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| -rw-r--r-- | PEDOT:PSS/chapter.tex | 246 | ||||
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3 files changed, 190 insertions, 56 deletions
diff --git a/PEDOT:PSS/agreement.png b/PEDOT:PSS/agreement.png Binary files differnew file mode 100644 index 0000000..1a7df33 --- /dev/null +++ b/PEDOT:PSS/agreement.png 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}
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