From cd162fef9d9f3145c1e29c63439759636ba62c41 Mon Sep 17 00:00:00 2001
From: Blaise Thompson <blaise@untzag.com>
Date: Mon, 26 Feb 2018 17:08:07 -0600
Subject: 2018-02-26 17:07

---
 PEDOT:PSS/chapter.tex | 246 ++++++++++++++++++++++++++++++++++++++------------
 1 file changed, 190 insertions(+), 56 deletions(-)

(limited to 'PEDOT:PSS/chapter.tex')

diff --git a/PEDOT:PSS/chapter.tex b/PEDOT:PSS/chapter.tex
index 9138972..8bb1510 100644
--- a/PEDOT:PSS/chapter.tex
+++ b/PEDOT:PSS/chapter.tex
@@ -67,8 +67,11 @@ processing.  %
 
 \section{Transmittance and reflectance}
 
-\afterpage{
-\begin{figure}
+\autoref{fig:PEDOTPSS_linear} shows the transmission, reflectance, and extinction spectrum of the
+thin film used in this work.  %
+
+\clearpage
+\begin{dfigure}
   \centering
 	\includegraphics[width=0.5\linewidth]{"PEDOT:PSS/linear"}
 	\caption[PEDOT:PSS transmission and reflectance spectra.]{
@@ -77,27 +80,10 @@ processing.  %
     Extinction is $\log_{10}{\mathsf{(transmission)}}$.  %
   }
 	\label{fig:PEDOTPSS_linear}
-\end{figure}
-\clearpage}
-
-\autoref{fig:PEDOTPSS_linear} shows the transmission, reflectance, and extinction spectrum of the
-thin film used in this work.  %
+\end{dfigure}
+\clearpage
 
-\section{Three-pulse echo spectroscopy}
-
-\afterpage{
-\begin{figure}
-  \centering
-	\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}
-\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}} =
@@ -110,60 +96,208 @@ All data was modeled using numerical integration of the Liouville-von Numann equ
 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.  %
 
-\afterpage{
-\begin{figure}
-  \centering
-	\includegraphics[width=0.5\linewidth]{"PEDOT:PSS/raw"}
+\autoref{fig:PEDOTPSS_mask} diagrams the phase matching mask used in this set of experiments.  %
+
+\begin{dfigure}
+	\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{dfigure}
+
+\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{dfigure}
+	\includegraphics[width=\linewidth]{"PEDOT:PSS/raw"}
 	\caption[PEDOT:PSS 3PE raw data.]{
     CAPTION TODO
   }
 	\label{fig:PEDOTPSS_raw}
-\end{figure}
-\clearpage}
+\end{dfigure}
 
+\subsection{Assignment of zero delay}  % ----------------------------------------------------------
 
-\afterpage{
-\begin{figure}
-  \centering
-	\includegraphics[width=0.5\linewidth]{"PEDOT:PSS/processed"}
+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{dfigure}
+	\includegraphics[width=\linewidth]{"PEDOT:PSS/delay space"}
+	\caption[PEDOT:PSS 3PE delay space.]{
+    CAPTION TODO
+  }
+	\label{fig:PEDOTPSS_delay_space}
+\end{dfigure}
+
+\begin{dfigure}
+	\includegraphics[width=\linewidth]{"PEDOT:PSS/processed"}
 	\caption[PEDOT:PSS 3PE processed data.]{
     CAPTION TODO
   }
 	\label{fig:PEDOTPSS_processed}
-\end{figure}
-\clearpage}
+\end{dfigure}
 
-\afterpage{
-\begin{figure}
-  \centering
-	\includegraphics[width=0.5\linewidth]{"PEDOT:PSS/delay_space"}
-	\caption[PEDOT:PSS 3PE delay space.]{
+\begin{dfigure}
+	\includegraphics[width=\linewidth]{"PEDOT:PSS/overtraces"}
+	\caption[PEDOT:PSS 3PE traces.]{
     CAPTION TODO
   }
-	\label{fig:PEDOTPSS_delay_space}
-\end{figure}
-\clearpage}
+	\label{fig:PEDOTPSS_overtraces}
+\end{dfigure}
 
-\afterpage{
-\begin{figure}
-  \centering
-	\includegraphics[width=0.5\linewidth]{"PEDOT:PSS/traces"}
+\begin{dfigure}
+	\includegraphics[width=\linewidth]{"PEDOT:PSS/traces"}
 	\caption[PEDOT:PSS 3PE traces.]{
     CAPTION TODO
   }
 	\label{fig:PEDOTPSS_traces}
-\end{figure}
-\clearpage}
+\end{dfigure}
 
-\afterpage{
-\begin{figure}
-  \centering
-	\includegraphics[width=0.5\linewidth]{"PEDOT:PSS/overtraces"}
+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{dfigure}
+	\includegraphics[width=\linewidth]{"PEDOT:PSS/parametric"}
 	\caption[PEDOT:PSS 3PE traces.]{
     CAPTION TODO
   }
-	\label{fig:PEDOTPSS_overtraces}
-\end{figure}
-\clearpage}
+	\label{fig:PEDOTPSS_parametric}
+\end{dfigure}
+
+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{dtable}
+ \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{dtable}
+
+\begin{dfigure}
+	\includegraphics[width=\linewidth]{"PEDOT:PSS/agreement"}
+	\caption[PEDOT:PSS 3PE traces.]{
+    CAPTION TODO
+  }
+	\label{fig:PEDOTPSS_agreement}
+\end{dfigure}
+
+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.  %
+
 
-\section{Frequency-domain transient grating spectroscopy}
\ No newline at end of file
-- 
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