+

+We want to go linearly in signal, meaning that we want to divide $S$ into even sections.  %

+If $S$ goes from 0 to 1 and we choose to acquire $N$ points,

+\begin{eqnarray}

+t_n &=& -\tau\log{\left(\frac{n}{N}\right)}.

+\end{eqnarray}

+Note that $t_n$ starts at long times and approaches zero delay.  %

+So the first $t_1$ is the smallest signal and $t_N$ is the largest.  %

+

+Now we can start to consider realistic cases, like where $\tau$ is not quite known and where some

+other longer dynamics persist (manifested as a static offset).  %

+Since these values are not separable in a general system, I'll keep $S$ normalized between 0 and

+1.  %

+\begin{eqnarray}

+S &=& (1-c)\me^{-\frac{t}{\tau_{\mathrm{actual}}}} + c \\

+S_n &=& (1-c)\me^{-\frac{-\tau_{\mathrm{step}}\log{\left(\frac{n}{N}\right)}}{\tau_{\mathrm{actual}}}} + c \\

+S_n &=& (1-c)\me^{-\frac{\tau_{\mathrm{step}}}{\tau_{\mathrm{actual}}} \log{\left(\frac{N}{n}\right)}} + c \\

+S_n &=& (1-c)\left(\frac{N}{n}\right)^{-\frac{\tau_{\mathrm{step}}}{\tau_{\mathrm{actual}}}} + c \\

+S_n &=& (1-c)\left(\frac{n}{N}\right)^{\frac{\tau_{\mathrm{step}}}{\tau_{\mathrm{actual}}}} + c

+\end{eqnarray}

+

+\begin{dfigure}[p!]

+	\includegraphics[scale=0.5]{"processing/PyCMDS/ideal axis positions/exponential"}

+	\caption[TODO]{TODO}

+  \label{fig:exponential_steps}

+\end{dfigure}

+

+\subsubsection{Gaussian}

+

+\subsubsection{Lorentzian}

+

+\subsubsection{Bimolecular}
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