2018-04-14 22:36

1 files changed, 182 insertions, 153 deletions

diff --git a/MX2/chapter.tex b/MX2/chapter.tex
index 6e5bdb1..98c37cc 100644
--- a/MX2/chapter.tex
+++ b/MX2/chapter.tex
@@ -91,26 +91,11 @@ frequencies $\omega_1$ and $\omega_2$.  %
 The $\omega_2$ beam is split into two beams, denoted by $\omega_2$ and $\omega_2^\prime$.  %
 These three beams are focused onto the MoS\textsubscript{2} thin film at angles, creating an output
 beam in the phase-matched direction
-$\mathbf{k}_{\mathrm{out}}=\mathbf{k}_1-\mathbf{k}_2+\mathbf{k}_{2^\prime}$ where $\mathbf{k}$ is the wave
-vector for each beam and the subscripts label the excitation frequencies.  %
+$\mathbf{k}_{\mathrm{out}}=\mathbf{k}_1-\mathbf{k}_2+\mathbf{k}_{2^\prime}$ where $\mathbf{k}$ is
+the wave vector for each beam and the subscripts label the excitation frequencies.  %
 Multidimensional spectra result from measuring the output intensity dependence on frequency and
 delay times.  %
 
-\begin{figure}
-  \includegraphics[width=0.5\textwidth]{MX2/01}
-  \caption[CMDS tutorial]{
-    (a) Example delays of the $\omega_1$, $\omega_2$, and $\omega_{2^\prime}$ excitation pulses.
-    (b) Dependence of the output intensity on the $\tau_{22^\prime}$ and $\tau_{21}$ time delays
-    for $\omega_1=\omega_2$.
-    The solid lines define the regions for the six different time orderings of the $\omega_1$,
-    $\omega_2$, and $\omega_{2^\prime}$ excitation pulses.
-    We have developed a convention for numbering these time orderings, as shown.
-    (c) Diagram of the band structure of MoS\textsubscript{2} at the $K$ point.
-    The A and B exciton transitions are shown. (d) Two dimensional frequency-frequency plot
-    labeling two diagonal and cross-peak features for the A and B excitons.}
-  \label{fig:Czech01}
-\end{figure}
-
 \autoref{fig:Czech01} introduces our conventions for representing multidimensional spectra.  %
 \autoref{fig:Czech01}b,d are simulated data.  %
 \autoref{fig:Czech01}a shows one of the six time orderings of the three excitation pulses where
@@ -155,14 +140,23 @@ The intensity of the cross-peaks depends on the importance of state filling and
 relaxation of hot A excitons as well as the presence of interband population trnasfer of the A and
 B exciton states.  %
 
-\section{Methods}  % ==============================================================================
-
 \begin{figure}
-  \includegraphics[width=\textwidth]{MX2/S1}
-  \caption{Schemiatic of the synthetic setup used for Mo thin film sulfidation reactions.}
-  \label{fig:CzechS1}
+  \includegraphics[width=0.5\textwidth]{MX2/01}
+  \caption[Introduction to CMDS.]{
+    (a) Example delays of the $\omega_1$, $\omega_2$, and $\omega_{2^\prime}$ excitation pulses.
+    (b) Dependence of the output intensity on the $\tau_{22^\prime}$ and $\tau_{21}$ time delays
+    for $\omega_1=\omega_2$.
+    The solid lines define the regions for the six different time orderings of the $\omega_1$,
+    $\omega_2$, and $\omega_{2^\prime}$ excitation pulses.
+    We have developed a convention for numbering these time orderings, as shown.
+    (c) Diagram of the band structure of MoS\textsubscript{2} at the $K$ point.
+    The A and B exciton transitions are shown. (d) Two dimensional frequency-frequency plot
+    labeling two diagonal and cross-peak features for the A and B excitons.}
+  \label{fig:Czech01}
 \end{figure}
 
+\section{Methods}  % ==============================================================================
+
 MoS\textsubscript{2} thin films were prepared \textit{via} a Mo film sulfidation reaction, similar
 to methods reported by \textcite{LaskarMasihhurR2013a}.  %
 A 1 nm amount of Mo (Kurt J. Lesker, 99.95\%) metal was electron-beam evaporated onto a fused
@@ -194,15 +188,11 @@ TEM experiments were performed on a FEI Titan aberration corrected (S)TEM under
 voltage.  %
 
 \begin{figure}
-  \includegraphics[width=\textwidth]{MX2/10}
-  \caption[Mask and epi vs transmissive.]{
-    (a) Mask.
-    (b) 2D delay spectra at the BB diagonal ($\omega_1=\omega_2\approx1.95$ eV) for transmissive
-    and reflective geometries.
-    Transmissive signal is a mixture of MoS\textsubscript{2} signal and a large amount of driven
-    signal from the substrate that only appears in the pulse overlap region. Reflective signal is
-    representative of the pure MoS\textsubscript{2} response.}
-  \label{fig:Czech10}
+  \includegraphics[width=\textwidth]{MX2/S1}
+  \caption[Schematic of sulfidation reaction chamber.]{
+    Schemiatic of the synthetic setup used for Mo thin film sulfidation reactions.
+  }
+  \label{fig:CzechS1}
 \end{figure}
 
 The coherent multidimensional spectroscopy system used a 35 fs seed pulse, centered at 800 nm and
@@ -216,12 +206,6 @@ Signal and idler were not filtered out, but played no role due to their low phot
 Pulse $\omega_2$ was split into pulses labeled $\omega_2$ and $\omega_{2^\prime}$ to create a total
 of three excitation pulses.  %
 
-\begin{figure}
-  \includegraphics[width=\textwidth]{MX2/S4}
-  \caption{OPA outputs at each color explored.}
-  \label{fig:CzechS4}
-\end{figure}
-
 In this experiment we use motorized OPAs which allow us to set the output color in software.  %
 OPA1 and OPA2 were used to create the $\omega_1$ and $\omega_2$ frequencies, respectively.  %
 In \autoref{fig:CzechS4} we compare the spectral envelope generated by the OPA at each set
@@ -237,12 +221,6 @@ focused onto the sample surface by a 1 meter focal length spherical mirror in a
 geometry to form a 630, 580, and 580 $\mu$m FWHM spot sizes for $\omega_1$, $\omega_2$, and
 $\omega_{2^\prime}$, respectively.  %
 
-\begin{figure}
-  \includegraphics[width=\textwidth]{MX2/S5}
-  \caption{Spectral delay correction.}
-  \label{fig:CzechS5}
-\end{figure}
-
 \autoref{fig:CzechS5} represents delay corrections applied for each OPA.  %
 The corrections were experimentally determined using driven FWM output from fused silica.  %
 Corrections were approximately linear against photon energy, in agreement with the normal dispersion
@@ -251,6 +229,18 @@ OPA2 required a relatively small correction along $\tau_{22^\prime}$ (middle sub
 for any dispersion experienced differently between the two split beams.  %
 OPA1 was not split and therefore needed no such correction.  %
 
+\begin{figure}
+  \includegraphics[width=\textwidth]{MX2/S4}
+  \caption{OPA outputs at each color explored.}
+  \label{fig:CzechS4}
+\end{figure}
+
+\begin{figure}
+  \includegraphics[width=\textwidth]{MX2/S5}
+  \caption{Spectral delay correction.}
+  \label{fig:CzechS5}
+\end{figure}
+
 \autoref{fig:Czech10}a represents the to-scale mask that defines our distorted BOXCARS
 configuration.  %
 Relative to the center of the BOXCARS mask (small black dot), $\omega_1$, $\omega_2$, and
@@ -276,6 +266,38 @@ signal id is the geometry chosen for this experiment.  %
 This discrimination between a film and the substrate was also seen in reflective and transmissive
 CARS microscopy experiments. \cite{VolkmerAndreas2001a}  %
 
+
+\begin{figure}
+  \includegraphics[width=\textwidth]{MX2/10}
+  \caption[Mask and epi vs transmissive.]{
+    (a) Mask.
+    (b) 2D delay spectra at the BB diagonal ($\omega_1=\omega_2\approx1.95$ eV) for transmissive
+    and reflective geometries.
+    Transmissive signal is a mixture of MoS\textsubscript{2} signal and a large amount of driven
+    signal from the substrate that only appears in the pulse overlap region. Reflective signal is
+    representative of the pure MoS\textsubscript{2} response.}
+  \label{fig:Czech10}
+\end{figure}
+
+Once measured, the FWM signal was sent through a four-stage workup process to create the data set
+shown here.  %
+This workup procedure is visualized in \autoref{fig:Czech11}.  %
+We use a chopper and boxcar in active background subtraction mode (averaging 100 laser shots) to
+extract the FWM signal from $\omega_1$ and $\omega_2$ scatter.  %
+We collect this differential signal (\autoref{fig:Czech11}b) in software with an additional 50
+shots of averaging.  %
+In post-process we subtract $\omega_2$ scatter and smooth the data using a 2D Kaiser window.  %
+Finally, we represent the homodyne collected data as (sig)$^{1/2}$ to make the dynamics and line
+widths comparable to heterodyne-collected techniques like absorbance and pump-probe spectra.  %
+Throughout this work, zero signal on the color bar is set to agree with the average rather than the
+minimum of noise.  %
+Values below zero due to measurement uncertainty underflow the color bar and are plotted in
+white.  %
+This is especially evident in lots such as +120 fs in \autoref{fig:Czech08}, where there is no real
+signal.  %
+IPython \cite{PerezFernando2007a} and matplotlib \cite{HunterJohnD2007a} were important for data
+processing and plotting in this work.
+
 \begin{figure}
   \includegraphics[width=\textwidth]{MX2/11}
   \caption[MoS\textsubscript{2} post processing.]{
@@ -303,42 +325,8 @@ CARS microscopy experiments. \cite{VolkmerAndreas2001a}  %
   \label{fig:Czech11}
 \end{figure}
 
-Once measured, the FWM signal was sent through a four-stage workup process to create the data set
-shown here.  %
-This workup procedure is visualized in \autoref{fig:Czech11}.  %
-We use a chopper and boxcar in active background subtraction mode (averaging 100 laser shots) to
-extract the FWM signal from $\omega_1$ and $\omega_2$ scatter.  %
-We collect this differential signal (\autoref{fig:Czech11}b) in software with an additional 50
-shots of averaging.  %
-In post-process we subtract $\omega_2$ scatter and smooth the data using a 2D Kaiser window.  %
-Finally, we represent the homodyne collected data as (sig)$^{1/2}$ to make the dynamics and line
-widths comparable to heterodyne-collected techniques like absorbance and pump-probe spectra.  %
-Throughout this work, zero signal on the color bar is set to agree with the average rather than the
-minimum of noise.  %
-Values below zero due to measurement uncertainty underflow the color bar and are plotted in
-white.  %
-This is especially evident in lots such as +120 fs in \autoref{fig:Czech08}, where there is no real
-signal.  %
-IPython \cite{PerezFernando2007a} and matplotlib \cite{HunterJohnD2007a} were important for data
-processing and plotting in this work.
-
 \section{Results and discussion}  % ===============================================================
 
-\begin{figure}
-  \includegraphics[width=0.75\textwidth]{MX2/02}
-  \caption[Few-layer MoS\textsubscript{2} thin film characterization.]{
-    Characterization of the few-layer MoS\textsubscript{2} film studied in this work.
-    Optical images of the
-    MoS\textsubscript{2} thin film on fused silica substrate in (a) transmission and (b)
-    reflection.
-    (c) Raman spectrum of the $E_{2g}^1$ and $A_{1g}$ vibrational modes.
-    (d) High-resolution TEM image and its corresponding FFT shown in the inset.
-    (e) Absorption (blue), photoluminescence (green), Gaussian fits to the A and B excitons, along
-    with the residules betwen the fits and absorbance (dotted), A and B exciton centers (dotted)
-    and representative excitation pulse shape (red).}
-  \label{fig:Czech02}
-\end{figure}
-
 The few-layer MoS\textsubscript{2} thin film sample studied in this work was prepared on a
 transparent fused silica substrate by a simple sufidation reaction of a Mo thin film using a
 procedure modified from a recent report. \cite{LaskarMasihhurR2013a}  %
@@ -356,15 +344,6 @@ corresponds to approximately four monolayers.  %
 and B excitonic line shapes that were extracted from the absorption spectrum. A representative
 excitation pulse profile is also shown in red for comparison.  %
 
-\begin{figure}
-  \includegraphics[width=\textwidth]{MX2/S3}
-  \caption[MoS\textsubscript{2} absorbance.]{Extraction of excitonic features from absorbance
-    spectrum. (a) Second derivative spectra of absorbance (black) and fit second derivative
-    spectrum (green). Gaussian fit parameters are shown in the inset table. (b) Absorption curve
-    (black), Gaussian fits (blue and red), and remainder (black dotted).}
-  \label{fig:CzechS3}
-\end{figure}
-
 Extracting the exciton absorbance spectrum is complicated by the large ``rising background'' signal
 from other MoS\textsubscript{2} bands.  %
 With this in mind, we fit the second derivative absorption spectrum to a sum of two second
@@ -374,6 +353,30 @@ the amplitude) of the remainder between the fit and the absorption spectrum.  %
 The fit parameters can be found in the inset table in \autoref{fig:CzechS3}.  %
 The Gaussians themselves and the remainder can be found in \autoref{fig:CzechS3}.  %
 
+\begin{figure}
+  \includegraphics[width=0.75\textwidth]{MX2/02}
+  \caption[Few-layer MoS\textsubscript{2} thin film characterization.]{
+    Characterization of the few-layer MoS\textsubscript{2} film studied in this work.
+    Optical images of the
+    MoS\textsubscript{2} thin film on fused silica substrate in (a) transmission and (b)
+    reflection.
+    (c) Raman spectrum of the $E_{2g}^1$ and $A_{1g}$ vibrational modes.
+    (d) High-resolution TEM image and its corresponding FFT shown in the inset.
+    (e) Absorption (blue), photoluminescence (green), Gaussian fits to the A and B excitons, along
+    with the residules betwen the fits and absorbance (dotted), A and B exciton centers (dotted)
+    and representative excitation pulse shape (red).}
+  \label{fig:Czech02}
+\end{figure}
+
+\begin{figure}
+  \includegraphics[width=\textwidth]{MX2/S3}
+  \caption[MoS\textsubscript{2} absorbance.]{Extraction of excitonic features from absorbance
+    spectrum. (a) Second derivative spectra of absorbance (black) and fit second derivative
+    spectrum (green). Gaussian fit parameters are shown in the inset table. (b) Absorption curve
+    (black), Gaussian fits (blue and red), and remainder (black dotted).}
+  \label{fig:CzechS3}
+\end{figure}
+
 The multiresonant CMDS experiment uses $\approx$70 fs excitation pulses created by two
 independently tunable optical parametric amplifiers (OPAs).  %
 Automated delay stages and neutral density filters set the excitation time delays over all values
@@ -387,23 +390,9 @@ In order to compare the FWM spectra with the absorption spectrum, the signal has
 the square root of the measured FWM signal since FWM depends quadratically on the sample
 concentration and path length.  %
 
-\begin{figure}
-  \includegraphics[width=\textwidth]{MX2/03}
-  \caption[MoS\textsubscript{2} frequency-frequency slices.]{2D frequency-frequency spectra of the
-    MoS\textsubscript{2} sample in the epi configuration. In all spectra $\tau_{22^\prime}=0$ fs,
-    while $\tau_{21}$ is designated in the bottom-right corner of each spectral panel. The color
-    bar defines the square root of the intensity normalized to the most intense feature in the
-    series of spectra. The integration of the signal onto the $\hbar\omega_1=\hbar\omega_m$ and
-    $\hbar\omega_2$ axes are represented ans the blue curves in the top and right side plots,
-    respectively. The side plots also contain the absorbance spectrum (black line) to aid
-    intepretation of the dynamics of the integrated 2D signals. The dashed lines mark the centers
-    of the A and B excitons, as designated from the absorption spectrum.}
-  \label{fig:Czech03}
-\end{figure}
-
 The main set of data presented in this work is an $\omega_1\omega_2\tau_{21}$ ``movie'' with
 $\tau_{22\prime}=0$. 
-\autoref{fig:Czec03} shows representative 2D frequency-frequency slices from this movie at
+\autoref{fig:Czech03} shows representative 2D frequency-frequency slices from this movie at
 increasingly negative $\tau_{21}$ times.  %
 Each 2D frequency spectrum contains side plots along both axes that compare the absorbance spectrum
 (black) to the projection of the integrated signal onto the axis (blue).  %
@@ -414,24 +403,19 @@ Instead, the signal amplitude increases toward bluer $\omega_2$ values.  %
 The decrease in FWM above 2.05 eV is caused by a drop in the $\omega_2$ OPA power.
 
 \begin{figure}
-  \includegraphics[width=0.75\textwidth]{MX2/04}
-  \caption[MoS\textsubscript{2} $\omega_1$ Wigner progression.]{Mixed $\omega_1$---$\tau_{21}$
-    time---frequency representations of the 3D data set at five ascending $\omega_2$ excitation
-    frequencies (solid black lines) showing the impact of the $\omega_2$ excitation frequency on
-    the $\omega_1$ spectral line shape as a function of time. The A and B exciton energies are
-    marked as dashed lines within each spectrum.}
-  \label{fig:Czech04}
+  \includegraphics[width=\textwidth]{MX2/03}
+  \caption[MoS\textsubscript{2} frequency-frequency slices.]{2D frequency-frequency spectra of the
+    MoS\textsubscript{2} sample in the epi configuration. In all spectra $\tau_{22^\prime}=0$ fs,
+    while $\tau_{21}$ is designated in the bottom-right corner of each spectral panel. The color
+    bar defines the square root of the intensity normalized to the most intense feature in the
+    series of spectra. The integration of the signal onto the $\hbar\omega_1=\hbar\omega_m$ and
+    $\hbar\omega_2$ axes are represented ans the blue curves in the top and right side plots,
+    respectively. The side plots also contain the absorbance spectrum (black line) to aid
+    intepretation of the dynamics of the integrated 2D signals. The dashed lines mark the centers
+    of the A and B excitons, as designated from the absorption spectrum.}
+  \label{fig:Czech03}
 \end{figure}
 
-\begin{figure}
-  \includegraphics[width=0.75\textwidth]{MX2/05}
-  \caption[MoS\textsubscript{2} $\omega_2$ Wigner progression.]{Mixed $\omega_2$---$\tau_{21}$
-    time---frequency representations of the 3D data set at five ascending $\omega_1$ probe
-    frequencies (solid black lines) showing the impact of the $\omega_1$ excitation frequency on
-    the $\omega_2$ spectral line shape as a function of time. The A and B exciton energies are
-    marked as dashed lines within each spectrum.}
-  \label{fig:Czech05}
-\end{figure}
 
 Figures \ref{fig:Czech04} and \ref{fig:Czech05} show representative 2D frequency-delay slices from
 this movie, where the absicissa is the $\omega_1$ or $\omega_2$ frequency, respectively, the
@@ -441,9 +425,9 @@ The color bar is normalized to the brightest feature in each subplot.  %
 This normalization allows comparison of the time dependence of the line shapes, positions, and
 relative signal amplitudes along the $\omega_1$ or $\omega_2$ axis directly.  %
 
-Each subplot in \autoref{fig:Czech04} is similar to published pump-probe, transient absorption, multidimensionaland
-transient reflection experiments that have measured the electronc dynamics of the A and B excitons.
-\cite{XiaoDi2012a, FangHui2014a, KumarNardeep2013a, NieZhaogang2014a, SunDezheng2014a,
+Each subplot in \autoref{fig:Czech04} is similar to published pump-probe, transient absorption,
+and transient reflection experiments that have measured the electronc dynamics of the A and B
+excitons. \cite{XiaoDi2012a, FangHui2014a, KumarNardeep2013a, NieZhaogang2014a, SunDezheng2014a,
   SimSangwan2013a, MakKinFai2012a, ThomallaMarkus2006a}  %
 These previous experiments measure relaxation dynamics on the same $\approx$400-600 fs time scale
 that is characteristic of \autoref{fig:Czech04}.  %
@@ -459,15 +443,26 @@ Both the line shapes and the dynamics of the spectral features are very similar.
 features do not depend strongly on the $\omega_1$ frequency.
 
 \begin{figure}
-  \includegraphics[width=0.5\textwidth]{MX2/06}
-  \caption[Pathway V, VI liouville pathways.]{Liouville pathways for \autoref{fig:Czech04}. gg and
-    ee designate ground- and excited-state populations, the eg, 2e,e, and e$^\prime$+e,e represent
-    the excitonic and biexcitonic output coherences, and the arrows are labeled with the
-    frequencies or population transfer responsible for the transitions. e and e$^\prime$ represent
-    either A or B excitonic states.}
-  \label{fig:Czech06}
+  \includegraphics[width=0.75\textwidth]{MX2/04}
+  \caption[MoS\textsubscript{2} $\omega_1$ Wigner progression.]{Mixed $\omega_1$---$\tau_{21}$
+    time---frequency representations of the 3D data set at five ascending $\omega_2$ excitation
+    frequencies (solid black lines) showing the impact of the $\omega_2$ excitation frequency on
+    the $\omega_1$ spectral line shape as a function of time. The A and B exciton energies are
+    marked as dashed lines within each spectrum.}
+  \label{fig:Czech04}
+\end{figure}
+
+\begin{figure}
+  \includegraphics[width=0.75\textwidth]{MX2/05}
+  \caption[MoS\textsubscript{2} $\omega_2$ Wigner progression.]{Mixed $\omega_2$---$\tau_{21}$
+    time---frequency representations of the 3D data set at five ascending $\omega_1$ probe
+    frequencies (solid black lines) showing the impact of the $\omega_1$ excitation frequency on
+    the $\omega_2$ spectral line shape as a function of time. The A and B exciton energies are
+    marked as dashed lines within each spectrum.}
+  \label{fig:Czech05}
 \end{figure}
 
+
 The spectral features in Figures \ref{fig:Czech03}, \ref{fig:Czech04} and \ref{fig:Czech05} depend
 on the quantum mechanical interference effects caused by the different pathways.  %
 \autoref{fig:Czech06} shows all of the Liouville pathways required to understand the spectral
@@ -506,6 +501,18 @@ A quantitative treatment of the cancellation effects between the GSB, SE, and ES
 knowledge of the transition moments and state degeneracies and is beyond the scope of this paper.
 \cite{WongCathyY2011a}  %
 
+\begin{figure}
+  \includegraphics[width=0.5\textwidth]{MX2/06}
+  \caption[Liouville pathways in time orderings V, VI.]{
+    Liouville pathways for \autoref{fig:Czech04}. gg and
+    ee designate ground- and excited-state populations, the eg, 2e,e, and e$^\prime$+e,e represent
+    the excitonic and biexcitonic output coherences, and the arrows are labeled with the
+    frequencies or population transfer responsible for the transitions. e and e$^\prime$ represent
+    either A or B excitonic states.
+  }
+  \label{fig:Czech06}
+\end{figure}
+
 The most important characteristic of the experimental spectra is the contrast between the absence
 of well-resolved excitonic features that depend on $\omega_2$ in Figures \ref{fig:Czech03} and
 \ref{fig:Czech05} and the well-defined excitonic features that depend on $\omega_1$.  %
@@ -557,17 +564,6 @@ The B/A ratio is higher when $\omega_2$ is resonant with the B excitonic transit
 $\omega_2$ is lower than the A exciton frequency (the top subplot).  %
 If population transfer of holes from the B to A valence bands occurred during temporal overlap, the
 B/A ratio would be independent of pump frequency at $\tau_21<0$.
-
-\begin{figure}
-  \includegraphics[width=\textwidth]{MX2/07}
-  \caption[MoS\textsubscript{2} transients.]{
-    Transients taken at the different $\omega_1$ and $\omega_2$ frequencies indicated by the
-    colored markers on the 2D spectrum.
-    The dynamics are assigned to a 680 fs fast time constant (black solid line) and a slow time
-    constant represented as an unchanging offset over this timescale (black dashed line).}
-  \label{fig:Czech07}
-\end{figure}
-
 \autoref{fig:Czech07} shows the delay transients at the different frequencies shown in the 2D
 spectrum.  %
 The colors of the dots on the 2D frequency-frequency spectrum match the colors of the
@@ -583,11 +579,13 @@ The 680 fs decay is similar to previously published pump-probe and transient abs
 experiments. \cite{NieZhaogang2014a, SunDezheng2014a, DochertyCallumJ2014a}  %
 
 \begin{figure}
-  \includegraphics[width=\textwidth]{MX2/08}
-  \caption[MoS\textsubscript{2} frequency-frequency slices near pulse overlap.]{2D
-    frequency-frquency spectra near zero $\tau_{21}$ delay times. The signal amplitude is
-    normalized to the brightest features in each spectrum.}
-  \label{fig:Czech08}
+  \includegraphics[width=\textwidth]{MX2/07}
+  \caption[MoS\textsubscript{2} transients.]{
+    Transients taken at the different $\omega_1$ and $\omega_2$ frequencies indicated by the
+    colored markers on the 2D spectrum.
+    The dynamics are assigned to a 680 fs fast time constant (black solid line) and a slow time
+    constant represented as an unchanging offset over this timescale (black dashed line).}
+  \label{fig:Czech07}
 \end{figure}
 
 The spectral features change quantitatively for delay times near temporal overlap.  %
@@ -600,14 +598,6 @@ The spectra also develop more diagonal character as the delay time moves from ne
 values.  %
 The AB cross-peak is also a strong feature in the spectrum at early times.  %
 
-\begin{figure}
-  \includegraphics[width=0.5\textwidth]{MX2/09}
-  \caption[Pathways I, III Liouville pathways.]{Liouville pathways for the $\omega_1$, $\omega_2$,
-    and $\omega_{2^\prime}$ time ordering of pulse interactions. e and e$^\prime$ represent either
-    A or B excitonic states.}
-  \label{fig:Czech09}
-\end{figure}
-
 The pulse overlap region is complicated by the multiple Liouville pathways that must be
 considered.  %
 Additionally, interference between scattered light from the $\omega_1$ excitation beam and the
@@ -634,6 +624,23 @@ More positie values of $\tau_{21}$ emphasize the \autoref{fig:Czech09} pathways
 \autoref{fig:Czech06} pathways, accounting for the increasing percentage of diagonal character at
 increasingly positive delays.  %
 
+\begin{figure}
+  \includegraphics[width=\textwidth]{MX2/08}
+  \caption[MoS\textsubscript{2} frequency-frequency slices near pulse overlap.]{2D
+    frequency-frquency spectra near zero $\tau_{21}$ delay times. The signal amplitude is
+    normalized to the brightest features in each spectrum.}
+  \label{fig:Czech08}
+\end{figure}
+
+\begin{figure}
+  \includegraphics[width=0.5\textwidth]{MX2/09}
+  \caption[Liouville pathways in time orderings I, III.]{
+    Liouville pathways for the $\omega_1$, $\omega_2$,
+    and $\omega_{2^\prime}$ time ordering of pulse interactions. e and e$^\prime$ represent either
+    A or B excitonic states.}
+  \label{fig:Czech09}
+\end{figure}
+
 \section{Conclusions}  % ==========================================================================
 
 This paper presents the first coherent multidimensional spectroscopy of MoS\textsubscript{2} thin
@@ -650,7 +657,7 @@ These spectra are complementary to previous work because they allow a direct com
 initially excited excitonic states and the states creating the final output coherence.  %
 The spectra show that the same hot A and B exciton continuum states that are observed in the
 absorption spectrum also dominate the CMDS excitation spectra.  %
-They also show that rapid, <70 fs intraband relaxation occurs to create the band-edge A and B
+They also show that rapid, $<70$ fs intraband relaxation occurs to create the band-edge A and B
 excitonic features observed in the CMSD spectrum.  %
 The relative intensity of the diagonal peak features depends on the relative absorption strength of
 the A and B excitons.  %
@@ -669,4 +676,26 @@ complex MoS\textsubscript{2} and other TMDC heterostructures with quantum-state
 The frequency domain based multiresonant CMDS methods described in this paper will play a central
 role in these measurements.  %
 They use longer, independently tunable pulses that provide state-selective excitation over a wide
-spectral range without the requirement for interferometric stability.  %
\ No newline at end of file
+spectral range without the requirement for interferometric stability.  %
+
+\section{Errata}  % ===============================================================================
+
+The following is an errata for the of this chapter, published in November 2015.
+\cite{CzechKyleJonathan2015a}  %
+
+\begin{ditemize}
+  \item Reference 13 is identical to reference 9.
+  \item In the last paragraph of the introduction the sentence ``The experimental spectra differ
+    from the simple 2D spectrum shown in Figure 1d and those of earlier CMDS experiments with model
+    systems'' appears. This sentence cites references 6 through 10. Instead, it should cite
+    references 15 through 20.
+  \item In the last paragraph begining on page 12148, the text ``Automated delay stages and neutral
+    density filters set the excitation time delays over all values of $\tau_{21}$ with
+    $\tau_{22}=0$''    appears. For the second $\tau$, the subscript should read $22^\prime$, not
+    $22$.
+  \item Caption of Figure 5 reads, in part: "showing the impact of the $\omega_1$ excitation
+    frequency on the $\omega_1$ spectral line shape". This should instead read "showing the impact
+    of the $\omega_1$ excitation frequency on the $\omega_2$ spectral line shape". The subscript on
+    the last $\omega$ should be a 2 and not a 1.
+  \item Figure 6 e$^\prime$+e,e$^\prime$ should read e$^\prime$+e,e and vice versa.
+\end{ditemize}
\ No newline at end of file