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-\chapter{PEDOT:PSS}
-
-\section{Introduction}
-
-Poly(3,4-ethylenedioxythiophene)-poly(styrenesulfonate) (PEDOT:PSS) is a transparent, electrically
-conductive (up to 4380 S cm$^{-1}$ \cite{KimNara2013a}) polymer.  %
-It has found widespread use as a flexible, cheap alternative to inorganic transparent electrodes
-such as indium tin oxide.  %
-
-As a polymer, PEDOT:PSS implicitly contains a large amount of structural inhomogeneity.  %
-On top of this, PEDOT:PSS is a two component material, composed of PEDOT (low molecular weight,
-p-doped, highly conductive) and PSS (high molecular-weight, insulating, stabilizing).  %
-These two components segment into domains of conductive and non-conductive material, leading to
-even more structural inhomogeneity.  %
-Nonlinear spectroscopy may be able to shed light on the microscopic environment of electronic
-states within PEDOT:PSS.  %
-
-\section{Background}
-
-Complex microstructure:
-\begin{enumerate}
-  \item PEDOT oligomers (6---18-mers)
-  \item these oligomers $\pi$-stack to form small nanocrystalites, 3 to 14 oligomers for pristine
-    films to as many as 13---14 oligomers for more conductive solvent treated films
-  \item nanocrystallites then arrange into globular conductive particles in a pancakge-like shape
-  \item these particles themselves are then linked via PSS-rich domains and assembled into
-    nanofibril geometry akin to a string of pearls
-  \item nanofibrils interweave to form thin films, with PSS capping layer at surface
-\end{enumerate}
-
-Prior spectroscopy (absorption anisotropy, X-ray scattering, condutivity).  %
-
-% TODO: absorption spectrum of thin film
-
-Broad in the infrared due to midgap states created during doping from charge-induced lattice
-relaxations.  %
-These electronic perturbations arise from injected holes producing a quinoidal distortion spread
-over 4-5 monomers of the CP aromatic backbone, collectively called a polaron.  %
-Energetically favorable to be spin-silent bipolaron.  %
-
-\section{Methods}
-
-PEDOT:PSS (Orgacon Dry, Sigma Aldrich) was dropcast onto a glass microscope slide at 1 mg/mL at a
-tilt to ensure homogeneous film formation.  %
-The sample was heated at 100 $^\circ$C for $\sim$15 min to evaporate water.  %
-
-An ultrafast oscillator (Spectra-Physics Tsunami) was used to prepare $\sim$35 fs seed pulses.  %
-These were amplified (Spectra-Physics Spitfire Pro XP, 1 kHz), split, and converted into 1300 nm 40
-fs pulses using two separate optical parametric amplifiers (Light Conversion TOPAS-C): ``OPA1'' and
-``OPA2''.  %
-Pulses from OPA2 were split again, for a total of three excitation pulses: $\omega_1$, $\omega_2$
-and $\omega_{2^\prime}$.  %
-These were passed through motorized (Newport MFA-CC) retroreflectors to control their relative
-arrival time (``delay'') at the sample: $\tau_{21} = \tau_2 - \tau_1$ and $\tau_{22^\prime} =
-\tau_2 - \tau_{2^\prime}$. The three excitation pulses were focused into the sample in a $1^\circ$
-right-angle isoceles triange, as in the BOXCARS configuration. \cite{EckbrethAlanC1978a}  %
-Each excitation beam was 67 nJ focused into a 375 $\mathsf{\mu m}$ symmetric Gaussian mode for an
-intensity of 67 $\mathsf{\mu J / cm^2}$.  %
-A new beam, emitted coherently from the sample, was isolated with apertures and passed into a
-monochromator (HORIBA Jobin Yvon MicroHR, 140 mm focal length) with a visible grating (500 nm blaze
-300 groves per mm).  %
-The monochromator was set to pass all colors (0 nm, 250 $\mathsf{\mu m}$ slits) to keep the
-measurement impulsive.  %
-Signal was detected using an InSb photodiode (Teledyne Judson J10D-M204-R01M-3C-SP28).  %
-Four wave mixing was isolated from excitation scatter using dual chopping and digital signal
-processing.  %
-
-\section{Transmittance and reflectance}
-
-\autoref{fig:PEDOTPSS_linear} shows the transmission, reflectance, and extinction spectrum of the
-thin film used in this work.  %
-
-\clearpage
-\begin{figure}
-  \centering
-	\includegraphics[width=0.5\linewidth]{"PEDOT:PSS/linear"}
-	\caption[PEDOT:PSS transmission and reflectance spectra.]{
-    Thin film spectra.
-    Transmission, reflectance, and extinction spectrum of the thin film used in this work.  %
-    Extinction is $\log_{10}{\mathsf{(transmission)}}$.  %
-  }
-	\label{fig:PEDOTPSS_linear}
-\end{figure}
-\clearpage
-
-\section{Three-pulse echo spectroscopy}  % --------------------------------------------------------
-
-Two dimensional $\tau_{21}, \tau_{22^\prime}$ scans were taken for two phase matching
-configurations: (1) $k_{\mathsf{out}} = k_1 - k_2 + k_{2^\prime}$ (3PE) and (2) $k_{\mathsf{out}} =
-k_1 + k_2 - k_{2^\prime}$ (3PE*).  %
-The rephasing and nonrephasing pathways exchange their time dependance between these two
-configurations.  %
-Comparing both pathways, rephasing-induced peak shifts can be extracted as in 3PE. [CITE]  %
-All data was modeled using numerical integration of the Liouville-von Numann equation.  %
-
-Continuously variable ND filters (THORLABS NDC-100C-4M, THORLABS NDL-10C-4) were used to ensure
-that all three excitation pulse powers were equal within measurement error.  %
-
-\autoref{fig:PEDOTPSS_mask} diagrams the phase matching mask used in this set of experiments.  %
-
-\begin{figure}
-	\includegraphics[width=0.5\linewidth]{"PEDOT:PSS/mask"}
-	\caption[PEDOT:PSS 3PE phase matching mask.]{
-    Phase matching mask used in this experiment.
-    Each successive ring subtends 1 degree, such that the excitation pulses are each angled one
-    degree relative to the mask center.
-    The two stars mark the two output poyntings detected in this work.
-  }
-	\label{fig:PEDOTPSS_mask}
-\end{figure}
-
-\autoref{fig:PEDOTPSS_raw} shows the ten raw 2D delay-delay scans that comprise the primary dataset
-described in this section.  %
-The rows correspond to the two phase matching conditions, as labeled.  %
-
-\begin{figure}
-	\includegraphics[width=\linewidth]{"PEDOT:PSS/raw"}
-	\caption[PEDOT:PSS 3PE raw data.]{
-    CAPTION TODO
-  }
-	\label{fig:PEDOTPSS_raw}
-\end{figure}
-
-\subsection{Assignment of zero delay}  % ----------------------------------------------------------
-
-The absolute position of complete temporal overlap of the excitation pulses (zero delay) is a
-crucial step in determining the magnitude of th epeak shift and therefore the total rephasing
-ability of the material.  %
-The strategy for assigning zero delay relies upon the intrinsic symmetry of the two-dimensional
-delay space.  %
-\autoref{fig:PEDOTPSS_delay_space} labels the six time-orderings (TOs) of the three pulses that are
-possible with two delays.  %
-The TO labeling scheme follow from a convention first defined my Meyer, Wright and Thompson.
-[CITE]  %
-[CITE] first discussed how these TOs relate to traditional 3PE experiments.  %
-Briefly, spectral peak shifts into the rephasing TOs \RomanNumeral{3} and \RomanNumeral{5} when
-inhomogeneous broadening creates a photon echo in the \RomanNumeral{3} and \RomanNumeral{5}
-rephasing pathways colored orange in \autoref{fig:PEDOTPSS_delay_space}.  %
-For both phase-matching conditions, there are two separate 3PE peak shift traces (represented as
-black arrows in \autoref{fig:PEDOTPSS_delay_space}), yielding four different measurements of the
-photon echo.  %
-Since both 3PE and 3PE* were measured using the same alignment on the same day, the zero delay
-position is identical for the four photon echo measurements.  %
-We focus on this signature when assigning zero delay---zero is correct only when all four peak
-shifts agree.  %
-Conceptually, this is the two-dimensional analogue to the traditional strategy of placing zero such
-that the two conjugate peak shifts (3PE and 3PE*) agree. [CITE]  %
-
-We found that the 3PEPS traces agree best when the data in \autoref{fig:PEDOTPSS_raw} is offset by
-19 fs in $\tau_{22^\prime}$ and 4 fs in $\tau_{21}$.  %
-\autoref{fig:PEDOTPSS_processed} shows the 3PEPS traces after correcting for the zero delay
-value.  %
-The entire 3PEPS trace ($\tau$ vs $T$) is show for regions \RomanNumeral{1}, \RomanNumeral{3}
-(purple and light green traces) and \RomanNumeral{5}, \RomanNumeral{6} (yellow and light blue
-traces) for the [PHASE MATCHING EQUATIONS] phase matching conditions, respectively.  %
-Peak-shift magnitudes were found with Gaussian figs on the intensity level, in accordance with
-3PEPS convention. [CITE]
-The bottom subplot of \autoref{fig:PEDOTPSS_overtraces} shows the agreement between the four traces
-for $T > 50$ fs where pulse-overlap effects become negligible.  %
-These pulse-overlap effects cause the 3PEPS at small $T$ even without inhomogeneous broadening.
-[CITE]  %
-At long $T$, the average static 3PEPS is 2.5 fs.  %
-
-\begin{figure}
-	\includegraphics[width=\linewidth]{"PEDOT:PSS/delay space"}
-	\caption[PEDOT:PSS 3PE delay space.]{
-    CAPTION TODO
-  }
-	\label{fig:PEDOTPSS_delay_space}
-\end{figure}
-
-\begin{figure}
-	\includegraphics[width=\linewidth]{"PEDOT:PSS/processed"}
-	\caption[PEDOT:PSS 3PE processed data.]{
-    CAPTION TODO
-  }
-	\label{fig:PEDOTPSS_processed}
-\end{figure}
-
-\begin{figure}
-	\includegraphics[width=\linewidth]{"PEDOT:PSS/overtraces"}
-	\caption[PEDOT:PSS 3PE traces.]{
-    CAPTION TODO
-  }
-	\label{fig:PEDOTPSS_overtraces}
-\end{figure}
-
-\begin{figure}
-	\includegraphics[width=\linewidth]{"PEDOT:PSS/traces"}
-	\caption[PEDOT:PSS 3PE traces.]{
-    CAPTION TODO
-  }
-	\label{fig:PEDOTPSS_traces}
-\end{figure}
-
-There is a deviation of the TO \RomanNumeral{1}-\RomanNumeral{3} 3PEPS* trace (green line) from the
-other traces.  %
-It is attributed to a combination of excitation pulse distortions and line shape differences
-between OPA1 and OPA2 (see \autoref{fig:PEDOTPSS_linear}) and small errors in the zero delay
-correction.  %
-\autoref{fig:PEDOTPSS_traces} shows what the four 3PEPS traces would llike like for different
-choices of zero-delay.  %
-The inset numbers in each subplot denote the offset (from chosen zero) in each delay axis.  %
-
-\subsubsection{Numerical model}  % ----------------------------------------------------------------
-
-We simulated the 3PEPS response of PEDOT:PSS through numerical integration of the Liouville-von
-Neumann Equation.  %
-Integration was performed on a homogeneous, three-level system with coherent dynamics described by
-
-\begin{equation}
-  \frac{1}{T_2} = \frac{1}{2T_1} + \frac{1}{T_2^*},
-\end{equation}
-
-where $T_2$, $T_1$ and $T_2^*$ are the net dephasing, population relaxation, and pure dephasing
-rates, respectively.  %
-A three-level system was used because a two-level system cannot explain the population relaxation
-observed at long populations times, $T$.  %
-This slow delcay may be the same as the slowly decaying optical nonlinearities in PEDOT:PSS.
-[CITE]  %
-Inhomogeneity was incorporated by convolving the homogeneous repsonse with a Gaussian distribution
-function of width $\Delta_{\mathsf{inhom}}$ and allowing the resultant polarization to interfere on
-the amplitude level.  %
-This strategy captures rephasing peak shifts and ensemble dephasing.  %
-
-It is difficult to determine the coherence dephasing and the inhomogeneous broadening using 3PE if
-both factors are large.  %
-To extract $T_2^*$ and $\Delta_{\mathsf{inhom}}$, we focused on two key components of the dataset,
-coherence duration and peak shift at large $T$.  %
-Since dephasing is very fast in PEDOT:PSS, we cannot directly respove an exponential free induction
-decay (FID).  %
-Instead, our model focuses on the FWHM of the $\tau$ trace to determine the coherence duration.  %
-At $T > 50$ fs, the transient has a FWHM of $\sim$ 80 fs (intensity level).  %
-For comparison, our instrumental response is estimated to be 70-90 fs, depending on the exact value
-of our puse duration $\Delta_t$ (35-45 fs FWHM, intensity level).  %
-An experimental peak shift of 2.5 fs was extracted using the strategy described above.  %
-Taken together, it is clear that both pure dephasing and ensemble dephasing influence FWHM and peak
-shift so it is important to find valuse of $T_2^*$ and $\Delta_{\mathsf{inhom}}$ that uniquely
-constrain the measured response.  %
-
-\begin{figure}
-	\includegraphics[width=\linewidth]{"PEDOT:PSS/parametric"}
-	\caption[PEDOT:PSS 3PE traces.]{
-    CAPTION TODO
-  }
-	\label{fig:PEDOTPSS_parametric}
-\end{figure}
-
-We simulated the $\tau$ trance for a variety of $\Delta_{\mathsf{inhom}}$ and $T_2$ values.  %
-The results for $\Delta_t = 40$ fs are summarized in \autoref{fig:PEDOTPSS_parametric}.  %
-The lines of constant $T_2$ span from $\Delta_{\mathsf{inhom}} = 0$ (green left ends of curves) to
-the limit $\Delta_{\mathsf{inhom}} \rightarrow \infty$ (blue right ends of curves).  %
-The lines of constant $T_2$ demonstrate that ensemble dephasing reduces the transient duration and
-introduces a peak shift.  %
-The influence of inhomogeneity on the observables vanishes as $T_2 \rightarrow \infty$. % 
-
-We preformed simulations analogus to those in \autoref{fig:PEDOTPSS_parametric} for pulse durations
-longer and smaller than $\Delta_t = 40$ fs.  %
-Longer pulse durations create solutions that do not intersect our experimental point (see
-right-most subplot of \autoref{fig:PEDOTPSS_parametric}), but shorter pulse durations do.  %
-[TABLE] summarizes the coherence dephasing time and inomogeneous broadening values that best
-matches the experimental FWHM and inhomogeneous broadening value for $\Delta_t = 35, 40$ and 45
-fs.  %
-Clearly, there is no upper limit that can provide an upper limit for the inhomogeneous
-broadening.  %
-
-\begin{table}
- \begin{tabular}{ c | c c c }
-  $\Delta_t$ (fs) & $T_2$ (fs) & $\hbar T_2^{-1}$ (meV) & $\Delta_{\mathsf{inhom}}$ (meV) \\ \hline
-  45 & --- & --- & --- \\
-  40 & 10 & 66 & $\infty$ \\
- \end{tabular}
- \caption[]{
-   CAPTION TODO
- }
- \label{tab:PEDOTPSS_table}
-\end{table}
-
-\begin{figure}
-	\includegraphics[width=\linewidth]{"PEDOT:PSS/agreement"}
-	\caption[PEDOT:PSS 3PE traces.]{
-    CAPTION TODO
-  }
-	\label{fig:PEDOTPSS_agreement}
-\end{figure}
-
-Our model system does ans excellent job of reproducing the entire 2D transient within measurement
-error (\autoref{fig:PEDOTPSS_agreement}).  %
-The most dramatic disagreement is in the upper right, where the experiment decays much slower than
-the simulation.  %
-Our system description does not account for signal contributions in TOs \RomanNumeral{2} and
-\RomanNumeral{4}, where double quantum coherence resonances are important.  %
-In additon, excitation pulse shapes may cause such distortions.  %
-Regardless, these contributions do not affect our analysis.  %
-
-Extremely fast (single fs) carrier scattering time constants have also been observed for PEDOT-base
-conductive films. [CITES]
-
-\section{Frequency-domain transient grating spectroscopy}  % --------------------------------------
-
-This section describes preliminary, unpublished work accomplished on PEDOT:PSS.  %
-
-