From 9d89c09dfe49aba4c68b6911600715add419babd Mon Sep 17 00:00:00 2001 From: Blaise Thompson 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{ - + \frac{B}{B_{PU}} + + \frac{D}{D_{PR}}}{} +\end{equation} + +\section{Light Generation} % --------------------------------------------------------------------- + +\subsection{Automated OPA Tuning} + +\section{Optomechanics} % ------------------------------------------------------------------------ + +\subsection{Automated Neutral Density Wheels} -- cgit v1.2.3