dual chopping normalization

1 files changed, 108 insertions, 18 deletions

diff --git a/instrument/chapter.tex b/instrument/chapter.tex
index 6eee6e8..ffb8eb5 100644
--- a/instrument/chapter.tex
+++ b/instrument/chapter.tex
@@ -2,7 +2,7 @@
 

 \chapter{Instrumental Development}

 

-\section{Hardware}

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

 

 \subsection{Delay Stages}

 

@@ -16,23 +16,30 @@ Old boxcar: 300 ns window, ~10 micosecond delay. Onset of saturation ~2 V.
 

 % TODO:

 

-

-

-\section{Artifacts and Noise}

+\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,

+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:

+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{BrixnerTobias2004.000}. The goal of any `scatter rejection' processing procedure is to isolate $|E_{\mathrm{4WM}}|^2$ from the other terms. 

+A similar expression in the case of heterodyne-detected 4WM is derived by

+\textcite{BrixnerTobias2004.000}.  %

+The goal of any `scatter rejection' processing procedure is to isolate $|E_{\mathrm{4WM}}|^2$ from

+the other terms.  %

 

 % TODO: verify derivation

 

@@ -40,16 +47,25 @@ A similar expression in the case of heterodyne-detected 4WM is derived by \textc
 

 \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.

+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.

+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

+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}

@@ -94,7 +110,12 @@ Figure \ref{fig:scatterinterferenceinTrEEcurrent} presents numerical simulations
 

 \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.

+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}

@@ -109,7 +130,22 @@ The effects of scatter can be entirely removed from CMDS signal by combining two
 	\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{FurutaKoichi2012.000} 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{AugulisRamunas2011.000} and \textcite{HeislerIsmael2014.000} 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:

+We use the dual chopping scheme developed by \textcite{FurutaKoichi2012.000} 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{AugulisRamunas2011.000} and

+\textcite{HeislerIsmael2014.000} 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} \\

@@ -125,9 +161,16 @@ 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.

+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{BrixnerTobias2004.000}.

+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{BrixnerTobias2004.000}.  %

 

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

 	\centering

@@ -135,15 +178,62 @@ An alternative to dual chopping is single-chopping and `leveling'... this techni
 	\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.

+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{SpectorIvanC2015.000,

+  McClainBrianL2004.000}.  %

+

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

+

+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}

 

-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{SpectorIvanC2015.000, McClainBrianL2004.000}.

+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}

+\section{Light Generation}  % ---------------------------------------------------------------------

 

 \subsection{Automated OPA Tuning}

 

-\section{Optomechanics}

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

 

 \subsection{Automated Neutral Density Wheels}