From 3bc8b4451803929d6a47f87a987947c37d95545d Mon Sep 17 00:00:00 2001
From: Blaise Thompson <blaise@untzag.com>
Date: Thu, 12 Apr 2018 21:00:37 -0500
Subject: 2018-04-12 21:00

---
 PbSe_susceptibility/chapter.tex | 378 ++++++++++++++++++++++------------------
 1 file changed, 213 insertions(+), 165 deletions(-)

(limited to 'PbSe_susceptibility/chapter.tex')

diff --git a/PbSe_susceptibility/chapter.tex b/PbSe_susceptibility/chapter.tex
index 873804f..0fff735 100644
--- a/PbSe_susceptibility/chapter.tex
+++ b/PbSe_susceptibility/chapter.tex
@@ -1,9 +1,13 @@
 \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
+\textit{This Chapter borrows extensively from a work-in-progress publication. The authors are:
+  \begin{denumerate}
+    \item Daniel D. Kohler
+    \item Blaise J. Thompson
+    \item John C. Wright
+  \end{denumerate}
+}
 
 Here we detail the extraction of quantitative information from ultrafast multiresonant CMDS
 spectra.  %
@@ -59,8 +63,8 @@ It is uncommon for CMDS spectra to obtain absolute units of susceptibility in th
 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.}
+limited in the multidimensional space they can explore.  %
+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}  % 
@@ -94,8 +98,8 @@ We then connect the well-known theory of optical bleaching of the 1S band to our
 \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}
+the non-linear susceptibility, $\chi^{(3)}$, by \textcite{MakerPD1965a}  %
+\begin{equation} \label{pss: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}),
@@ -104,14 +108,13 @@ the non-linear susceptibility, $\chi^{(3)}$, by\cite{MakerPD1965a}  %
 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}  %
+the interference of $n$ distinguishable excitation fields.  %
+$D = 6$ 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
+Equation \ref{pss: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.  %
@@ -120,8 +123,8 @@ This complications arising from impulsive aspects of our experiment are addresse
 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}
+to \cite{CarlsonRogerJohn1989a}  %
+\begin{equation} \label{pss: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 \\
@@ -131,24 +134,24 @@ to\cite{CarlsonRogerJohn1989a}  %
 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}
+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
+Equation \ref{pss: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}  %
+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.
+Rather than Eqn. \ref{pss: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}
+\begin{equation} \label{pss: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)} 
@@ -156,7 +159,7 @@ The (absorption-corrected) output intensity is proportional to:
 \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}
+\begin{equation} \label{pss:eq:hyperpolarizability}
 	\chi_i^{(3)} = 
 		f(\omega_1)^2 f(\omega_2)^2 N_i \gamma_i^{(3)},
 \end{equation}
@@ -166,8 +169,8 @@ Both $n$ and $f$ are frequency dependent, but both vary small amounts ($\sim 0.1
 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}
+Equation \ref{pss:eq:fwm_intensity2} can be expressed as the classic signal-local oscillator interference, 
+\begin{equation} \label{pss: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 $E_\text{LO} = \alpha_2 \ell_\text{w} \frac{1 + e^{-\alpha_2 \ell_s}}{1 - e^{-\alpha_2 \ell_s}} \chi_\text{w}^{(3)} + \chi_\text{sol}^{(3)}$.
@@ -185,19 +188,20 @@ The local oscillator and signal fields are non-additive unless the phase differe
 %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}
+\begin{gather} \label{pss:eq:beta_to_chi}
+	\beta = \frac{32 \pi^2 D \omega}{n_0^2 c^2} \text{Im}\left[ \chi^{(3)} \right]
+  \\	\label{pss: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, $\langle n \rangle$.  
+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}
+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{equation} \label{pss: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 
@@ -206,13 +210,14 @@ We can then write the non-linear change in absorptivity as
 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}
+By combining Eqns. \ref{pss:eq:hyperpolarizability}, \ref{pss:eq:beta_to_chi}, and
+\ref{pss:eq:Delta_alpha1} we can relate the bleach factor directly to the hyperpolarizability:
+\begin{equation} \label{pss: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
+Equation \ref{pss: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}  % =========================================================================
@@ -226,7 +231,7 @@ Aliquots were stored in 1 mm path length fused silica cuvettes with 1.25 mm thic
 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}
+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}
@@ -272,17 +277,17 @@ The nuclear response depends on the vibrational dephasing times (ps and longer).
 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$_4$.
+Fig. \ref{pss: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
+When all pulses are overlapped (Fig. \ref{pss: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
+The horizontal and vertical structure observed in Fig. \ref{pss: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.  %
+\ref{pss: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 $\nu1$
@@ -293,7 +298,7 @@ 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.
+Figure \ref{pss: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.  %
@@ -301,9 +306,9 @@ The oscillations in the exponential decay are quantum beating between Raman mode
 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).
+At most colors, the ratio will be much less (confer Fig. \ref{pss: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:  $\text{Re} \left[ E_\text{LO} \gamma_\text{QD}^* \right]  = E_\text{LO} \text{Re} \left[ \gamma_\text{QD} \right]$.  
+This simplifies Eqn. \ref{pss: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"}
@@ -318,63 +323,70 @@ This simplifies Eqn. \ref{eq:fwm_intensity2} because the dispersion of the inter
 		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}
+	\label{pss:fig:ccl4}
 \end{figure}
 
 \subsection{Concentration-dependent corrections}  % ------------------------------------------------
 
+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{pss: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{pss:eq:fwm_intensity2}).  %
+Power-normalized output amplitudes (Fig. \ref{pss:fig:mfactors}a) are positively correlated with QD
+concentration.  %
+Density-normalized ($N_\text{QD}$) output amplitudes (Fig. \ref{pss: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{pss: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
+[CHAPTER].  %
+
 \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.
+		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 $10^{16} \ \text{cm}^{-3}$).  
+		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 ($M$).  
+		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}
+	\label{pss: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}
+\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). 
+Figure \ref{pss: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{pss: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}
+\cite{KohlerDanielDavid2017a, BritoCruzCH1988a, JoffreM1988a}  %
+These line shapes are not easily related to material properties, such as inhomogeneous broadening
+and pure dephasing.  %
 
-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}.  %
+The concentration dependence in Fig. \ref{pss:fig:dilution_integral} can be understood with our
+knowledge of the solvent/window character and Eqn. \ref{pss: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.  %
@@ -386,10 +398,10 @@ fitting.  %
 
 \subsubsection{Spectral integration}
 
-If we integrate Eqn. \ref{eq:fwm_intensity2}, the integral of the solvent-QD interference term
+If we integrate Eqn. \ref{pss: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{equation} \label{pss:eq:fit_integral}
 \begin{split}
 	\int_a^{a+\Delta} 
 	\frac{I}{I_1 I_2 I_{2^\prime} M^2} \ d\omega_1	
@@ -404,11 +416,11 @@ where $A$ is a proportionality factor and $f(N_\text{QD}) = \sigma_2 N_\text{QD}
 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
+Figure \ref{pss: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).
+Our data fit well to Eqn. \ref{pss: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
@@ -435,14 +447,18 @@ This gives a peak hyperpolarizability of $|\gamma_\text{QD, peak}| = 1.2 \times
 \gamma_\text{sol}$.  %
 
 \begin{figure}
-	\includegraphics[width=\linewidth]{"PbSe_susceptibility/dilution_fits"}
+	\includegraphics[width=\linewidth]{"PbSe_susceptibility/dilution_integral"}
 	\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.  
+		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:dilution2}
+	\label{pss:fig:dilution_integral}
 \end{figure}
 
 \subsubsection{Global line shape fitting}  % ------------------------------------------------------
@@ -453,16 +469,16 @@ 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
+To account for this feature, we perform a global fit of Eqn. \ref{pss:eq:fwm_intensity2} with the QD
 line shape definition  %
-\begin{equation}\label{eq:fit_lineshape}
+\begin{equation} \label{pss: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
+The results of the fit are overlaid with our data in Fig. \ref{pss:fig:dilution2}.
+The data is normalized by $N_\text{QD}^2$ (as in Fig. \ref{pss: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}.  
+The fit parameters are listed in Table \ref{pss: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
@@ -473,12 +489,24 @@ 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{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{pss:eq:fwm_intensity2}) for the different QD
+    concentrations.  
+		The thick, lighter lines are the result of a global fit.  
+	}
+	\label{pss:fig:dilution2}
+\end{figure}
+
 \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}.
+    \ref{pss:eq:fwm_intensity2} and \ref{pss:eq:fit_lineshape}.
     Bold items were extracted by least squares minimization.  All other values were fixed parameters.  }
-	\label{tab:lineshape_fit}
+	\label{pss:tab:lineshape_fit}
 	\begin{tabular}{l|c}
 		variable & value \\
 		\hline
@@ -503,104 +531,124 @@ This yields a peak QD hyperpolarizability to be $3 \cdot 10^{-31}  \text{cm}^6 /
 \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 $\Delta_t$. 
+Equation \ref{pss: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, $\Delta_t$. 
+Since $\gamma^{(3)}$ assumes the driven limit for field-matter interactions (see Eqn.
+\ref{pss: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}).
+Conversely, $\gamma^{(3)}$ factors out the degeneracy of the experiment, but non-linear absorptivity does not (Eqn. \ref{pss: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.  
+Equation \ref{pss: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.
+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.
+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).  %
+The values are shown in Table \ref{pss: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{pss: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.
+Note that $\gamma_\text{QD} / \Delta_t$ is similar between Yurs and our measurement, as expected
+from the pulse duration dependence.  %
+
+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{pss:tab:litcompare}).  %
+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 ($\phi = 0.15$).  %
+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 $\phi = 0.3$).  %
 
-\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}
+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.  %
+
+We can check our measured susceptibility with the accepted $\phi$ value using Equation
+\ref{pss: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{pss:eq:gamma_to_phi} predicts
+$\text{Im}\left[ \gamma^{(3)} \right] = -1.6 \cdot 10^{-31} \text{cm}^6 \text{erg}^{-1}$ for $\phi
+= 0.25$.
+Our method gives agreement with the $\phi=0.25$ bleach factor.  %
+
+\begin{table}
 	\begin{tabular}{r|ccc}
-		&  this work &  Yurs et. al. & ratio \\
-		\hline
+		&  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}
+  \caption{Non-linear parameters relative to CCl$_4$ hyperpolarizability.  $\gamma_{\nu 1}$:
+    hyperpolarizability of the $\nu_1$ Raman transition.}
+	\label{pss:tab:gamma_ratio}
 \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}
+\begin{table}
+  \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]$ 				
+		$ \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]$ 	
+		$ \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 \\
+		$\phi$ (Eqn. \ref{pss: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.  %
+  \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{pss:tab:litcompare}
+\end{table}
 
 \section{Conclusion}  % ===========================================================================
 
-- 
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