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-% TODO: BerkTobyS1975.000 people trust computers too much

-

-\chapter{Instrumental Development}

-

-\section{Hardware}  % -----------------------------------------------------------------------------

-

-\subsection{Delay Stages}

-

-% TODO: discuss _all 3_ delay configurations.... implications for sign conventions etc

-

-\section{Signal Acquisition}

-

-Old boxcar: 300 ns window, ~10 micosecond delay. Onset of saturation ~2 V.

-

-\subsection{Digital Signal Processing}

-

-% TODO:

-

-\section{Artifacts and Noise}  % ------------------------------------------------------------------

-

-\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}[p!] \label{fig:scatterinterferenceinTrEEold}

-	\centering

-	\includegraphics[scale=0.5]{"instrument/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.}

-\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\gls{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}[p!] \label{fig:scatterinterferenceinTrEEcurrent}

-	\centering

-	\includegraphics[width=7in]{"instrument/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.}

-\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}[p!] \label{fig:ta-chopping-comparison}

-	\centering

-	\includegraphics[scale=0.5]{"instrument/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.}

-\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{Light Generation}  % ---------------------------------------------------------------------

-

-\subsection{Automated OPA Tuning}

-

-\section{Optomechanics}  % ------------------------------------------------------------------------

-

-\subsection{Automated Neutral Density Wheels}