\chapter{Active Correction in MR-CMDS}

\begin{dquote}
  Calibrate, \\
  calibrate, \\
  calibrate, \\
  calibrate, \\
  calibrate, \\
  run.

  \dsignature{Long-hanging poster in Wright Group laser lab.}
\end{dquote}  % TODO: verify this quote

\clearpage

\section{Introduction}  % =========================================================================

MR-CMDS is subject to a number of possible artifacts, many of them stemming from the imperfect
nature of the frequency-tunable light sources we currently have.  %
It is self-evidently desirable to correct these artifacts, when possible.  %
Indeed many of these artifacts, such as OPA power, phase mismatch and absorption effects have
regularly been corrected for.  % TODO: link to examples in applications section
These corrections are applied after measurement, typically including information from other sources
(such as absorption spectra, in the case of absorption effect corrections).

A more interesting class of corrections are ``active'' corrections---that is, corrections that must
be actively applied during acquisition and cannot be applied in post processing.  %
These corrections are more insidious, and they are often neglected because the hardware and/or
software does not allow for them.  %

In this chapter I explore some of these active correction strategies that are useful in the context
of MR-CMDS.  %
Some of these strategies have already been implemented, others are partially implemented, and
others are still just ideas.  %
I hope to show that active correction is a particularly useful strategy in MR-CMDS.  %

\section{Spectral delay correction}  % ============================================================

\section{Poynting correction}  % ==================================================================

\section{Excitation power correction}  % ==========================================================

\section{Chopping}  % =============================================================================

\subsection{Scatter}  % ---------------------------------------------------------------------------

Scatter is a complex microscopic process whereby light traveling through a material elastically
changes its propagation direction.  %
In CMDS we use propagation direction to isolate signal.  %
Scattering samples defeat this isolation step and allow some amount of excitation light to reach
the detector.  %
In homodyne-detected 4WM experiments,
\begin{equation}
I_{\mathrm{detected}} = |E_{\mathrm{4WM}} + E_1 + E_2 + E_{2^\prime}|^2
\end{equation}
Where $E$ is the entire time-dependent complex electromagnetic field.  %
When expanded, the intensity will be composed of diagonal and cross terms:
\begin{equation}
\begin{split}
I_{\mathrm{detected}} = \overline{(E_1+E_2)}E_{2^\prime} + (E_1+E_2)\overline{E_{2^\prime}} + |E_1+E_2|^2 + (E_1+E_2)\overline{E_{\mathrm{4WM}}} \\ + (E_1+E_2)\overline{E_{\mathrm{4WM}}} + \overline{E_{2^\prime}}E_{\mathrm{4WM}} + E_{2^\prime}\overline{E_{\mathrm{4WM}}} + |E_{\mathrm{4WM}}|^2
\end{split}
\end{equation}
A similar expression in the case of heterodyne-detected 4WM is derived by
\textcite{BrixnerTobias2004a}.  %
The goal of any `scatter rejection' processing procedure is to isolate $|E_{\mathrm{4WM}}|^2$ from
the other terms.  %

% TODO: verify derivation

\subsubsection{Abandon the Random Phase Approximation}

\subsubsection{Interference Patterns in TrEE}

TrEE is implicitly homodyne-detected.  %
Scatter from excitation fields will interfere on the amplitude level with TrEE signal, causing
interference patterns that beat in delay and frequency space.  %
The pattern of beating will depend on which excitation field(s) reach(es) the detector, and the
parameterization of delay space chosen.  %

First I focus on the interference patterns in 2D delay space where all excitation fields and the
detection field are at the same frequency.  %

\begin{figure}
	\includegraphics[scale=0.5]{"active_correction/scatter/scatter interference in TrEE old"}
	\caption[Simulated interference paterns in old delay parameterization.]{Numerically simulated
    interference patterns between scatter and TrEE for the old delay parametrization. Each column
    has scatter from a single excitation field. The top row shows the measured intensities, the
    bottom row shows the 2D Fourier transform, with the colorbar's dynamic range chosen to show the
    cross peaks.}
  label{fig:scatterinterferenceinTrEEold}
\end{figure}

Here I derive the slopes of constant phase for the old delay space, where
$\mathrm{d1}=\tau_{2^\prime1}$ and $\mathrm{d2}=\tau_{21}$.  %
For simplicity, I take $\tau_1$ to be $0$, so that $\tau_{21}\rightarrow\tau_2$ and
$\tau_{2^\prime1}\rightarrow\tau_{2^\prime}$.  %
The phase of signal is then
\begin{equation}
\Phi_{\mathrm{sig}} = \mathrm{e}^{-\left((\tau_{2^\prime}-\tau_2)\omega\right)}
\end{equation}
The phase of each excitation field can also be written:
\begin{eqnarray}
\Phi_{1} &=& \mathrm{e}^0 \\
\Phi_{2} &=& \mathrm{e}^{-\tau_2\omega} \\
\Phi_{2^\prime} &=& \mathrm{e}^{-\tau_{2^\prime}\omega}
\end{eqnarray}
The cross term between scatter and signal is the product of $\Phi_\mathrm{sig}$ and $\Phi_\mathrm{scatter}$. The cross terms are:
\begin{eqnarray}
\Delta_{1} = \Phi_{\mathrm{sig}} &=& \mathrm{e}^{-\left((\tau_{2^\prime}-\tau_2)\omega\right)} \\
\Delta_{2} = \Phi_{\mathrm{sig}}\mathrm{e}^{-\tau_2\omega} &=&  \mathrm{e}^{-\left((\tau_{2^\prime}-2\tau_2)\omega\right)}\\
\Delta_{2^\prime} = \Phi_{\mathrm{sig}}\mathrm{e}^{-\tau_{2^\prime}\omega} &=& \mathrm{e}^{-\tau_{2}\omega}
\end{eqnarray}
Figure \ref{fig:scatterinterferenceinTrEEold} presents numerical simulations of scatter interference as a visual aid. See Yurs 2011 \cite{YursLenaA2011a}.
% TODO: Yurs 2011 Data

\begin{figure}
	\includegraphics[width=7in]{"active_correction/scatter/scatter interference in TrEE current"}
	\caption[Simulated interference paterns in current delay parameterization.]{Numerically simulated
    interference patterns between scatter and TrEE for the current delay parametrization. Each
    column has scatter from a single excitation field. The top row shows the measured intensities,
    the bottom row shows the 2D Fourier transform, with the colorbar's dynamic range chosen to show
    the cross peaks.}
  \label{fig:scatterinterferenceinTrEEcurrent}
\end{figure}

Here I derive the slopes of constant phase for the current delay space, where $\mathrm{d1}=\tau_{22^\prime}$ and $\mathrm{d2}=\tau_{21}$. I take $\tau_2$ to be $0$, so that $\tau_{22^\prime}\rightarrow\tau_{2^\prime}$ and $\tau_{21}\rightarrow\tau_1$. The phase of the signal is then
\begin{equation}
\Phi_{\mathrm{sig}} = \mathrm{e}^{-\left((\tau_{2^\prime}+\tau_1)\omega\right)}
\end{equation}
The phase of each excitation field can also be written:
\begin{eqnarray}
\Phi_{1} &=& \mathrm{e}^{-\tau_1\omega} \\
\Phi_{2} &=& \mathrm{e}^{0} \\
\Phi_{2^\prime} &=& \mathrm{e}^{-\tau_{2^\prime}\omega}
\end{eqnarray}
The cross term between scatter and signal is the product of $\Phi_\mathrm{sig}$ and $\Phi_\mathrm{scatter}$. The cross terms are:
\begin{eqnarray}
\Delta_{1} = \Phi_{\mathrm{sig}}\mathrm{e}^{-\tau_1\omega} &=& \mathrm{e}^{-\tau_{2^\prime}\omega} \\
\Delta_{2} = \Phi_{\mathrm{sig}} &=& \mathrm{e}^{-\left((\tau_2+\tau_1)\omega\right)} \\
\Delta_{2^\prime} = \Phi_{\mathrm{sig}}\mathrm{e}^{-\tau_{2^\prime}\omega} &=& \mathrm{e}^{-\tau_1\omega}
\end{eqnarray}
Figure \ref{fig:scatterinterferenceinTrEEcurrent} presents numerical simulations of scatter interference for the current delay parameterization.

\subsubsection{Instrumental Removal of Scatter}

The effects of scatter can be entirely removed from CMDS signal by combining two relatively
straight-forward instrumental techniques: \textit{chopping} and \textit{fibrillation}.  %
Conceptually, chopping removes intensity-level offset terms and fibrillation removes
amplitude-level interference terms.  %
Both techniques work by modulating signal and scatter terms differently so that they may be
separated after light collection.  %

\begin{table}[h] \label{tab:phase_shifted_parallel_modulation}
	\begin{center}
		\begin{tabular}{ r | c | c | c | c }
			& A       & B          & C          & D                       \\
			signal    &            &            & \checkmark &            \\
			scatter 1 &            & \checkmark & \checkmark &            \\
			scatter 2 &            &            & \checkmark & \checkmark \\
			other     & \checkmark & \checkmark & \checkmark & \checkmark
		\end{tabular}
	\end{center}
	\caption[Shot-types in phase shifted parallel modulation.]{Four shot-types in a general phase shifted parallel modulation scheme. The `other' category represents anything that doesn't depend on either chopper, including scatter from other excitation sources, background light, detector voltage offsets, etc.}
\end{table}

We use the dual chopping scheme developed by \textcite{FurutaKoichi2012a} called `phase shifted
parallel modulation'.  %
In this scheme, two excitation sources are chopped at 1/4 of the laser repetition rate (two pulses
on, two pulses off).  %
Very similar schemes are discussed by \textcite{AugulisRamunas2011a} and
\textcite{HeislerIsmael2014a} for two-dimensional electronic spectroscopy.  %
The two chop patterns are phase-shifted to make the four-pulse pattern represented in Table
\ref{tab:phase_shifted_parallel_modulation}.  %
In principle this chopping scheme can be achieved with a single judiciously placed mechanical
chopper - this is one of the advantages of Furuta's scheme.  %
Due to practical considerations we have generally used two choppers, one on each OPA.  %
The key to phase shifted parallel modulation is that signal only appears when both of your chopped
beams are passed.  %
It is simple to show how signal can be separated through simple addition and subtraction of the A,
B, C, and D phases shown in Table \ref{tab:phase_shifted_parallel_modulation}.  %
First, the components of each phase:
\begin{eqnarray}
A &=& I_\mathrm{other} \\
B &=& I_\mathrm{1} + I_\mathrm{other} \\
C &=& I_\mathrm{signal} + I_\mathrm{1} + I_\mathrm{2} + I_\mathrm{other} \\
D &=& I_\mathrm{2} + I_\mathrm{other}
\end{eqnarray}
Grouping into difference pairs,
\begin{eqnarray}
A-B &=& -I_\mathrm{1} \\
C-D &=& I_\mathrm{signal} + I_\mathrm{1} 
\end{eqnarray}
So:
\begin{equation} \label{eq:dual_chopping}
A-B+C-D = I_\mathrm{signal}
\end{equation}
I have ignored amplitude-level interference terms in this treatment because they cannot be removed
via any chopping strategy.  %
Interference between signal and an excitation beam will only appear in `C'-type shots, so it will
not be removed in Equation \ref{eq:dual_chopping}.  %
To remove such interference terms, you must \textit{fibrillate} your excitation fields.

An alternative to dual chopping is single-chopping and `leveling'...  %
this technique was used prior to May 2016 in the Wright Group...  %
`leveling' and single-chopping is also used in some early 2DES work...
\cite{BrixnerTobias2004a}.  %

\begin{figure} 
	\includegraphics[scale=0.5]{"active_correction/scatter/TA chopping comparison"}
	\caption[Comparison of single, dual chopping.]{Comparison of single and dual chopping in a
    MoS\textsubscript{2} transient absorption experiment. Note that this data has not been
    processed in any way - the colorbar represents changes in intensity seen by the detector. The
    grey line near 2 eV represents the pump energy. The inset labels are the number of laser shots
    taken and the chopping strategy used.}
  \label{fig:ta-chopping-comparison}
\end{figure}

Figure \ref{fig:ta-chopping-comparison} shows the effects of dual chopping for some representative
MoS\textsubscript{2} TA data.  %
Each subplot is a probe wigner, with the vertical grey line representing the pump energy.  %
Note that the single chopper passes pump scatter, visible as a time-invariant increase in intensity
when the probe and monochromator are near the pump energy.  %
Dual chopping efficiently removes pump scatter, but at the cost of signal to noise for the same
number of laser shots.  %
Taking twice as many laser shots when dual chopping brings the signal to noise to at least as good
as the original single chopping.  %

Fibrillation is the intentional randomization of excitation phase during an experiment.  %
Because the interference term depends on the phase of the excitation field relative to the signal,
averaging over many shots with random phase will cause the interference term to approach zero.  %
This is a well known strategy for removing unwanted interference terms \cite{SpectorIvanC2015a,
  McClainBrianL2004a}.  %

\subsection{Normalization of dual-chopped self-heterodyned signal}

%\begin{table}[!htb]
%  \centering
%  \renewcommand{\arraystretch}{1.5}
%\begin{array}{r | c | c | c | c }
%  & A           & B & C & D \\ \hline
%  \text{signal}   & S_A=V_A^S   & S_B=R^SC^1I_B^S & S_C=R^S(C^1) & S_D=R^SC^2I_D^S \\ \hline
%  \text{source 1} & M_A^1=V_A^1 & M_B=R^1I_B^1    & M_C^1=R^1I_C^1 & M_D^1=V_D^1 \\ \hline
%  \text{source 2} & M_A^2=V_A^2 & M_B^2=V_B^2     & M_C^2=R^2I_C^2 & M_D^2=R^2I_D^2
%\end{array}
%  \caption{CAPTION}
%\end{table}

Shot-by-shot normalization is not trivial for these experiments.  %
As in table above, with 1 as pump and 2 as probe.  %

Starting with $\Delta I$ from \ref{eq:dual_chopping}, we can normalize by probe intensity to get
the popular $\Delta I / I$ representation.  %
Using the names defined above:
\begin{equation}
  \frac{\Delta I}{I} = \frac{A-B+C-D}{D-A}
\end{equation}
Now consider the presence of excitation intensity monitors, indicated by subscripts PR for probe
and PU for pump.

We can further normalize by the pump intensity by dividing the entire expression by $C_{PU}$:
\begin{equation}
  \frac{\Delta I}{I} = \frac{A-B+C-D}{(D-A)*C_{PU}}
\end{equation}

Now, substituting in BRAZARD formalism:

\begin{eqnarray}
  A &=& constant \\
  B &=& S I_{PU}^B (1+\delta_{PU}^B) \\
  C &=& I_{PR}^C(1+\delta_{PR}^C) + S I_{PU}^C(1+\delta_{PR}^C) \\
  D &=& I_{PR}^D(1+\delta_{PR}^D)
\end{eqnarray}

\begin{equation}
  \frac{\Delta I}{I} = \frac{<A> -
    \frac{<B_{PU}>B}{B_{PU}} +
    \frac{<C_{PU}><C_{PR}C}{C_{PU}C_{PR}} -
    \frac{<D_{PR}>D}{D_{PR}}}{<PR><PU>}
\end{equation}

\section{Fibrillation}  % =========================================================================

\section{Conclusions}  % ==========================================================================