2018-04-09 00:24

36 files changed, 4089 insertions, 83 deletions

diff --git a/PbSe/chapter.tex b/PbSe/chapter.tex
deleted file mode 100644
index 466a7d9..0000000
--- a/PbSe/chapter.tex
+++ /dev/null
@@ -1,3 +0,0 @@
-\chapter{PbSe} \label{cha:pbx}
-
-Relevant content from PbSe publications goes here.
\ No newline at end of file
diff --git a/PbSe_global_analysis/.gitignore b/PbSe_global_analysis/.gitignore
new file mode 100644
index 0000000..0e5cf51
--- /dev/null
+++ b/PbSe_global_analysis/.gitignore
@@ -0,0 +1,234 @@
+*.png
+*.#*
+
+## Simulation files
+simulations/ellipticity/*
+simulations/NISE_output/Batch A/*
+simulations/NISE_output/Batch B/*
+
+*.h5
+
+## Core latex/pdflatex auxiliary files:
+*.aux
+*.lof
+*.log
+*.lot
+*.fls
+*.out
+*.toc
+*.fmt
+*.fot
+*.cb
+*.cb2
+
+## Intermediate documents:
+*.dvi
+*-converted-to.*
+*.pdf
+
+## Generated if empty string is given at "Please type another file name for output:"
+.pdf
+
+## Bibliography auxiliary files (bibtex/biblatex/biber):
+*.bbl
+*.bcf
+*.blg
+*-blx.aux
+*-blx.bib
+*.run.xml
+
+## Build tool auxiliary files:
+*.fdb_latexmk
+*.synctex
+*.synctex(busy)
+*.synctex.gz
+*.synctex.gz(busy)
+*.pdfsync
+
+## Auxiliary and intermediate files from other packages:
+# algorithms
+*.alg
+*.loa
+
+# achemso
+acs-*.bib
+
+# amsthm
+*.thm
+
+# beamer
+*.nav
+*.pre
+*.snm
+*.vrb
+
+# changes
+*.soc
+
+# cprotect
+*.cpt
+
+# elsarticle (documentclass of Elsevier journals)
+*.spl
+
+# endnotes
+*.ent
+
+# fixme
+*.lox
+
+# feynmf/feynmp
+*.mf
+*.mp
+*.t[1-9]
+*.t[1-9][0-9]
+*.tfm
+
+#(r)(e)ledmac/(r)(e)ledpar
+*.end
+*.?end
+*.[1-9]
+*.[1-9][0-9]
+*.[1-9][0-9][0-9]
+*.[1-9]R
+*.[1-9][0-9]R
+*.[1-9][0-9][0-9]R
+*.eledsec[1-9]
+*.eledsec[1-9]R
+*.eledsec[1-9][0-9]
+*.eledsec[1-9][0-9]R
+*.eledsec[1-9][0-9][0-9]
+*.eledsec[1-9][0-9][0-9]R
+
+# glossaries
+*.acn
+*.acr
+*.glg
+*.glo
+*.gls
+*.glsdefs
+
+# gnuplottex
+*-gnuplottex-*
+
+# gregoriotex
+*.gaux
+*.gtex
+
+# hyperref
+*.brf
+
+# knitr
+*-concordance.tex
+# TODO Comment the next line if you want to keep your tikz graphics files
+*.tikz
+*-tikzDictionary
+
+# listings
+*.lol
+
+# makeidx
+*.idx
+*.ilg
+*.ind
+*.ist
+
+# minitoc
+*.maf
+*.mlf
+*.mlt
+*.mtc[0-9]*
+*.slf[0-9]*
+*.slt[0-9]*
+*.stc[0-9]*
+
+# minted
+_minted*
+*.pyg
+
+# morewrites
+*.mw
+
+# nomencl
+*.nlo
+
+# pax
+*.pax
+
+# pdfpcnotes
+*.pdfpc
+
+# sagetex
+*.sagetex.sage
+*.sagetex.py
+*.sagetex.scmd
+
+# scrwfile
+*.wrt
+
+# sympy
+*.sout
+*.sympy
+sympy-plots-for-*.tex/
+
+# pdfcomment
+*.upa
+*.upb
+
+# pythontex
+*.pytxcode
+pythontex-files-*/
+
+# thmtools
+*.loe
+
+# TikZ & PGF
+*.dpth
+*.md5
+*.auxlock
+
+# todonotes
+*.tdo
+
+# easy-todo
+*.lod
+
+# xindy
+*.xdy
+
+# xypic precompiled matrices
+*.xyc
+
+# endfloat
+*.ttt
+*.fff
+
+# Latexian
+TSWLatexianTemp*
+
+## Editors:
+# WinEdt
+*.bak
+*.sav
+
+# Texpad
+.texpadtmp
+
+# Kile
+*.backup
+
+# KBibTeX
+*~[0-9]*
+
+# emacs
+/auto/*
+_*.tex
+
+# vim
+*.swp
+
+# expex forward references with \gathertags
+*-tags.tex
+
+# python
+*__pycache__*
diff --git a/PbSe_global_analysis/README.txt b/PbSe_global_analysis/README.txt
new file mode 100644
index 0000000..22ecdb5
--- /dev/null
+++ b/PbSe_global_analysis/README.txt
@@ -0,0 +1,4 @@
+To replicate data processing you will need Python 3, WrightTools 2.13.7 and dependencies thereof.
+See http://wright.tools/en/master/install.html for more information.
+
+To run everything, simply call "./run.sh all".
diff --git a/PbSe_global_analysis/SI.tex b/PbSe_global_analysis/SI.tex
new file mode 100644
index 0000000..d3cb783
--- /dev/null
+++ b/PbSe_global_analysis/SI.tex
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+% document
+\documentclass[11pt, full]{article}
+\usepackage[letterpaper, portrait, margin=0.75in]{geometry}
+\usepackage{setspace}
+\usepackage{color}
+
+% text
+\usepackage[utf8]{inputenc}
+\setlength\parindent{0pt}
+\setlength{\parskip}{1em}
+\usepackage{enumitem}
+\renewcommand{\familydefault}{\sfdefault}
+\newcommand{\RomanNumeral}[1]{\textrm{\uppercase\expandafter{\romannumeral #1\relax}}}
+
+% math
+\usepackage{amssymb}
+\usepackage{amsmath}
+\usepackage[cm]{sfmath}
+\usepackage{commath}
+\usepackage{multirow}
+\DeclareMathAlphabet{\mathpzc}{OT1}{pzc}{m}{it}
+
+% graphics
+\usepackage{graphics}
+\usepackage{graphicx}
+\usepackage{epsfig}
+\usepackage{epstopdf}
+\usepackage{xpatch}
+\graphicspath{{./figures/}}
+
+% "S" prefix
+\renewcommand{\theequation}{S\arabic{equation}}
+\renewcommand{\thefigure}{S\arabic{figure}}
+\renewcommand{\thetable}{S\arabic{table}}
+
+% bibliography
+\usepackage[backend=biber, natbib=true, url=false, sorting=none, maxbibnames=99]{biblatex}
+\bibliography{mybib}
+
+% hyperref
+\usepackage[colorlinks=true, linkcolor=black, urlcolor=blue, citecolor=black, anchorcolor=black]{hyperref}
+\usepackage[all]{hypcap}  % helps hyperref work properly
+
+\begin{document}
+\pagenumbering{gobble}
+
+\begin{center}
+	\LARGE
+	
+	Supplementary Information
+
+	Global Analysis of Transient Grating and Transient Absorption \\ of PbSe Quantum Dots
+	
+	\normalsize
+	
+	\textit{Daniel D. Kohler, Blaise J. Thompson, John C. Wright*}
+
+	Department of Chemistry, University of Wisconsin--Madison\\
+	1101 University Ave., Madison, Wisconsin 53706
+\end{center}
+
+\vspace{\fill}
+
+*Corresponding Author \\
+\hspace*{2ex} email: wright@chem.wisc.edu \\
+\hspace*{2ex} phone: (608) 262-0351 \\
+\hspace*{2ex} fax: (608) 262-0381
+
+\pagebreak
+\renewcommand{\baselinestretch}{0.75}\normalsize
+\tableofcontents
+\renewcommand{\baselinestretch}{1.0}\normalsize
+
+\pagebreak
+\setcounter{page}{1}
+\pagenumbering{arabic}
+
+\section{Absorbance}  % ---------------------------------------------------------------------------
+
+\autoref{figure:absorbance} displays the absorbance spectra of the two batches considered in this
+work.
+The lower spectra are plotted relative to each batches 1S peak center,
+emphasizing the peak-shape differences around the 1S.
+
+\begin{figure}[!htb]
+  \centering
+  \includegraphics[scale=0.5]{absorbance}
+  \label{figure:absorbance}
+  \caption{Normalized absorbance spectra of the two baches considered in this
+    work. In the upper plot, the spectra are plotted directly against energy. In
+  the lower plot the spectra are plotted relative to the 1S peak center.}
+\end{figure}
+
+To extract peak parameters from the rising continuum absorption, the data was fitted on the second
+derivative level, as described in the supplementary information of \textcite{Czech2015}.
+The script used to accomplish this fit, full parameter output, and additional figures showing the
+separate excitonic features and fit remainder are contained in the supplementary repository, as
+described in \autoref{section:repository}.
+
+Note that the aliquots used for each of the two Batch A experiments were at
+slightly different concentrations, a crucial detail for m-factor corrections
+(see \autoref{section:m-factors}). The two Batch B experiments were done using
+the same aliquot. The absorbance spectrum of each sample is kept in an
+associated ``cal'' directory in the supplementary repository (see \autoref{section:repository}).
+
+
+\pagebreak
+\section{Artifact correction}  % ------------------------------------------------------------------
+
+\subsection{Spectral delay correction}
+
+\begin{figure}[!htb]
+  \centering
+  \includegraphics[scale=0.5]{spectral_delay_correction}
+  \label{figure:spectral_delay_correction}
+  \caption{Frequency dependent delay calibration using CCl$_4$. (a) Measurement
+    of the pulse overlap position in $\tau_{21}$ space with respect to
+    $\omega_1$ ($\omega_2$ = 7500 cm$^{-1}$). The thick black line shows the
+    center of the temporal profile, as determined by Gaussian fits. (b) Same as
+    (a), but now $\omega_1$ is kept static while $\omega_2$ is scanned. (c) Same
+    as (a), but now active spectral delay corrections have been applied. (d)
+    Two-dimensional frequency-frequecy scan of CCl$_4$ with spectral delay
+    correction applied.}
+\end{figure}
+
+\pagebreak
+\subsection{Power factors}
+
+\begin{figure}[!htb]
+  \centering
+  \includegraphics[scale=0.5]{power_factors}
+  \label{figure:power_factors}
+  \caption{TODO}
+\end{figure}
+
+\pagebreak
+\subsection{m factors} \label{section:m-factors}
+
+\begin{figure}[!htb]
+  \centering
+  \includegraphics[scale=0.5]{m_factors}
+  \label{figure:power_factors}
+  \caption{TODO}
+\end{figure}
+
+\pagebreak
+\subsection{Processing}
+
+\begin{figure}[!htb]
+  \centering
+  \includegraphics[scale=0.5]{TG_artifacts}
+  \label{figure:power_factors}
+  \caption{TODO}
+\end{figure}
+
+\begin{figure}[!htb]
+  \centering
+  \includegraphics[scale=0.5]{TA_artifacts}
+  \label{figure:power_factors}
+  \caption{TODO}
+\end{figure}
+
+
+\pagebreak
+\section{Auger recombination dynamics}  % ---------------------------------------------------------
+
+%\begin{figure}[!htb]
+%	\centering
+%	\includegraphics[scale=0.5]{"fsb19-3"}
+%	\label{fig:matrix_flow_diagram}
+%	\caption{$S_{\mathsf{TG}}$ measured before and after multiexciton relaxation dynamics.}
+%\end{figure}
+
+Using a Poisson distribution to model the effects here: keep in mind that
+Poisson is only valid when excitation probability is "low". m
+Scholes thinks an equations of motion approach might be more fitting.
+Others have approached this by truncating the Poisson model so that dots are effectively "off" at
+high fluence (this is when pumping the continuum, so no SE contributions from the pump).
+
+According to the Poisson distribution, initial population created by pump is given by
+
+\begin{equation}
+P(k;\lambda) = \frac{\lambda^k e^{-\lambda}}{k!}.
+\end{equation}
+
+Assumes all absorption events have equal probability.
+The absorption of the pumped sample will be proportional to
+
+\begin{eqnarray}
+a_{\mathsf{NL}} &=& a_0 \left(1-e^{-\lambda}\sum_{k=1}\frac{\lambda^k}{k!}\right)
++ e^{-\lambda}\sum_{k=1}a_k\frac{\lambda^k}{k!} \\
+&=& a_0 - e^{-\lambda}\sum_{k=1} (a_0 - a_k)\frac{\lambda^k}{k!}.
+\end{eqnarray}
+
+So the difference in the absorption is
+
+\begin{equation}
+S(T=0) = a_{\mathsf{NL}} - a_0 = -e^{-\lambda}\sum_{k=1}(a_0-a_k)\frac{\lambda^k}{k!}.
+\end{equation}
+
+We will assume that absorption is proportional to the number of ground state excitons: $a_k = ck$
+for all $k$.
+Then
+
+\begin{eqnarray}
+S(T=0) &=& ce^{-\lambda}\sum_{k=1}k\frac{\lambda^k}{k!} \\
+&=& c\lambda e^{-\lambda}\sum_{k=0}\frac{\lambda^k}{k!} \\
+&=& c \lambda,
+\end{eqnarray}
+
+and the mean value corresponds to the response (as we expect when the relationship between
+occupation and signal is linear i.e. $<ck> = c\lambda$).
+
+After Auger recombination, the excited state distribution has homogenized to $k=1$.
+Signal is thus given by
+
+\begin{eqnarray}
+S &=& ce^{-\lambda}(a_0-a_1)\sum_{k=1}\frac{\lambda^k}{k!} \\
+&=& ce^{-\lambda}(e^\lambda-1) \\
+&=& c(1-e^{-\lambda}).
+\end{eqnarray}
+
+Previous work has analyzed this.
+
+Comparing the distribution theory with our results.
+
+The mean number of excitations should be proportional to our fluence: $\lambda = mI$.
+This predicts the linear scaling of intensity close to zero delay, and it also predicts the
+exponential saturation observed at longer delays.
+Both observations qualitatively agree with our results.
+Quantitatively, however, our two delays suggest different scaling constants with respect to pump
+fluence: the long-time $m$-value is roughly 40\% larger than the short time scaling.
+This means that our initial scaling underestimates how quickly the band saturates.
+
+Philosophically, there are two problems with this distribution: (1) I should use the equations of
+motion for degenerate pumping, and (2) the pump is filtered by $k$-vector conservation (two pumps).
+My strategy: come up with an expression for the distribution using coupled equations of motion.
+Assume the driven limit, so that a steady state is reached.
+We can account for these issues by utilizing the more general Conway-Maxwell-Poisson distribution.
+
+Estimate spot size as 300 um: 1 um ~ 1 mJ per cm squared.
+
+I think I should revisit the scaling of my exciton signal---I do not expect it to be the same as a
+Poisson distribution because of the stimulated emission channels.
+
+\begin{eqnarray}
+\frac{d \rho_{00}}{dt} &=& \frac{i}{\hbar} E \rho_{01} + \Gamma\rho_{11} \\
+\frac{d \rho_{00}}{dt} &=& \frac{i}{\hbar} E \rho_{01} + \Gamma\rho_{11} \\
+\frac{d \rho_{00}}{dt} &=& \cdots
+\end{eqnarray}
+
+\pagebreak
+\section{Supplementary repository} \label{section:repository}  % ----------------------------------
+
+All scripts and raw data used in this work have been uploaded to the Open Science Framework (OSF),
+a project of the Center for Open Science.
+These can be found at DOI: \href{http://dx.doi.org/10.17605/OSF.IO/N9CDP}{10.17605/OSF.IO/N9CDP}.
+
+To download the contents of this repository from your command line...  % TODO
+
+To completely reproduce this work, simply execute \texttt{./run.sh all} from your terminal.
+You will require the following:
+
+\begin{enumerate}
+  \item python 3.6
+  \item WrightTools VERSION TODO (and dependencies)
+  \item latex
+\end{enumerate}
+
+You can replace \texttt{all} with one of \texttt{data}, \texttt{simulations}, \texttt{figures},
+or \texttt{documents}.
+
+Otherwise, the OSF repository attempts to be generally self-explanatory.
+README files and comments are used to explain what was done.
+
+\pagebreak
+\printbibliography
+
+\end{document}
diff --git a/PbSe_global_analysis/__init__.py b/PbSe_global_analysis/__init__.py
new file mode 100644
index 0000000..e69de29
--- /dev/null
+++ b/PbSe_global_analysis/__init__.py
diff --git a/PbSe_global_analysis/chapter.tex b/PbSe_global_analysis/chapter.tex
new file mode 100644
index 0000000..2f017b4
--- /dev/null
+++ b/PbSe_global_analysis/chapter.tex
@@ -0,0 +1,5 @@
+\chapter{PbSe} \label{cha:psg}
+
+\textit{This Chapter borrows extensively from a work-in-progress publication.}
+
+\clearpage
\ No newline at end of file
diff --git a/PbSe_global_analysis/conclusion.tex b/PbSe_global_analysis/conclusion.tex
new file mode 100644
index 0000000..e7491d1
--- /dev/null
+++ b/PbSe_global_analysis/conclusion.tex
@@ -0,0 +1,7 @@
+By combining TA and TG measurements, we have described the complex third-order, 2D susceptibility of the 1S resonance of PbSe quantum dots. 

+We have demonstrated a parameter extraction procedure that is reproducible for different quantum dot samples, and that some of the parameters, such as the pure dephasing time, are batch dependent. 

+Inhomogeneity, exciton-induced broadening, exciton-exciton coulombic coupling shifts, and intraband absorption are all required to reconcile both datasets. 

+TA features about 1S exciton band are not exclusively assigned as 1S transitions, which can have important consequences for interpreting the evolution of the 1S bleach.

+

+While the TA spectra show prominent 1S-resonant features, the intraband absorption and its associated refractive index signature are most visible in the TG dataset, so that disentangling the 1S resonant response and the broadband response is a more well-defined problem when both datasets are used together. 

+This approach is thus useful for characterization of non-linear signals in spectrally congested systems.

diff --git a/PbSe_global_analysis/discussion.tex b/PbSe_global_analysis/discussion.tex
new file mode 100644
index 0000000..97ea143
--- /dev/null
+++ b/PbSe_global_analysis/discussion.tex
@@ -0,0 +1,176 @@
+\subsection{Comparison of TA and TG line shapes}

+

+We first attempted simple fits on a subset of the data to reduce the parameter complexity. 

+We consider our experimental data at $T=120$ fs to remove contributions from probe-pump and pump-probe-pump time-ordered processes. 

+By further restricting our considerations to a single probe slice ($\omega_2 = \omega_{\text{1S}}$), we discriminate against inhomogeneous broadening and thus neglect ensemble effects for initial considerations. 

+We fit our probe spectrum with Equation \ref{eq:chi3_lorentz} along with the added treatment of convolving the response with our monochromator instrumental function. 

+Proper accounting of all time-ordered pathways, finite pulse bandwidth, and inhomogeneity are treated later.

+

+\begin{figure}

+	\includegraphics[scale=0.5]{kramers_kronig}

+	\caption{Kramers-Kronig analysis of TA spectra compared with TG spectra.}

+	\label{fig:kramers_kronig}

+\end{figure}

+

+We find that the TA spectra are more sensitive to the model parameters than TG, and that the parameter interplay necessary to reproduce the spectra can be easily described. 

+We note three features of the TA spectra that are crucial to reproduce in simulation: (1) the net bleach; (2) the photon energy of the bleach feature minimum is blue of the 1S absorption peak; (3) the probe spectrum decays quickly to zero away from the bleach resonance, leaving a minimal ESA feature to the red. 

+These features are consistent with the vast majority of published TA spectra of the 1S exciton,\cite{Trinh2013,Schins2009,Gesuele2012,Gdor2013a,Kraatz2014,DeGeyter2012} and can only be reproduced when all three of our nonlinearities (state-filling, excitation-induced broadening, and Coulombic coupling) are invoked (exposition of this result is found in supporting information). 

+The extracted fit parameters are listed in Table \ref{tab:fit1}.

+

+\begin{table}[]

+	\centering

+	\caption{Parameters used in fitting experimental probe slices using Equation \ref{eq:chi3_lorentz}; $S=0.75$, $\omega_2 = \omega_\text{1S}$.}

+	\label{tab:fit1}

+	\begin{tabular}{l|cc}

+		& \multicolumn{2}{l}{Batch} \\

+													   		   &  A   &  B        \\

+		\hline

+		$ \varepsilon_\text{Coul} \left(\text{cm}^{-1}\right)$ & 81   &  53        \\

+		$ \Gamma_{10} \left(\text{cm}^{-1}\right)$ 			   & 380  & 200         \\

+		$ \xi $ 								   			   & 1.35 & 1.39        

+	\end{tabular}

+\end{table}

+

+With the 1S band TA well-characterized, we now consider applying the fitted parameters to the TG signal.

+Transferring this simulation to the TG data poses technical challenges. 

+A critical factor is appropriately scaling the TG signals relative to TA signals.

+The TA simulation can define the scaling through the Kramers-Kronig transform, which gives the transient refraction. 

+The computed transient refraction is unique to within an arbitrary offset; for a single resonant TA/TG feature, the transient refraction offset is zero.  

+We take the offset to be zero now and address this assumption later.

+The transient refraction (Figure \ref{fig:cw_sim1}, third column) shows highly dispersive character with a node near resonance. 

+This means that there is a point in our spectrum at which $\left| \chi^{(3)} \right| = \left| \text{Im}\left[ \chi^{(3)} \right] \right|$.  

+Of course, we also have the constraint $ \left| \chi^{(3)} \right| \geq \left| \text{Im} \left[ \chi^{(3)} \right]  \right|$ for every probe color. 

+These two constraints uniquely determine the appropriate scaling factor as the minimum scalar $c_0$ that satisfies $c_0 S_{\text{TG}} \geq \left| S_{\text{TA}} \right|$ for all probe colors. 

+%Such a scaling of the experimental data is consistent with our TA fit because the peak TA component is nearly equal to the peak TG amplitude (when the arbitrary offset of the KK-transform is zero).

+

+As we alluded, the arbitrary offset of the Kramers-Kronig transform deserves special consideration.  

+%A single TA resonance should not cause an offset in the transient reflection spectrum, but it is conceivable that states outside our spectral range are strongly coupled to the 1S band and produce strong refractive signals at these colors.  

+%While the peaked TA line shape might seem to imply a dispersively shaped $\text{Re} \left[ \chi^{(3)} \right]$ with a node near the bleach center, this is not guaranteed by the Kramers-Kronig relations.  

+The physical origin for this offset would be coupling between the 1S band and states outside our spectral range. 

+If the coupling is sufficiently strong, the $\text{Re}\left[ \chi^{(3)} \right] $ offset may be large enough to remove the node, invalidating the minimum scaling factor method.  

+We believe such a large offset is not viable for several reasons.

+From a physical standpoint, it seems very unlikely a non-resonant state would have coupling stronger coupling to the 1S band than the 1S band itself.  % DK: elaborate?

+Furthermore, TA studies with larger spectral ranges see no evidence of coupling features with strength comparable to the 1S bleach.\cite{Gdor2013a,Trinh2013}  

+Finally, a sufficiently large refractive offset would produce a dip in the $\left| \chi^{(3)} \right|$ line shape near the FHWM points; such features are definitively absent in the TG spectra.  

+

+% DK:  a more direct topic sentence might be nice--jump to the fact that equation 4 is inadequate

+%While we have confidence in relating the TG and TA measurements using the minimum scaling factor, we find two inconsistencies when fitting Equation \ref{eq:chi3_lorentz} with experiment. 

+%Firstly, the TA simulation gives a resonant bleach factor that is much greater than that predicted by state-filling alone: for instance, with $1-S=0.25$, we see $\phi>0.5$ for both batches (Figure \ref{fig:cw_sim1}, first column). 

+%While our parameters successfully recreate the features of the TA line shapes, the simulation grossly overshoots the magnitude of the non-linearity. 

+While we have confidence in relating the TG and TA measurements using the minimum scaling factor, Equation \ref{eq:chi3_lorentz} fails to accurately reproduce the TG spectrum (Figure \ref{fig:cw_sim1}, third column). 

+The errors are systematic: in both batches, our simulation misses the characteristic red skew of our experimental TG and instead skews signal to the blue. 

+Based on the excellent agreement with $S_{\text{TA}}$ (Figure \ref{fig:cw_sim1}, second column), it follows that the chief source of error in our simulation lies in $\text{Re} \left[ \chi^{(3)} \right]$.  

+The experimentally-consistent $\text{Re} \left[ \chi^{(3)} \right]$ can be calculated from Equation \ref{eq:chi_real} (Figure \ref{fig:cw_sim1}, fourth column). 

+The dark green curve highlights which of the two roots of Equation \ref{eq:chi_real} is closest to our simulation.

+The choice of a negative non-linear refractive index value on resonance is in agreement with z-scan measurements.\cite{Moreels2006} 

+The discrepancy between the experimental and simulated real components is well-approximated by a constant offset.

+

+\begin{figure}

+	\includegraphics[width=0.5\linewidth]{"CW_sim2"}

+	\caption{Top row: Global fits of $S_\text{TA}$ (blue), $S_\text{TG}$ (red), and the associated real projection (green) using Equation \ref{eq:offset_fit}. Light colors indicate the simulations and the darker lines indicate the experimental data. Bottom row:  Final simulated absorption spectra for the excited state and the ground state.}

+	\label{fig:cw_sim2}

+\end{figure}

+

+The presence of this offset forced a re-evaluation of the model.

+By revising our simulation to include a complex-valued offset term, $\Delta = |\Delta| e^{i\theta}$, so that

+\begin{equation}\label{eq:offset_fit}

+	\chi^{(3)} = \text{Im} \left[ L_0(\omega_2) \right] \left[ SL_1(\omega_1) - L_0(\omega_1) + \Delta \right],

+\end{equation}

+the discrepancy between $S_{\text{TA}}$ and $S_{\text{TG}}$ can be resolved. 

+It was found, however, that minimizing error between Equation \ref{eq:offset_fit} and the two datasets alone does not confine all variables uniquely. 

+Specifically, the effects of the EID non-linearity ($\xi$) on the line shapes were highly correlated with those of the broadband photoinduced absorption (PA) term, $\text{Im}\left[ \Delta \right]$, so that getting a unique parameter combination was not possible. 

+The fitting routine was robust,\footnote{

+	Robustness here is defined as the ability to permute the fitting parameter order when minimizing the residual. For example, $\tau_{10}$ can be fit either before or after $\Delta$ is fit without significantly changing the resulting parameters.

+} however, when the resonant bleach magnitude was pinned to the state-filling: $\phi \approx 1-S$.  

+The resulting parameters are shown in Table \ref{tab:fit2}, and the results of the fit are shown in Figure \ref{fig:cw_sim2}. 

+As both $\phi=0.25$ and $\phi_{\text{int}}=0.25$ have been measured, this added constraint has a reasonable precedence. 

+As mentioned earlier, EID and Coulombic coupling prevent this equality (as in equation \ref{eq:chi3_lorentz}), but PA can counteract these terms and keep the resonant bleach near $1-S$. 

+In addition, since EID and PA are correlated, decreasing the resonant bleach also decreases the need for EID to influence the line shape, which results in a reduced EID non-linearity in this fit (compare $\xi$ in Table \ref{tab:fit1} and Table \ref{tab:fit2}).

+

+\begin{table}[]

+	\centering

+	\caption{Parameters of the simulated $\chi^{(3)}$ response extracted by global fits of TA and TG at $T=120$ fs using Equation \ref{eq:offset_fit} and with $S=0.75$.  Numbers in parentheses refer to fits at $T=300$ fs}

+	\label{tab:fit2}

+	\begin{tabular}{l|cc}

+		Batch & A          & B         \\

+		\hline

+		$ \Gamma_{10} \left( \text{cm}^{-1}  \right) $ 							 & 340 (320)    &  210 (210)    \\

+		$ \xi $ 												 				 & 1.07 (1.04)  & 1.05 (1.02)   \\

+		$ \epsilon_\text{Coul} \left( \text{cm}^{-1} \right)$ 			     	 &  54 (46) 	& 28 (26)       \\

+		$ \left|\Delta \right| / \text{Im}\left[ L_0(\omega_\text{1S}) \right] $ &  0.07 (0.06) & 0.06 (0.06)   \\

+		$ \theta \left( \text{deg} \right)$ 					 			     & 151 (156)    & 146 (148)

+	\end{tabular}

+\end{table}

+

+The nature of $\Delta$ may be associated with resonant and non-resonant transitions from the 1S excitonic state to the continuum of intraband states involving the electron and/or hole. 

+The magnitude and phase of this contribution would then depend on the ensemble average from all transitions. 

+This contribution has been identified in previous TA studies.

+DeGeyter et.al. isolated a net absorption at sub-bandgap probe

+frequencies.\cite{DeGeyter2012}

+Geigerat et.al. found an absorptive contribution was needed to explain the fluence dependence of the 1S-resonant bleach.\cite{Geiregat2014}  

+The absorptive contribution was $5\%$ of the 1S absorbance, a value that agrees with this work (see Table \ref{tab:fit2}). 

+Our data unifies both observations by showing that additional contribution persists at both bandgap and sub-bandgap frequencies. 

+In addition, our data provides the spectral phase of the contribution. 

+It also shows that the red skew of the TG line shape is very sensitive to the relative importance of the 1S resonance and the additional contribution.

+

+There may also be a relationship to studies on CdSe quantum dots where an additional broad ESA feature was observed for $\omega_1 < \omega_{\text{1S}}$. 

+The feature had separate narrow and broad components.

+The narrow component closest to the band edge bleach corresponded to the Coulombically shifted biexciton transition. 

+Since the broad component correlated with inadequate surface passivation, it was attributed to the surface inducing ESA transitions to the broad band of continuum states that would normally be forbidden. 

+In addition to creating additional ESA transitions, it also created a short-lived transient that was similar to the transients attributed to multiexciton relaxation and multiexcion generation.

+

+\subsection{Determination of State Filling Factor}

+

+% Given the 

+%We measured the peak $chi^{(3)}$ hyperpolarizability of Batch B via standard additions (SI) and found good agreement with the TA hyperpolarizability from Equation \ref{eq:gamma3_state_filling} (for $\phi=0.25$).

+%A offset that would remove the node would create a difference between the peak $\chi^{(3)}$ values measured from TA and TG.  

+%To check this possibility we measured the absolute susceptibility of the TG response and compared it to the susceptibility due to TA. 

+Our results show that the peak susceptibility is almost entirely imaginary, which means we can attribute our TG peak $\left|\chi^{(3)}\right|$ susceptibility to the TA state-filling factor.  

+A standard addition method was used to extract the peak TG hyperpolarizability of $\left| \gamma^{(3)} \right|=7\pm 2 \cdot 10^{-31} \text{cm}^6 \ \text{erg}^{-1}$ for Batch A, while Equation \ref{eq:gamma3_state_filling} predicts a peak imaginary hyperpolarizability of $\text{Im}\left[ \gamma^{(3)} \right] =2.1\phi \cdot 10^{-30} \text{cm}^6 \ \text{erg}^{-1}$. 

+The two values are similar for $\phi=0.25$, while $\phi=0.125$ gives an imaginary component that is much smaller than the total susceptibility.  

+We conclude that only the $\phi=0.25$ bleach factor is consistent with our observations. % DK: also discuss what S likely is

+

+\subsection{Inhomogeneity and the Pulse Overlap Response}

+

+Our parameter extraction above gives plausible parameters to explain the observed photophysics of a small slice of our multidimensional data. 

+We now apply a more rigorous simulation of the model system to address the entire dataset and consider the broader experimental space. 

+This rigorous simulation is meant to account for the complex signals that arise at temporal pulse overlap, the pulsed nature of our excitation beams, and sample inhomogeneity. 

+We calculate signal through numerical integration techniques. % DK: cite paper 1

+The homogeneous and inhomogeneous broadening were constrained to compensate each other so that the total ensemble line shape was kept constant and equal to that extracted from absorption measurements (Table \ref{tab:QD_abs}).\footnote{

+	For a Lorentzian of FWHM $2\Gamma_{10}$ and a Gaussian line shape of standard deviation $\sigma_{\text{inhom}}$, the resulting Voigt line shape has a FWHM well-approximated by $\text{FWHM}_{\text{tot}} \left[ \text{cm}^{-1} \right] \approx 5672 \Gamma_{10}\left[ \text{fs}^{-1} \right] + \sqrt{2298 \Gamma_{10}\left[ \text{fs}^{-1} \right] + 8 \ln 2 \left(\sigma_{\text{inhom}}\left[ \text{cm}^{-1} \right]\right) }$.\cite{Olivero1977}

+} 

+\begin{table}[]

+	\centering

+	\caption{}

+	\label{tab:fit3}

+	\begin{tabular}{l|cc}

+		Batch & A          & B         \\

+		\hline

+		$ \Gamma_{10} \left( \text{cm}^{-1}  \right) $ 					& 220 & 130    \\

+		$\text{FWHM}_\text{inhom} \left( \text{cm}^{-1} \right)$		& 520 & 360

+	\end{tabular}

+\end{table}

+

+Rather than a complete global fit of all parameters, we accept the non-linear parameters extracted earlier (Table \ref{tab:fit2}) and optimize the inhomogeneous-homogeneous ratio only, using the ellipticity of the 2D peak shape\cite{Okumura1999} at late population times as the figure of merit. 

+Table \ref{tab:fit3} shows the degree of inhomogeneity in the sample from matching the ellipticity of the peak shape. 

+As expected, Batch B fits to a smaller Gaussian distribution of transition energies than Batch A, but it also exhibits a longer homogeneous dephasing time, suggesting different system-bath coupling strengths for both samples. 

+Previous research on the coherent dynamics of PbSe 1S exciton feature noted homogeneous lifetimes are a significant source of broadening on the 1S exciton;\cite{Kohler2014} our results demonstrate that the relationship between exciton size distribution and 1S exciton linewidth is further complicated by sample-dependent system-bath coupling.

+

+\begin{figure}

+	\includegraphics[width=\linewidth]{"movies_fitted"}

+	\caption{

+		Global simulation using numerical integration and comparison with experiment. 

+		Batches A (left block) and B (right block) are shown, with the TG experimental (top), the simulated TG (2nd row), the experimental TA (3rd row), and the simulated TA (bottom row) data. 

+		Pump probe delay times of $T=0$, and $120$ fs are shown in each case (see

+    column labels). For each pair, the colors are globally normalized and the

+    contours are locally normalized.}

+	\label{fig:nise_fits}

+\end{figure}

+

+The results of this final simulation are compared with the experimental data in Figure \ref{fig:nise_fits}. 

+It is important to note that the simulations get many details of the rise-time spectra correct. 

+Particularly, the transition from narrow, diagonal line shape to a broad, less diagonal line shape is reproduced very well in both TA and TG simulations. 

+Such behavior is expected for responses from excitonic peaks of material systems; the rise time behavior for such systems was studied in detail previously.\cite{Kohler2017} 

+%One qualitative disagreement between the experiment and the simulation is the amount of excited-state absorption in 𝑆𝑆TA at $\omega_1 < $𝜔𝜔1 < 𝜔𝜔1S, 𝑇𝑇=120fs. 

+%We attribute this to the aforementioned broadening of the signal from CW simulations, from which most parameters are taken. 

+Because these simulations do not account for hot-exciton creation from the pump, simulations differ from experiment increasingly as the pump becomes bluer than the 1S center.

diff --git a/PbSe_global_analysis/introduction.tex b/PbSe_global_analysis/introduction.tex
new file mode 100644
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--- /dev/null
+++ b/PbSe_global_analysis/introduction.tex
@@ -0,0 +1,50 @@
+Lead chalcogenide nanocrystals are among the simplest manifestations of quantum confinement\cite{Wise2000} and provide a foundation for the rational design of nano-engineered photovoltaic materials. 

+The time and frequency resolution capabilities of the different types of ultrafast pump-probe methods have provided the most detailed understanding of quantum dot (QD) photophysics.

+Transient absorption (TA) studies have dominated the literature.

+In a typical TA experiment, the pump pulse induces a change in the transmission of the medium that is measured by a subsequent probe pulse.

+The change in transmission is described by the change in the dissipative (imaginary) part of the complex refractive index, which is linked to the dynamics and structure of photoexcited species.

+TA does not provide information on the real-valued refractive index changes. 

+Although the real component is less important for photovoltaic performance, it is an equal indicator of underlying structure and dynamics. 

+In practice, having both real and imaginary components is often helpful.  

+For example, the fully-phased response is crucial for correctly interpreting spectroscopy when interfaces are important, which is common in evaluation of materials.\cite{Price2015,Yang2015,Yang2017} 

+The real and imaginary responses are directly related by the Kramers-Kronig relation, but it is experimentally difficult to measure the ultrafast response over the range of frequencies required for a Hilbert transform. 

+Interferometric methods, such as two-dimensional eletronic spectroscopy (2DES), can resolve both components, but they are demanding methods and not commonly used.

+% note that they often use TA to phase spectra

+

+Transient grating (TG) is a pump-probe method closely related to TA.

+Figures \ref{fig:tg_vs_ta} illustrates both methods. 

+In TG, two pulsed and independently tunable excitation fields, $E_1$ and $E_2$, are incident on a sample. 

+The TG experiment modulates the optical properties of the sample by creating a population grating from the interference between the two crossed beams, $E_2$ and $E_{2^\prime}$. 

+The grating diffracts the $E_1$ probe field into a new direction defined by the phase matching condition $\vec{k}_{\text{sig}} = \vec{k}_1 - \vec{k}_2 + \vec{k}_{2^\prime}$.

+In contrast, the TA experiment creates a spatially uniform excited population, but temporally modulates the ground and excited state populations with a chopper. 

+TA can be seen as a special case of a TG experiment in which the grating fringes

+become infinitely spaced ($\vec{k}_2-\vec{k}_{2^\prime} \rightarrow \vec{0}$)

+and, instead of being diffracted, the nonlinear field overlaps and interferes with the probe beam.

+% BJT: we might consider introducing TA first, since it is more familiar

+

+\begin{figure}

+	\includegraphics[width=\linewidth]{"tg vs ta"}

+	\caption{The similarities between transient grating and transient absorption measurements. 

+		Both signals are derived from creating a population difference in the sample. 

+			(a) A transient grating experiment crosses two pump beams of the same optical frequency ($E_2$, $E_{2^\prime}$) to create an intensity grating roughly perpendicular to the direction of propagation. 

+			(b) The intensity grating consequently spatially modulates the balance of ground state and excited state in the sample. 

+				The probe beam ($E_1$) is diffracted, and the diffracted intensity is measured. 

+			In transient absorption (c), the probe creates a monolithic population difference, which changes the attenuation the probe beam experiences through the sample. 

+			(d) The pump is modulated by a chopper, which facilitates measurement of the population difference.}

+	\label{fig:tg_vs_ta}

+\end{figure}

+

+Like TA, TG does not fully characterize the non-linear response.  

+Both imaginary and real parts of the refractive index spatially modulate in the TG experiment. 

+The diffracted probe is sensitive only to the total grating contrast (the response \textit{amplitude}), and not the phase relationships of the grating.

+Since both techniques are sensitive to different components of the non-linear response, however, the combination of both TA and TG can solve the fully-phased response.  

+%A local oscillator beam can act as a phase-sensitive reference and is often used to provide that resolution in time-domain techniques. 

+%In this paper, we demonstrate that one can discern the complete, fully-phased optically-induced refractive from frequency domain techniques.

+

+Here we report the results of dual 2DTA-2DTG experiments of PbSe quantum dots at the 1S exciton transition. 

+We explore the three-dimensional experimental space of pump color, probe color, and population delay time.

+We define the important experimental factors that must be taken into account for accurate comparison of the two methods.

+We show that both methods exhibit reproducible spectra across different batches of different exciton sizes.

+Finally, we show that the methods can be used to construct a phased third-order response spectrum. Both experiments can be reproduced via simulations using the standard theory of PbSe excitons.

+Interestingly, the combined information reveals broadband contributions to the quantum dots non-linearity, barely distinguishable with transient absorption spectra alone.

+This work demonstrates TG and TA serve as complementary methods for the study of exciton structure and dynamics.

diff --git a/PbSe_global_analysis/main.tex b/PbSe_global_analysis/main.tex
new file mode 100644
index 0000000..f7f5e83
--- /dev/null
+++ b/PbSe_global_analysis/main.tex
@@ -0,0 +1,60 @@
+\documentclass[journal=jpccck]{achemso}

+%\usepackage[version=3]{mhchem} % Formula subscripts using \ce{}

+%\usepackage[T1]{fontenc}       % Use modern font encodings

+

+\usepackage{amsmath}

+\usepackage{amsfonts}

+\usepackage{amssymb}

+\usepackage{graphicx}

+%\usepackage{multirow}

+\usepackage{bm} % bold mathtype

+%\usepackage{tabularx}

+

+\graphicspath{{"figures/"}}

+

+\title{Global Analysis of Transient Grating and Transient Absorption of PbSe Quantum Dots}

+\author{Daniel D. Kohler}

+\author{Blaise J. Thompson}

+\author{John C. Wright}

+\affiliation{Department of Chemistry,

+	           University of Wisconsin-Madison,

+           	 1101 University Ave, 

+	           Madison, WI 53706, United States}

+\keywords{CMDS, FWM}  % BJT: we need more / better keywords

+

+\date{\today}

+\begin{document}

+

+\begin{abstract}

+We examine the non-linear response of PbSe quantum dots about the 1S exciton using two-dimensional transient absorption and transient grating techniques.

+The combined analysis of both methods provides the complete amplitude and phase of the non-linear susceptibility.

+The phased spectra reconcile questions about the relationships between the PbSe quantum dot electronic states and the nature of nonlinearities measured by two-dimensional absorption and transient grating methods.

+The fits of the combined dataset reveal and quantify the presence of continuum transitions.

+\end{abstract}

+

+\maketitle

+\section{Introduction}

+	\input{introduction}

+\section{Theory}

+	\input{theory}

+\section{Methods}

+	\input{methods}

+\section{Results}

+	\input{results}

+\section{Discussion}

+	\input{discussion}

+\section{Conclusion}

+	\input{conclusion}

+

+\begin{acknowledgement}

+The authors grateful acknowledge staff scientist Andrei Pakoulev (1960-2014) for insightful conversations and devoted tutelage. We thank Qi Ding for her TEM characterizations of the quantum dot samples.

+\end{acknowledgement}

+

+\begin{suppinfo}

+lots of data is online

+\end{suppinfo}

+

+%\bibliographystyle{achemso}

+\bibliography{mybib}

+

+\end{document}

diff --git a/PbSe_global_analysis/methods.tex b/PbSe_global_analysis/methods.tex
new file mode 100644
index 0000000..a62d590
--- /dev/null
+++ b/PbSe_global_analysis/methods.tex
@@ -0,0 +1,38 @@
+Quantum dot samples used in this study were synthesized using the hot injection method.\cite{Wehrenberg2002} 

+Samples were kept in a glovebox after synthesis and exposure to visible and UV light was minimized.  

+These conditions preserved the dots for several months.  

+Two samples, Batch A and Batch B, are presented in this study, in an effort to show the robustness of the results. 

+Properties of their optical characterization are shown in Table \ref{tab:QD_abs}.  

+The 1S band of Batch A is broader than Batch B, an effect which is usually attributed to a wider size distribution and therefore greater inhomogeneous broadening.

+

+\begin{table}[]

+	\centering

+	\caption{Batch Parameters extracted from absorption spectra.  $\langle d \rangle$: average QD diameter, as inferred by the 1S transition energy. }

+	\label{tab:QD_abs}

+	\begin{tabular}{l|cc}

+		 													  & A    & B \\

+		\hline

+		$ \omega_{10} \left( \text{cm}^{-1} \right)$		  & 7570 & 6620 \\

+		$ \text{FWHM} \left(\text{cm}^{-1}\right) $  		  & 780  & 540 \\

+		$ \langle d \rangle  \left(\text{nm}\right)$ 		  & 4    & 4.8 \\

+		$ \sigma_0 \left( \times 10^{16} \text{cm}^2 \right)$ & 1.7  & 2.9

+	\end{tabular}

+\end{table}

+

+The experimental system for the TG experiment has been previously explained.\cite{Kohler2014,Czech2015} 

+Briefly, two independently tunable OPAs are used to make pulses $E_1$ and $E_2$ with colors $\omega_1$ and $\omega_2$. 

+The third beam, $E_{2^\prime}$, is split off from $E_2$. The TG experiment utilized here uses temporally overlapped $E_2$ and $E_{2^\prime}$. 

+Previous ultrafast TG work has characterized the delay of $E_1$ as $\tau_{21}=\tau_2-\tau_1$; to connect the experimental space with the TA measurements, we will report the population delay time between the probe and the pump as $T(=-\tau_{21})$. 

+Pulse timing is controlled by a motorized stage that adjusts the arrival time of $E_1$ relative to $E_2$ and $E_{2^\prime}$.

+

+All three beams are focused onto the sample in a BOXCARS geometry and the direction $\vec{k}_1-\vec{k}_2+\vec{k}_{2^\prime}$ is isolated and sent to a monochromator to isolate the $\omega_1$ frequency with $\sim 120 \text{cm}^{-1}$ detection bandwidth. 

+The signal, $N_{\text{TG}}$, was detected with an InSb photodiode. Reflective neutral density filters (Inconel) limit the pulse fluence to avoid multi-photon absorption. 

+To control for frequency-dependent changes in pulse arrival time due to the OPAs and the neutral density, a calibration table was established to assign a correct zero delay for each color combination (see supporting information for more details).

+

+The TA experiments were designed to minimally change the TG experimental conditions.  

+The $E_{2^\prime}$ beam was blocked and signal in the $\vec{k}_1$ direction was measured.  

+$E_2$ was chopped and the differential signal and the average signal were measured to define $T_0$ and $T$ needed to compute $\Delta A$.  

+Just as in TG experiments, the excitation frequencies were scanned while the monochromator was locked at $\omega_m=\omega_1$.

+% DK: perhaps leave this part out

+%Finally, fluence studies resonant with the 1S band were performed to test for indications of intensity-dependent relaxation. 

+%These studies showed no indication of accelerated Auger recombination rates (see supporting info).

diff --git a/PbSe_global_analysis/mybib.bib b/PbSe_global_analysis/mybib.bib
new file mode 100644
index 0000000..b195cce
--- /dev/null
+++ b/PbSe_global_analysis/mybib.bib
@@ -0,0 +1,1268 @@
+@article{Besemann2004,
+  author =       {Besemann, Daniel M. and Meyer, Kent A. and Wright, John C.},
+  title =        {{Spectroscopic Characteristics of Triply Vibrationally
+                  Enhanced Four-Wave Mixing Spectroscopy †}},
+  journal =      {The Journal of Physical Chemistry B},
+  volume =       108,
+  number =       29,
+  pages =        {10493--10504},
+  year =         2004,
+  doi =          {10.1021/jp049597l},
+  url =          {http://pubs.acs.org/doi/abs/10.1021/jp049597l},
+  abstract =     {Triply vibrationally enhanced (TRIVE) four-wave mixing is a
+                  fully resonant, frequency domain spectroscopy that is capable
+                  of coherent multidimensional vibrational spectroscopy. TRIVE
+                  has 12 different coherence pathways that differ in their time
+                  ordering and resonances. The pathways are the coherent
+                  analogue to two-color pump-probe pathways. Specific pathways
+                  or sets of pathways can be chosen by appropriate selection of
+                  time delays and resonance conditions. The pathways have
+                  characteristic positions and line shapes in three-dimensional
+                  frequency space and their coherent interference has
+                  consequences in interpreting the spectra. The line shapes and
+                  the relative intensities of different pathways are dependent
+                  on the population relaxation and dephasing rates. The
+                  different pathways also have different capabilities for
+                  line-narrowing inhomogeneously broadened transitions. The
+                  narrowing is controlled by the interference between pathways
+                  and the quantum level interference between different parts of
+                  the inhomogeneously broadened envelope. We also show that
+                  selection of the output frequency in two-color TRIVE methods
+                  constrains the selection rules that control the relative
+                  transition probabilities of the four transitions.},
+  file =         {::},
+  issn =         {1520-6106},
+  month =        {jul},
+}
+
+@article{Block2012,
+  author =       {Block, Stephen B. and Yurs, Lena a. and Pakoulev, Andrei V.
+                  and Selinsky, Rachel Sarah and Jin, Song and Wright, John C.},
+  title =        {{Multiresonant Multidimensional Spectroscopy of
+                  Surface-Trapped Excitons in PbSe Quantum Dots}},
+  journal =      {The Journal of Physical Chemistry Letters},
+  volume =       3,
+  number =       18,
+  pages =        {2707--2712},
+  year =         2012,
+  doi =          {10.1021/jz300599b},
+  url =          {http://pubs.acs.org/doi/abs/10.1021/jz300599b},
+  file =         {::},
+  issn =         {1948-7185},
+  month =        {sep},
+}
+
+@article{BritoCruz1988,
+  author =       {{Brito Cruz}, C.H. and Gordon, J.P. and Becker, P.C. and Fork,
+                  R.L. and Shank, C.V.},
+  title =        {{Dynamics of spectral hole burning}},
+  journal =      {IEEE Journal of Quantum Electronics},
+  volume =       24,
+  number =       2,
+  pages =        {261--269},
+  year =         1988,
+  doi =          {10.1109/3.122},
+  url =          {http://ieeexplore.ieee.org/document/122/},
+  file =         {::},
+  issn =         {0018-9197},
+  month =        {feb},
+}
+
+@article{Carlson1989,
+  author =       {Carlson, Roger J. and Wright, John C.},
+  title =        {{Absorption and Coherent Interference Effects in Multiply
+                  Resonant Four-Wave Mixing Spectroscopy}},
+  journal =      {Applied Spectroscopy},
+  volume =       43,
+  number =       7,
+  pages =        {1195--1208},
+  year =         1989,
+  doi =          {10.1366/0003702894203408},
+  url =
+                  {http://openurl.ingenta.com/content/xref?genre=article{\&}issn=0003-7028{\&}volume=43{\&}issue=7{\&}spage=1195},
+  abstract =     {The effects of coherent interference between resonant and
+                  nonresonant signals and the effects of absorption on
+                  multiresonant four-wave mixing spectra are investigated both
+                  theoretically and experimentally in azu- iene-doped
+                  naphthalene crystals at 2 K. Both effects can strongly alter
+                  line shapes and intensities. Line splittings and negative
+                  spectral features are demonstrated. Coherent interference,
+                  although measurable in this system, is weak. Absorption,
+                  however, plays an important role, and its effects on phase
+                  matching, peak shapes, and peak intensities are pre- dicted
+                  theoretically and are confirmed by the experimental measure-
+                  ments. A new expression for the four-wave mixing is derived in
+                  terms of sample absorption coefficients and absorption cross
+                  sections, and gen- eral conditions for maximum efficiency are
+                  determined.},
+  file =         {::},
+  issn =         00037028,
+  keywords =     {M-factors},
+  mendeley-tags ={M-factors},
+  month =        {sep},
+}
+
+@article{Czech2015,
+  author =       {Czech, Kyle J. and Thompson, Blaise J. and Kain, Schuyler and
+                  Ding, Qi and Shearer, Melinda J. and Hamers, Robert J. and
+                  Jin, Song and Wright, John C.},
+  title =        {{Measurement of Ultrafast Excitonic Dynamics of Few-Layer MoS
+                  2 Using State-Selective Coherent Multidimensional
+                  Spectroscopy}},
+  journal =      {ACS Nano},
+  volume =       9,
+  number =       12,
+  pages =        {12146--12157},
+  year =         2015,
+  doi =          {10.1021/acsnano.5b05198},
+  url =          {http://pubs.acs.org/doi/abs/10.1021/acsnano.5b05198},
+  abstract =     {We report the first coherent multidimensional spectroscopy
+                  study of a MoS2 film. A four-layer sample of MoS2 was
+                  synthesized on a silica substrate by a simplified sulfidation
+                  reaction and characterized by absorption and Raman
+                  spectroscopy, atomic force microscopy, and transmission
+                  electron microscopy. State-selective coherent multidimensional
+                  spectroscopy (CMDS) on the as-prepared MoS2 film resolved the
+                  dynamics of a series of diagonal and cross-peak features
+                  involving the spin-orbit split A and B excitonic states and
+                  continuum states. The spectra are characterized by striped
+                  features that are similar to those observed in CMDS studies of
+                  quantum wells where the continuum states contribute strongly
+                  to the initial excitation of both the diagonal and cross-peak
+                  features, while the A and B excitonic states contributed
+                  strongly to the final output signal. The strong contribution
+                  from the continuum states to the initial excitation shows that
+                  the continuum states are coupled to the A and B excitonic
+                  states and that fast intraband relaxation is occurring on a
+                  sub-70 fs time scale. A comparison of the CMDS excitation
+                  signal and the absorption spectrum shows that the relative
+                  importance of the continuum states is determined primarily by
+                  their absorption strength. Diagonal and cross-peak features
+                  decay with a 680 fs time constant characteristic of exciton
+                  recombination and/or trapping. The short time dynamics are
+                  complicated by coherent and partially coherent pathways that
+                  become important when the excitation pulses are temporally
+                  overlapped. In this region, the coherent dynamics create
+                  diagonal features involving both the excitonic states and
+                  continuum states, while the partially coherent pathways
+                  contribute to cross-peak features.},
+  file =         {::},
+  issn =         {1936-0851},
+  keywords =     {2D,2d,are layered,molybdenum sul fi de,molybdenum
+                  sulfide,multidimensional,nonlinear,ransition metal
+                  dichalcogenides,semiconductors with strong spin,such as mos
+                  2,tmdcs,transition metal dichalcogenides,ultrafast
+                  dynamics,{\`{a}}},
+  month =        {dec},
+  pmid =         26525496,
+}
+
+@article{Dai2009,
+  author =       {Dai, Quanqin and Wang, Yingnan Yiding and Li, Xinbi and Zhang,
+                  Yu and Pellegrino, Donald J. and Zhao, Muxun and Zou, Bo and
+                  Seo, JaeTae and Wang, Yingnan Yiding and Yu, William W.},
+  title =        {{Size-dependent composition and molar extinction coefficient
+                  of PbSe semiconductor nanocrystals.}},
+  journal =      {ACS nano},
+  volume =       3,
+  number =       6,
+  pages =        {1518--24},
+  year =         2009,
+  doi =          {10.1021/nn9001616},
+  url =          {http://www.ncbi.nlm.nih.gov/pubmed/19702314},
+  abstract =     {Atomic compositions and molar extinction coefficients of PbSe
+                  semiconductor nanocrystals were determined by atomic
+                  absorption spectrometry, UV-vis-NIR spectrophotometry, and
+                  transmission electron microscopy. The Pb/Se atomic ratio was
+                  found to be size-dependent with a systematic excess of Pb
+                  atoms in the PbSe nanocrystal system. Experimental results
+                  indicated that the individual PbSe nanocrystal was
+                  nonstoichiometric, consisting of a PbSe core and an extra
+                  layer of Pb atoms. For these nonstoichiometric PbSe
+                  semiconductor nanocrystals, we proposed a new computational
+                  approach to calculate the total number of Pb and Se atoms in
+                  different sized particles. This calculation played a key role
+                  on the accurate determination of the strongly size-dependent
+                  extinction coefficient, which followed a power law with an
+                  exponent of approximately 2.5.},
+  file =         {::},
+  isbn =         {1936-0851},
+  issn =         {1936-086X},
+  keywords =     {Composition,Molar extinction coefficient,PbSe,Semiconductor
+                  nanocrystal,Size dependence,informative,size-exciton
+                  correlation},
+  mendeley-tags ={informative},
+  month =        {jun},
+  pmid =         19435305,
+}
+
+@article{DeGeyter2012,
+  author =       {{De Geyter}, Bram and Geiregat, Pieter and Gao, Yunan and {Ten
+                  Cate}, Sybren and Houtepen, Arjan J. and Schins, Juleon M. and
+                  {Van Thourhout}, Dries and Siebbeles, Laurens D A and Hens,
+                  Zeger},
+  title =        {{Broadband and picosecond intraband absorption in lead based
+                  colloidal quantum dots}},
+  journal =      {ACS Nano},
+  number =       7,
+  pages =        {6067--6074},
+  year =         2012,
+  doi =          {10.1109/ICTON.2012.6254469},
+  abstract =     {Using femtosecond transient absorption spectroscopy we
+                  demonstrate that lead chalcogenide nanocrystals show
+                  considerable, photoinduced absorption (PA) in a broad
+                  wavelength range just below the bandgap. The time-dependent
+                  decay of the PA signal correlates with the recovery of the
+                  band gap absorption, indicating that the same carriers are
+                  involved. Based on this, we assign this PA signal to intraband
+                  absorption, i.e., the excitation of photogenerated carriers
+                  from the bottom of the conduction band or the top of the
+                  valence band to higher energy levels in the conduction and
+                  valence band continuum. We confirm our experiments with
+                  tight-binding calculations. This broadband response in the
+                  commercially interesting near to mid-infrared range is very
+                  relevant for ultra high speed all optical signal processing.
+                  We benchmark the performance with bulk-Si and
+                  Si-nanocrystals.},
+  file =         {::},
+  isbn =         9781467322270,
+  issn =         21627339,
+  keywords =     {colloidal nanocrystals,free carrier absorption,intraband
+                  absorption,lead chalcogenide,optical signal
+                  processing,tight-binding,transient absorption spectroscopy},
+  pmid =         22686663,
+}
+
+@article{DelCoso2004,
+  author =       {del Coso, Ra{\'{u}}l and Solis, Javier},
+  title =        {{Relation between nonlinear refractive index and third-order
+                  susceptibility in absorbing media}},
+  journal =      {Journal of the Optical Society of America B},
+  volume =       21,
+  number =       3,
+  pages =        640,
+  year =         2004,
+  doi =          {10.1364/JOSAB.21.000640},
+  url =          {http://www.opticsinfobase.org/abstract.cfm?URI=JOSAB-21-3-640
+                  https://www.osapublishing.org/abstract.cfm?URI=josab-21-3-640},
+  abstract =     {Expressions relating complex third-order optical
+                  susceptibility ($\chi$(3)=$\chi$R(3)+i$\chi$I(3)) with
+                  nonlinear refractive index (n2) and nonlinear absorption
+                  coefficient ($\beta$) have been formulated that eliminate the
+                  commonly used approximation of a negligible linear absorption
+                  coefficient. The resulting equations do not show the
+                  conventional linear dependence of $\chi$R(3) with n2 and
+                  $\chi$I(3) with $\beta$. Nonlinear refraction and absorption
+                  result instead from the interplay between the real and
+                  imaginary parts of the first- and third-order susceptibilities
+                  of the material. This effect is illustrated in the case of a
+                  metal–dielectric nanocomposite for which n2 and $\beta$ values
+                  were experimentally obtained by Z-scan measurements and for
+                  which the use of the new formulas for $\chi$R(3) and
+                  $\chi$I(3) yield a large correction and a sign reversal for
+                  $\chi$I(3).},
+  file =         {::},
+  issn =         {0740-3224},
+  month =        {mar},
+  pmid =         220057300022,
+}
+
+@article{Gdor2012,
+  author =       {Gdor, Itay and Sachs, Hanan and Roitblat, Avishy and
+                  Strasfeld, David B. and Bawendi, Moungi G. and Ruhman,
+                  Sanford},
+  title =        {{Exploring exciton relaxation and multiexciton generation in
+                  PbSe nanocrystals using hyperspectral near-IR probing.}},
+  journal =      {ACS nano},
+  volume =       6,
+  number =       4,
+  pages =        {3269--77},
+  year =         2012,
+  doi =          {10.1021/nn300184n},
+  url =          {http://www.ncbi.nlm.nih.gov/pubmed/22390473},
+  abstract =     {Hyperspectral femtosecond transient absorption spectroscopy is
+                  employed to record exciton relaxation and recombination in
+                  colloidal lead selenide (PbSe) nanocrystals in unprecedented
+                  detail. Results obtained with different pump wavelengths and
+                  fluences are scrutinized with regard to three issues: (1)
+                  early subpicosecond spectral features due to "hot" excitons
+                  are analyzed in terms of suggested underlying mechanisms; (2)
+                  global kinetic analysis facilitates separation of the
+                  transient difference spectra into single, double, and triple
+                  exciton state contributions, from which individual band
+                  assignments can be tested; and (3) the transient spectra are
+                  screened for signatures of multiexciton generation (MEG) by
+                  comparing experiments with excitation pulses both below and
+                  well above the theoretical threshold for multiplication. For
+                  the latter, a recently devised ultrafast pump-probe
+                  spectroscopic approach is employed. Scaling sample
+                  concentrations and pump pulse intensities inversely with the
+                  extinction coefficient at each excitation wavelength overcomes
+                  ambiguities due to direct multiphoton excitation,
+                  uncertainties of absolute absorption cross sections, and low
+                  signal levels. As observed in a recent application of this
+                  method to InAs core/shell/shell nanodots, no sign of MEG was
+                  detected in this sample up to photon energy 3.7 times the band
+                  gap. Accordingly, numerous reports of efficient MEG in other
+                  samples of PbSe suggest that the efficiency of this process
+                  varies from sample to sample and depends on factors yet to be
+                  determined.},
+  file =         {::},
+  isbn =         {1936-086X (Electronic)$\backslash$r1936-0851 (Linking)},
+  issn =         {1936-086X},
+  keywords =     {SADS,carrier multiplication,ciency in nanocrystals has,exciton
+                  cooling,in accord,led,meg effi-,multiexciton
+                  generation,quantum dots,the apparent enhancement of,ultrafast
+                  spectroscopy},
+  mendeley-tags ={SADS},
+  month =        {apr},
+  pmid =         22390473,
+}
+
+@article{Gdor2013a,
+  author =       {Gdor, Itay and Yang, Chunfan and Yanover, Diana and Sachs,
+                  Hanan and Lifshitz, Efrat and Ruhman, Sanford},
+  title =        {{Novel Spectral Decay Dynamics of Hot Excitons in PbSe
+                  Nanocrystals: A Tunable Femtosecond Pump–Hyperspectral Probe
+                  Study}},
+  journal =      {The Journal of Physical Chemistry C},
+  volume =       117,
+  number =       49,
+  pages =        {26342--26350},
+  year =         2013,
+  doi =          {10.1021/jp409530z},
+  url =          {http://pubs.acs.org/doi/abs/10.1021/jp409530z},
+  abstract =     {Ultrafast exciton cooling in highly monodisperse PbSe
+                  nanocrystals is followed with tunable pump?hyperspectral
+                  near-IR probe spectroscopy. Unexpected kinetic and spectral
+                  correlations between induced bleach and absorption features
+                  are revealed, which are incompatible with standard models for
+                  excited nanocrystal absorption. Interband optical excitation
+                  immediately generates a sharp bleach feature near the 1Sh1Se
+                  transition which is unchanged during exciton thermalization,
+                  while pumping well above the band edge induces an intense
+                  absorption at frequencies just below the band edge which
+                  decays concurrently with a buildup of renewed absorbance at
+                  the 1Ph1Pe peak during exciton cooling. Transient spectra of
+                  hot single and double excitons are nearly indistinguishable,
+                  arguing against the controversial involvement of Auger cooling
+                  in the rapid dissipation of excess energy in excited PbSe QDs.
+                  Finally, quantitative signal analysis shows no signs of
+                  multiexciton generation up to photon energies four times the
+                  sample band gap. Ultrafast exciton cooling in highly
+                  monodisperse PbSe nanocrystals is followed with tunable
+                  pump?hyperspectral near-IR probe spectroscopy. Unexpected
+                  kinetic and spectral correlations between induced bleach and
+                  absorption features are revealed, which are incompatible with
+                  standard models for excited nanocrystal absorption. Interband
+                  optical excitation immediately generates a sharp bleach
+                  feature near the 1Sh1Se transition which is unchanged during
+                  exciton thermalization, while pumping well above the band edge
+                  induces an intense absorption at frequencies just below the
+                  band edge which decays concurrently with a buildup of renewed
+                  absorbance at the 1Ph1Pe peak during exciton cooling.
+                  Transient spectra of hot single and double excitons are nearly
+                  indistinguishable, arguing against the controversial
+                  involvement of Auger cooling in the rapid dissipation of
+                  excess energy in excited PbSe QDs. Finally, quantitative
+                  signal analysis shows no signs of multiexciton generation up
+                  to photon energies four times the sample band gap.},
+  file =         {::},
+  isbn =         1557522790,
+  issn =         {1932-7447},
+  month =        {dec},
+}
+
+@article{Gdor2015,
+  author =       {Gdor, Itay and Shapiro, Arthur and Yang, Chunfan and Yanover,
+                  Diana and Lifshitz, Efrat and Ruhman, Sanford},
+  title =        {{Three-pulse femtosecond spectroscopy of PbSe nanocrystals: 1S
+                  bleach nonlinearity and sub-band-edge excited-state absorption
+                  assignment}},
+  journal =      {ACS Nano},
+  volume =       9,
+  number =       2,
+  pages =        {2138--2147},
+  year =         2015,
+  doi =          {10.1021/nn5074868},
+  abstract =     {Above band-edge photo-excitation of PbSe nanocrystals induces
+                  strong below band-gap absorption as well as a multi-phased
+                  buildup of bleaching in the 1Se1Sh transition. The amplitudes
+                  and kinetics of these features deviate from expectations based
+                  on bi-exciton shifts and state filling which are the
+                  mechanisms usually evoked to explain them. To clarify these
+                  discrepancies, the same transitions are investigated here by
+                  double-pump probe spectroscopy. Re-exciting in the below
+                  band-gap induced absorption characteristic of hot excitons is
+                  shown to produce additional excitons with high probability. In
+                  addition, pump-probe experiments on a sample saturated with
+                  single relaxed excitons proves that the resulting 1Se1Sh
+                  bleach is not linear with the number of excitons per
+                  nanocrystal. This finding holds for two samples differing
+                  significantly in size, demonstrating its generality. Analysis
+                  of the results suggest that below band edge induced absorption
+                  in hot exciton states is due to excited state absorption and
+                  not to shifted absorption of cold carriers, and that 1Se1Sh
+                  bleach signals are not an accurate counter of sample excitons
+                  when their distribution includes multi-exciton states.},
+  file =         {::},
+  isbn =         {1936-0851},
+  issn =         {1936086X},
+  keywords =     {exciton cooling,multiexciton generation,nanocrystals,quantum
+                  dots,ultrafast spectroscopy},
+}
+
+@article{Gdor2015a,
+  author =       {Gdor, Itay and Yanover, Dianna and Yang, Chunfan and Shapiro,
+                  Arthur and Lifshitz, Efrat and Ruhman, Sanford},
+  title =        {{Three Pulse Femtosecond Spectroscopy of PbSe Nano-Crystals ;
+                  1S Bleach Nonlinearity and Sub Band-Edge Excited State
+                  Absorption Assignment Three Pulse Femtosecond Spectroscopy of
+                  PbSe nano-Crystals ; 1S Bleach Nonlinearity and Sub Band-edge
+                  Excited State A}},
+  journal =      {ACS Nano},
+  pages =        {2--5},
+  year =         2015,
+  doi =          {10.1021/nn5074868},
+  file =         {::},
+}
+
+@article{Geiregat2014,
+  author =       {Geiregat, Pieter and Houtepen, Arjan J. and Justo, Yolanda and
+                  Grozema, Ferdinand C. and {Van Thourhout}, Dries and Hens,
+                  Zeger},
+  title =        {{Coulomb Shifts upon Exciton Addition to Photoexcited PbS
+                  Colloidal Quantum Dots}},
+  journal =      {The Journal of Physical Chemistry C},
+  volume =       118,
+  number =       38,
+  pages =        {22284--22290},
+  year =         2014,
+  doi =          {10.1021/jp505530k},
+  url =          {http://pubs.acs.org/doi/abs/10.1021/jp505530k},
+  abstract =     {Using ultrafast hyperspectral transient absorption (TA)
+                  spectroscopy, we determine the biexciton addition energies in
+                  PbS quantum dots (QDs) with different sizes when either a cold
+                  or a hot electron−hole pair is added to a QD already
+                  containing a cooled exciton. The observed dependence of this
+                  so-called biexciton addition energy on the QD diameter and the
+                  exciton energy can be rationalized by interpreting the
+                  addition energies as the result of an imbalance in the Coulomb
+                  interactions between the newly created carriers and the
+                  carriers already present in a QD. The obtained results are
+                  therefore relevant from both a fundamental and practical point
+                  of view. They provide experimental data on Coulomb interaction
+                  between charge carriers in confined semiconductors that can be
+                  compared with theoretical estimates. Moreover, understanding
+                  the way hot−cold biexciton addition energies influence the
+                  transient absorption spectrum adds a new element to the
+                  transient absorption toolbox for the optoelectronic properties
+                  of colloidal QDs. ■ INTRODUCTION Colloidal semiconductor
+                  nanocrystals or quantum dots (QDs) are an increasingly
+                  prominent class of low-dimensional nanomaterials that combine
+                  size-tunable electronic and optical properties with a
+                  suitability for solution-based processing. Starting from
+                  fundamental research and theoretical modeling on their unique
+                  physical properties, QDs are now applied in a variety of
+                  domains such as solar energy harvesting, photo-detection, and
+                  light-emitting diodes or displays. 1−5 These applications
+                  typically rely on the linear optical properties of QDs, i.e.,
+                  light absorption by unexcited QDs and light emission by
+                  radiative recombination in excited QDs. On the other hand,
+                  various studies have shown that the spectral and
+                  time-dependent properties of excited QDs can strongly enhance
+                  the performance of QD-based devices in the above mention
+                  applications or enable QDs to be used in completely different
+                  applications. Quantum dots excited with photons having
+                  energies exceeding twice that of the QD bandgap transition
+                  can, for example, dissipate their excess energy by forming
+                  biexcitons in a process called multiple exciton generation
+                  (MEG) that can considerably enhance the short circuit current
+                  of single junction, QD-based solar cells. 2,6−8 Controlling
+                  the recombination rate of biexcitons by nonradiative Auger
+                  processes allowed for the formation of blinking-free QDs and
+                  facilitated the formation of QD-based lasers. 9,10 Moreover,
+                  it was proven that excited QDs exhibit a broadband and
+                  ultrafast photoinduced absorption related to intraband
+                  transitions of either the excited electron or hole, which
+                  could be used for optical modulation. 11,12},
+  file =         {::;::},
+  issn =         {1932-7447},
+  month =        {sep},
+}
+
+@article{Gesuele2012,
+  author =       {Gesuele, F and Sfeir, M Y and Murray, C B and Heinz, T F and
+                  Wong, C W},
+  title =        {{Biexcitonic Effects in Excited Ultrafast Supercontinuum
+                  S{\aa}ectroscopy of Carrier Multiplication and Biexctionic
+                  Effects in Excited States of PbS Quantum Dots}},
+  journal =      {Nano letters},
+  volume =       12,
+  pages =        2658,
+  year =         2012,
+  file =         {::},
+  keywords =     {1,2 the strong spatial,based-materials for third-generation
+                  photovol-,carrier multiplication,confinement of electronic
+                  wave,here is great interest,in the properties
+                  of,multiple-exciton generation,qd,quantum dot,solar
+                  cells,taics,ultrafast spectroscopy},
+}
+
+@article{Hogemann1996,
+  author =       {H{\"{o}}gemann, Claudia and Pauchard, Marc and Vauthey, Eric},
+  title =        {{Picosecond transient grating spectroscopy: The nature of the
+                  diffracted spectrum}},
+  journal =      {Review of Scientific Instruments},
+  volume =       67,
+  number =       10,
+  pages =        {3449--3453},
+  year =         1996,
+  doi =          {10.1063/1.1147157},
+  url =          {http://link.aip.org/link/RSINAK/v67/i10/p3449/s1{\&}Agg=doi
+                  http://aip.scitation.org/doi/10.1063/1.1147157},
+  file =         {::},
+  issn =         {0034-6748},
+  month =        {oct},
+}
+
+@article{Hutchings1992,
+  author =       {Hutchings, D C and Sheik-Bahae, M and Hagan, D J and {Van
+                  Stryland}, E W},
+  title =        {{Kramers-Kronig relations in nonlinear optics}},
+  journal =      {Optical and Quantum Electronics},
+  volume =       24,
+  number =       1,
+  pages =        {1--30},
+  year =         1992,
+  doi =          {10.1007/BF01234275},
+  url =
+                  {http://dx.doi.org/10.1007/BF01234275{\%}5Cnhttp://www.springerlink.com/index/10.1007/BF01234275},
+  abstract =     {We review dispersion relations, which relate the real part of
+                  the optical susceptibility (refraction) to the imaginary part
+                  (absorption). We derive and discuss these relations as applied
+                  to nonlinear optical systems. It is shown that in the
+                  nonlinear case, for self-action effects the correct form for
+                  such dispersion relations is nondegenerate, i.e. it is
+                  necessary to use multiple frequency arguments. Nonlinear
+                  dispersion relations have been shown to be very useful as they
+                  usually only require integration over a limited frequency
+                  range (corresponding to frequencies at which the absorption
+                  changes), unlike the conventional linear Kramers-Kr{\"{o}}nig
+                  relation which requires integration over all absorbing
+                  frequencies. Furthermore, calculation of refractive index
+                  changes using dispersion relations is easier than a direct
+                  calculation of the susceptibility, as transition rates (which
+                  give absorption coefficients) are, in general, far easier to
+                  calculate than the expectation value of the optical
+                  polarization. Both resonant (generation of some excitation
+                  that is long lived compared with an optical period) and
+                  nonresonant ‘instantaneous' optical nonlinearities are
+                  discussed, and it is shown that the nonlinear dispersion
+                  relation has a common form and can be understood in terms of
+                  the linear Kramers-Kr{\"{o}}nig relation applied to a new
+                  system consisting of the material plus some ‘perturbation'. We
+                  present several examples of the form of this external
+                  perturbation, which can be viewed as the pump in a pump-probe
+                  experiment. We discuss the two-level saturated atom model and
+                  bandfilling in semiconductors among others for the resonant
+                  case. For the nonresonant case some recent work is included
+                  where the electronic nonlinear refractive coefficient,n2, is
+                  determined from the nonlinear absorption processes of
+                  two-photon absorption, Raman transitions and the a.c. Stark
+                  effect. We also review how the dispersion relations can be
+                  extended to give alternative forms for frequency summation
+                  which, for example, allows the real and imaginary parts of?(2)
+                  to be related.},
+  file =         {::;::},
+  isbn =         {0306-8919},
+  issn =         03068919,
+}
+
+@article{Kang1997,
+  author =       {Kang, Inuk and Wise, Frank W.},
+  title =        {{Electronic structure and optical properties of PbS and PbSe
+                  quantum dots}},
+  journal =      {Journal of the Optical Society of America B},
+  volume =       14,
+  number =       7,
+  pages =        1632,
+  year =         1997,
+  doi =          {10.1364/JOSAB.14.001632},
+  url =
+                  {http://www.opticsinfobase.org/abstract.cfm?URI=josab-14-7-1632},
+  file =         {:C$\backslash$:/Users/Dan/AppData/Local/Mendeley Ltd./Mendeley
+                  Desktop/Downloaded/Kang, Wise - 1997 - Electronic structure
+                  and optical properties of PbS and PbSe quantum dots.pdf:pdf},
+  issn =         {0740-3224},
+  month =        {jul},
+}
+
+@article{Karki2013,
+  author =       {Karki, Khadga J and Ma, Fei and Zheng, Kaibo and Zidek, Karel
+                  and Mousa, Abdelrazek and Abdellah, Mohamed and Messing, Maria
+                  E and Wallenberg, L Reine and Yartsev, Arkadi and Pullerits,
+                  T{\~{o}}nu},
+  title =        {{Multiple exciton generation in nano-crystals revisited:
+                  consistent calculation of the yield based on pump-probe
+                  spectroscopy.}},
+  journal =      {Scientific reports},
+  volume =       3,
+  pages =        2287,
+  year =         2013,
+  doi =          {10.1038/srep02287},
+  url =
+                  {http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=3724175{\&}tool=pmcentrez{\&}rendertype=abstract},
+  abstract =     {Multiple exciton generation (MEG) is a process in which more
+                  than one exciton is generated upon the absorption of a high
+                  energy photon, typically higher than two times the band gap,
+                  in semiconductor nanocrystals. It can be observed
+                  experimentally using time resolved spectroscopy such as the
+                  transient absorption measurements. Quantification of the MEG
+                  yield is usually done by assuming that the bi-exciton signal
+                  is twice the signal from a single exciton. Herein we show that
+                  this assumption is not always justified and may lead to
+                  significant errors in the estimated MEG yields. We develop a
+                  methodology to determine proper scaling factors to the signals
+                  from the transient absorption experiments. Using the
+                  methodology we find modest MEG yields in lead chalcogenide
+                  nanocrystals including the nanorods.},
+  file =         {::},
+  issn =         {2045-2322},
+  month =        {jan},
+  pmid =         23887181,
+}
+
+@article{Kohler2014,
+  author =       {Kohler, Daniel D. and Block, Stephen B. and Kain, Schuyler and
+                  Pakoulev, Andrei V. and Wright, John C.},
+  title =        {{Ultrafast Dynamics within the 1S Exciton Band of Colloidal
+                  PbSe Quantum Dots Using Multiresonant Coherent
+                  Multidimensional Spectroscopy}},
+  journal =      {The Journal of Physical Chemistry C},
+  volume =       118,
+  number =       9,
+  pages =        {5020--5031},
+  year =         2014,
+  doi =          {10.1021/jp412058u},
+  url =          {http://pubs.acs.org/doi/abs/10.1021/jp412058u},
+  file =         {::},
+  issn =         {1932-7447},
+  month =        {mar},
+}
+
+@article{Kohler2017,
+  author =       {Kohler, Daniel D and Thompson, Blaise J and Wright, John C},
+  title =        {{Frequency-domain coherent multidimensional spectroscopy when
+                  dephasing rivals pulsewidth: Disentangling material and
+                  instrument response}},
+  journal =      {The Journal of Chemical Physics},
+  volume =       147,
+  number =       8,
+  pages =        084202,
+  year =         2017,
+  doi =          {10.1063/1.4986069},
+  url =          {http://dx.doi.org/10.1063/1.4986069
+                  http://aip.scitation.org/toc/jcp/147/8
+                  http://aip.scitation.org/doi/10.1063/1.4986069},
+  abstract =     {Ultrafast spectroscopy is often collected in the mixed
+                  frequency/time domain, where pulse durations are similar to
+                  system dephasing times. In these experiments, expectations
+                  derived from the familiar driven and impulsive limits are not
+                  valid. This work simulates the mixed-domain four-wave mixing
+                  response of a model system to develop expectations for this
+                  more complex field-matter interaction. We explore frequency
+                  and delay axes. We show that these line shapes are exquisitely
+                  sensitive to excitation pulse widths and delays. Near pulse
+                  overlap, the excitation pulses induce correlations that
+                  resemble signatures of dynamic inhomogeneity. We describe
+                  these line shapes using an intuitive picture that connects to
+                  familiar field-matter expressions. We develop strategies for
+                  distinguishing pulse-induced correlations from true system
+                  inhomogeneity. These simulations provide a founda-tion for
+                  interpretation of ultrafast experiments in the mixed domain.},
+  file =         {::},
+  issn =         {0021-9606},
+  month =        {aug},
+}
+
+@article{Kraatz2014,
+  author =       {Kraatz, Ingvar T and Booth, Matthew and Whitaker, Benjamin J
+                  and Nix, Michael G D and Critchley, Kevin},
+  title =        {{Sub-Bandgap Emission and Intraband Defect-Related
+                  Excited-State Dynamics in Colloidal CuInS 2 /ZnS Quantum Dots
+                  Revealed by Femtosecond Pump − Dump − Probe Spectroscopy}},
+  year =         {2014},
+  file =         {::},
+}
+
+@article{Lucarini2008,
+  author =       {Lucarini, Valerio},
+  title =        {{Response theory for equilibrium and non-equilibrium
+                  statistical mechanics: Causality and generalized
+                  kramers-kronig relations}},
+  journal =      {Journal of Statistical Physics},
+  volume =       131,
+  number =       3,
+  pages =        {543--558},
+  year =         2008,
+  doi =          {10.1007/s10955-008-9498-y},
+  abstract =     {We consider the general response theory proposed by Ruelle for
+                  describing the impact of small perturbations to the
+                  non-equilibrium steady states resulting from Axiom A dynamical
+                  systems. We show that the causality of the response functions
+                  allows for writing a set of Kramers-Kronig relations for the
+                  corresponding susceptibilities at all orders of nonlinearity.
+                  Nonetheless, only a special class of observable
+                  susceptibilities obey Kramers-Kronig relations. Specific
+                  results are provided for arbitrary order harmonic response,
+                  which allows for a very comprehensive Kramers-Kronig analysis
+                  and the establishment of sum rules connecting the asymptotic
+                  behavior of the susceptibility to the short-time response of
+                  the system. These results generalize previous findings on
+                  optical Hamiltonian systems and simple mechanical models, and
+                  shed light on the general impact of considering the principle
+                  of causality for testing self-consistency: the described
+                  dispersion relations constitute unavoidable benchmarks for any
+                  experimental and model generated dataset. In order to connect
+                  the response theory for equilibrium and non equilibrium
+                  systems, we rewrite the classical results by Kubo so that
+                  response functions formally identical to those proposed by
+                  Ruelle, apart from the measure involved in the phase space
+                  integration, are obtained. We briefly discuss how these
+                  results, taking into account the chaotic hypothesis, might be
+                  relevant for climate research. In particular, whereas the
+                  fluctuation-dissipation theorem does not work for
+                  non-equilibrium systems, because of the non-equivalence
+                  between internal and external fluctuations, Kramers-Kronig
+                  relations might be more robust tools for the definition of a
+                  self-consistent theory of climate change.},
+  archivePrefix ={arXiv},
+  arxivId =      {0710.0958},
+  eprint =       {0710.0958},
+  file =         {::},
+  issn =         00224715,
+  keywords =     {Axiom A dynamical systems,Chaotic hypothesis,Climate,Harmonic
+                  generation,Kramers-Kronig relations,Kubo response
+                  theory,Non-equilibrium steady states,Ruelle response
+                  theory,SRB measure},
+}
+
+@article{Moreels2006,
+  author =       {Moreels, I. and Hens, Z. and Kockaert, P. and Loicq, J. and
+                  {Van Thourhout}, D.},
+  title =        {{Spectroscopy of the nonlinear refractive index of colloidal
+                  PbSe nanocrystals}},
+  journal =      {Applied Physics Letters},
+  volume =       89,
+  number =       19,
+  pages =        {0--4},
+  year =         2006,
+  doi =          {10.1063/1.2385658},
+  abstract =     {A spectroscopic study of the optical nonlinearity of PbSe
+                  colloidal solutions was performed with the Z-scan technique at
+                  wavelength intervals of 1200-1350 and 1540-1750 nm. While
+                  nonlinear absorption remains below the detection threshold,
+                  the third order nonlinear refractive index n(2) shows clear
+                  resonances, somewhat blueshifted relative to the exciton
+                  transitions in the absorbance spectrum. The occurrence of
+                  thermal effects is ruled out by time-resolved measurements. At
+                  1.55 mu m, measured (resonant) n(2) values exceed typical bulk
+                  semiconductor values by two orders of magnitude. At high
+                  optical intensity, the refractive index change saturates,
+                  indicating that statefilling lies at the origin of the
+                  observed effect.},
+  file =         {::},
+  isbn =         {0003-6951},
+  issn =         00036951,
+}
+
+@article{Moreels2007,
+  author =       {Moreels, Iwan and Lambert, Karel and {De Muynck}, David and
+                  Vanhaecke, Frank and Poelman, Dirk and Martins, Jos{\'{e}} C.
+                  and Allan, Guy and Hens, Zeger},
+  title =        {{Composition and Size-Dependent Extinction Coefficient of
+                  Colloidal PbSe Quantum Dots}},
+  journal =      {Chemistry of Materials},
+  volume =       19,
+  number =       25,
+  pages =        {6101--6106},
+  year =         2007,
+  doi =          {10.1021/cm071410q},
+  url =          {http://pubs.acs.org/doi/abs/10.1021/cm071410q},
+  file =         {::},
+  issn =         {0897-4756},
+  keywords =     {reference,size-exciton correlation},
+  mendeley-tags ={reference},
+  month =        {dec},
+}
+
+@article{Moreels2008,
+  author =       {Moreels, Iwan and Hens, Zeger},
+  title =        {{On the interpretation of colloidal quantum-dot absorption
+                  spectra.}},
+  journal =      {Small (Weinheim an der Bergstrasse, Germany)},
+  volume =       4,
+  number =       11,
+  pages =        {1866--8; author reply 1869--70},
+  year =         2008,
+  doi =          {10.1002/smll.200800068},
+  url =          {http://www.ncbi.nlm.nih.gov/pubmed/18855972},
+  file =         {::},
+  issn =         {1613-6829},
+  keywords =     {Absorption,Colloids,Colloids: chemistry,Quantum Dots,Spectrum
+                  Analysis,Spectrum Analysis: methods},
+  month =        {nov},
+  pmid =         18855972,
+}
+
+@article{Moreels2009,
+  author =       {Moreels, Iwan and Lambert, Karel and Muynck, David De and
+                  Vanhaecke, Frank and Poelman, Dirk and Martins, Jos{\'{e}} C
+                  and Allan, Guy and Hens, Zeger},
+  title =        {{Size-Dependent Optical Properties of Colloidal {\{}PbS{\}}
+                  Quantum Dots}},
+  journal =      {ACS Nano},
+  volume =       3,
+  number =       10,
+  pages =        {3023--3030},
+  year =         2009,
+  doi =          {10.1021/nn900863a},
+  abstract =     {We quantitatively investigate the size-dependent optical
+                  properties of colloidal {\{}PbS{\}} nanocrystals or quantum
+                  dots (Qdots), by combining the Qdot absorbance spectra with
+                  detailed elemental analysis of the Qdot suspensions. At high
+                  energies, the molar extinction coefficient ?? increases with
+                  the Qdot volume d3 and agrees with theoretical calculations
+                  using the Maxwell???Garnett effective medium theory and bulk
+                  values for the Qdot dielectric function. This demonstrates
+                  that quantum confinement has no influence on ?? in this
+                  spectral range, and it provides an accurate method to
+                  calculate the Qdot concentration. Around the band gap, ?? only
+                  increases with d1.3, and values are comparable to the ?? of
+                  {\{}PbSe{\}} Qdots. The data are related to the oscillator
+                  strength fif of the band gap transition and results agree well
+                  with theoretical tight-binding calculations, predicting a
+                  linear dependence of fif on d. For both {\{}PbS{\}} and
+                  {\{}PbSe{\}} Qdots, the exciton lifetime ?? is calculated from
+                  fif. We find values ranging between 1 and 3 ??s, in agreement
+                  with experimental literature data from time-resolved
+                  luminescence spectroscopy. Our results provide a thorough
+                  general framework to calculate and understand the optical
+                  properties of suspended colloidal quantum dots. Most
+                  importantly, it highlights the significance of the local field
+                  factor in these systems.},
+  file =         {::},
+  issn =         {1936-0851},
+  keywords =     {exciton lifetime,extinction coefficient,lead
+                  chalcogenide,molar,oscillator strength,pbse,semiconductor
+                  nanocrystals},
+}
+
+@article{Nootz2011,
+  author =       {Nootz, Gero and Padilha, Lazaro A. and Levina, Larissa and
+                  Sukhovatkin, Vlad and Webster, Scott and Brzozowski, Lukasz
+                  and Sargent, Edward H. and Hagan, David J. and {Van Stryland},
+                  Eric W.},
+  title =        {{Size dependence of carrier dynamics and carrier
+                  multiplication in PbS quantum dots}},
+  journal =      {Physical Review B - Condensed Matter and Materials Physics},
+  volume =       83,
+  number =       15,
+  pages =        {1--7},
+  year =         2011,
+  doi =          {10.1103/PhysRevB.83.155302},
+  abstract =     {The time dynamics of the photoexcited carriers and
+                  carrier-multiplication efficiencies in PbS quantum dots (QDs)
+                  are investigated. In particular, we report on the carrier
+                  dynamics, including carrier multiplication, as a function of
+                  QD size and compare them to the bulk value. We show that the
+                  intraband 1P -{\textgreater} 1S decay becomes faster for
+                  smaller QDs, in agreement with the absence of a phonon
+                  bottleneck. Furthermore, as the size of the QDs decreases, the
+                  energy threshold for carrier multiplication shifts from the
+                  bulk value to higher energies. However, the energy threshold
+                  shift is smaller than the band-gap shift and, therefore, for
+                  the smallest QDs, the threshold approaches 2.35 E(g), which is
+                  close to the theoretical energy conservation limit of twice
+                  the band gap. We also show that the carrier-multiplication
+                  energy efficiency increases with decreasing QD size. By
+                  comparing to theoretical models, our results suggest that
+                  impact ionization is sufficient to explain carrier
+                  multiplication in QDs.},
+  file =         {::},
+  isbn =         {1098-0121},
+  issn =         10980121,
+}
+
+@article{Okumura1999,
+  author =       {Okumura, Ko and Tokmakoff, Andrei and Tanimura, Yoshitaka},
+  title =        {{Two-dimensional line-shape analysis of photon-echo signal}},
+  journal =      {Chemical Physics Letters},
+  volume =       314,
+  number =       {5-6},
+  pages =        {488--495},
+  year =         1999,
+  doi =          {10.1016/S0009-2614(99)01173-2},
+  url =          {http://linkinghub.elsevier.com/retrieve/pii/S0009261499011732},
+  abstract =     {We analyze the two-dimensional (2D) line shape obtained by 2D
+                  Fourier transforming the time-domain response of a photon-echo
+                  signal as a function of the two coherence periods, t(1) and
+                  t(3). The line shape obtained for a two-level system with
+                  homogeneous and inhomogeneous broadening is shown to be
+                  sensitive to the magnitude of both of these line-broadening
+                  mechanisms. It is shown that the ellipticity of the 2D line
+                  shape can be related to the ratio of homogeneous to
+                  inhomogeneous broadening. (C) 1999 Elsevier Science B.V. All
+                  rights reserved.},
+  file =         {:C$\backslash$:/Users/Dan/AppData/Local/Mendeley Ltd./Mendeley
+                  Desktop/Downloaded/Okumura, Tokmakoff, Tanimura - 1999 -
+                  Two-dimensional line-shape analysis of photon-echo
+                  signal.pdf:pdf},
+  isbn =         {0009-2614},
+  issn =         00092614,
+  keywords =     {3rd-order,dynamics,glasses,phase,probes,spectroscopy},
+  month =        {dec},
+}
+
+@article{Olivero1977,
+  author =       {Olivero, J.J. and Longbothum, R.L.},
+  title =        {{Empirical fits to the voigt line width: A brief review}},
+  journal =      {J. Quant. Spectrosc. Radiat. Transfer},
+  volume =       17,
+  pages =        {233--236},
+  year =         1977,
+  file =         {::},
+}
+
+@article{Omari2012,
+  author =       {Omari, Abdoulghafar and Moreels, Iwan and Masia, Francesco and
+                  Langbein, Wolfgang and Borri, Paola and {Van Thourhout}, Dries
+                  and Kockaert, Pascal and Hens, Zeger},
+  title =        {{Role of interband and photoinduced absorption in the
+                  nonlinear refraction and absorption of resonantly excited PbS
+                  quantum dots around 1550 nm}},
+  journal =      {Physical Review B},
+  volume =       85,
+  number =       11,
+  pages =        115318,
+  year =         2012,
+  doi =          {10.1103/PhysRevB.85.115318},
+  url =          {http://link.aps.org/doi/10.1103/PhysRevB.85.115318},
+  file =         {::},
+  issn =         {1098-0121},
+  month =        {mar},
+}
+
+@article{Pang1991,
+  author =       {Pang, Yang and Samoc, Marek and Prasad, Paras N.},
+  title =        {{Third‐order nonlinearity and two‐photon‐induced molecular
+                  dynamics: Femtosecond time‐resolved transient absorption, Kerr
+                  gate, and degenerate four‐wave mixing studies in poly ( p
+                  ‐phenylene vinylene)/sol‐gel silica film}},
+  journal =      {The Journal of Chemical Physics},
+  volume =       94,
+  number =       8,
+  pages =        {5282--5290},
+  year =         1991,
+  doi =          {10.1063/1.460512},
+  url =
+                  {http://scitation.aip.org/content/aip/journal/jcp/94/8/10.1063/1.460512
+                  http://aip.scitation.org/doi/10.1063/1.460512},
+  abstract =     {Femtosecond response and relaxation of the third-order optical
+                  nonlinearity in a newly developed poly (p-phenylene
+                  vinylene)/sol-gel silica composite are investigated by time-
+                  resolved forward wave degenerate four-wave mixing, Kerr gate,
+                  and transient absorption techniques using 60 fs pulses at 620
+                  nm. Using a theoretical description of two- and four-wave
+                  mixing in optically nonlinear media, it is shown that the
+                  results obtained from simultaneous use of these techniques
+                  yield valuable information on the real and imaginary
+                  components of the third-order susceptibility. In the composite
+                  material investigated here, the imaginary component is derived
+                  from the presence of a two-photon resonance at the wavelength
+                  of 620 nm used for the present study. This two-photon
+                  resonance is observed as transient absorption of the probe
+                  beam induced by the presence of a strong pump beam. It also
+                  provides fifth-order nonlinear response both in transient
+                  absorption and in degenerate four-wave mixing. The fifth-
+                  order contributions are derived from the two-photon generated
+                  excited species which can absorb at the measurement wavelength
+                  and therefore modify both the absorption coefficient and the
+                  refractive index of the medium.},
+  file =         {::},
+  issn =         {0021-9606},
+  month =        {apr},
+}
+
+@article{Peterson2007,
+  author =       {Peterson, JJ and Huang, Libai and Delerue, C. and Allan, Guy},
+  title =        {{Uncovering forbidden optical transitions in PbSe
+                  nanocrystals}},
+  journal =      {nano Letters},
+  year =         2007,
+  url =          {http://pubs.acs.org/doi/abs/10.1021/nl072487g},
+  file =         {:C$\backslash$:/Users/Dan/AppData/Local/Mendeley Ltd./Mendeley
+                  Desktop/Downloaded/Peterson et al. - 2007 - Uncovering
+                  forbidden optical transitions in PbSe nanocrystals.pdf:pdf},
+  keywords =     {1P exciton,2 photon absorption,new,relevant},
+  mendeley-tags ={new,relevant},
+}
+
+@article{Price2015,
+  author =       {Price, Michael B. and Butkus, Justinas and Jellicoe, Tom C.
+                  and Sadhanala, Aditya and Briane, Anouk and Halpert, Jonathan
+                  E. and Broch, Katharina and Hodgkiss, Justin M. and Friend,
+                  Richard H. and Deschler, Felix},
+  title =        {{Hot-carrier cooling and photoinduced refractive index changes
+                  in organic–inorganic lead halide perovskites}},
+  journal =      {Nature Communications},
+  volume =       6,
+  number =       {May},
+  pages =        8420,
+  year =         2015,
+  doi =          {10.1038/ncomms9420},
+  url =          {http://dx.doi.org/10.1038/ncomms9420
+                  http://www.nature.com/doifinder/10.1038/ncomms9420},
+  archivePrefix ={arXiv},
+  arxivId =      {arXiv:1504.07508},
+  eprint =       {arXiv:1504.07508},
+  file =         {::},
+  issn =         {2041-1723},
+  publisher =    {Nature Publishing Group},
+}
+
+@article{Schaller2003,
+  author =       {Schaller, Richard D and Petruska, M. a. and Klimov, Victor I.},
+  title =        {{Tunable Near-Infrared Optical Gain and Amplified Spontaneous
+                  Emission Using PbSe Nanocrystals}},
+  journal =      {The Journal of Physical Chemistry B},
+  volume =       107,
+  number =       50,
+  pages =        {13765--13768},
+  year =         2003,
+  doi =          {10.1021/jp0311660},
+  url =          {http://pubs.acs.org/doi/abs/10.1021/jp0311660},
+  file =         {::;::},
+  issn =         {1520-6106},
+  month =        {dec},
+}
+
+@article{Schins2009,
+  author =       {Schins, Juleon and Trinh, M. and Houtepen, Arjan and
+                  Siebbeles, Laurens},
+  title =        {{Probing formally forbidden optical transitions in PbSe
+                  nanocrystals by time- and energy-resolved transient absorption
+                  spectroscopy}},
+  journal =      {Physical Review B},
+  volume =       80,
+  number =       3,
+  pages =        035323,
+  year =         2009,
+  doi =          {10.1103/PhysRevB.80.035323},
+  url =          {http://link.aps.org/doi/10.1103/PhysRevB.80.035323},
+  file =         {::},
+  issn =         {1098-0121},
+  month =        {jul},
+}
+
+@article{Svirko1999,
+  author =       {{P. Svirko}, Yuri and Shirane, Masayuki and Suzuura, Hidekatsu
+                  and Kuwata-Gonokami, Makoto},
+  title =        {{Four-Wave Mixing Theory at the Excitonic Resonance: Weakly
+                  Interacting Boson Model}},
+  journal =      {Journal of the Physical Society of Japan},
+  volume =       68,
+  number =       2,
+  pages =        {674--682},
+  year =         1999,
+  doi =          {10.1143/JPSJ.68.674},
+  url =          {http://journals.jps.jp/doi/10.1143/JPSJ.68.674},
+  file =         {::},
+  issn =         {0031-9015},
+  keywords =     {Exciton,Four-particle correlation,Four-wave mixing,Normal mode
+                  splitting,Semiconductor microcavity},
+  month =        {feb},
+}
+
+@article{Trinh2008,
+  author =       {Trinh, M Tuan and Houtepen, Arjan J and Schins, Juleon M and
+                  Piris, Jorge and Siebbeles, Laurens D a},
+  title =        {{Nature of the second optical transition in PbSe
+                  nanocrystals.}},
+  journal =      {Nano letters},
+  volume =       8,
+  number =       7,
+  pages =        {2112--7},
+  year =         2008,
+  doi =          {10.1021/nl8010963},
+  url =          {http://www.ncbi.nlm.nih.gov/pubmed/18510369},
+  abstract =     {The second peak in the optical absorption spectrum of PbSe
+                  nanocrystals is arguably the most discussed optical transition
+                  in semiconductor nanocrystals. Ten years of scientific debate
+                  have produced many theoretical and experimental claims for the
+                  assignment of this feature as the 1P e1P h as well as the 1S
+                  h,e1P e,h transitions. We studied the nature of this
+                  absorption feature by pump-probe spectroscopy, exactly
+                  controlling the occupation of the states involved, and present
+                  conclusive evidence that the optical transition involves
+                  neither 1S e nor 1S h states. This suggests that it is the 1P
+                  h1P e transition that gives rise to the second peak in the
+                  absorption spectrum of PbSe nanocrystals.},
+  file =         {::},
+  issn =         {1530-6984},
+  month =        {jul},
+  pmid =         18510369,
+}
+
+@article{Trinh2013,
+  author =       {Trinh, M. Tuan and Sfeir, Matthew Y. and Choi, Joshua J. and
+                  Owen, Jonathan S. and Zhu, Xiaoyang},
+  title =        {{A Hot Electron–Hole Pair Breaks the Symmetry of a
+                  Semiconductor Quantum Dot}},
+  journal =      {Nano Letters},
+  volume =       13,
+  number =       12,
+  pages =        {6091--6097},
+  year =         2013,
+  doi =          {10.1021/nl403368y},
+  url =          {http://pubs.acs.org/doi/abs/10.1021/nl403368y},
+  file =         {::},
+  issn =         {1530-6984},
+  keywords =     {google the phrase,hot carriers,nanocrystals or,near the
+                  bandgap is,now well-,one,ptical excitation of
+                  semiconductor,qds,quantum dots,selection rules,stark
+                  effect,symmetry breaking,transient absorption,understood},
+  month =        {dec},
+}
+
+@article{Wehrenberg2002,
+  author =       {Wehrenberg, Brian L. and Wang, Congjun and Guyot-Sionnest,
+                  Philippe},
+  title =        {{Interband and Intraband Optical Studies of PbSe Colloidal
+                  Quantum Dots}},
+  journal =      {The Journal of Physical Chemistry B},
+  volume =       106,
+  number =       41,
+  pages =        {10634--10640},
+  year =         2002,
+  doi =          {10.1021/jp021187e},
+  url =          {http://pubs.acs.org/doi/abs/10.1021/jp021187e},
+  file =         {::},
+  issn =         {1520-6106},
+  month =        {oct},
+}
+
+@article{Wise2000,
+  author =       {Wise, Frank W.},
+  title =        {{Lead salt quantum dots: the limit of strong quantum
+                  confinement.}},
+  journal =      {Accounts of chemical research},
+  volume =       33,
+  number =       11,
+  pages =        {773--80},
+  year =         2000,
+  url =          {http://www.ncbi.nlm.nih.gov/pubmed/11087314},
+  abstract =     {Nanocrystals or quantum dots of the IV-VI semiconductors PbS,
+                  PbSe, and PbTe provide unique properties for investigating the
+                  effects of strong confinement on electrons and phonons. The
+                  degree of confinement of charge carriers can be many times
+                  stronger than in most II-VI and III-V semiconductors, and lead
+                  salt nanostructures may be the only materials in which the
+                  electronic energies are determined primarily by quantum
+                  confinement. This Account briefly reviews recent research on
+                  lead salt quantum dots.},
+  file =         {:C$\backslash$:/Users/Dan/AppData/Local/Mendeley Ltd./Mendeley
+                  Desktop/Downloaded/Wise - 2000 - Lead salt quantum dots the
+                  limit of strong quantum confinement.pdf:pdf},
+  issn =         {0001-4842},
+  keywords =     {Crystallization,Electrochemistry,Lead,Lead:
+                  chemistry,Semiconductors,Temperature},
+  month =        {nov},
+  pmid =         11087314,
+}
+
+@article{Yang2015,
+  author =       {Yang, Ye and Yan, Yong and Yang, Mengjin and Choi, Sukgeun and
+                  Zhu, Kai and Luther, Joseph M and Beard, Matthew C},
+  title =        {{Low surface recombination velocity in}},
+  journal =      {Nature Communications},
+  volume =       6,
+  pages =        {1--6},
+  year =         2015,
+  doi =          {10.1038/ncomms8961},
+  url =          {http://dx.doi.org/10.1038/ncomms8961},
+  file =         {::},
+  publisher =    {Nature Publishing Group},
+}
+
+@article{Yang2017,
+  author =       {Yang, Ye and Yang, Mengjin and Moore, David?T. and Yan, Yong
+                  and Miller, Elisa?M. and Zhu, Kai and Beard, Matthew?C.},
+  title =        {{Top and bottom surfaces limit carrier lifetime in lead iodide
+                  perovskite films}},
+  journal =      {Nature Energy},
+  volume =       2,
+  number =       2,
+  pages =        16207,
+  year =         2017,
+  doi =          {10.1038/nenergy.2016.207},
+  url =          {http://www.nature.com/articles/nenergy2016207},
+  abstract =     {Carrier recombination at defects is detrimental to the
+                  performance of solar energy conversion systems, including
+                  solar cells and photoelectrochemical devices. Point defects
+                  are localized within the bulk crystal while extended defects
+                  occur at surfaces and grain boundaries. If not properly
+                  managed, surfaces can be a large source of carrier
+                  recombination. Separating surface carrier dynamics from bulk
+                  and/or grain-boundary recombination in thin films is
+                  challenging. Here, we employ transient reflection spectroscopy
+                  to measure the surface carrier dynamics in methylammonium lead
+                  iodide perovskite polycrystalline films. We find that surface
+                  recombination limits the total carrier lifetime in perovskite
+                  polycrystalline thin films, meaning that recombination inside
+                  grains and/or at grain boundaries is less important than top
+                  and bottom surface recombination. The surface recombination
+                  velocity in polycrystalline films is nearly an order of
+                  magnitude smaller than that in single crystals, possibly due
+                  to unintended surface passivation of the films during
+                  synthesis.},
+  file =         {::},
+  issn =         {2058-7546},
+  month =        {jan},
+}
+
+@article{Yurs2011,
+  author =       {Yurs, Lena A and Block, Stephen B and Pakoulev, Andrei V. and
+                  Selinsky, Rachel S. and Jin, Song and Wright, John},
+  title =        {{Multiresonant Coherent Multidimensional Electronic
+                  Spectroscopy of Colloidal PbSe Quantum Dots}},
+  journal =      {The Journal of Physical Chemistry C},
+  volume =       115,
+  number =       46,
+  pages =        {22833--22844},
+  year =         2011,
+  doi =          {10.1021/jp207273x},
+  url =          {http://pubs.acs.org/doi/abs/10.1021/jp207273x},
+  file =         {::},
+  issn =         {1932-7447},
+  month =        {nov},
+}
+
+@article{Yurs2012,
+  author =       {Yurs, Lena A and Block, Stephen B and Pakoulev, Andrei V. and
+                  Selinsky, Rachel S. and Jin, Song and Wright, John},
+  title =        {{Spectral Isolation and Measurement of Surface-Trapped State
+                  Multidimensional Nonlinear Susceptibility in Colloidal Quantum
+                  Dots}},
+  journal =      {The Journal of Physical Chemistry C},
+  volume =       116,
+  number =       9,
+  pages =        {5546--5553},
+  year =         2012,
+  doi =          {10.1021/jp3014139},
+  url =          {http://pubs.acs.org/doi/abs/10.1021/jp3014139},
+  file =         {::},
+  issn =         {1932-7447},
+  month =        {mar},
+}
\ No newline at end of file
diff --git a/PbSe_global_analysis/results.tex b/PbSe_global_analysis/results.tex
new file mode 100644
index 0000000..0bb0f03
--- /dev/null
+++ b/PbSe_global_analysis/results.tex
@@ -0,0 +1,47 @@
+\subsection{Pump-Probe 3D acquisitions for TA and TG}

+

+\begin{figure}

+	\includegraphics[scale=0.5]{"movies_combined"}

+	\caption{$S_{\text{TG}}$ (left) and $S_{\text{TA}}$ 2D spectra (see colorbar

+    labels) of Batch A (top) and Batch B (bottom) as a function of T delay. The

+    colors of each 2D spectrum are normalized to the global maximum of the 3D

+    acquisition, while the contour lines are normalized to each particular 2D

+    spectrum.}

+	\label{fig:movies}

+\end{figure}

+

+For both samples, 2D spectra were collected for increments along the population rise time. 

+For these acquisitions, concentrated samples ($\text{OD}_{\text{1S}} \sim 0.6, 0.8$) were used to minimize contributions from non-resonant background. 

+Both samples maintained constant signal amplitude for at least hundreds of picoseconds after initial excitation, indicating multiexcitons and trapping were negligible effects in these studies. 

+The TA and TG results for both batches are shown in Figure \ref{fig:movies}. For $T<0$ (probe arrives before pump), both collections show spectral line-narrowing in the anti-diagonal direction. 

+This highly correlated line shape is indicative of an inhomogeneous distribution, but the correlation is enhanced by pulse overlap effects. When the probe arrives before or at the same time as the pump, the typical pump-probe pathways are suppressed and more unconventional pathways with probe-pump and pump-probe-pump pulse orderings are enhanced.

+Such pathways exhibit resonant enhancement when $\omega_1=\omega_2$, even in the absence of inhomogeneity. 

+The pulse overlap effect is well-understood in both TA\cite{BritoCruz1988} and TG\cite{Kohler2017} experiments. 

+

+After the initial excitation rise time ($T > 50$ fs), the signal reaches a maximum, followed by a slight loss of signal ($\sim 10\%$) over the course of ~150 fs, after which the signal converges to a line shape that remains static over the dynamic range of our experiment ($200$ ps). 

+This signal loss occurs in both samples in both TA and TG; in TA measurements, the loss of amplitude occurred on both the ESA feature and the bleach feature, so that the band integral\cite{Gdor2013a} did not appreciably change. We do not know the cause of this loss, but speculate it could be a signature of bandgap renormalization.

+

+The static line shape distinguishes the homogeneous and inhomogeneous contributions to the 1S band. 

+The elongation of the peak along the diagonal, relative to the antidiagonal, demonstrates a persistent correlation between the pumped state and excited state; we attribute this correlation to the size distribution of the synthesized quantum dots. 

+The diagonal elongation is much more noticeable in the TA spectrum; the TG spectra is much more elongated along the $\omega_1$ axis, which makes discerning the antidiagonal and diagonal widths more difficult. 

+The TG spectrum is elongated along $\omega_1$ because it measures both the absorptive and refractive components of the probe spectrum, while it is sensitive only to the absorptive components along the pump axis. 

+At all delays, Batch A exhibits a much broader diagonal line shape than that of Batch B, indicative of its larger size distribution.

+

+Our spectra show that the 2D line shape of the 1S exciton is significantly distorted by contributions from hot carrier excitation just above the 1S state. 

+These hot carriers arise from transitions between the 1S and 1P resonances, which have been attributed to either the “rising edge” of the continuum or the pseudo-forbidden 1S-1P exciton transition\cite{Schins2009,Peterson2007}. 

+Contributions from these hot carriers distort the 1S 2D line shape for $\omega_2 > \omega_{\text{1S}}$, resulting in a bleach feature centered at $\omega_1=\omega_{\text{1S}}$ and containing bleach contributions from the unresolved ensemble. 

+The rise time of this feature is indistinguishable from the 1S rise time, indicating either extremely fast ($\leq 50$ fs) relaxation or direct excitation of a hot 1S exciton. 

+Since the ensemble is inhomogeneous, these hot exciton contributions are presumably also present within the 1S band due to the larger (lower energy bandgap) members of the ensemble. 

+Such contributions would not be recognized or resolved without scanning the pump frequency.

+

+\subsection{The skewed TG probe spectrum}

+The most surprising spectral feature presented here is the skew of the TG probe spectrum towards the red of $\omega_1=\omega_{\text{1S}}$. 

+If 1S state-filling completely describes the nonlinear response, the TG signal will mimic the absorptive bleach behavior of TA and show a line shape symmetric about $\omega_1$. 

+Although the spectral range of our experimental system limits the measurement of the red skew of Batch A, this feature was reproducible across many batches and system alignments. 

+We find no grounds to discount the red skew based on our experimental procedures or sample reproducibility issues.

+

+As $T$ is scanned, the skewed part rises in concert with the 1S-resonant signal that has the pump-probe pulse sequence. 

+We therefore explain the skewness as either an instantaneous spectral signature of the photoexcited population or a feature with dynamics much faster than our pulses. 

+For all pump colors, the skew maintains a magnitude of $30-40\%$ of maximum TG signal for each probe slice.  % BJT: we should show this in the SI

+In contrast, TA signal red of the 1S exction is no more than $10\%$ of the maximum amplitude of the bleach. 

+The difference in prominence shows that the redshifted feature is primarily refractive in character.

diff --git a/PbSe_global_analysis/run.sh b/PbSe_global_analysis/run.sh
new file mode 100755
index 0000000..a38e6ff
--- /dev/null
+++ b/PbSe_global_analysis/run.sh
@@ -0,0 +1,75 @@
+set -e  # force exit upon error
+
+function printColor {
+  YELLOW='\033[0;33m'
+  NC='\033[0m' # No Color
+  printf "${YELLOW}$*${NC}\n"
+}
+
+function printLine {
+  printColor "%`tput cols`s"|tr ' ' '#'
+}
+
+function printThenPython {
+  printColor "python $*"
+  python "$*"
+}
+
+if [[ $# -eq 0 ]] ; then
+    echo 'please provide an argument in [data, simulations, figures, documents, all]'
+    exit 1
+fi
+
+# process data
+if [[ "$1" = "all" ]] || [[ "$1" = "data" ]] ; then
+  printLine
+  printColor data
+  printThenPython "data/TA A.py"
+  printThenPython "data/TA B.py"
+  printThenPython "data/TG A.py"
+  printThenPython "data/TG B.py"  
+fi
+
+# run simulations
+if [[ "$1" = "all" ]] || [[ "$1" = "simulations" ]] ; then
+  printLine
+  printColor simulations
+  #printThenPython "simulations/fit fsb19.py"
+  #printThenPython "simulations/fit fsb25.py"
+fi
+
+# make figures
+if [[ "$1" = "all" ]] || [[ "$1" = "figures" ]] ; then
+  printLine
+  printColor figures
+  printThenPython "figures/absorbance.py"
+  printThenPython "figures/kramers_kronig.py"
+  printThenPython "figures/m_factors.py"
+  #printThenPython "figures/movies_fitted.py"
+  printThenPython "figures/movies_combined.py"
+  printThenPython "figures/power_factors.py"
+  #printThenPython "figures/driven_initial.py"
+  printThenPython "figures/TA_artifacts.py"
+  printThenPython "figures/TG_artifacts.py"
+  printThenPython "figures/ta_vs_tg.py"
+fi
+
+# render documents
+if [[ "$1" = "all" ]] || [[ "$1" = "documents" ]] ; then
+  printLine
+  printColor documents
+  # main
+  printColor main
+  pdflatex --interaction=nonstopmode main
+  bibtex main
+  pdflatex --interaction=nonstopmode main
+  pdflatex --interaction=nonstopmode main
+  # SI
+  printColor SI
+  pdflatex --interaction=nonstopmode SI
+  biber SI
+  pdflatex --interaction=nonstopmode SI
+  pdflatex --interaction=nonstopmode SI
+fi
+
+printColor finished
diff --git a/PbSe_global_analysis/theory.tex b/PbSe_global_analysis/theory.tex
new file mode 100644
index 0000000..d20d844
--- /dev/null
+++ b/PbSe_global_analysis/theory.tex
@@ -0,0 +1,188 @@
+\subsection{Nonlinear Band Edge Response}

+

+\begin{figure}

+	\includegraphics[width=\linewidth]{"model_system"}

+	\caption{

+		Model system for the 1S band of PbSe quantum dots. 

+			(a) The ground state shown in the electron-hole basis. 

+				All electrons (holes) are in the valence (conduction) band. 

+				There are two electrons and holes in each of the four degenerate $L$ points. 

+			(b) The excitonic basis and the transitions accessible in this experiment. 

+				The arrows illustrate the available absorptive or emissive transitions that take place in the $\chi^{(3)}$ experiment, and are labeled by parameters that control the cross-sectional strength (arrow width qualitatively indicates transition strength).

+			}

+	\label{fig:model_system}

+\end{figure}

+

+The optical non-linearity of near-bandgap QD excitons has been extensively investigated.%[CITE]

+The response derives largely from state-filling and depends strongly on the exciton occupancy of the dots.

+In a PbSe quantum dot, the 1S peak is composed of transitions between 8 1S electrons and 8 1S holes.\cite{Kang1997}  

+%In PbSe, near-bandgap excitons arise from confinement of direct transitions at the four $L$-points of the FCC lattice, yielding an 8-fold degeneracy within the 1S band.\cite{Kang1997}  

+%Both the electron states and hole states are split by exchange and Coulombic coupling but these splittings are small.  

+Figure \ref{fig:model_system}a shows the ground state configuration for a PbSe quantum dot.  The energy levels

+The 8-fold degeneracy means there are $8 \times 8 = 64$ states in the single-exciton ($|1\rangle$) manifold and $7 \times 7 = 49$ states in the biexciton ($|2\rangle$) manifold, so $1/4$ of optical transitions are lost upon single exciton creation.    

+

+Occupancy reduces the number of available transitions and 

+%A microscopic description of the optical properties of each state is outside the scope of this work.  

+

+%The 8-fold degenerate lead chalcogenide 1S exciton peak is composed of 8 electrons and 8 holes, which gives 64 states in the single exciton ($|1\rangle$) manifold and 49 states in the biexciton ($|2\rangle$) manifold.  

+Figure \ref{fig:model_system} shows the model system used in this study and the parameters that control the third-order response. 

+

+We assign all electron-hole transitions the same dipole moment, $\mu_{eh}$, so that the total cross-section between manifolds, $N_i \mu_{eh}^2$, is determined by the number optically active transitions available, $N_i$. 

+Although this assumption % state more correctly about what we are doing--there is the assumption that all dipoles are the same, and there is the observable that cross-sections correspond to the number of optically active transitions.  

+has come under scrutiny\cite{Karki2013,Gdor2015} it remains valid for the perturbative fluence used in this study. 

+This model corresponds to the weakly interacting boson model, used to describe the four-wave mixing response of quantum wells,\cite{Svirko1999} in the limit of small quantum well area.

+

+With this excitonic structure, we now describe the resulting non-linear polarization. 

+We restrict ourselves to field-matter interactions in which the pump, $E_2$, precedes the probe, $E_1$ (the ``true'' pump-probe time-ordering).

+\footnote{Note that $E_2$ and $E_{2^\prime}$ represent the same pulse in TA, but they are distinct fields ($\vec{k}_2 \neq \vec{k}_{2^\prime}$) in TG. For brevity, we will write equations assuming these pulse parameters are interchangeable.}  

+We consider the limit of low pump fluence, so that only single absorption events need be considered: $\text{Tr}\left[ \rho \right]  = (1-\bar{n})\rho_{00} + \bar{n} \rho_{11}$, where $\bar{n}\ll 1$ is the (average) fractional conversion of population.  

+In this limit, $\bar{n} = \frac{\sigma_0(\omega_2)}{\hbar \omega_2} \int{I_2(t) dt}$, where $I_2(t)$ is the pump intensity and $\sigma_0$ is the ground state absorptive cross-section.  

+%In TA, $I_2(t) = \frac{nc}{8\pi}\left| E_2(t) \right|^2$, while in TG, the average pump intensity is $I_2(t) = \frac{nc}{4\pi}\left| E_2(t) \right|^2$

+%The pump induces a non-equilibrium population difference closely approximated by the Poisson distribution; in the case of low fluence ($\hbar \omega_2 \ll \sigma_{QD} \int{I_2 dt}$), 

+%The expected population conversion is $\langle n \rangle = \frac{\sigma_0(\omega_2)}{\hbar \omega_2} \int{I_2(t) dt}$, where $ I_2(t) = \frac{nc}{8\pi}\left| E_2(t) \right|^2 $ and $\sigma_0$ is the ground state absorptive cross-section.  

+For a Gaussian temporal profile of standard deviation $\Delta_t$, $I_2(t) = I_{\text{2,peak}} \exp \left( -t^2 / 	2\Delta_t^2 \right)$, the exciton population is

+\begin{equation}\label{eq:n}

+	\bar{n} = \frac{\sigma_0(\omega_2)\sqrt{2\pi}}{\hbar \omega_2} \Delta_t I_{\text{2,peak}}.

+\end{equation}

+When the probe interrogates this ensemble; each population will interact linearly:

+\begin{equation}\label{eq:ptot}

+\begin{split}

+	P_{\text{tot}} &= \left( 1-\bar{n} \right)\chi_0^{(1)}(\omega_1)E_1 + \bar{n} \chi_1^{(1)} (\omega_1)E_1 \\ 

+		&= \chi_0^{(1)}E_1 + \bar{n}\left( \chi_1^{(1)}(\omega_1) - \chi_0^{(1)}(\omega_1) \right)E_1.

+\end{split}

+\end{equation}

+Here $\chi_0^{(1)}$ ($\chi_1^{(1)}$) denotes the linear susceptibility of the pure state $|0\rangle$ ($|1\rangle$). The third-order field scales as $I_2 E_1$, so from Equation \ref{eq:ptot}

+\begin{equation}\label{eq:chi3}

+	\chi^{(3)} \propto \sigma_0 (\omega_2) \left( \chi_1^{(1)} - \chi_0^{(1)} \right).

+\end{equation}

+This expression accounts for the familiar population-level pathways such as excited state absorption/emission and ground state depletion. 

+Conforming the linear susceptibilities to our model, the non-linear portion of Equation \ref{eq:chi3} can be written as:

+\begin{gather}

+	\chi^{(3)} \propto \mu_{eh}^4\text{Im}\left[L_0(\omega_2)\right]\left[ SL_1(\omega) - L_0(\omega_1) \right], \label{eq:chi3_lorentz}\\

+	L_0(\omega) = \frac{1}{\omega - \omega_{10} + i\Gamma_{10}} ,\\

+	L_1(\omega) = \frac{1}{\omega - (\omega_{10} - \epsilon) + i\xi\Gamma_{10}} ,

+\end{gather}

+where $S = \frac{N_2}{N_1 + 1}$. The denominator of $S$ is larger than $N_1$ due to the contribution of stimulated emission; this contribution is often neglected. 

+From Equation \ref{eq:chi3_lorentz} we can see that a finite response can result from three conditions: $S\neq 1$, $\xi \neq 1$, and/or $\epsilon \neq 0$. 

+The first inequality is the model's manifestation of state-filling, $S < 1$. 

+If we assume that all 64 ground state transitions are optically active, then $S = 0.75$. 

+The second condition is met by exciton-induced dephasing (EID), $\xi > 1$, 

+% EID has also been attributed to stark splitting of exciton states

+and the third from the net attractive Coulombic coupling of excitons, $ \epsilon > 0 $. 

+The finite bandwidth of the monochromator can be accounted for by convolving equation \ref{eq:chi3_lorentz} with the monochromator instrumental function.

+

+\subsection{The Bleach Nonlinearity}

+The bleach of the 1S band is the most studied nonlinear signature of the PbSe quantum dots. 

+Experiments keep track of the bleach factor, $\phi$, which is the proportionality factor that relates the relative change in the absorption coefficient at the exciton resonance, $\alpha_\text{FWM}(\omega_{10})$, with the average exciton occupation:

+\begin{equation}\label{eq:bleach_factor}

+	\frac{\alpha_\text{FWM}(\omega_{10})}{\alpha_0(\omega_{10})} = -\phi \bar{n}

+\end{equation}

+where $\alpha_0$ is the linear absorption coefficient.  

+If QDs are completely bleached by the creation of a single exciton, then $\phi = 1$; if QDs are unperturbed by the exciton, then $\phi=0$. 

+For PbSe, 1S-resonant values of $\phi=0.25$ and $0.125$ have been reported in literature \cite{Gdor2013a, Schaller2003, Nootz2011, Omari2012, Geiregat2014}, each with supporting theories on how state-filling should behave in an 8-fold degenerate system.  

+Inspection of Equation \ref{eq:chi3_lorentz} shows that $\phi = \frac{\text{Im} \left[ L_0 - SL_1 \right]}{L_0}$; if the 1S nonlinearity is dominated by state-filling ($\xi=1$ and $\epsilon=0$), then the bleach fraction has perfect correspondence with the change in the number of optically active states: $\phi = 1-S$. 

+Because Coulombic shifts and EID act to decrease the resonant absorption of the excited state, we have the strict relation $\phi \geq 1-S$.

+

+More recently, a bleach factor metric has been adopted\cite{Trinh2008,Trinh2013} as the proportionality between the spectrally integrated probe and the carrier concentration:

+\begin{equation}\label{eq:bleach_factor_int}

+	\frac{\int{\alpha_\text{FWM}(\omega) d\omega}}{\int{\alpha_0(\omega)d\omega}} = -\phi_{\text{int}} \bar{n}.

+\end{equation}

+This metric is a more robust description of state filling, because it is unaffected by Coulomb shifts or EID: Equation \ref{eq:chi3_lorentz} gives $\phi_{\text{int}}=1-S$ regardless of $\xi$ and $\epsilon$. 

+An experimental value of $\phi_{\text{int}}=0.25$ has been reported\cite{Trinh2013} which consequently supports the measurement of $\phi = 0.25$.

+

+\subsection{TG/TA scaling}

+TG is often quantified in the framework of non-linear spectroscopy, which measures the non-linear susceptibility, $\chi^{(3)}$, while TA often uses metrics of absorption spectroscopy.  

+The global study of both TA and TG requires relating the typical metrics of both experiments. 

+Here we outline how the measured signals from both methods compare. We assume perfect phase matching and collinear beams, and we neglect frequency dispersion of the linear refractive index.

+

+When absorption is important ($\alpha_0 \ell \gg 0$), the spatial dependence of the electric field amplitudes must be considered. 

+For TG, the polarization modulated in the phase-matched direction is given by

+\begin{equation}

+	P_{\text{TG}}(z) = E_2^* E_{2^\prime} e^{-\alpha_0(\omega_2)z} E_1 e^{-\alpha_0(\omega_1)z / 2} \chi^{(3)}

+\end{equation}

+The TG electric field propagation can be solved using the slowly varying envelope approximation, which yields an output intensity of\cite{Carlson1989}

+\begin{gather}

+	I_{\text{TG}} \propto \omega_1^2 M_{\text{TG}}^2 I_1 I_2 I_{2^\prime} \left| \chi^{(3)} \right|^2 \ell^2 \\

+	M_{\text{TG}}(\omega_1, \omega_2) = e^{-\alpha_0(\omega_1)\ell / 2} \frac{1-e^{-\alpha_0(\omega_2)\ell}}{\alpha_0(\omega_2)\ell}.

+\end{gather}

+%$A_0(\omega)= \alpha_0 \ell \ln 10$ is the absorbance spectrum of the sample with path length $\ell$. 

+This motivates the following metric for TG:

+\begin{equation}

+\begin{split}\label{eq:S_TG}

+	S_{\text{TG}} &\equiv \frac{1}{M_{\text{TG}}\omega_1}\sqrt{\frac{N_{\text{TG}}}{I_1 I_2^2}} \\

+	&\propto \left| \chi^{(3)}\right|

+\end{split}

+\end{equation}

+Here $N_{\text{TG}}$ is the measured four-wave mixing signal from a squared-law detector ($~I_{\text{TG}} / \omega_1$). 

+Again, the third-order response amplitude is extracted from this measurement.

+

+We now derive a comparable metric for TA measurements. Because the third-order field is spatially overlapped with the probe field, $E_1$, the relevant polarization includes the first- and third-order susceptibility:

+\begin{equation}

+	P_{\text{TA}} = \left( \chi^{(3)} (z;\omega_1) + e^{-\alpha_0(\omega_2)z} \left| E_2 \right|^2 \chi^{(3)}(z;\bm{\omega})\right)E_1 .

+\end{equation}

+Maxwell's equations show that the imaginary component of this polarization changes the intensity of the $E_1$ beam with a total absorption coefficient that is a composite of the linear and non-linear propagation:

+\begin{equation}

+\begin{split}

+	\alpha_{\text{tot}} &= \frac{2\omega_1}{c} 

+	\text{Im}\left[\sqrt{

+		1 + 4\pi \left( 

+			\chi^{(1)} + \frac{1-e^{-\alpha_0(\omega_2)\ell}}{\alpha_0(\omega_2)\ell} \left| E_2 \right|^2 \chi^{(3)} 

+		\right) 

+	} \right] \\

+	& \approx \frac{4\pi\omega_1}{c} \left( \text{Im}\left[ \chi^{(1)} \right]  + 

+		\left[ \frac{1-e^{-\alpha_0(\omega_2)\ell}}{\alpha_0(\omega_2)\ell} \left( \frac{8\pi}{nc}I_2 \right) \right] \text{Im}\left[ \chi^{(3)} \right] \right)

+\end{split}

+\end{equation}

+The $\chi^{(3)}$ response term can be isolated by chopping the pump beam, which yields a measured transmittance indicative of the nominal absorption, $T_0 = I_1 e^{-\alpha_0(\omega_1)\ell}$, and the change in the absorption $T = I_1 e^{-\alpha_{\text{tot}}\ell}$. 

+We quantify the non-linear absorption by $\alpha_\text{FWM}=1/\ell \ln \frac{T-T_0}{T_0}= \alpha_0 - \alpha_{\text{tot}}$, which can now be written as

+\begin{gather}

+	\alpha_\text{FWM}(\omega_1; \omega_2) = \frac{32\pi^2 \omega_1}{nc^2}M_{\text{TA}}(\omega_2) I_2 \text{Im} \left[\chi^{(3)}\right], \label{eq:alpha_fwm} \\

+	M_{\text{TA}}(\omega_2) = \frac{1-e^{-\alpha_0(\omega_2)\ell}}{\alpha_0(\omega_2) \ell}.

+\end{gather}

+Similar to TG propagation, $\alpha_\text{FWM}$ suffers from absorption effects that distort the proportional relationship to $\text{Im} \left[ \chi^{(3)} \right] $. 

+It is notable that in this case distortions are only from the pump beam. 

+The signal field heterodynes with the probe, which takes the absorption losses into account automatically. 

+Note that $M_{\text{TA}}$ simply accounts for the fraction of pump light absorbed in the sample, and consequently is closely related to the average exciton occupation across the entire path length of the sample, $\bar{n}_\ell = \ell^{-1} \int_0^\ell \bar{n}(z)dz$, which can be written using Equation \ref{eq:n} as:

+\begin{equation}\label{eq:n_tot}

+	\bar{n}_\ell = \frac{\sqrt{2} \sigma_0 (\omega_2) \Delta_t}{\hbar \omega_2} I_{\text{2,peak}} M_{\text{TA}}

+\end{equation}

+We define an experimental metric that isolates the $\chi^{(3)}$ tensor:

+\begin{equation}

+\begin{split}

+	S_{\text{TA}} &= \frac{\alpha_\text{FWM}}{\omega_1 I_2 M_{\text{TA}}} \\

+		&\propto \text{Im} \left[ \chi^{(3)} \right]

+\end{split}

+\end{equation}

+For the general $\chi^{(3)}$ response, the imaginary and real components of the spectrum satisfy complicated relations owing to the causality of all three laser interactions.

+For the pump-probe time-ordered processes, the probe causality is separable from the pump excitation event, which makes the causality relation of the pump and probe separable.\cite{Hutchings1992} 

+The causality relations of the probe spectrum are the famous Kramers-Kronig equations that relate ground state absorption to the index of refraction. 

+This relation is foundational in analysis of TG \cite{Hogemann1996} and TA measurements.

+% DK: need better citations for this

+

+Theoretically, TA probe spectra alone could be transformed to generate the real spectrum. 

+In practice, such a transform is difficult because the spectral breadth needed to accurately calculate the integral is experimentally difficult to achieve. 

+When both the absolute value and the imaginary projection of $\chi^{(3)}$ are known, however, the real part can also be defined by the much simpler relation:

+\begin{equation}\label{eq:chi_real}

+	\text{Re} \left[ \chi^{(3)} \right] = \pm \sqrt{\left| \chi^{(3)} \right|^2 - \text{Im} \left[ \chi^{(3)} \right]^2}

+\end{equation}

+% DK: concluding sentence

+

+\subsection{The Absorptive Third-Order Susceptibility}

+Though the bleach factor is defined within the context of absorptive measurements, it can be converted into the form of a third-order susceptibility as well. Equations \ref{eq:ptot} and \ref{eq:bleach_factor} motivate alternative expressions for differential absorptivity of the probe:

+\begin{equation}\label{eq:alpha_fwm_to_bleach1}

+\begin{split}

+	\alpha_\text{FWM} &= \bar{n} \left( \alpha_1(\omega_1) - \alpha_0(\omega_1) \right) \\

+		& =-\phi \bar{n} \alpha_0(\omega_1).

+\end{split}

+\end{equation}

+Substituting Equation \ref{eq:n_tot} into \ref{eq:alpha_fwm_to_bleach1}, we can write the non-linear absorption as

+\begin{equation}

+	\alpha_\text{FWM}(\omega_1) = -\phi \frac{\sqrt{2\pi}}{\hbar \omega_2} \alpha_0(\omega_1)\alpha_0(\omega_2) \Delta_t I_{\text{2,peak}} M_{\text{TA}}

+\end{equation}

+By Equation \ref{eq:alpha_fwm}, $\chi^{(3)}$ and the molecular hyperpolarizability, $\gamma_{\text{QD}}^{(3)} = \chi_{\text{QD}}^{(3)} / F^4 N_{\text{QD}}$, where $F$ is the local field factor, can also be related to the bleach fraction:

+\begin{gather}\label{eq:chi3_state_filling}

+	\text{Im}\left[ \chi^{(3)}_{\text{QD}} \right] = - \phi \frac{\sqrt{2\pi} n c^2 \Delta_t N_{\text{QD}}}{32\pi^2 \hbar \omega_1 \omega_2} \sigma_0(\omega_2)\sigma_0(\omega_1) \\

+	\text{Im}\left[ \gamma^{(3)}_{\text{QD}} \right] = - \phi \frac{\sqrt{2\pi} n c^2 \Delta_t }{32\pi^2 \hbar \omega_1 \omega_2 F^4} \sigma_0(\omega_2)\sigma_0(\omega_1). \label{eq:gamma3_state_filling} 

+\end{gather}

+Because this formula only predicts the imaginary component of the signal, its magnitude gives an approximate lower limit for the peak susceptibility and hyperpolarizability. 

+Absorptive cross-sections have been experimentally determined for PbSe QDs. \cite{Dai2009,Moreels2007}

diff --git a/PbSe_susceptibility/appendix.tex b/PbSe_susceptibility/appendix.tex
new file mode 100644
index 0000000..898086b
--- /dev/null
+++ b/PbSe_susceptibility/appendix.tex
@@ -0,0 +1,42 @@
+\subsection{Relationship between $\chi^{(3)}$ and to non-linear pulse propagation constants}
+
+At low field intensities, there are familiar relations between the refractive index, $n_0$, the absorptivity coefficient, $\alpha$, the linear susceptibility, $\chi$, and the complex wave-vector, $k$:
+\begin{equation}
+\begin{split}
+	k &= \frac{\omega n_0}{c} + i\frac{\alpha}{2} \\
+		&= \frac{\omega}{c} \sqrt{1 + \kappa \chi},
+\end{split}
+\end{equation}
+where $\kappa= 1$ in the SI unit system and $\kappa=4\pi$ in the cgs unit system.  
+$E_i$ is the electric field amplitude of pulse $m$, so $I_i = \frac{nc}{8\pi} |E_i|^2$.
+For higher fluences, there are three important phenomenologies, each based on a different optical constant:
+\begin{gather}
+	k = \frac{\omega n^*}{c} + i \frac{\alpha^*}{2} \\
+	\alpha^* = \alpha + \beta I \\
+	n^* = n_0 + n_2 I \\
+	\chi^* = \left[ \chi^{(1)} + D \chi^{(3)} \left| E_2 \right|^2 \right]
+\end{gather}
+where $D$ is the permutation degeneracy factor:\cite{Maker1965} if a weak probe pulse is distorted by perturbations from a separate pump (as in transient absorption measurements), $D=6$, while if a lone probe pulse is intense enough to cause its own distortions (i.e. $E_2 = E_1$, as in $z$-scan measurements), $D=3$.
+%where $M_2 = \frac{1 - e^{-\alpha_0(\omega_2) \ell}}{\alpha_0(\omega_2) \ell}$ accounts for absorptive effects of the pump beam $E_2$. 
+For small perturbations ($\left|\kappa D \chi^{(3)}|E_2|^2 \right| \ll |1+\kappa \chi^{(1)}|$), we can use a first-order Taylor expansion of $\mathbf{n}^*$ about $1 + \kappa \chi^{(1)}$ to write
+\begin{equation}
+	\begin{split}
+		\mathbf{n}^* &= \sqrt{1 + \kappa \left( \chi^{(1)} + D \chi^{(3)} |E_2|^2 \right)} \\
+		&\approx \mathbf{n} + \frac{1}{2 \mathbf{n}} \kappa D \chi^{(3)}|E_2|^2 
+	\end{split}
+\end{equation}
+and in the common case (as is herein) where $\frac{\alpha c}{2\omega} \ll n_0$, $\mathbf{n}^{-1} \approx n_0^{-1}$. 
+Ignoring terms independent of $I$, we arrive at
+\begin{equation}
+\begin{split}
+	\left(n_2 + i\frac{c \beta}{2\omega}\right)I &= \frac{\kappa D \chi^{(3)}}{2 n_0} |E_2|^2 \\
+	&= \frac{4 \pi \kappa D \chi^{(3)}}{n_0^2 c} I
+\end{split}
+\end{equation}
+Eqns. \ref{eq:beta_to_chi} and \ref{eq:n2_to_chi} then follow with Eqn. \ref{eq:hyperpolarizability}.
+%Relations between non-linear constants and $\chi^{(3)}$ are summarized by:
+%\begin{gather}
+%	\beta = \frac{8 \pi \kappa D \omega}{n_0^2 c^2} \text{Im}\left[ \chi^{(3)} \right]	\\		
+%	n_2 = \frac{4 \pi \kappa D }{n_0^2 c} \text{Re}\left[ \chi^{(3)} \right]	
+%\end{gather}
+%From this it is easily seen that $\beta_\text{TA} = 2 \beta_\text{z-scan}$.
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diff --git a/PbSe_susceptibility/chapter.tex b/PbSe_susceptibility/chapter.tex
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+\chapter{Resonant third-order susceptibility of PbSe quantum dots determined by standard dilution
+  and transient grating spectroscopy} \label{cha:pss}
+
+\textit{This Chapter borrows extensively from a work-in-progress publication.}
+
+\clearpage
+
+Here we detail the extraction of quantitative information from ultrafast multiresonant CMDS
+spectra.  %
+PbSe quantum dots are used as a model system.  %
+We investigate the severity of pulse propagation effects, as well as the increased prominence of
+solvent background contributions to signals.  %
+We characterize ultrafast Four-Wave Mixing signals of colloidal quantum dots using non-resonant
+solvent response as a standard.  %
+We show that dilution studies can be quantitatively described, and apply a robust, few-parameter
+fit to extract the peak susceptibility.  %  
+These method should be generally applicable to solution phase systems and other CMDS
+techniques.    %
+
+\section{Introduction}  % =========================================================================
+
+Coherent multidimensional spectroscopy (CMDS) provides a wealth of information on the structure,
+energetics, and dynamics of solution phase systems.  %
+By using multiple excitations to probe resonances, CMDS elucidates correlations and couplings
+between electronic, vibrational, and vibronic states.  %
+It is now commonplace to interrogate such structures with femtosecond time resolution.  %  
+The time resolved nature of the measurement in this regime allows characterization of transient
+states that are unresolvable in steady-state methods.  %
+Multiresonant CMDS (MR-CMDS) is a frequency-domain technique, whereby tunable lasers are scanned to
+obtain multidimensional spectra.  %
+It has been demonstrated on material systems with femtosecond pulses. \cite{KohlerDanielDavid2014a,
+  CzechKyleJonathan2015a}
+
+Traditionally CMDS employs line shape analysis (peak center, width, sign, homogeneous and
+inhomogeneous line width) and dynamics analysis (time constants, amplitudes) to give insight into
+the material of interest.  %
+The magnitudes of optical non-linearities, though commonly ignored, are intrinsic properties that
+also inform on microscopic properties.  %
+The microscopic mechanisms for optical nonlinearities are determined by a sequence of field-matter
+interactions (Liouville pathways) that depend on \textit{linear} properties (cross-sections, etc.)
+with each interaction.  %
+Some methods (TSF) isolate single Liouville pathways so that the magnitude of the non-linearity is
+a simple product of cross-sections.  %
+Some cross sections are easily measured with conventional experiments (Raman, absorption), but
+others depend on exotic/inaccessible transitions.  %
+Techniques like CMDS must be employed to measure the cross-sections of these inaccessible
+transitions.  %
+
+Other methods (2DES, TG, TA) have multiple similar Liouville pathways, such that the nonlinearity
+arises from imperfect cancellation between them.  %
+This lack of cancellation also is connected to the microscopic properties. 
+For example, saturation effects, such as state-filling, have to do with the degeneracy of a
+transition.  %
+CMDS can explicitly measure the strengths of these saturation effects.  
+This paper describes the measurement of state-filling.  
+
+Absolute susceptibility is not an observable with most experimental configurations.  
+It is uncommon for CMDS spectra to obtain absolute units of susceptibility in the spectra they
+report.  %
+Measurements such as the $z$-scan \cite{SheikBahaeMansoor1989a, SheikBahaeMansoor1990a} and
+transient absorption, specialize in quantifying optical non-linearities, but these methods are
+limited in the multidimensional space they can explore. \footnote{TA cannot do 3-color
+  non-linearities, and $z$-scan cannot interrogate dynamics.}
+
+Internal standards are a convenient means to quantify the non-linearity magnitude.
+\cite{LevensonMD1974a}  % 
+The method has also been demonstrated on MR-CMDS of quantum dots under picosecond excitation
+pulses. \cite{YursLenaA2012a}  %
+These methods typically require a full characterization (phase and amplitude) of analyte spectral
+properties in order to relate the two signals.  %
+For CMDS methods that possess a multitude of Liouville pathways, this characterization requires a
+large number of parameters and can be quite complex to solve.  %
+
+This work details a few-parameter extraction of the third-order susceptibility of the 1S band of
+PbSe quantum dots (QDs).  %
+We utilize standard additions for characterizing the absolute third-order susceptibility of
+resonant signals.  %
+Applying justified approximations, we extract the absolute susceptibility without explicitly
+modeling Liouville pathways of an excitonic manifold.  %
+We connect the common phenomenologies of optical non-linearities to the theory of state-filling.  %
+Finally, we compare our measurements with the diverse and numerous values of literature.  
+Once pulse-length factors and propagation distortions are accounted for, we find that the measured
+non-linear susceptibility of these femtosecond experiments is in good agreement with previously
+published values.  %
+
+\section{Theory}  % ===============================================================================
+
+These experiments consider the CMDS signal resulting from a chromophore resonance in a transparent
+solvent.  %
+We first formulate the CMDS intensity in terms of the separate contributions of solvent and
+solute.  %
+We then connect the well-known theory of optical bleaching of the 1S band to our measurements.  %
+
+\subsection{Extraction of susceptibility}  % ------------------------------------------------------
+
+In the Maker-Terhune convention, the relevant third-order polarization, $P^{(3)}$, is related to
+the non-linear susceptibility, $\chi^{(3)}$, by\cite{MakerPD1965a}  %
+\begin{equation}\label{eq:Maker_Terhune}
+	\begin{split}
+		P^{(3)}(z, \omega) =& D \chi^{(3)}(\omega; \omega_1, -\omega_2,	\omega_{2^\prime})  \\
+			& \times E_1(z, \omega_1) E_2(z, -\omega_2) E_{2^\prime}(z, \omega_{2^\prime}),
+	\end{split}
+\end{equation}
+where $z$ is the optical axis coordinate (the experiment is approximately collinear), $E_i$ is the
+real-valued electric field of pulse $i$, and $\omega_i$ is the frequency of pulse $i$.  %  
+The degeneracy factor $D = 3! / (3 - n)!$ accounts for the permutation symmetry that arises from
+the interference of $n$ distinguishable excitation fields.\footnote{$D = 6$ for transient
+  absorption and transient grating, and $D = 3$ for $z$-scan}  %
+Permutation symmetry reflects the strength of the excitation fields and not the intrinsic
+non-linearity of the sample.  %
+Including $D$ in our convention makes $\chi^{(3)}$ invariant to different beam geometries.  %
+
+
+Equation \ref{eq:Maker_Terhune} is the steady-state solution to the Liouville Equation, valid when
+excitation fields are greatly detuned from resonance and/or much longer than coherence times.  %
+This convention is invalid for impulsive excitation, where $\chi^{(3)}$ will be sensitive to pulse
+duration.  %
+This complications arising from impulsive aspects of our experiment are addressed in section...
+
+The non-linear polarization launches an output field.
+The intensity of this output depends on the accumulation of polarization throughout the sample.  
+For a homogeneous material, the output intensity, $I$, is proportional
+to\cite{CarlsonRogerJohn1989a}  %
+\begin{equation}\label{eq:fwm_intensity}
+	\begin{split}
+		I &\propto \left| \int P^{(3)} (z, \omega) dz \right|^2 \\
+		  &\propto \left| M P^{(3)}(0, \omega) \ell \right|^2 \\
+		  &= |M D \chi^{(3)} E_1(0,\omega_1) E_2(0, -\omega_2) E_3(0, \omega_{2^\prime})|^2. 
+	\end{split}
+\end{equation}
+Here $\ell$ is the sample length and $M$ is a frequency-dependent factor that accounts for phase
+mismatch and absorption effects.  %
+Phase mismatch is negligible in these experiments (see Supplementary Materials). 
+For purely absorptive effects, $M$ may be written as\cite{CarlsonRogerJohn1989a, YursLenaA2011a}
+\begin{equation}
+	M(\omega_1, \omega_2) = \frac{e^{-\alpha_1 \ell /2}\left(1 - e^{-\alpha_2 \ell} \right)}{\alpha_2 \ell}
+\end{equation}
+where $\alpha_i = \sigma_i N_\text{QD}$ is the absorptivity of the sample at frequency
+$\omega_i$.  %
+Absorption effects disrupt the proportional relationship between $I$ and $\chi^{(3)}$.  %
+Equation \ref{eq:fwm_intensity} shows that we can derive spectra free of pulse propagation effects
+by normalizing the output intensity by $M^2$.  %
+The distortions incurred by optically thick samples are well-known and have been treated in similar
+CMDS experiments. \cite{SpencerAustinP2015a, YetzbacherMichaelK2007a, ChoByungmoon2009a,
+  KeustersDorine2004a}  %
+
+For cuvettes, the sample solution is sandwiched between two transparent windows.  
+Rather than Eqn. \ref{eq:fwm_intensity}, the total polarization has three distinct homogeneous regions:  the front window, the solution, and the back window.
+The windows each have the same thickness, $\ell_\text{w}$, and susceptibility, $\chi_\text{w}^{(3)}$.
+The (absorption-corrected) output intensity is proportional to:
+\begin{equation}\label{eq:fwm_intensity2}
+	\frac{I}{I_1 I_2 I_{2^\prime} M^2} \propto 
+		\left| \alpha_2 \ell_\text{w} \frac{1 + e^{-\alpha_2 \ell_s}}{1 - e^{-\alpha_2 \ell_s}} \chi_\text{w}^{(3)} 
+			+ \chi_\text{sol}^{(3)} + \chi_\text{QD}^{(3)} 
+		\right|^2
+\end{equation}
+where $\chi^{(3)}_\text{QD}$ is the QD susceptibility and the $\chi^{(3)}_\text{sol}$ is the solvent susceptibility.
+Each susceptibility depends on the chromophore number density and local field enhancements for each wave:
+\begin{equation}\label{eq:hyperpolarizability}
+	\chi_i^{(3)} = 
+		f(\omega_1)^2 f(\omega_2)^2 N_i \gamma_i^{(3)},
+\end{equation}
+where $\gamma_i^{(3)}$ is the intrinsic (per-QD/per-molecule, \textit{in vacuo}) hyperpolarizability, $N_i$ is the number density of species $i$, and $f(\omega)$ is the local field enhancement factor.
+Since QDs constitute a negligible number/volume fraction of the solution, the field enhancement is derived entirely from the solvent:  $f(\omega) = \left( \frac{n_\text{sol}(\omega)^2 + 2}{3} \right)$, where $n_\text{sol}$ is the solvent refractive index.  
+Both $n$ and $f$ are frequency dependent, but both vary small amounts ($\sim 0.1 \%$) over the frequency ranges considered here.  
+We approximate both as constants, and remove the frequency argument from further equations.
+
+
+Equation \ref{eq:fwm_intensity2} can be expressed as the classic signal-local oscillator interference, 
+\begin{equation}\label{eq:LO}
+	I \propto \left| E_\text{LO} \right|^2 + N_\text{QD}^2 f^8 \left| \gamma_\text{QD} \right|^2 + 2 N_\text{QD} f^4 \text{Re}\left[ E_\text{LO} \gamma_\text{QD}^* \right]
+\end{equation}
+where we have used the substitutions $E_\text{LO} = \alpha_2 \ell_\text{w} \frac{1 + e^{-\alpha_2 \ell_s}}{1 - e^{-\alpha_2 \ell_s}} \chi_\text{w}^{(3)} + \chi_\text{sol}^{(3)}$.
+The character of the interference depends both on the amplitude of the QD field and on the phase relationship between the two fields.  
+The QD field amplitude can be controlled by $N_\text{QD}$.  
+At low concentrations there is a linear dependence on $N_\text{QD}$, but this changes at high optical densities due to an $\alpha_2$ dependence on the window contribution.
+The phase relationship cannot be externally controlled and is frequency dependent:  it is defined by the resonant character of each material.  
+The phase is defined by electronic resonances in QD and by Raman resonances in the solvent and the windows.
+The local oscillator and signal fields are non-additive unless the phase difference is $\pm \pi / 2$.  
+%Solving for $|\chi_\text{QD}^{(3)}|$ in Eqn. \ref{eq:interference} requires knowledge of $\theta$. 
+
+\subsection{Optical bleaching and dependencies on experimental conditions}
+% Note:  perhaps split up optical bleaching paragraph and separate from the rest of thsi section (to be called "connections")
+
+%Though it is convenient to express our experiment in terms of the non-linear polarization, 
+Most non-linear experiments on QDs extract pulse propagation parameters, such as the non-linear absorptivity, $\beta$ or non-linear index of refraction, $n_2$.  
+These parameters are connected to the third-order susceptibility (in the cgs unit system) by
+\begin{gather}\label{eq:beta_to_chi}
+	\beta = \frac{32 \pi^2 D \omega}{n_0^2 c^2} \text{Im}\left[ \chi^{(3)} \right]	\\	\label{eq:n2_to_chi}
+	n_2 = \frac{16 \pi^2 D }{n_0^2 c} \text{Re}\left[ \chi^{(3)} \right].
+\end{gather}
+These relations are derived in the Appendix.
+
+At the band edge, the non-linear absorptivity of semiconductor QDs is dominated by state-filling\cite{Klimov2000}, where absorption is reduced in proportion to the average number of excitons pumped into the band, $\langle n \rangle$.  
+Due to an 8-fold degeneracy, lead chalcogenide (PbX) QDs are bleached fractionally by the presence of an exciton. 
+%Under low intensities, this bleach fraction, $\phi$ is considered to be 0.25.  
+An 8-fold degeneracy predicts a bleach fraction of $\phi = 0.25$.\cite{Trinh2008,Trinh2013,Gdor2013a,Spoor,Schaller2003}
+For a Gaussian pump pulse of peak intensity $I$, frequency $\omega$, and full-width at half-maximum (FWHM) of $\Delta_t$, $\langle n \rangle = \frac{\sqrt{2 \pi} \sigma}{\hbar \omega} \Delta_t I$ where $\sigma$ is the QD absorptive cross-section at frequency $\omega$.
+We can then write the non-linear change in absorptivity as
+\begin{equation}\label{eq:Delta_alpha1}
+\begin{split}
+	\beta I_2 &= -\phi \langle n \rangle \alpha \\
+		&= - \phi N_\text{QD} \frac{\sqrt{2 \pi} \sigma_1 \sigma_2}{\hbar \omega} \Delta_t I_2 
+\end{split}
+\end{equation}
+where the indexes $1$ and $2$ denote properties of the probe and pump fields, respectively.
+In some techniques (e.g. $z$-scan), both probe and pump fields are the same, in which case the subscripts become unnecessary.
+
+By combining Eqns. \ref{eq:hyperpolarizability}, \ref{eq:beta_to_chi}, and \ref{eq:Delta_alpha1} we can relate the bleach factor directly to the hyperpolarizability:
+\begin{equation}\label{eq:gamma_to_phi}
+	\text{Im}\left[ \gamma^{(3)} \right] = 
+		-\phi \frac{\sqrt{2\pi} n^2 c^2}{32 \pi^2 D f^4 \hbar \omega_1 \omega_2} \sigma_1 \sigma_2 \Delta_t.
+\end{equation}
+% getting ahead of myself; the 1S bleach is a little complex at zero delay
+Equation \ref{eq:gamma_to_phi} will be useful for benchmarking our results because it connects our
+observable, $\gamma_\text{QD}$, with the nonlinearity of the microscopic model, $\phi$.  %
+
+\section{Experimental}  % =========================================================================
+
+\subsubsection{Sample preparation and Characterization}
+QDs were created using a standard solution-phase technique. \cite{WehrenbergBrianL2002a}  %
+QDs were washed in ethanol-toluene before being immersed in carbon tetrachloride (CCl$_4$) and stored in a nitrogen-pumped glovebox.  
+Successive dilutions created the aliquots of different concentration used here.  
+Aliquots were stored in 1 mm path length fused silica cuvettes with 1.25 mm thick windows.  
+
+Each aliquot was characterized by absorption spectroscopy (JASCO).
+The spectra are consistent between all dilutions (no agglomeration, see Supplementary Info).
+The 1S feature peaks at 0.937 eV and has a FWHM of 92 meV.
+Concentrations were extracted using Beer's law and published cross-sections.\cite{Moreels2007,Dai2009}
+The peak ODs range from 0.06 to 0.86 (QD densities of $10^{16} - 10^{17} \ \text{cm}^{-3}$). 
+
+\subsubsection{Four-wave Mixing}
+
+An ultrafast oscillator (Tsunami, Spectra-Physics) produced a 80 MHz train of 35 fs pulses, which
+were amplified (Spitifire Pro XP, Spectra-Physics, 1kHz) and split to pump two independently
+tunable OPAs (TOPAS-C, Light Conversion): OPA1 and OPA2.  %
+The frequency-dependent OPA power output was measured (407-A Thermopile, Spectra-Physics) and used
+to normalize the non-linear spectra.  %
+Pulses from OPA2 were split again, for a total of three excitation pulses:  $E_1$, $E_2$, and
+$E_{2^\prime}$.  %
+These pulses were passed through motorized (Newport MFA-CC) retroreflectors to control their
+relative delays, defined as $\tau_{21} = \tau_2 - \tau_1$ and $\tau_{22^\prime} = \tau_2 -
+\tau_{2^\prime}$.  %
+ 
+The three excitation pulses were focused (1m FL spherical mirror) into the sample using a BOXCARS
+geometry ($\sim 1 \deg$ angle of incidence for all beams).  %
+All input fields were co-polarized.  %
+The coherent output at $k_1 - k_2 + k_{2^\prime}$ was isolated using apertures and passed into a
+monochromator, with an InSb photodiode (Teledyne-Judson) at the exit slit.  %
+The monochromator was scanned with the FWM output frequency:  $\omega_m = \omega_1$.  %
+
+\section{Results}  % ==============================================================================
+
+In this section we describe the extraction of the QD susceptibility through standard dilution.
+First, we examine the window and solvent response, which will be our local oscillator, in frequency
+and time.  % 
+Next, we isolate the pure QD response, using temporal discrimination, to validate the correction
+factors used to account for concentration dependence.  %
+Finally, we consider the interference between the solvent and QDs at pulse overlap, extracting the
+QD susceptibility by ratio.  %
+
+\subsection{Solvent response}  % ------------------------------------------------------------------
+
+Carbon tetrachloride is an ideal solvent because of the high QD solubility, transparency in the
+near infra-red, and its well-studied non-linear properties.  %
+The FWM response of transparent solvents, like carbon tetrachloride, has components from nuclear
+and electronic non-linearities. \cite{HellwarthRW1971a, HellwarthRW1977a}  
+The electronic perturbations renormalize nearly instantaneously and thus are only present with
+pulse overlap.  %
+The nuclear response depends on the vibrational dephasing times (ps and longer). \cite{HoPP1979a,
+  MatsuoShigeki1997a}  %
+Vibrational features appear in the 2D spectra when stimulated Raman pathways resonantly enhance the
+FWM at constant ($\omega_1 - \omega_2$) frequencies.  %
+
+Fig. \ref{fig:ccl4} summarizes the nonlinear measurements performed on neat CCl$_4$.
+In general, our results corroborate with impulsive stimulated Raman experiments.
+\cite{MatsuoShigeki1997a, VoehringerPeter1995a}  %
+When all pulses are overlapped (Fig. \ref{fig:ccl4}a), the electronic response creates a
+featureless 2D spectrum.  %
+The horizontal and vertical structure observed in Fig. \ref{fig:ccl4}a is believed to reflect the
+power levels of our OPAs, which were not accounted for in these scans.  %
+The weak diagonal enhancement observed may result from overdamped nuclear libration.  
+The broad spectrum tracks with temporal pulse overlap, quickly disappearing at finite delays.  
+If pulses $E_1$ and $E_2$ are kept overlapped and the $E_{2^\prime}$ is delayed (Fig.
+\ref{fig:ccl4}b), the contributions from the Raman resonances can be resolved.  %
+These ``TRIVE-Raman'' \cite{MeyerKentA2004a} resonances have been observed in carbon tetrachloride
+previously. \cite{KohlerDanielDavid2014a}  %
+The bright mode seen at approximately $\omega_1 - \omega_2 = \pm 50 \ \text{meV}$ is the $\nu1$
+symmetric stretch ($459 \ \text{cm}^{-1}$). \cite{ShimanouchiT}
+
+Characterization of the solvent response at pulse overlap can be simplified if Raman resonances are
+negligible.  %
+If Raman resonances are important, their spectral phase needs to be characterized and included in
+modeling. \cite{YursLenaA2012a}  %
+To estimate the relative magnitude of Raman components at pulse overlap, we consider a delay trace.  
+Figure \ref{fig:ccl4}c shows the signal dependence on $\tau_{22^\prime}$ with pulse frequencies resonant with the large $\nu 1$ resonance.
+%The largest Raman contributions occur with $|\omega_1 - \omega_2|$ tuned to resonance with the $\nu 1$ Raman mode.  
+The transient was fit to two components:  a fast Gaussian (electronic) component and an exponential
+decay (Raman) component.  %
+The oscillations in the exponential decay are quantum beating between Raman modes of CCl$_4$ and
+are well-understood. \cite{KohlerDanielDavid2014a}  %
+We determined the fast (non-resonant) component to be $4.0 \pm 0.7$ times larger than the long
+(Raman) contributions (amplitude level).  %
+At most colors, the ratio will be much less (confer Fig. \ref{fig:ccl4}b).
+Since the Raman features are small in magnitude and spectrally sparse, we assume the CCl$_4$ spectrum near pulse overlap is well-approximated by non-resonant response ($\gamma_\text{sol}$ is constant and real-valued).  
+This simplifies Eqn. \ref{eq:fwm_intensity2} because the dispersion of the interference term is determined completely by the real component of quantum dot response:  $\text{Re} \left[ E_\text{LO} \gamma_\text{QD}^* \right]  = E_\text{LO} \text{Re} \left[ \gamma_\text{QD} \right]$.  
+
+\begin{figure}
+	\includegraphics[width=\linewidth]{"PbSe_susceptibility/ccl4_raman"}
+	\caption{
+		CMDS amplitude of neat CCl$_4$.  In all plots, $E_1$ and $E_2$ are coincident ($\tau_{21}= 0$ fs).
+		Spectra are not normalized by the frequency-dependent OPA input powers.  
+		(a)  The 2D frequency response at pulse overlap ($\tau_{21}=\tau_{22^\prime} = 0$).  
+		(b)  Same as (a), but $E_{2^\prime}$ is latent by 200 fs.  
+		(c)  The $\tau_{22^\prime}$ dependence on CMDS amplitude (thin blue line) is tracked at
+    $\left(\hbar\omega_1, \hbar\omega_2 \right) = \left( .905, 0.955 \right)$ eV, so that the $\nu
+    1$ Raman mode is resonantly excited.  %  
+		The fit to the measured transient (thick blue line) is described further in the text.  
+		The $\omega_1, \omega_2$ frequency combination is represented in (a) and (b) as a blue dot.
+	}
+	\label{fig:ccl4}
+\end{figure}
+
+\subsection{Concentration-dependent corrections}  % ------------------------------------------------
+
+\begin{figure}
+	\includegraphics[width=\linewidth]{"PbSe_susceptibility/mfactors_check"}
+	\caption{
+		The three panels show the changes in the FWM spectra of the five QD concentrations when corrected for concentration and absorption effects.
+		%Ultrafast four-wave mixing spectra of solution phase QD at different concentrations.  
+		The legend at the top identifies each QD loading level by the number density (units of $10^{16} \ \text{cm}^{-3}$).  
+		In all plots a representative QD absorption spectrum is overlaid (gray).
+		Top:  $I / I_1 I_2 I_{2^\prime}$ spectra (intensity level).  
+		Middle:  FWM amplitude spectra after normalizing by the carrier concentration ($\sqrt{I / \left( I_1 I_2 I_{2^\prime} N_\text{QD}^2 \right)}$).
+		Bottom:  same as middle, but with the additional normalization by the absorptive correction factor ($M$).  
+	}
+	\label{fig:mfactors}
+\end{figure}
+
+It is important to address concentration effects on the CMDS output intensity because the resulting absorption dependence can dramatically change the signal features.
+Fig. \ref{fig:mfactors} shows $\omega_1$ spectra gathered at the QD concentrations explored in this work.  
+All spectra were gathered at delay values $\tau_{21} = -200$ fs, $\tau_{22^\prime} = 0$ fs, and $\omega_2$ is tuned to the exciton resonance.  
+The pulse delays are chosen to remove all solvent and window contributions; the signal is due entirely to QDs ($\chi_\text{w},\chi_\text{solvent}=0$ in Eqn. \ref{eq:fwm_intensity2}).  
+Power-normalized output amplitudes (Fig. \ref{fig:mfactors}a) are positively correlated with QD concentration.  
+Density-normalized ($N_\text{QD}$) output amplitudes (Fig. \ref{fig:mfactors}b) are negatively correlated with concentration because of absorption effects.
+This normalization is adopted because the QD intensity term remains constant for any dilution level.
+This loss in efficiency is due to both absorption of the unresolved $\omega_2$ axis (loss across all $\omega_1$ values) and the plotted $\omega_1$ axis (losses correlate to sample absorbance, thick grey line).
+After normalizing by $M$ (Fig. \ref{fig:mfactors}c), the density-normalized output amplitudes agree for all QD concentrations.
+The robustness of these corrections (derived from accurate absorption spectra) implies that data can be taken at large concentrations and corrected to reveal clean signal with large dynamic range.
+The nature of the corrected line shape, including the tail to lower energies, will be addressed in a future publication.
+
+\subsection{Quantum dot response}
+
+We now consider the behavior at pulse overlap, where solvent and window contributions are important.  
+Figure \ref{fig:dilution_integral}a shows the (absorption-corrected) spectra for all samples at zero delay.
+The spectrum changes qualitatively with concentration, from peaked symmetric at high concentration (purple), to dispersed and antisymmetric at low concentration (yellow).  
+This behavior contrasts with the signals at finite $\tau_{21}$ delays, where the sample spectra are independent of concentration (Fig. \ref{fig:mfactors}c). 
+Pulse overlap is complicated by the interference of multiple time-orderings and pulse effects.
+\cite{KohlerDanielDavid2017a, BritoCruzCH1988a, JoffreM1988a}
+These line shapes are not easily related to material properties, such as inhomogeneous broadening and pure dephasing.
+
+\begin{figure}
+	\includegraphics[width=\linewidth]{"PbSe_susceptibility/dilution_integral"}
+	\caption{
+		FWM at temporal pulse overlap ($\tau_{21}=\tau_{22^\prime} = 0$ fs), with $\omega_2 = \omega_\text{1S}$.
+		(a) Absorption-corrected $\omega_1$ spectra for each of the concentrations, offset for clarity. 
+			Yellow is most dilute, purple is most concentrated. 
+			Each spectrum is individually normalized (amplification factors are shown by each spectrum).  
+		(b) The integrals of the FWM line shapes in part (a) are plotted against the QD concentration.  
+			The dashed black line is the result of a linear fit (the $x$-axis is logarithmic).
+			%Integrated FWM intensities with different concentrations of PbSe.  
+	}
+	\label{fig:dilution_integral}
+\end{figure}
+
+The concentration dependence in Fig. \ref{fig:dilution_integral} can be understood with our
+knowledge of the solvent/window character and Eqn. \ref{eq:LO}.  %
+We approximate the solvent and window susceptibilities as real and constant, such that the
+frequency dependence of the interference is solely from the real projecton of the QD
+nonlinearity.  %
+As a consequence, the interference term will be the Kramers-Kronig counterpart of the peaked
+transient absorption spectrum.  % 
+This explains the observed antisymmetric, dispersed line shape at low concentrations.  %
+We analyze these spectra through two different methods:  spectral integration and global line shape
+fitting.  %
+
+\subsubsection{Spectral integration}
+
+If we integrate Eqn. \ref{eq:fwm_intensity2}, the integral of the solvent-QD interference term
+disappears and the contributions are additive again.  %
+We can write
+\begin{equation}\label{eq:fit_integral}
+\begin{split}
+	\int_a^{a+\Delta} 
+	\frac{I}{I_1 I_2 I_{2^\prime} M^2} \ d\omega_1	
+		= & A \Delta \left( 1 + 
+			%\frac{2 \ell_w}{\ell_\text{s}} 
+			\frac{\chi_\text{w}^{(3)}}{\chi_\text{sol}^{(3)}} f(N_\text{QD}) \right)^2 \\
+		& + \frac{A N_\text{QD}^2}{\gamma_\text{sol}^2 N_\text{sol}^2} \int_a^{a+\Delta} |\gamma_\text{QD}^{(3)}|^2  \ d\omega_1
+\end{split}
+\end{equation}
+where $A$ is a proportionality factor and $f(N_\text{QD}) = \sigma_2 N_\text{QD} \ell_\text{w}
+\frac{1 + e^{-\sigma_2 N_\text{QD} \ell_s}}{1 - e^{-\sigma_2 N_\text{QD} \ell_s}}$.  %
+Care must be taken when choosing integral bounds $a$ and $a + \Delta$ so that the odd character of
+the interference is adequately destroyed.  %
+
+Figure \ref{fig:dilution_integral}b shows the integral values for all five concentrations
+considered in this work (colored circles).  %
+At high concentrations the QD intensity dominates and we see quadratic scaling with $N_\text{QD}$.  
+The lower intensities converge to a fixed offset due to the solvent and window contributions.  
+Our data fit well to Eqn. \ref{eq:fit_integral} (black dashed line).
+
+Notably, our fit fails to distinguish between window and solvent contributions.
+The solvent integral is invariant to $N_\text{QD}$, while the window contribution changes only
+moderately over the concentrations studied ($f(N_\text{QD})$ varies by $\sim 0.3x$).  %
+In contrast, the QD integral will change by $\sim 100x$ over these concentration ranges,
+overwhelming the changes in window behavior.  %
+The approximation of $f(N_\text{QD})$ as constant produces equally good fits.  %
+
+In order to distinguish between window and solvent contributions, we take literature values from
+Kerr lensing $z$-scan measurements of $\chi_\text{w} / \chi_\text{sol} \approx 0.13$.
+\cite{Rau2008}  % 
+This value agrees with our comparisons of FWM in cuvette windows and CCl$_4$-filled cuvettes (data
+not shown).  %
+
+The peak QD susceptibility can now be determined by assuming a line shape function;
+if $\gamma_\text{QD}$ is a causal Lorentzian with half-width at half-maximum (HWHM) $\Gamma$,
+\begin{equation}
+	\gamma_\text{QD} = \gamma_\text{QD,peak} \frac{\Gamma}{\omega_\text{1S} - \omega_1 - i \Gamma},
+\end{equation}
+then $\gamma_\text{QD, peak} = 2 \sqrt{\Gamma^{-1} \pi^{-1} \int |\gamma_\text{QD}|^2 d\omega_1}$.
+The peak width can be inferred, for instance, from spectra with high QD concentration ($\sim 25$
+meV HWHM).  %
+This gives a peak hyperpolarizability of $|\gamma_\text{QD, peak}| = 1.2 \times 10^{6}
+\gamma_\text{sol}$.  %
+
+\begin{figure}
+	\includegraphics[width=\linewidth]{"PbSe_susceptibility/dilution_fits"}
+	\caption{
+		CMDS signal with different concentrations of PbSe.  
+		In all spectra $\omega_2 = \omega_\text{1S}$.
+		Calculated $\gamma^{(3)}$ spectra (Eqn. \ref{eq:fwm_intensity2}) for the different QD concentrations.  
+		The thick, lighter lines are the result of a global fit.  
+	}
+	\label{fig:dilution2}
+\end{figure}
+
+\subsubsection{Global line shape fitting}  % ------------------------------------------------------
+
+The integration approach provides a simple means to separate the contributions to the non-linearity, 
+but it relies on QDs having a purely resonant line shape.  %
+This approximation may not be appropriate for PbX QDs.  %
+Many studies have reported a broadband contribution, attributed to excited state absorption of
+excitons, in addition to the narrow 1S bleach feature. \cite{YursLenaA2012a, GeiregatPieter2014a,
+  DeGeyterBram2012a}  %
+To account for this feature, we perform a global fit of Eqn. \ref{eq:fwm_intensity2} with the QD
+line shape definition  %
+\begin{equation}\label{eq:fit_lineshape}
+	\gamma_\text{QD}^{(3)} = \gamma_\text{QD,peak}^{(3)} \frac{\Gamma}{\omega_\text{1S} - \omega_1 - i \Gamma} + B,
+\end{equation}
+where $\Gamma$ is a line width parameter and $B$ is the broadband QD contribution.  %
+The results of the fit are overlaid with our data in Fig. \ref{fig:dilution2}.
+The data is normalized by $N_\text{QD}^2$ (as in Fig. \ref{fig:mfactors}c) so that least-squares
+fitting weighs all samples on similar scales.  %
+The fit parameters are listed in Table \ref{tab:lineshape_fit}.  
+Again, we use a literature value for $\chi_\text{w} / \chi_\text{sol}$.
+The extracted value of $\gamma_\text{QD}$ is $\sim 35\%$ smaller than in the integral analysis
+because the integral method did not distinguish between the broadband contribution and the 1S
+state-filling feature.  %
+
+Unlike the integral method, the line shape fit also gives signed information:  we find that the
+sign of $\gamma_\text{QD}$ is in fact negative, consistent with a photobleach.  %
+The broadband contribution has a positive imaginary component, consistent with excited state
+absorption.  %
+
+\begin{table}
+	\centering
+	\caption{Parameters and extracted values from the global line shape fit using Eqns.
+    \ref{eq:fwm_intensity2} and \ref{eq:fit_lineshape}.
+    Bold items were extracted by least squares minimization.  All other values were fixed parameters.  }
+	\label{tab:lineshape_fit}
+	\begin{tabular}{l|c}
+		variable & value \\
+		\hline
+		$N_\text{sol} \ (\text{cm}^{-3})$ & $6.23 \times 10^{21}$ \\
+		$\hbar \omega_\text{1S} \ (\text{eV})$ & 0.945 \\
+		$\sigma_2 \ (\text{cm}^2)$ & $1.47 \times 10^{-16}$ \cite{Moreels2007, Dai2009} \\
+		$\chi_\text{w} / \chi_\text{sol}$ & 0.13 \cite{Rau2008} \\
+		$\mathbf{ \Gamma \ (\text{meV})}$ & 25 \\
+		$\mathbf{\gamma_\text{QD,peak} / \gamma_\text{sol}}$ & $-7.7 \times 10^5$  \\
+		$\mathbf{B / \gamma_\text{QD}}$ & $0.10 - 0.13i$ \\
+	\end{tabular}
+\end{table}
+
+\subsubsection{Choice of $\gamma_\text{sol}$}
+To translate our result into an absolute susceptibility, $|\gamma_\text{sol}|$ must be known.
+There are numerous measurements in the literature, \cite{Hellwarth1971, LevensonMD1974a, Levine1975,Cherlow1976,Ho1979,Thalhammer1983,Etchepare1985,Nibbering1995,Rau2004} but unfortunately, the variation between values is quite large ($\pm 50\% $) for quantitative analysis (see Supplementary Materials).  
+This is the largest uncertainty in the determination of $\gamma_\text{QD}^{(3)}$.  %
+With this concern noted, we adopt the median susceptibility of $\gamma_\text{sol} = 4 \cdot
+10^{-37} \frac{\text{cm}^{6}}{\text{erg}}$ as our value to give comparisons to literature.  %
+This yields a peak QD hyperpolarizability to be $3 \cdot 10^{-31}  \text{cm}^6 / \text{erg}$.  
+
+\section{Discussion}  % ===========================================================================
+
+We now consider the agreement of our non-linearity with those of literature. 
+%Table \ref{tab:litcompare} gives the values from this work as well as values from literature and theory. 
+% two other works measuring PbX nonlinearities, and theory for state-filling.
+%We describe the elements of this table throughout this discussion.  
+Comparison between different measured non-linearities is difficult because the effects of the
+excitation sources are often intertwined with the non-linear response. \cite{Kohler2017}  
+Equation \ref{eq:gamma_to_phi} shows that the metrics of non-linear response, $\phi$ and $\gamma^{(3)}$, are connected not just by intrinsic properties, but by experimental parameters $D$ and $\Delta_t$. 
+
+Since $\gamma^{(3)}$ assumes the driven limit for field-matter interactions (see Eqn. \ref{eq:Maker_Terhune}), signal scales with pulse intensity and not fluence. 
+The third-order susceptibility will be proportional to the pulse duration of the experiment, $\Delta_t$. 
+
+Conversely, $\gamma^{(3)}$ factors out the degeneracy of the experiment, but non-linear absorptivity does not (Eqn. \ref{eq:beta_to_chi}).
+Since $\phi$ is defined by the non-linear absorptivity, it is also proportional to $D$.  
+
+Equation \ref{eq:Delta_alpha1} is valid for a temporally separated pump and probe, so that the probe sees the entire population created by the pump.  
+Our experiments examine the non-linearity for temporally overlapped pump and probe pulses. 
+The differences due to these effects can be calculated under reasonable assumptions (see the Supplementary Materials); we find the population seen at temporal overlap about $80\%$ that of the excited state probed after the pump.
+This factor is needed for comparisons between our measurements and transient absorption with well separated pulses.
+This correction factor is small compared to our uncertainty, so we neglect it.  
+It may be important in more precise measurements.
+
+
+%The non-linear optical properties of the 1S band of PbX quantum dots are well-studied, with a variety of techniques and excitation sources used.  
+%In the TA community, however, there is a heavy reliance on the A:B ratio for quantifying the state-filling fraction.  .  
+The most direct comparison of our measurements with literature is Yurs et.al. \cite{YursLenaA2012a}, who performed the picosecond pulse analogue of the work herein.  
+Both studies used CCl$_4$ as an internal standard, so we can compare the ratios directly to avoid uncertainty from the value of $\gamma_\text{sol}$.
+Note, however, that the picosecond study does not account for window contributions, which could mean their reported ratios are under-reported (the solvent field is actually the solvent and window fields).
+%Though picosecond pulses are narrow-band relative to the 1S transition, extensive modeling of the multidimensional spectra accounted for the inhomogeneous distribution.  
+The values are shown in Table \ref{tab:gamma_ratio}. 
+Both Raman and 1S band susceptibilities differ by approximately the ratio of pulse durations, consistent with a intermediate state with lifetime longer than 1 ps (a Raman coherence and a 1S population, respectively).  
+
+The broadband QD hyperpolarizability ($B$) is similar with both pulse durations, indicating that this contribution originates not from a 1S population, but something very fast (driven limit).  
+Possible explanations are double/zero quantum coherences, ultrafast relaxation, or simply a non-resonant polarization.  
+This broadband feature may be different from that observed in transient absorption because temporal pulse overlap isolates the fastest observable features (most TA features are analyzed at finite delays from pulse overlap).  
+
+Table \ref{tab:litcompare} compares various non-linear quantities for this work, Yurs et. al., and a PbS experiment.
+We will continue to refer to this table for the rest of this discussion.
+Note that $\gamma_\text{QD} / \Delta_t$ is similar between Yurs and our measurement, as expected from the pulse duration dependence.
+
+\begin{table}[]
+	\centering
+	\caption{Non-linear parameters relative to CCl$_4$ hyperpolarizability.  $\gamma_{\nu 1}$:  hyperpolarizability of the $\nu_1$ Raman transition.}
+	\label{tab:gamma_ratio}
+	\begin{tabular}{r|ccc}
+		&  this work &  Yurs et. al. & ratio \\
+		\hline
+		$ |\gamma_\text{QD,peak}| / |\gamma_\text{sol}| $ & $7.3 \cdot 10^5$ & $1.1 \cdot 10^7$ & 15 \\
+		$ |B| / |\gamma_\text{sol}| $ & $1.3 \cdot 10^5$ & $1.6 \cdot 10^5$ & 1.3 \\
+		$ |\gamma_{\nu1}| / |\gamma_\text{sol}| $ & $0.25 \pm 0.04 $ & $5.1$ & 20.4 \\
+	\end{tabular}
+\end{table}
+
+The sample studied by Yurs et. al. was significantly degraded, and the authors described their QD spectra using mechanisms other than state-filling.  
+The relative similarity of the absolute susceptibility, given such extraordinary spectral differences, is noteworthy.  
+Yurs et. al. also uses a different $\gamma_\text{sol}$ value to calculate their absolute susceptibility, which gives more disagreement in reported values than the literature suggests.  
+
+Omari et. al. \cite{OmariAbdoulghafar2012a} performed $z$-scan measurements of PbS QDs to quantify the non-linear parameters (see right-hand column of Table \ref{tab:litcompare}).  
+%They report a hyperpolarizability of $(-10^4 - 300i) \cdot 10^{-30} \text{cm}^6 \text{erg}^{-1}$\footnote{
+%	Derive from Eqn. 11 in their text.  The reported value of $\beta$ does not account for the rep rate or the inhomogenoeus excitation}. 
+In contrast to our measurements, their degenerate susceptibility is primarily real in character and much larger than that reported here or in Yurs.  
+While we cannot reconcile the real component, the imaginary component agrees with the standard bleach theory ($\phi = 0.15$\footnote{explain where this comes from}. 
+Omari et. al. report that their results do not agree with the $\phi = 0.25$ bleach theory of transient absorption, but we note that their observed bleach fractions is actually in great agreement once the experimental degeneracy is accounted for (a transient absorption measurement of their sample would give $\phi = 0.3$).
+% TODO:  consider deriving in SI
+%$\beta_\text{TA} = 2\beta_\text{z-scan}$).  
+
+\begin{table*}
+	\centering
+	\caption{Comparison of these measurements with PbX measurements in literature.  $\gamma_\text{raman}^{(3)}$ refers to the 465 $\text{cm}^{-1}$ (symmetric stretch) mode of $\text{CCl}_4$}
+	\label{tab:litcompare}
+	\begin{tabular}{l|ccc}
+		&  this work   &  Yurs et. al. & Omari et. al.\footnote{samples D (imaginary) and K (real)} \\
+		QD & PbSe & PbSe & PbS \\
+		measurement & $|\gamma|$ & $|\gamma|$ & $\gamma$ \\
+		\hline
+		$ \Delta_t \left[ \text{fs} \right]$
+		& $\sim 50 $ & $\sim 1250 $	& $\sim 2500$ \\
+		$ \left| \gamma_\text{QD}^{(3)} \right| \left[ 10^{-30} \frac{\text{cm}^6}{\text{erg}} \right]$ 				
+		& 0.2 & 8.8 & $-(1 + .03i) \cdot 10^4$ \\
+		$ \left| \gamma_\text{QD}^{(3)} / \Delta_t \right| \left[ 10^{-18} \frac{\text{cm}^6}{\text{erg s}} \right]$ 	
+		& 4 & 7 & 120 \\
+		%$ \gamma_\text{raman}^{(3)} / \Delta_t \left[ 10^{-24} \frac{\text{cm}^3}{\text{erg s}} \right]$
+		%	& 2.9 & 3.0 & -- & \\
+		$\phi$ (Eqn. \ref{eq:gamma_to_phi}) & 0.3 & 0.6 & 0.15 \\
+	\end{tabular}
+\end{table*}
+
+We now turn our focus to comparison between our measurement and $\phi$.
+There is some variance in the value of $\phi$ reported for PbX quantum dots.
+The commonly accepted value for the bleach fraction of $\phi = 0.25$ is approximate, and runs
+counter to saturated absorption studies, where a fully inverted 1S band reduces by a factor of 1/8
+after Auger recombination yields single-exciton species. \cite{Nootz2011, Istrate2008} 
+Only a few transient absorption studies address the photobleach magnitude explicitly, rather than the more common state-filling analysis via the A:B ratio. 
+%It also seemingly runs counter to $z$-scan determinations of the state-filling, which found $\phi \approx 0.1$\footnote{
+%	Derived from Eqn. 11 in their text.  The reported value of $\beta$ does not account for the repetition rate or the inhomogenoeus excitation; do not assume inhomogeneously broadened is the limit we are in}. 
+
+We can check our measured susceptibility with the accepted $\phi$ value using Equation \ref{eq:gamma_to_phi}.
+If the peak susceptibility is mostly imaginary, we can attribute our TG peak $\left|\chi^{(3)}\right|$ susceptibility to the TA state-filling factor.  
+Our peak TG hyperpolarizability measurements give values of $\left| \gamma^{(3)} \right| = 3 \pm 2 \cdot 10^{-31} \text{cm}^6 \ \text{erg}^{-1}$, while Eqn. \ref{eq:gamma_to_phi} predicts $\text{Im}\left[ \gamma^{(3)} \right] = -1.6 \cdot 10^{-31} \text{cm}^6 \text{erg}^{-1}$ for $\phi = 0.25$.   %and $-8.0 \cdot 10^{-32} \text{cm}^6  \text{erg}^{-1}$ for $\phi = 0.125$.
+Our method gives agreement with the $\phi=0.25$ bleach factor.  %
+
+\section{Conclusion}  % ===========================================================================
+
+We have shown that ultrafast CMDS can isolate the non-linearities of resonant signal and background
+solvent in manner similar to classic three-wave mixing analyses of the past.  %
+At pulse overlap, featureless solvent contributions can be especially large.  %
+The resonant Raman contributions decrease in prominence when pulses are shorter than the Raman
+lifetimes.  %
+Solvent contributions can also be suppressed by using large sample concentrations, in which case
+spectra have to be corrected by well-defined absorptive correction factors.  %
+The solute and solvent interference can be separated using simple, few parameter models, as we have
+demonstrated here with quantum dots.  %
+We have employed simple, few-parameter fits to easily disentangle the role of solvent and solute.  
+These methods should be applicable to other CMDS spectroscopies, but the description of the solvent
+may change, especially when non-co-polarized excitations are used. \cite{DeegFW1989a}  %  
+In fact, the polarization behavior provides a useful way to alter the balance of solvent and solute
+contributions in a predictable way, and is likely a viable method for separating solvent and solute
+contributions.  %
+
+Solvent may be used as an internal standard to measure the solute non-linearity, but there are
+still large uncertainties in the non-linear susceptibility that propagate to the solute optical
+constants.  %  
+More work should be done to reduce this uncertainty and characterize the dispersion of non-linear
+susceptibility of solvents.  %
+
+Absolute nonlinearities are an important property to study in material systems because their
+relation to linear susceptibilities informs on the underlying physics.  %
+For MR-CMDS, it is important to identify how to extract these non-linearities.  %  
+We have demonstrated ultrafast MR-CMDS as a viable method to extract the absolute non-linear
+susceptibility by using CCl$_4$ as an internal standard.  %
+The extraction requires accounting for the impulsive population creation, as well as absorptive
+propagation effects within the sample.  %
+These absorptive effects are also crucial factors for general analysis of MR-CMDS spectra.  %
diff --git a/PbSe_susceptibility/conclusion.tex b/PbSe_susceptibility/conclusion.tex
new file mode 100644
index 0000000..6ff88c6
--- /dev/null
+++ b/PbSe_susceptibility/conclusion.tex
@@ -0,0 +1,27 @@
+%In addition, in situ characterization of dipole strength can also be very important in the case of quantum dots, because states have been known to degrade by bleaching, thus mainly destroying the dipole moment strength of the sample.  
+%Our measurements use solution-phase CMDS to quantitate the non-linear magnitude of quantum dot resonance.  
+%We first discuss how our measurements in the context of the photobleach mechanism.  
+%We then discuss how our measurements compare to the other measurements of the photobleach.  
+%These measurements provide a framework to describe the photobleach mechanism in the same language as TA (the imaginary analogue of our measurement).  
+%This measurement can be quantitatively connected to the non-linear mechanism, which in this case is a fractional photobleach.  
+%The method could also be applied to TSF to extract displacements responsible for Franck-Condon vibronic structure. 
+
+We have shown that ultrafast CMDS can isolate the non-linearities of resonant signal and background solvent in manner similar to classic three-wave mixing analyses of the past.  
+%Ultrafast CMDS spectroscopy can be contaminated by partially resonant and non-resonant contributions of the solvent.  
+At pulse overlap, featureless solvent contributions can be especially large.  
+The resonant Raman contributions decrease in prominence when pulses are shorter than the Raman lifetimes.  
+Solvent contributions can also be suppressed by using large sample concentrations, in which case spectra have to be corrected by well-defined absorptive correction factors.  
+The solute and solvent interference can be separated using simple, few parameter models, as we have demonstrated here with quantum dots.  
+We have employed simple, few-parameter fits to easily disentangle the role of solvent and solute.  
+These methods should be applicable to other CMDS spectroscopies, but the description of the solvent may change, especially when non-co-polarized excitations are used.\cite{Deeg1989}  
+In fact, the polarization behavior provides a useful way to alter the balance of solvent and solute contributions in a predictable way, and is likely a viable method for separating solvent and solute contributions.
+
+Solvent may be used as an internal standard to measure the solute non-linearity, but there are still large uncertainties in the non-linear susceptibility that propagate to the solute optical constants.  
+More work should be done to reduce this uncertainty and characterize the dispersion of non-linear susceptibility of solvents.  
+
+Absolute nonlinearities are an important property to study in material systems because their relation to linear susceptibilities informs on the underlying physics.  For MR-CMDS, it is important to identify how to extract these non-linearities.  
+We have demonstrated ultrafast MR-CMDS as a viable method to extract the absolute non-linear susceptibility by using CCl$_4$ as an internal standard.  
+The extraction requires accounting for the impulsive population creation, as well as absorptive propagation effects within the sample.  
+These absorptive effects are also crucial factors for general analysis of MR-CMDS spectra.
+
+
diff --git a/PbSe_susceptibility/dilution_fits.png b/PbSe_susceptibility/dilution_fits.png
new file mode 100644
index 0000000..54b5a27
--- /dev/null
+++ b/PbSe_susceptibility/dilution_fits.png
diff --git a/PbSe_susceptibility/dilution_integral.png b/PbSe_susceptibility/dilution_integral.png
new file mode 100644
index 0000000..abee739
--- /dev/null
+++ b/PbSe_susceptibility/dilution_integral.png
diff --git a/PbSe_susceptibility/mfactors_check.png b/PbSe_susceptibility/mfactors_check.png
new file mode 100644
index 0000000..0c781c8
--- /dev/null
+++ b/PbSe_susceptibility/mfactors_check.png
diff --git a/PbSe_susceptibility/partially_coherent_fwm_scaling.png b/PbSe_susceptibility/partially_coherent_fwm_scaling.png
new file mode 100644
index 0000000..e109604
--- /dev/null
+++ b/PbSe_susceptibility/partially_coherent_fwm_scaling.png
diff --git a/PbSe_susceptibility/pump_delay_factor.png b/PbSe_susceptibility/pump_delay_factor.png
new file mode 100644
index 0000000..d2f5647
--- /dev/null
+++ b/PbSe_susceptibility/pump_delay_factor.png
diff --git a/PbSe_susceptibility/raman_traces.png b/PbSe_susceptibility/raman_traces.png
new file mode 100644
index 0000000..3a67fde
--- /dev/null
+++ b/PbSe_susceptibility/raman_traces.png
diff --git a/abstract.tex b/abstract.tex
index 64d230f..5ef1404 100644
--- a/abstract.tex
+++ b/abstract.tex
@@ -24,6 +24,7 @@ The pulses interact with the material, and due to the specific interference betw
 fields the material is driven to emit a new pulse: the MR-CMDS signal.  %
 This signal may be different in frequency from the input pulses, and it may travel in a new
 direction depending on the exact experiment being performed.  %
+% BJT: flip proceeding sentence around
 The MR-CMDS experiment involves tracking the intensity of this output signal as a function of
 different properties of the excitation pulses.  %
 These properties include
@@ -33,10 +34,11 @@ These properties include
 4. polarization \cite{FournierFrederic2009a}, among others.  %
 Thus MR-CMDS can be thought of as a multidimensional experimental space, where experiments
 typically involve explorations in one to four of the properties above.  %
+% BJT: emphasize UP TO four dimensions, practically
 
 Because MR-CMDS is a family of related-but-separate experiments, each of them a multidimensional
 space, there are special challenges that must be addressed when designing a general-purpose MR-CMDS
-instrument.  %
+instrument.  % BJT: be more specific about the challenge at hand
 These issues require development of software, hardware, and theory.  %
 Five different improvements to MR-CMDS are presented in this dissertation: 1. processing software
 (\autoref{cha:pro}), 2. acquisition software (\autoref{cha:acq}) 3, active artifact correction
diff --git a/active_correction/chapter.tex b/active_correction/chapter.tex
index e40ff9d..d7b6a4e 100644
--- a/active_correction/chapter.tex
+++ b/active_correction/chapter.tex
@@ -34,12 +34,35 @@ Some of these strategies have already been implemented, others are partially imp
 others are still just ideas.  %

 I hope to show that active correction is a particularly useful strategy in MR-CMDS.  %

 

+[SPECTRAL DELAY PARAGRAPH]

+

+[POYNTING CORRECTION PARAGRAPH]

+

+[EXCITATION POWER CORRECTION PARAGRAPH]

+

+[CHOPPING PARAGRAPH]

+

+[FIBRILLATION]

+

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

 

+\begin{figure}

+  \caption[CAPTION TODO]{

+    CAPTION TODO: SPECTRAL DELAY CORRECTION FIGURE

+  }

+\end{figure}

+

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

 

+\begin{figure}

+  \caption[CAPTION TODO]{

+    CAPTION TODO: POYNTING CORRECTION FIGURE

+  }

+\end{figure}

+

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

 

+\clearpage

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

 

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

@@ -225,12 +248,6 @@ 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]

@@ -280,4 +297,10 @@ Now, substituting in BRAZARD formalism:
 

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

 

+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}.  %

+

 \section{Conclusions}  % ==========================================================================
\ No newline at end of file
diff --git a/bibliography.bib b/bibliography.bib
index b633eae..956092a 100644
--- a/bibliography.bib
+++ b/bibliography.bib
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   month =        {jul},

 }

 

+@article{Asunskis2008,

+  author =       {Asunskis, Daniel J and Bolotin, Igor L and Hanley, Luke},

+  title =        {{Nonlinear Optical Properties of PbS Nanocrystals Grown in Polymer Solutions}},

+  journal =      {The Journal of Physical Chemistry C},

+  volume =       112,

+  number =       26,

+  pages =        {9555--9558},

+  year =         2008,

+  doi =          {10.1021/jp8037076},

+  issn =         {1932-7447},

+  month =        {jul},

+}

+

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   author =       {Aub{\"{o}}ck, Gerald and Consani, Cristina and van Mourik, Frank and Chergui,

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@@ -242,6 +255,20 @@
   url =          {https://cran.r-project.org/},

 }

 

+@article{Cademartiri2006,

+  author =       {Cademartiri, Ludovico and Montanari, Erica and Calestani, Gianluca and Migliori,

+                  Andrea and Guagliardi, Antonietta and Ozin, Geoffrey A},

+  title =        {{Size-Dependent Extinction Coefficients of PbS Quantum Dots}},

+  journal =      {Journal of the American Chemical Society},

+  volume =       128,

+  number =       31,

+  pages =        {10337--10346},

+  year =         2006,

+  doi =          {10.1021/ja063166u},

+  issn =         {0002-7863},

+  month =        {aug},

+}

+

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   author =       {David N. Card and Victor E. Church and William W. Agresti},

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@@ -374,6 +401,20 @@
   publisher =    {{AIP} Publishing},

 }

 

+@article{Cheng2001,

+  author =       {Cheng, Ji-xin and Volkmer, Andreas and Book, Lewis D and Xie, X Sunney},

+  title =        {{An Epi-Detected Coherent Anti-Stokes Raman Scattering (E-CARS) Microscope with

+                  High Spectral Resolution and High Sensitivity}},

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+  year =         2001,

+  doi =          {10.1021/jp003774a},

+  isbn =         {1520-6106},

+  issn =         {1520-6106},

+}

+

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+@article{Cherlow1976,

+  author =       {Cherlow, J. and Yang, T. and Hellwarth, R.},

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+  issn =         {0018-9197},

+  month =        {oct},

+}

+

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+                  R and Jonas, David M.},

+  title =        {{Propagation and Beam Geometry Effects on Two-Dimensional Fourier Transform

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+  doi =          {10.1021/jp904504z},

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+}

+

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@@ -401,6 +470,19 @@
   month =        {apr},

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+@article{Crisp1970,

+  author =       {Crisp, M. D.},

+  title =        {{Propagation of Small-Area Pulses of Coherent Light through a Resonant Medium}},

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+}

+

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   month =        {dec},

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+@article{Dai2009,

+  author =       {Dai, Quanqin and Wang, Yingnan Yiding and Li, Xinbi and Zhang, Yu and Pellegrino,

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+  doi =          {10.1021/nn9001616},

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+  issn =         {1936-086X},

+  keywords =     {Composition,Molar extinction coefficient,PbSe,Semiconductor nanocrystal,Size

+                  dependence,informative,size-exciton correlation},

+  month =        {jun},

+  pmid =         19435305,

+}

+

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 year = {1998}

 }

 

+@article{DeGeyterBram2012a,

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+                  Siebbeles, Laurens D A and Hens, Zeger},

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+                  dots}},

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+  issn =         21627339,

+}

+

+@article{DeegFW1989a,

+  author =       {Deeg, F. W. and Fayer, M. D.},

+  title =        {{Analysis of complex molecular dynamics in an organic liquid by polarization

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+  year =         1989,

+  doi =          {10.1063/1.457034},

+  issn =         {0021-9606},

+  month =        {aug},

+}

+

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@@ -599,6 +730,21 @@ year = {1998}
   issn =         {0302-0738},

 }

 

+@article{Du2002,

+  author =       {Du, Hui and Chen, Chialing and Krishnan, Rishikesh and Krauss, Todd D. and

+                  Harbold, Jeffrey M. and Wise, Frank W. and Thomas, Malcolm G. and Silcox, John},

+  title =        {{Optical Properties of Colloidal PbSe Nanocrystals}},

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+  doi =          {10.1021/nl025785g},

+  isbn =         {1530-6984},

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+  month =        {nov},

+}

+

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   title =        {{BOXCARS}: Crossed-beam phase-matched {CARS} generation in

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   month =        {may},

 }

 

+@article{Etchepare1985,

+  author =       {Etchepare, J and Grillon, G and Thomazeau, I and Migus, A and Antonetti, A},

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+  issn =         {0740-3224},

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+}

+

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   note =         {Accessed: 2018-03-27},

   title =        {FITS: Flexable image transport},

@@ -771,6 +931,37 @@ year = {1998}
   month =        {aug},

 }

 

+@article{Gdor2013a,

+  author =       {Gdor, Itay and Yang, Chunfan and Yanover, Diana and Sachs, Hanan and Lifshitz,

+                  Efrat and Ruhman, Sanford},

+  title =        {{Novel Spectral Decay Dynamics of Hot Excitons in PbSe Nanocrystals: A Tunable

+                  Femtosecond Pump–Hyperspectral Probe Study}},

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+}

+

+@article{GeiregatPieter2014a,

+  author =       {Geiregat, Pieter and Houtepen, Arjan J. and Justo, Yolanda and Grozema, Ferdinand

+                  C. and {Van Thourhout}, Dries and Hens, Zeger},

+  title =        {{Coulomb Shifts upon Exciton Addition to Photoexcited PbS Colloidal Quantum

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+}

+

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   publisher =    {{AIP} Publishing},

 }

 

+@article{Gubler2000,

+  author =       {Gubler, Ulrich and Bosshard, Christian},

+  title =        {{Optical third-harmonic generation of fused silica in gas atmosphere: Absolute

+                  value of the third-order nonlinear optical susceptibility {\textless}math

+                  display="inline"{\textgreater} {\textless}mrow{\textgreater}

+                  {\textless}msup{\textgreater} {\textless}mrow{\textgreater}

+                  {\textless}mi{\textgreater}$\chi${\textless}/mi{\textgreater}

+                  {\textless}/mrow{\textgreater} {\textless}mrow{\textgreater}

+                  {\textless}mn/{\textgreater}

+                  {\textless}mo{\textgreater}({\textless}/mo{\textgreater}

+                  {\textless}mn{\textgreater}3{\textless}/mn{\textgreater}

+                  {\textless}mo{\textgreater}){\textless}/mo{\textgreater}

+                  {\textless}mn/{\textgreater} {\textless}/m}},

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+}

+

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   author =       {Hamm, Peter and Lim, Manho and DeGrado, W. F. and Hochstrasser, Robin M},

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@@ -857,7 +1071,6 @@ year = {1998}
   month =        {mar},

 }

 

-

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   month =        {jun},

 }

 

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+}

+

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+}

+

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+@article{Istrate2008,

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+}

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@@ -1067,6 +1334,20 @@ year = {1998}
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+}

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 }

 

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+  author =       {Klimov, Victor I.},

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+

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+}

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+}

+

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+}

+

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+  author =       {Levine, B. F. and Bethea, C. G.},

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+}

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   publisher =    {American Chemical Society ({ACS})},

 }

 

+

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+  author =       {Maker, P D and Terhune, R. W.},

+  title =        {{Study of Optical Effects Due to an Induced Polarization Third Order in the

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+  year =         1965,

+  doi =          {10.1103/PhysRev.137.A801},

+  issn =         {0031-899X},

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+}

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+@article{MatsuoShigeki1997a,

+  author =       {Matsuo, Shigeki and Tahara, Tahei},

+  title =        {{Phase-stabilized optical heterodyne detection of impulsive stimulated Raman

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+                  Poelman, Dirk and Martins, Jos{\'{e}} C. and Allan, Guy and Hens, Zeger},

+  title =        {{Composition and Size-Dependent Extinction Coefficient of Colloidal PbSe Quantum

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+  mendeley-tags ={reference},

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+}

+

+@article{Moreels2009,

+  author =       {Moreels, Iwan and Lambert, Karel and Muynck, David De and Vanhaecke, Frank and

+                  Poelman, Dirk and Martins, Jos{\'{e}} C and Allan, Guy and Hens, Zeger},

+  title =        {{Size-Dependent Optical Properties of Colloidal {\{}PbS{\}} Quantum Dots}},

+  journal =      {ACS Nano},

+  volume =       3,

+  number =       10,

+  pages =        {3023--3030},

+  year =         2009,

+  doi =          {10.1021/nn900863a},

+  issn =         {1936-0851},

+}

+

 @book{MukamelShaul1995a,

   author =       {Mukamel, Shaul},

   title =        {{Principles of nonlinear optical spectroscopy}},

@@ -1401,7 +1826,6 @@ year = {1998}
   isbn =         9780195092783,

   series =       {Oxford series in optical and imaging sciences},

 }

-

 @article{MukamelShaul2000a,

   author =       {Mukamel, Shaul},

   title =        {Multidimensional Femtosecond Correlation Spectroscopies of Electronic and

@@ -1451,6 +1875,21 @@ year = {1998}
   publisher =    {OSA},

 }

 

+@article{Nibbering1995,

+  author =       {Nibbering, Etj and Franco, M.A. and Prade, BS and Grillon, G. and {Le Blanc}, C

+                  and Mysyrowicz, A},

+  title =        {{Measurement of the nonlinear refractive index of transparent materials by

+                  spectral analysis after nonlinear propagation}},

+  journal =      {Optics Communications},

+  volume =       119,

+  number =       {5-6},

+  pages =        {479--484},

+  year =         1995,

+  doi =          {10.1016/0030-4018(95)00394-N},

+  issn =         00304018,

+  month =        {sep},

+}

+

 @article{NieZhaogang2014a,

   author =       {Zhaogang Nie and Run Long and Linfeng Sun and Chung-Che Huang

                   and Jun Zhang and Qihua Xiong and Daniel W. Hewak and Zexiang

@@ -1466,6 +1905,23 @@ year = {1998}
   month =        {oct},

 }

 

+@article{NootzGero2011a,

+  author =       {Nootz, Gero and Padilha, Lazaro A. and Levina, Larissa and Sukhovatkin, Vlad and

+                  Webster, Scott and Brzozowski, Lukasz and Sargent, Edward H. and Hagan, David J.

+                  and {Van Stryland}, Eric W.},

+  title =        {{Size dependence of carrier dynamics and carrier multiplication in PbS quantum

+                  dots}},

+  journal =      {Physical Review B},

+  volume =       83,

+  number =       15,

+  pages =        155302,

+  year =         2011,

+  doi =          {10.1103/PhysRevB.83.155302},

+  isbn =         {1098-0121},

+  issn =         {1098-0121},

+  month =        {apr},

+}

+

 @article{OkumuraKo1999a,

   author =       {Okumura, Ko and Tokmakoff, Andrei and Tanimura, Yoshitaka},

   title =        {{Two-dimensional line-shape analysis of photon-echo signal}},

@@ -1528,6 +1984,19 @@ year = {1998}
   month =        {sep},

 }

 

+@article{Oughstun1996,

+  author =       {Oughstun, Kurt E. and Balictsis, Constantinos M.},

+  title =        {{Gaussian Pulse Propagation in a Dispersive, Absorbing Dielectric}},

+  journal =      {Physical Review Letters},

+  volume =       77,

+  number =       11,

+  pages =        {2210--2213},

+  year =         1996,

+  doi =          {10.1103/PhysRevLett.77.2210},

+  issn =         {0031-9007},

+  month =        {sep},

+}

+

 @article{PakoulevAndreiV2006a,

   author =       {Pakoulev, Andrei V. and Rickard, Mark a and Meyer, Kent a. and Kornau, Kathryn

                   and Mathew, Nathan a. and Thompson, David E. and Wright, John C.},

@@ -1747,6 +2216,35 @@ year = {1998}
   url =          {https://doc.qt.io/archives/qt-4.8/qmutex.html},

 }

 

+@article{Rau2004,

+  author =       {Rau, A R P},

+  title =        {{Perspectives on the Fano Resonance Formula}},

+  journal =      {Physica Scripta},

+  volume =       69,

+  number =       1,

+  pages =        {C10--C13},

+  year =         2004,

+  doi =          {10.1238/Physica.Regular.069a00C10},

+  issn =         {0031-8949},

+  month =        {jan},

+}

+

+@article{Rau2008,

+  author =       {Rau, Ileana and Kajzar, Fran{\c{c}}ois and Luc, J{\'{e}}r{\^{o}}me and Sahraoui,

+                  Bouchta and Boudebs, Georges},

+  title =        {{Comparison of Z-scan and THG derived nonlinear index of refraction in selected

+                  organic solvents}},

+  journal =      {Journal of the Optical Society of America B},

+  volume =       25,

+  number =       10,

+  pages =        1738,

+  year =         2008,

+  doi =          {10.1364/JOSAB.25.001738},

+  issn =         {0740-3224},

+  keywords =     {1904400,1904710,OCIS codes},

+  month =        {oct},

+}

+

 @article{RentzepisPM1970a,

   author =       {Rentzepis, P. M.},

   title =        {{Ultrafast Processes}},

@@ -1850,6 +2348,21 @@ year = {1998}
   month =        07,

 }

 

+@article{Schaller2003,

+  author =       {Schaller, Richard D and Petruska, M. a. and Klimov, Victor I.},

+  title =        {{Tunable Near-Infrared Optical Gain and Amplified Spontaneous Emission Using PbSe

+                  Nanocrystals}},

+  journal =      {The Journal of Physical Chemistry B},

+  volume =       107,

+  number =       50,

+  pages =        {13765--13768},

+  year =         2003,

+  doi =          {10.1021/jp0311660},

+  url =          {http://pubs.acs.org/doi/abs/10.1021/jp0311660},

+  issn =         {1520-6106},

+  month =        {dec},

+}

+

 @article{SchweigertIgorV2008a,

   author =       {Schweigert, Igor V. and Mukamel, Shaul},

   title =        {{Simulating multidimensional optical wave-mixing signals with finite-pulse

@@ -1921,6 +2434,19 @@ year = {1998}
   publisher =    {Royal Society of Chemistry ({RSC})},

 }

 

+@article{SheikBahaeMansoor1989a,

+  author =       {Sheik-bahae, M. and Said, A A and {Van Stryland}, E. W.},

+  title =        {{High-sensitivity, single-beam n{\_}2 measurements}},

+  journal =      {Optics Letters},

+  volume =       14,

+  number =       17,

+  pages =        955,

+  year =         1989,

+  doi =          {10.1364/OL.14.000955},

+  issn =         {0146-9592},

+  month =        {sep},

+}

+

 @article{SheikBahaeMansoor1990a,

   author =       "Sheik-Bahae, M and Said, AA and Wei, T-H and Hagan, DJ and Van Stryland, EW",

   title =        "Sensitive measurement of optical nonlinearities using a single beam",

@@ -1934,6 +2460,14 @@ year = {1998}
   publisher =    "Institute of Electrical and Electronics Engineers",

 }

 

+@incollection{ShimanouchiT,

+  author =       {Shimanouchi, T.},

+  booktitle =    {NIST Standard Reference Database Number 69},

+  doi =          {10.18434/T4D303},

+  editor =       {Linstrom, P.J. and Mallard, W.G.},

+  title =        {{Molecular Vibrational Frequencies}},

+}

+

 @article{SimSangwan2013a,

   author =       {Sangwan Sim and Jusang Park and Jeong-Gyu Song and Chihun In

                   and Yun-Shik Lee and Hyungjun Kim and Hyunyong Choi},

@@ -1990,6 +2524,20 @@ year = {1998}
 	doi = {10.1021/acs.jpca.5b00001},

 }

 

+@article{Spoor,

+  author =       {Spoor, Frank C M and Tomi{\'{c}}, Stanko and Houtepen, Arjan J and Siebbeles,

+                  Laurens D A},

+  title =        {{Broadband Cooling Spectra of Hot Electrons and Holes in PbSe Quantum Dots}},

+  journal =      {ACS Nano},

+  volume =       11,

+  number =       6,

+  pages =        {6286--6294},

+  year =         2017,

+  doi =          {10.1021/acsnano.7b02506},

+  issn =         {1936-0851},

+  month =        {jun},

+}

+

 @article{SteehlerJK1985a,

   author =       {Steehler, J. K. and Wright, J. C.},

   title =        {{Parametric and nonparametric four‐wave mixing in pentacene: p ‐terphenyl}},

@@ -2031,6 +2579,30 @@ year = {1998}
   month =        {oct},

 }

 

+@book{Sutherland,

+  author =       {Sutherland, Richard L and Mclean, Daniel G and Kirkpatrick, Sean},

+  title =        {{Handbook of Nonlinear Optics}},

+  booktitle =    {New York},

+  file =         {:C$\backslash$:/Users/Dan/AppData/Local/Mendeley Ltd./Mendeley

+                  Desktop/Downloaded/Sutherland, Mclean, Kirkpatrick - Unknown - Handbook of

+                  Nonlinear Optics with contributions by.pdf:pdf},

+  isbn =         {0824742435},

+}

+

+@article{Thalhammer1983,

+  author =       {Thalhammer, M and Penzkofer, A},

+  title =        {{Measurement of third-order nonlinear susceptibilities by non-phase matched

+                  third-harmonic generation}},

+  journal =      {Applied Physics B Photophysics and Laser Chemistry},

+  volume =       32,

+  number =       3,

+  pages =        {137--143},

+  year =         1983,

+  doi =          {10.1007/BF00688819},

+  issn =         {0721-7269},

+  month =        {nov},

+}

+

 @article{ThomallaMarkus2006a,

   author =       {Markus Thomalla and Helmut Tributsch},

   title =        {Photosensitization of Nanostructured {TiO}2 with {WS}2 Quantum Sheets},

@@ -2070,6 +2642,35 @@ year = {1998}
   publisher =    {Wiley-Blackwell},

 }

 

+@article{Trinh2008,

+  author =       {Trinh, M Tuan and Houtepen, Arjan J and Schins, Juleon M and Piris, Jorge and

+                  Siebbeles, Laurens D a},

+  title =        {{Nature of the second optical transition in PbSe nanocrystals.}},

+  journal =      {Nano letters},

+  volume =       8,

+  number =       7,

+  pages =        {2112--7},

+  year =         2008,

+  doi =          {10.1021/nl8010963},

+  issn =         {1530-6984},

+  month =        {jul},

+  pmid =         18510369,

+}

+

+@article{Trinh2013,

+  author =       {Trinh, M. Tuan and Sfeir, Matthew Y. and Choi, Joshua J. and Owen, Jonathan S.

+                  and Zhu, Xiaoyang},

+  title =        {{A Hot Electron–Hole Pair Breaks the Symmetry of a Semiconductor Quantum Dot}},

+  journal =      {Nano Letters},

+  volume =       13,

+  number =       12,

+  pages =        {6091--6097},

+  year =         2013,

+  doi =          {10.1021/nl403368y},

+  issn =         {1530-6984},

+  month =        {dec},

+}

+

 @article{TurnerDanielB2009a,

   author =       {Daniel B. Turner and Katherine W. Stone and Kenan Gundogdu and

                   Keith A. Nelson},

@@ -2103,6 +2704,20 @@ year = {1998}
   url =          {http://sm.stanford.edu/archive/stanmed/2012summer/article4.html},

 }

 

+@article{VoehringerPeter1995a,

+  author =       {Voehringer, Peter and Scherer, Norbert F},

+  title =        {{Transient Grating Optical Heterodyne Detected Impulsive Stimulated Raman

+                  Scattering in Simple Liquids}},

+  journal =      {The Journal of Physical Chemistry},

+  volume =       99,

+  number =       9,

+  pages =        {2684--2695},

+  year =         1995,

+  doi =          {10.1021/j100009a027},

+  issn =         {0022-3654},

+  month =        {mar},

+}

+

 @article{VolkmerAndreas2001a,

   author =       {Andreas Volkmer and Ji-Xin Cheng and X. Sunney Xie},

   title =        {Vibrational Imaging with High Sensitivity via Epidetected Coherent Anti-Stokes

@@ -2144,6 +2759,20 @@ year = {1998}
   journal = {Reviews of Modern Physics}

 }

 

+@article{Wawrzynczyk2016,

+  author =       {Wawrzynczyk, Dominika and Szeremeta, Janusz and Samoc, Marek and Nyk, Marcin},

+  title =        {{Third-Order Nonlinear Optical Properties of Infrared Emitting PbS and PbSe

+                  Quantum Dots}},

+  journal =      {The Journal of Physical Chemistry C},

+  volume =       120,

+  number =       38,

+  pages =        {21939--21945},

+  year =         2016,

+  doi =          {10.1021/acs.jpcc.6b05956},

+  issn =         {1932-7447},

+  month =        {sep},

+}

+

 @article{WehnerMU1996a,

   author =       {M. U. Wehner and D. Steinbach and M. Wegener},

   title =        {Ultrafast coherent transients due to exciton-continuum scattering in bulk {GaAs}},

@@ -2157,6 +2786,32 @@ year = {1998}
   publisher =    {American Physical Society ({APS})},

 }

 

+@article{WehrenbergBrianL2002a,

+  author =       {Wehrenberg, Brian L. and Wang, Congjun and Guyot-Sionnest, Philippe},

+  title =        {{Interband and Intraband Optical Studies of PbSe Colloidal Quantum Dots}},

+  journal =      {The Journal of Physical Chemistry B},

+  volume =       106,

+  number =       41,

+  pages =        {10634--10640},

+  year =         2002,

+  doi =          {10.1021/jp021187e},

+  issn =         {1520-6106},

+  month =        {oct},

+}

+

+@article{Weiner1991,

+  author =       {Weiner, A M and Wiederrecht, Gary P and Nelson, Keith A and Leaird, D E},

+  title =        {{Femtosecond multiple-pulse impulsive stimulated Raman scattering spectroscopy}},

+  journal =      {Journal of the Optical Society of America B},

+  volume =       8,

+  number =       6,

+  pages =        1264,

+  year =         1991,

+  doi =          {10.1364/JOSAB.8.001264},

+  issn =         {0740-3224},

+  month =        {jun},

+}

+

 @article{WeinerAM1985a,

   author =       {Weiner, A M and {De Silvestri}, S. and Ippen, E P},

   title =        {{Three-pulse scattering for femtosecond dephasing studies: theory and

@@ -2251,7 +2906,6 @@ year = {1998}
   month =        {oct},

 }

 

-

 @article{WrightJohnCurtis1997a,

   author =       {Wright, John C.},

   title =        {{New selective nonlinear vibrational spectroscopies}},

@@ -2295,6 +2949,7 @@ year = {1998}
   month =        {oct},

 }

 

+

 @article{WrightJohnCurtis2011a,

   author =       {John C. Wright},

   title =        {Multiresonant Coherent Multidimensional Spectroscopy},

@@ -2374,6 +3029,22 @@ year = {1998}
   month =        {oct},

 }

 

+@article{YetzbacherMichaelK2007a,

+  author =       {Yetzbacher, Michael K and Belabas, Nadia and Kitney, Katherine a and Jonas, David

+                  M.},

+  title =        {{Propagation, beam geometry, and detection distortions of peak shapes in

+                  two-dimensional Fourier transform spectra}},

+  journal =      {The Journal of Chemical Physics},

+  volume =       126,

+  number =       4,

+  pages =        044511,

+  year =         2007,

+  doi =          {10.1063/1.2426337},

+  issn =         {0021-9606},

+  month =        {jan},

+  pmid =         17286491,

+}

+

 @article{YoonSangwoon2005a,

   author =       {Yoon, Sangwoon and McCamant, David W. and Kukura, Philipp and Mathies, Richard A

                   and Zhang, Donghui and Lee, Soo-Y},

@@ -2389,6 +3060,21 @@ year = {1998}
   month =        {jan},

 }

 

+@article{YursLenaA2012a,

+  author =       {Yurs, Lena A and Block, Stephen B and Pakoulev, Andrei V. and Selinsky, Rachel S.

+                  and Jin, Song and Wright, John},

+  title =        {{Spectral Isolation and Measurement of Surface-Trapped State Multidimensional

+                  Nonlinear Susceptibility in Colloidal Quantum Dots}},

+  journal =      {The Journal of Physical Chemistry C},

+  volume =       116,

+  number =       9,

+  pages =        {5546--5553},

+  year =         2012,

+  doi =          {10.1021/jp3014139},

+  issn =         {1932-7447},

+  month =        {mar},

+}

+

 @article{YursLenaA2011a,

   author =       {Yurs, Lena A and Block, Stephen B. and Pakoulev, Andrei V and Selinsky, Rachel S.

                   and Jin, Song and Wright, John},

diff --git a/dissertation.tex b/dissertation.tex
index 23d5ad6..f84f38f 100644
--- a/dissertation.tex
+++ b/dissertation.tex
@@ -64,18 +64,19 @@ This dissertation is approved by the following members of the Final Oral Committ
 \include{introduction/chapter}

 

 \part{Background} \label{prt:background}

-\include{spectroscopy/chapter}

-\include{software/chapter}

+%\include{spectroscopy/chapter}

+%\include{software/chapter}  % sof

 

 \part{Development} \label{prt:development}

-\include{processing/chapter}

-%\include{acquisition/chapter}

-%\include{active_correction/chapter}

+%\include{processing/chapter}  % pro

+%\include{acquisition/chapter}  % acq

+%\include{active_correction/chapter}  % act

 %\include{opa/chapter}

 %\include{mixed_domain/chapter}

 

 \part{Applications} \label{prt:applications}

-%\include{PbSe/chapter}

+\include{PbSe_susceptibility/chapter}

+\include{PbSe_global_analysis/chapter}

 %\include{MX2/chapter}

 %\include{PEDOT_PSS/chapter}

 

@@ -92,6 +93,7 @@ This dissertation is approved by the following members of the Final Oral Committ
 %\include{irf/chapter}

 %\include{quantitative_ta/chapter}

 %\include{tg/chapter}

+%\include{opa_phase/chapter}  % rpa

 %\include{errata/chapter}

 %\include{colophon/chapter}

 \end{appendix}

diff --git a/introduction/chapter.tex b/introduction/chapter.tex
index 42943a5..285bd14 100644
--- a/introduction/chapter.tex
+++ b/introduction/chapter.tex
@@ -28,8 +28,9 @@ dimensionality and selection rules.  %
 With the advent of ultrafast lasers, CMDS can resolve dynamics in excited states and the coupling

 between them. \cite{RentzepisPM1970a}  %

 

-CMDS is most often performed in the time domain, where multiple broadband pulses are scanned in

-time to collect a multidimensional interferogram. \cite{MukamelShaul2009a, GallagherSarahM1998a}  %

+CMDS is most often performed in the time domain, where multiple broadband pulses are scanned to

+collect a multidimensional interferogram. \cite{MukamelShaul2009a, GallagherSarahM1998a}  %

+% BJT: scanned in WHAT? delay? time?

 This technique is fast and robust---it has even been performed on a single shot.

 \cite{HarelElad2010a}  %

 However time-domain CMDS has some fundamental limitations:

diff --git a/opa/chapter.tex b/opa/chapter.tex
index a57847a..2918e94 100644
--- a/opa/chapter.tex
+++ b/opa/chapter.tex
@@ -16,6 +16,8 @@
 

 \clearpage

 

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

+

 In frequency-domain Multi-Resonant Coherent Multidimensional Spectroscopy (MR-CMDS), automated

 Optical Parametric Amplifiers (OPAs) are used to actively scan excitation color axes. %

 To accomplish these experiments, exquisite OPA performance is required. %

@@ -36,22 +38,20 @@ The major challenges in automated OPA tuning are:
   \item Optimization metrics are not necessarily separable along motor dimensions.

 \end{enumerate}

 

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

-

 \section{TOPAS-C}  % ==============================================================================

 

-% TODO: introduction to the internal design of the OPA

+[INTRODUCTION TO THE TOPAS-C]

 

-\section{Preamp}  % ===============================================================================

+% TODO: introduction to the internal design of the OPA

 

 \begin{figure}

-  \includegraphics[width=\textwidth]{opa/preamp}

-  \caption{

-    CAPTION TODO

+  \caption[CAPTION TODO]{

+    CAPTION TODO: TOPAS-C FACTORY POWER CURVES

   }

-  \label{opa:fig:preamp}

 \end{figure}

 

+\section{Preamp}  % ===============================================================================

+

 In TOPAS-C OPAs, a small portion of input light is used to generate a signal seed in a BBO crystal

 ``C1''.  %

 A motorized delay stage ``D1'' is used to temporally overlap a particular color in chirped white

@@ -66,14 +66,6 @@ shown below.  %
 

 % TODO: procedure

 

-\begin{figure}

-  \includegraphics[width=\linewidth]{opa/autotune_preamp}

-  \caption{

-    CAPTION TODO

-  }

-  \label{opa:fig:autotune_preamp}

-\end{figure}

-

 A representative preamp tune procedure output image is shown in \autoref{fig:autotune_preamp}.  %

 The thick black line is the final output curve.  %

 The dark grey lines are the contours of constant color.  %

@@ -82,6 +74,22 @@ The colorbar shows the Delaunay-interpolated intensity values for each motor pos
 Preamp tuning takes less than 20 minutes, in large part due to a NIR array detector which collects

 the full spectrum at each motor position.  %

 

+\begin{figure}

+  \includegraphics[width=\textwidth]{opa/preamp}

+  \caption{

+    CAPTION TODO

+  }

+  \label{opa:fig:preamp}

+\end{figure}

+

+\begin{figure}

+  \includegraphics[width=\linewidth]{opa/autotune_preamp}

+  \caption{

+    CAPTION TODO

+  }

+  \label{opa:fig:autotune_preamp}

+\end{figure}

+

 \section{Poweramp}  % =============================================================================

 

 \begin{figure}

@@ -120,6 +128,14 @@ In total, poweramp tuning typically takes less than 1 hour. %
 Representative procedure output images for D2 (\autoref{op:fig:d2}) and C2 (\autoref{opa:fig:c2})

 are shown.  %

 

+For the D2 figure, the lower panel shows the intensity of the data taken.  %

+Note the thick grey line, which represents the chosen points before the final spline step.  %

+The top panel compares the old tuning curve (thin) with the output tuning curve (thick).  %

+For the C2 image, the bottom panel represents the color of each fit mapped onto detuning.  %

+Each separate marker color represents a different setpoint.  %

+As with D2, the C2 upper panel compares the old tuning curve (thin black) with the output tuning

+curve (colored X's).  %

+

 \begin{figure}

   \includegraphics[width=\textwidth]{opa/d2}

   \caption{

@@ -136,18 +152,19 @@ are shown.  %
   \label{opa:fig:c2}

 \end{figure}

 

-For the D2 figure, the lower panel shows the intensity of the data taken.  %

-Note the thick grey line, which represents the chosen points before the final spline step.  %

-The top panel compares the old tuning curve (thin) with the output tuning curve (thick).  %

-For the C2 image, the bottom panel represents the color of each fit mapped onto detuning.  %

-Each separate marker color represents a different setpoint.  %

-As with D2, the C2 upper panel compares the old tuning curve (thin black) with the output tuning

-curve (colored X's).  %

-

 \section{Mixers}  % ===============================================================================

 

+[DESCRIPTION OF MIXERS]

+

 \section{Generalizability}  % =====================================================================

 

+\begin{figure}

+  \includegraphics[width=\textwidth]{"opa/signal_and_idler_motortune"}

+  \caption[CAPTION TODO]{

+    CAPTION TODO

+  }

+\end{figure}

+

 \section{Future directions}  % ====================================================================

 

 % TODO: discuss Attune
\ No newline at end of file
diff --git a/opa_phase/430 seconds.png b/opa_phase/430 seconds.png
new file mode 100644
index 0000000..e9c5cf5
--- /dev/null
+++ b/opa_phase/430 seconds.png
diff --git a/opa_phase/chapter.tex b/opa_phase/chapter.tex
new file mode 100644
index 0000000..774ab45
--- /dev/null
+++ b/opa_phase/chapter.tex
@@ -0,0 +1,35 @@
+\chapter{Abandon the random phase approximation} \label{cha:rpa}
+
+Recently we've made some measurements that seem to imply phase stability between the fs OPAs.  %
+We've typically assumed that the OPAs have random phase on every shot, making coherent heterodyne
+processes average to zero.  %
+These measurements show that this is a very bad assumption.  %
+
+I've taken the interferogram of OPA1 vs OPA2, Clearly the OPAs remain phase locked for many shots.  %
+In 'over time' I show the spectral phase pattern (D = 500 fs) for 1000 single shot acquisitions
+over 430 seconds in lab time.  %
+The phase does drift, but it is certainly not quickly randomized.  %
+
+I have more data showing:
+How the spectral phase changes over the course of hours.
+How the phase evolves as we scan the OPAs against each-other in color.
+The reproducibility of phase as the OPA motors move away and then return to a given color.
+
+I'll work this data up and send out another email with many more details and thoughts once I have
+time.  %
+This quick note is just to let the group know that we must abandon the 'random phase' assumption
+when thinking about what heterodyne processes can happen as coherent artifacts.  %
+
+\begin{figure}
+  \includegraphics[width=\textwidth]{"opa_phase/cross interference"}
+  \caption[CAPTION TODO]{
+    CAPTION TODO
+  }
+\end{figure}
+
+\begin{figure}
+  \includegraphics[width=\textwidth]{"opa_phase/430 seconds"}
+  \caption[CAPTION TODO]{
+    CAPTION TODO
+  }
+\end{figure}
diff --git a/opa_phase/cross interference.png b/opa_phase/cross interference.png
new file mode 100644
index 0000000..8891214
--- /dev/null
+++ b/opa_phase/cross interference.png
diff --git a/processing/chapter.tex b/processing/chapter.tex
index 1ec2c37..997b297 100644
--- a/processing/chapter.tex
+++ b/processing/chapter.tex
@@ -1084,44 +1084,31 @@ dataset—it’s really useful for inspecting the quality of your fit procedure.
 \section{Construction, maintenance, and distribution}  % ==========================================

 

 While WrightTools has already been useful to the work done in the WrightGroup over the last 3

-years, the true potential of the package is in potential future uses.  %

-WrightTools is designed to be extended and continuously enhanced to serve an ever-wider set of

-users and spectroscopies.  %

-Despite it's name, WrightTools is built to be used even by those outside of the Wright Group.  %

+years, the true potential of the package lies in its future.  %

+WrightTools is designed in a modular way so that it can be continuously enhanced to serve an

+ever-wider set of users and spectroscopies.  %

+Despite its name, WrightTools is built to be used even by those outside of the Wright Group.  %

 Currently WrightTools may be only 75\% of what a typical multidimensional spectroscopist needs, but

 if those scientists work to enhance the package with what \emph{they} need, they may also solve

 problems for others such that the usefulness of the software gradually increases.  %

 

 In order for this dream to come true, WrightTools must be constructed and maintained by

 collaborative tools such that users feel comfortable contributing to future enhancements.  %

-It has not been easy to build this software as a group of 3 to 5 contributors, and the coordination

-problems will be harder as more users and developers join.  %

-To this end, the following section discusses strategies for keeping WrightTools collaborative and

-useful.  %

+All of the challenges to collaboration discussed in \autoref{cha:sof} certainly apply to

+WrightTools, so it is important that we follow best practices now in order to make WrightTools as

+maintainable and future-proof as possible.  %

+To this end, this section discusses strategies that I have employed in the construction,

+maintenance, and distribution of WrightTools.  %

 

 \subsection{Licensing}  % ------------------------------------------------------------------------

 

-In order to even talk about collaborative development, one must at least have an open source

-license.  %

-According to US copyright law, software is fully copyright protected by default.  % TODO: cite

+As discussed in \autoref{cha:sof}, open source licenses are an important part of scientific

+software development.  %

 Those writing software must explicitly license their project in order to ensure that users have

 basic rights to copy edit and distribute code.  %

-A whole set of licenses have been written to use for this purpose.  %

-Briefly, strong copyleft licenses do not allow for software to be modified or enhancements without

-sharing those enhancements under the same licenses.  %

-These ``viral'' licenses are meant to force companies and individuals who would otherwise not open

-source to share their code.  %

-The strongest copyleft licenses do not even allow others to link against the licensed software

-without themselves being copyleft.  %

-This strategy has been moderately successful, but is becoming less popular.  % TODO: cite

-Open source licenses are less opinionated, merely granting users the right to copy, modify,

-distribute, and publish code without restriction except perhaps credit to the source.  %

-

-% TODO: link GPL, LGPL etc

-

-WrightTools is MIT licensed (otherwise called the Expat license).  %

+WrightTools is licensed under the hugely popular Expat/MIT license.  %

 This license is incredibly permissive and puts as few restrictions as possible on the end users.  %

-Because the license is short, it is reproduced below.  %

+The following is the WrightTools license, reproduced in its entirety.  %

 

 \begin{dquote}

 

@@ -1154,30 +1141,139 @@ However many Python libraries end up being interfaces to compiled code that coul
 closed-source.  %

 The Scientific Python Stack have MIT-compatible licenses, including BSD-like licenses.  %

 

-\subsection{Distribution} \label{pro:sec:distribution}  % ------------------------------------------

+\subsection{Version control}  % -------------------------------------------------------------------

+

+As mentioned several times in \autoref{cha:sof}, having software under source control is probably

+the most important recommendation in scientific software development.  %

+Source control allows developers to create ``checkpoints'' for their software package that can be

+returned to again and again.  %

+Developers can collaborate together to edit the software by making incremental changes that are

+easy to review.  %

+

+WrightTools uses git \cite{git} for source control, and the package is hosted on GitHub

+\cite{GitHub, WrightToolsGitHub}.  %

+As of 2018-04-08, WrightTools has 1,018 commits from seven developers, as shown in

+\autoref{pro:tab:commits}.  %

+In addition to simply hosting the git repo, GitHub gives us issue tracking and automated

+integration with zenodo.  %

+

+The WrightTools package has a developer controlled version as well, following the semantic

+versioning convention \cite{SemanticVersioning}.  %

+The current distributed version of WrightTools is \bash{3.0.1}, with \bash{3.0.2} under active

+development.  %

+The wt5 file also has a semantic version, currently \bash{1.0.0}.  %

+These attributes can be accessed through python: \python{wt.__version__} and

+\python{wt.__wt5_version__}.  %

 

-How does WrightTools get onto end-users machines?  %

-Distribution...

+\begin{table}

+  \begin{tabular}{c | c | c | c}

+    person & number of commits & lines added & lines removed \\ \hline

+    Blaise Thompson & 478 & 621,918 & 507,938 \\ \hline

+    Kyle Sunden & 208 & 19,293 & 9,218 \\ \hline

+    Darien Morrow & 29 & 1,589 & 127 \\ \hline

+    Nathan Neff-Mallon & 20 & 2,880 & 910 \\ \hline

+    Kyle Czech & 5 & 1,150 & 25 \\ \hline

+    Daniel Kohler & 3 & 113 & 29 \\ \hline

+    Rachel Swedin & 1 & 5,197 & 0 \\ \hline

+  \end{tabular}

+  \caption[CAPTION TODO]{

+    Commits to WrightTools by individual, ordered by number of commits.

+    Wright Stuff is...

+    Note that datasets...

+  }

+  \label{pro:tab:commits}

+\end{table}

 

-WrightTools is distributed on PyPI and conda-forge.  %

+\subsection{Unit tests}  % ------------------------------------------------------------------------

 

-WrightTools uses semantic versioning.  %

+Maintainable code must be tested, so that future developers can use tests to ensure that they do

+not break necessary functionality.  %

+Unit testing is a very simple testing paradigm in which small, separate tests are written to

+address each ``unit'' of the software package.  %

+As an example, the following is one of WrightTools' tests:

+\begin{codefragment}{python}

+# part of WrightTools/tests/data/convert_data.py

+def test_wigner():

+    p = datasets.COLORS.v2p2_WL_wigner

+    data = wt.data.from_COLORS(p)

+    assert data.d1.units == 'fs'

+    data.convert('ns')

+    assert data.d1.units == 'ns'

+    assert data.wm.units == 'nm'

+    data.close()

+\end{codefragment}

+This test loads one of the distributed COLORS datasets and makes sure that the \python{convert}

+method works as intended.  %

+To do this, it uses the \python{assert} statement which raises an exception when a condition is

+\python{False}.  %

+This particular test is pretty humble, but there is strength in numbers: as of 2018-04-08 there are

+224 unit tests within WrightTools.  %

+Using the built in \python{pytest} library (\bash{python setup.py test}), a programmer can run all

+of the tests and receive a report on what failed and why.  %

+If a future programmer unintentionally breaks \python{convert}, the above test will fail and alert

+her to the unexpected side effect of her modification.  %

+

+[PARAGRAPH ABOUT CONTINUOUS INTEGRATION]

 

-% TODO: citations to WrightTools on PyPI and conda-forge

+\subsection{Distribution} \label{pro:sec:distribution}  % ------------------------------------------

+

+WrightTools is on GitHub, which is a fantastic way for developers to get software onto their

+computers.  %

+But how does software get onto everybody elses machine?  %

+Developers call this process ``distribution''.  %

+Luckily for us, distribution is fairly simple within the Python ecosystem.  %

+The same tools that are used to distribute hugely popular packages like numpy are also available

+for anyone else, including WrightTools.  %

+

+The Python Package Index (PyPI), affectionately known as the cheese shop, is the official third

+party software repository for Python.  %

+It is community maintained, and supported by the Python Software Foundation and The Python

+Packaging Group.  %

+As of 2018-04-08 PyPI hosts 134,758 Python packages, all for free.  %

+WrightTools is also hosted on PyPI.  %

+Every time we change our version, we ``release'' by uploading the newest version to PyPI.  %

+pip (``pip installs packages'', ``pip installs python'', or ``preferred installer program'') [CITE]

+can be used to install packages directly from PyPI:  %

+\begin{codefragment}{bash}

+pip install WrightTools

+\end{codefragment} 

+

+Conda is a multilingual package manager that handles virtual environments and dependencies, even

+binary dependencies, in a hassle-free way.  %

+Since the scientific Python ecosystem has so many non-Python binary dependencies, Conda is a

+popular choice---especially on Windows where the necessary compilers are not typically

+pre-installed.  %

+Unlike pip, conda is not tied to a single repository.  %

+There is the official repository, maintained by Anaconda, the company that develops conda.

+[CITE]  %

+There is also the popular conda-forge repository, which is maintained by the community via

+GitHub. [CITE]  %

+WrightTools is on conda-forge:  %

+\begin{codefragment}{bash}

+conda config --add channels conda-forge

+conda install WrightTools

+\end{codefragment}

 

-\subsection{Collaborative development}  % ---------------------------------------------------------

+\subsection{Documentation}  % ---------------------------------------------------------------------

 

-Git...

+WrightTools is a piece of scripted software, and many spectroscopists many not be comfortable with

+using such a thing immediately.  %

+To this end, it is important to have easy to use, searchable documentation with end-users in

+mind.  %

 

-As of 2018-04-08, WrightTools has ........ commits from ..... developers

+wright.tools

 

-\subsection{Unit tests}  % ------------------------------------------------------------------------

+Built and hosted by readthedocs

 

-Unit testing...

+Both master and development versions...

 

 \clearpage

 \section{Future directions}  % ====================================================================

 

 Single variable decomposition.  %

 

-Usage in next-generation simulation packages.  %
\ No newline at end of file
+Usage in next-generation simulation packages.  %

+

+More tests.  %

+

+Usage by multiple groups.  %
\ No newline at end of file
diff --git a/software/chapter.tex b/software/chapter.tex
index 14693ee..e6707dd 100644
--- a/software/chapter.tex
+++ b/software/chapter.tex
@@ -1,3 +1,14 @@
+%A whole set of licenses have been written to use for this purpose.  %

+%Briefly, strong copyleft licenses do not allow for software to be modified or enhancements without

+%sharing those enhancements under the same licenses.  %

+%These ``viral'' licenses are meant to force companies and individuals who would otherwise not open

+%source to share their code.  %

+%The strongest copyleft licenses do not even allow others to link against the licensed software

+%without themselves being copyleft.  %

+%This strategy has been moderately successful, but is becoming less popular.  % TODO: cite

+%Open source licenses are less opinionated, merely granting users the right to copy, modify,

+%distribute, and publish code without restriction except perhaps credit to the source.  %

+

 \chapter{Software} \label{cha:sof}

 

 \begin{dquote}