From ca98a888e2e9bee0db59b7f6ebf21aa48d5892a8 Mon Sep 17 00:00:00 2001
From: Blaise Thompson <blaise@untzag.com>
Date: Sat, 14 Apr 2018 17:09:39 -0500
Subject: 2018-04-14 17:09

---
 PbSe_susceptibility/chapter.tex | 285 +++++++++++++++++++++++-----------------
 1 file changed, 162 insertions(+), 123 deletions(-)

(limited to 'PbSe_susceptibility/chapter.tex')

diff --git a/PbSe_susceptibility/chapter.tex b/PbSe_susceptibility/chapter.tex
index 0fff735..8bacebf 100644
--- a/PbSe_susceptibility/chapter.tex
+++ b/PbSe_susceptibility/chapter.tex
@@ -100,25 +100,22 @@ We then connect the well-known theory of optical bleaching of the 1S band to our
 In the Maker-Terhune convention, the relevant third-order polarization, $P^{(3)}$, is related to
 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}),
-	\end{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{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.  %
-$D = 6$ for transient absorption and transient grating, and $D = 3$ for $z$-scan}  %
+$D = 6$ 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{pss:eq:Maker_Terhune} is the steady-state solution to the Liouville Equation, valid when
+\autoref{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.  %
-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.  
@@ -133,7 +130,7 @@ to \cite{CarlsonRogerJohn1989a}  %
 \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). 
+Phase mismatch is negligible in these experiments.  %
 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}
@@ -141,14 +138,14 @@ For purely absorptive effects, $M$ may be written as \cite{CarlsonRogerJohn1989a
 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{pss:eq:fwm_intensity} shows that we can derive spectra free of pulse propagation effects
+\autoref{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}.  %
 
 For cuvettes, the sample solution is sandwiched between two transparent windows.  
-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.
+Rather than \autoref{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{pss:eq:fwm_intensity2}
@@ -169,37 +166,45 @@ 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{pss:eq:fwm_intensity2} can be expressed as the classic signal-local oscillator interference, 
+\autoref{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 $E_\text{LO} = \alpha_2 \ell_\text{w} \frac{1 + e^{-\alpha_2 \ell_s}}{1 - e^{-\alpha_2 \ell_s}} \chi_\text{w}^{(3)} + \chi_\text{sol}^{(3)}$.
-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, $n_2$.  
+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 $\pm \pi /
+2$.  %  
+
+\subsection{Optical bleaching and dependencies on experimental conditions}  % ---------------------
+
+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{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, $\langle n \rangle$.  
-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 $\omega$.
+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.  %
+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{pss:eq:Delta_alpha1}
 \begin{split}
@@ -210,30 +215,58 @@ 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{pss:eq:hyperpolarizability}, \ref{pss:eq:beta_to_chi}, and
+By combining Equations \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{pss:eq:gamma_to_phi} will be useful for benchmarking our results because it connects our
+\autoref{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}  % =========================================================================
 
-\subsubsection{Sample preparation and Characterization}
+\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}
+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 $10^{16} - 10^{17} \ \text{cm}^{-3}$). 
 
+\autoref{pss:fig:absorbance}, top, shows the absorption spectra gathered for each of the aliquots
+used in this study.  %
+All spectra were baseline-subtracted to account for reflection of the cuvette.  %
+Differences in spectral properties between aliquots can be observed by normalizing each spectra to
+the 1S band (\autoref{pss:fig:absorbance}, middle).  %
+No significant differences were observed; the small differences between aliquots near $\hbar\omega
+\approx 0.88$ eV is attributed to a small absorption feature of the fused silica cuvette, typically
+attributed to OH stretches).  %
+For computation of absorptive losses, the effective absorptivity of the pules were computed.  
+The effective absorptivity spectrum is achieved by convolving the absorption spectra with the pulse
+bandwidth.  %
+\autoref{pss:fig:absorbance}, bottom, shows the differences between the effective absorptivity and
+the absorptivity of the darkest aliquot used.  %
+
+\begin{figure}
+	\includegraphics[width=0.5\linewidth]{"PbSe_susceptibility/absorbance"}
+	\caption[Absorption spectra of QD aliquots used in this study.]{
+		The absorption spectra of the QD aliquots used in this study.  
+		Top:  the raw absorbance spectra of each aliquot used.
+		Middle:  the absorbance spectra normalized by the 1S peak.  
+		Bottom:  Sample absorptivity (solid) and the effective absorption (assuming 50 meV FWHM pulse
+    bandwidth).
+	}
+	\label{pss:fig:absorbance}
+\end{figure}
+
 \subsubsection{Four-wave Mixing}
 
 An ultrafast oscillator (Tsunami, Spectra-Physics) produced a 80 MHz train of 35 fs pulses, which
@@ -264,7 +297,7 @@ 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}  % ------------------------------------------------------------------
+\subsection{Solvent and window 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.  %
@@ -277,17 +310,17 @@ 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{pss:fig:ccl4} summarizes the nonlinear measurements performed on neat CCl$_4$.
+\autoref{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{pss:fig:ccl4}a), the electronic response creates a
+When all pulses are overlapped (\autoref{pss:fig:ccl4}a), the electronic response creates a
 featureless 2D spectrum.  %
-The horizontal and vertical structure observed in Fig. \ref{pss:fig:ccl4}a is believed to reflect the
+The horizontal and vertical structure observed in \autoref{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{pss:fig:ccl4}b), the contributions from the Raman resonances can be resolved.  %
+If pulses $E_1$ and $E_2$ are kept overlapped and the $E_{2^\prime}$ is delayed
+(\autoref{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 $\nu1$
@@ -298,21 +331,25 @@ 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{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.  
+\autoref{pss:fig:ccl4}c shows the signal dependence on $\tau_{22^\prime}$ with pulse frequencies
+resonant with the large $\nu 1$ resonance.  %
 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{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{pss:eq:fwm_intensity2} because the dispersion of the interference term is determined completely by the real component of quantum dot response:  $\text{Re} \left[ E_\text{LO} \gamma_\text{QD}^* \right]  = E_\text{LO} \text{Re} \left[ \gamma_\text{QD} \right]$.  
+At most colors, the ratio will be much less (confer \autoref{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 \autoref{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"}
-	\caption{
+	\includegraphics[width=0.5\linewidth]{"PbSe_susceptibility/ccl4_raman"}
+	\caption[CMDS amplitude of neat carbon tetrachloride.]{
 		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 ($\tau_{21}=\tau_{22^\prime} = 0$).  
@@ -330,35 +367,34 @@ This simplifies Eqn. \ref{pss:eq:fwm_intensity2} because the dispersion of the i
 
 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
+\autoref{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
+entirely to QDs ($\chi_\text{w},\chi_\text{solvent}=0$ in \autoref{pss:eq:fwm_intensity2}).  %
+Power-normalized output amplitudes (\autoref{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
+Density-normalized ($N_\text{QD}$) output amplitudes (\autoref{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
+After normalizing by $M$ (\autoref{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].  %
+\autoref{cha:psg}.  %
 
 \begin{figure}
-	\includegraphics[width=\linewidth]{"PbSe_susceptibility/mfactors_check"}
-	\caption{
+	\includegraphics[width=0.5\linewidth]{"PbSe_susceptibility/mfactors_check"}
+	\caption[Absorption effects in QD dilution study.]{
 		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).
@@ -374,19 +410,19 @@ The nature of the corrected line shape, including the tail to lower energies, wi
 \subsection{Quantum dot response}  % --------------------------------------------------------------
 
 We now consider the behavior at pulse overlap, where solvent and window contributions are important.  
-Figure \ref{pss:fig:dilution_integral}a shows the (absorption-corrected) spectra for all samples at
+\autoref{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).  %
+independent of concentration (\autoref{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.  %
 
-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}.  %
+The concentration dependence in \autoref{pss:fig:dilution_integral} can be understood with our
+knowledge of the solvent/window character and \autoref{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.  %
@@ -398,29 +434,26 @@ fitting.  %
 
 \subsubsection{Spectral integration}
 
-If we integrate Eqn. \ref{pss:eq:fwm_intensity2}, the integral of the solvent-QD interference term
+If we integrate \autoref{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{pss:eq:fit_integral}
-\begin{split}
-	\int_a^{a+\Delta} 
+  \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}
+	=  A \Delta \left( 1 + 
+	\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{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{pss:fig:dilution_integral}b shows the integral values for all five concentrations
+\autoref{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{pss:eq:fit_integral} (black dashed line).
+Our data fit well to \autoref{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
@@ -447,8 +480,8 @@ This gives a peak hyperpolarizability of $|\gamma_\text{QD, peak}| = 1.2 \times
 \gamma_\text{sol}$.  %
 
 \begin{figure}
-	\includegraphics[width=\linewidth]{"PbSe_susceptibility/dilution_integral"}
-	\caption{
+	\includegraphics[width=0.5\linewidth]{"PbSe_susceptibility/dilution_integral"}
+	\caption[FWM at pulse overlap, and integral thereof.]{
 		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. 
@@ -456,12 +489,11 @@ This gives a peak hyperpolarizability of $|\gamma_\text{QD, peak}| = 1.2 \times
 			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{pss:fig:dilution_integral}
 \end{figure}
 
-\subsubsection{Global line shape fitting}  % ------------------------------------------------------
+\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.  %
@@ -469,16 +501,16 @@ 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{pss:eq:fwm_intensity2} with the QD
+To account for this feature, we perform a global fit of \autoref{pss:eq:fwm_intensity2} with the QD
 line shape definition  %
 \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{pss:fig:dilution2}.
-The data is normalized by $N_\text{QD}^2$ (as in Fig. \ref{pss:fig:mfactors}c) so that least-squares
+The results of the fit are overlaid with our data in \autoref{pss:fig:dilution2}.
+The data is normalized by $N_\text{QD}^2$ (as in \autoref{pss:fig:mfactors}c) so that least-squares
 fitting weighs all samples on similar scales.  %
-The fit parameters are listed in Table \ref{pss:tab:lineshape_fit}.  
+The fit parameters are listed in \autoref{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
@@ -489,12 +521,24 @@ 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.  %
 
+\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{HellwarthRW1971a, LevensonMD1974a,
+  Levine1975, Cherlow1976, HoPP1979a, Thalhammer1983, Etchepare1985, Nibbering1995, Rau2004} but
+unfortunately, the variation between values is quite large ($\pm 50\% $) for quantitative
+analysis.  % 
+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 of $3 \cdot 10^{-31}  \text{cm}^6 / \text{erg}$.  
+
 \begin{figure}
-	\includegraphics[width=\linewidth]{"PbSe_susceptibility/dilution_fits"}
-	\caption{
+	\includegraphics[width=0.5\linewidth]{"PbSe_susceptibility/dilution_fits"}
+	\caption[CMDS signal with different concentrations of PbSe.]{
 		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
+		Calculated $\gamma^{(3)}$ spectra (\autoref{pss:eq:fwm_intensity2}) for the different QD
     concentrations.  
 		The thick, lighter lines are the result of a global fit.  
 	}
@@ -502,12 +546,7 @@ absorption.  %
 \end{figure}
 
 \begin{table}
-	\centering
-	\caption{Parameters and extracted values from the global line shape fit using Eqns.
-    \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{pss:tab:lineshape_fit}
-	\begin{tabular}{l|c}
+  \begin{tabular}{l|c}
 		variable & value \\
 		\hline
 		$N_\text{sol} \ (\text{cm}^{-3})$ & $6.23 \times 10^{21}$ \\
@@ -518,34 +557,34 @@ absorption.  %
 		$\mathbf{\gamma_\text{QD,peak} / \gamma_\text{sol}}$ & $-7.7 \times 10^5$  \\
 		$\mathbf{B / \gamma_\text{QD}}$ & $0.10 - 0.13i$ \\
 	\end{tabular}
+  \caption[Parameters and extracted values from the gloabl line shape fit.]{
+    Parameters and extracted values from the global line shape fit using Equations
+    \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{pss:tab:lineshape_fit}
 \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. 
 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{pss:eq:gamma_to_phi} shows that the metrics of non-linear response, $\phi$ and
+excitation sources are often intertwined with the non-linear response.
+\cite{KohlerDanielDavid2017a}  %  
+\autoref{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{pss:eq:Maker_Terhune}), signal scales with pulse intensity and not fluence.  %
+Since $\gamma^{(3)}$ assumes the driven limit for field-matter interactions (see
+\autoref{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{pss:eq:beta_to_chi}).
+Conversely, $\gamma^{(3)}$ factors out the degeneracy of the experiment, but non-linear
+absorptivity does not (\autoref{pss:eq:beta_to_chi}).  %
 Since $\phi$ is defined by the non-linear absorptivity, it is also proportional to $D$.  
 
-Equation \ref{pss:eq:Delta_alpha1} is valid for a temporally separated pump and probe, so that the
+\autoref{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
@@ -556,14 +595,14 @@ separated pulses.  %
 This correction factor is small compared to our uncertainty, so we neglect it.  
 It may be important in more precise measurements.
 
-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.  %
+The most direct comparison of our measurements with literature is \textcite{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}. 
+The values are shown in \autoref{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).  %
@@ -576,8 +615,8 @@ This broadband feature may be different from that observed in transient absorpti
 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.  %
+\autoref{pss:tab:litcompare} compares various non-linear quantities for this work,
+\textcite{YursLenaA2012a}, and a PbS experiment \cite{OmariAbdoulghafar2012a}.  %
 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.  %
@@ -589,8 +628,8 @@ 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}).  %
+\textcite{OmariAbdoulghafar2012a} performed $z$-scan measurements of PbS QDs to quantify the
+non-linear parameters (see right-hand column of \autoref{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
@@ -604,16 +643,16 @@ 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} 
+after Auger recombination yields single-exciton species. \cite{NootzGero2011a, 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}.  %
+We can check our measured susceptibility with the accepted $\phi$ value using
+\autoref{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
+\cdot 10^{-31} \text{cm}^6 \ \text{erg}^{-1}$, while \autoref{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.  %
@@ -642,7 +681,7 @@ Our method gives agreement with the $\phi=0.25$ bleach factor.  %
 		& 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 \\
-		$\phi$ (Eqn. \ref{pss:eq:gamma_to_phi}) & 0.3 & 0.6 & 0.15 \\
+		$\phi$ (\autoref{pss:eq:gamma_to_phi}) & 0.3 & 0.6 & 0.15 \\
 	\end{tabular}
   \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
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