From 9d89c09dfe49aba4c68b6911600715add419babd Mon Sep 17 00:00:00 2001
From: Blaise Thompson <blaise@untzag.com>
Date: Tue, 27 Feb 2018 23:58:32 -0600
Subject: 2018-02-27 23:58

---
 active_correction/chapter.tex | 263 ++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 263 insertions(+)
 create mode 100644 active_correction/chapter.tex

(limited to 'active_correction/chapter.tex')

diff --git a/active_correction/chapter.tex b/active_correction/chapter.tex
new file mode 100644
index 0000000..6d429a0
--- /dev/null
+++ b/active_correction/chapter.tex
@@ -0,0 +1,263 @@
+% TODO: BerkTobyS1975.000 people trust computers too much
+
+\chapter{Active Correction in MR-CMDS}
+
+\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{dfigure}
+	\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{dfigure}
+
+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{dfigure}
+	\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{dfigure}
+
+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{dfigure} 
+	\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{dfigure}
+
+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}
-- 
cgit v1.2.3

