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+(TeX-add-style-hook
+ "chapter"
+ (lambda ()
+   (LaTeX-add-labels
+    "fig:ta_and_tr_setup"
+    "eq:ta_complete"
+    "eq:generic"))
+ :latex)
+
diff --git a/spectroscopy/chapter.tex b/spectroscopy/chapter.tex
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+% TODO: discuss and cite CerulloGiulio2003.000

+% TODO: discuss and cite BrownEmilyJ1999.000

+% TODO: cite and discuss Sheik-Bahae 1990 (first z-scan)

+

+\chapter{Spectroscopy}

+

+In this chapter I lay out the foundations of spectroscopy.

+

+\section{Light}

+

+% TODO: add reference to HuygensChristiaan1913.000

+

+% TODO: add reference to MaimanTheodore.000

+

+\section{Light-Matter Interaction}

+

+Spectroscopic experiments all derive from the interaction of light and matter. Many material

+properties can be deduced by measuring the nature of this interaction.  %

+

+Nonlinear spectroscopy relies upon higher-order terms in the light-matter interaction. In a generic

+system, each term is roughly ten times smaller than the last.  % TODO: cite?

+

+% TODO: Discuss dephasing induced resonance. Example: florescence

+

+\subsection{Representations}

+

+Many strategies have been introduced for diagrammatically representing the interaction of multiple

+electric fields in an experiment.  %

+

+\subsubsection{Circle Diagrams}

+

+% TODO: add reference to YeeTK1978.000

+

+% TODO: Discuss circle diagrams from a historical perspective

+

+\subsubsection{Double-sided Feynman Diagrams}

+

+% TODO: Discuss double-sided Feynman diagrams from a historical perspective

+

+\subsubsection{WMEL Diagrams}

+

+So-called wave mixing energy level (\gls{WMEL}) diagrams are the most familiar way of representing

+spectroscopy for Wright group members.  %

+\gls{WMEL} diagrams were first proposed by Lee and Albrecht in an appendix to their seminal work

+\emph{A Unified View of Raman, Resonance Raman, and Fluorescence Spectroscopy}

+\cite{LeeDuckhwan1985.000}.  %

+\gls{WMEL} diagrams are drawn using the following rules.  %

+\begin{enumerate}

+	\item The energy ladder is represented with horizontal lines - solid for real states and dashed

+    for virtual states.

+	\item Individual electric field interactions are represented as vertical arrows. The arrows span

+    the distance between the initial and final state in the energy ladder.

+	\item The time ordering of the interactions is represented by the ordering of arrows, from left

+    to right.

+	\item Ket-side interactions are represented with solid arrows.

+	\item Bra-side interactions are represented with dashed arrows.

+	\item Output is represented as a solid wavy line.

+\end{enumerate}

+

+\subsubsection{Mukamel Diagrams}

+

+% TODO: Discuss Mukamel diagrams from a historical perspective

+

+\section{Linear Spectroscopy}

+

+\subsection{Reflectivity}

+

+This derivation adapted from \textit{Optical Processes in Semiconductors} by Jacques I. Pankove

+\cite{PankoveJacques1975.000}.  %

+For normal incidence, the reflection coefficient is

+\begin{equation}

+R = \frac{(n-1)^2+k^2}{(n+1)^2+k^2}

+\end{equation}

+% TODO: finish derivation

+

+Further derivation adapted from \cite{KumarNardeep2013.000}.  %

+To extend reflectivity to a differential measurement

+% TODO: finish derivation

+

+\section{Coherent Multidimensional Spectroscopy}

+

+% TODO: (maybe) include discussion of photon echo famously discovered in 1979 in Groningen

+

+\gls{multiresonant coherent multidimensional spectroscopy}

+

+

+\subsection{Three Wave}

+

+\subsection{Four Wave}

+

+Fluorescence

+

+Raman

+

+\subsection{Five Wave}

+

+\subsection{Six Wave}

+

+\gls{multiple population-period transient spectroscopy} (\Gls{MUPPETS})

+

+\section{Strategies for CMDS}

+

+\subsection{Homodyne vs. Heterodyne Detection}

+

+Two kinds of spectroscopies: 1) \gls{heterodyne} 2) \gls{homodyne}.

+Heterodyne techniques may be \gls{self heterodyne} or explicitly heterodyned with a local

+oscillator.

+

+In all heterodyne spectroscopies, signal goes as $\gls{N}$.  %

+In all homodyne spectroscopies, signal goes as $\gls{N}^2$.  %

+This literally means that homodyne signals go as the square of heterodyne signals, which is what we

+mean when we say that homodyne signals are intensity level and heterodyne signals are amplitude

+level.

+

+\Gls{transient absorption}, \gls{TA}

+

+\subsection{Frequency vs. Time Domain}

+

+Time domain techniques become more and more difficult when large frequency bandwidths are

+needed.  %

+With very short, broad pulses:  %

+\begin{itemize}

+	\item Non-resonant signal becomes brighter relative to resonant signal

+	\item Pulse distortions become important.

+\end{itemize}

+

+This epi-CARS paper might have some useful discussion of non-resonant vs resonant for shorter and

+shorter pulses \cite{ChengJixin2001.000}.  %

+

+An excellent discussion of pulse distortion phenomena in broadband time-domain experiments was

+published by \textcite{SpencerAustinP2015.000}.  %

+

+Another idea in defense of frequency domain is for the case of power studies.  %

+Since time-domain pulses in-fact possess all colors in them they cannot be trusted as much at

+perturbative fluence.  %

+See that paper that Natalia presented...  %

+

+\subsection{Triply Electronically Enhanced Spectroscopy}

+

+Triply Electronically Enhanced (TrEE) spectroscopy has become the workhorse homodyne-detected 4WM

+experiment in the Wright Group.  %

+

+% TODO: On and off-diagonal TrEE pathways

+

+% TODO: Discussion of old and current delay space

+

+\subsection{Transient Absorbance Spectroscopy} 

+

+\Gls{transient absorption} (\gls{TA})

+

+\subsubsection{Quantitative TA}

+

+Transient absorbance (TA) spectroscopy is a self-heterodyned technique.  %

+Through chopping you can measure nonlinearities quantitatively much easier than with homodyne

+detected (or explicitly heterodyned) experiments.

+

+\begin{figure}[p!]

+	\centering

+	\includegraphics[width=\textwidth]{"spectroscopy/TA setup"}

+	\label{fig:ta_and_tr_setup}

+	\caption{CAPTION TODO}

+\end{figure}

+

+\autoref{fig:ta_and_tr_setup} diagrams the TA measurement for a generic sample.  %

+Here I show measurement of both the reflected and transmitted probe beam \dots not important in

+opaque (pyrite) or non-reflective (quantum dot) samples \dots  %

+

+Typically one attempts to calculate the change in absorbance $\Delta A$ \dots  %

+

+\begin{eqnarray}

+\Delta A &=& A_{\mathrm{on}} - A_{\mathrm{off}} \\

+&=& -\log_{10}\left(\frac{I_\mathrm{T}+I_\mathrm{R}+I_{\Delta\mathrm{T}} + I_{\Delta\mathrm{R}}}{I_0}\right) + \log\left(\frac{I_\mathrm{T}+I_\mathrm{R}}{I_0}\right) \\

+&=& -\left(\log_{10}(I_\mathrm{T}+I_\mathrm{R}+I_{\Delta\mathrm{T}}+ I_{\Delta\mathrm{R}})-\log_{10}(I_0)\right)+\left(\log_{10}(I_\mathrm{T}+I_\mathrm{R})-\log_{10}(I_0)\right) \\

+&=& -\left(\log_{10}(I_\mathrm{T}+I_\mathrm{R}+I_{\Delta\mathrm{T}}+ I_{\Delta\mathrm{R}})-\log_{10}(I_\mathrm{T}+I_\mathrm{R})\right) \\

+&=& -\log_{10}\left(\frac{I_\mathrm{T}+I_\mathrm{R}+I_{\Delta\mathrm{T}}+ I_{\Delta\mathrm{R}}}{I_\mathrm{T}+I_\mathrm{R}}\right) \label{eq:ta_complete}

+\end{eqnarray}

+

+\autoref{eq:ta_complete} simplifies beautifully  if reflectivity is negligible \dots

+

+Now I define a variable for each experimental measurable:

+\begin{center}

+	\begin{tabular}{c | l}

+		$V_\mathrm{T}$ & voltage recorded from transmitted beam, without pump \\

+		$V_\mathrm{R}$ & voltage recorded from reflected beam, without pump \\

+		$V_{\Delta\mathrm{T}}$ & change in voltage recorded from transmitted beam due to pump \\

+		$V_{\Delta\mathrm{R}}$ & change in voltage recorded from reflected beam due to pump

+	\end{tabular}

+\end{center}

+

+We will need to calibrate using a sample with a known transmisivity and reflectivity constant:

+\begin{center}

+	\begin{tabular}{c | l}

+		$V_{\mathrm{T},\,\mathrm{ref}}$ & voltage recorded from transmitted beam, without pump \\

+		$V_{\mathrm{R},\,\mathrm{ref}}$ & voltage recorded from reflected beam, without pump \\

+		$\mathcal{T}_\mathrm{ref}$ & transmissivity \\

+		$\mathcal{R}_\mathrm{ref}$ & reflectivity

+	\end{tabular}

+\end{center}

+

+Define two new proportionality constants...

+\begin{eqnarray}

+C_\mathrm{T} &\equiv& \frac{\mathcal{T}}{V_\mathrm{T}} \\

+C_\mathrm{R} &\equiv& \frac{\mathcal{R}}{V_\mathrm{R}}

+\end{eqnarray}

+These are explicitly calibrated (as a function of probe color) prior to the experiment using the

+calibration sample.  %

+

+Given the eight experimental measurables ($V_\mathrm{T}$, $V_\mathrm{R}$, $V_{\Delta\mathrm{T}}$,

+$V_{\Delta\mathrm{R}}$, $V_{\mathrm{T},\,\mathrm{ref}}$, $V_{\mathrm{R},\,\mathrm{ref}}$,

+$\mathcal{T}_\mathrm{ref}$, $\mathcal{R}_\mathrm{ref}$) I can express all of the intensities in

+\autoref{eq:ta_complete} in terms of $I_0$.  %

+

+\begin{eqnarray}

+C_\mathrm{T} &=& \frac{\mathcal{T}_\mathrm{ref}}{V_{\mathrm{T},\,\mathrm{ref}}} \\

+C_\mathrm{R} &=& \frac{\mathcal{R}_\mathrm{ref}}{V_{\mathrm{R},\,\mathrm{ref}}} \\

+I_\mathrm{T} &=& I_0 C_\mathrm{T} V_\mathrm{T} \\

+I_\mathrm{R} &=& I_0 C_\mathrm{R} V_\mathrm{R} \\

+I_{\Delta\mathrm{T}} &=& I_0 C_\mathrm{T} V_{\Delta\mathrm{T}} \\

+I_{\Delta\mathrm{R}} &=& I_0 C_\mathrm{R} V_{\Delta\mathrm{R}}

+\end{eqnarray} 

+

+Wonderfully, the $I_0$ cancels when plugged back in to \autoref{eq:ta_complete}, leaving a final

+expression for $\Delta A$ that only depends on my eight measurables.  %

+

+\begin{equation}

+\Delta A = - \log_{10} \left(\frac{C_\mathrm{T}(V_\mathrm{T} + V_{\Delta\mathrm{T}}) + C_\mathrm{R}(V_\mathrm{R} + V_{\Delta\mathrm{R}})}{C_\mathrm{T} V_\mathrm{T} + C_\mathrm{R} V_\mathrm{R}}\right)

+\end{equation}

+

+\subsection{Cross Polarized TrEE}

+

+\subsection{Pump-TrEE-Probe}

+

+\Gls{pump TrEE probe} (\gls{PTP}).

+

+\section{Instrumental Response Function}

+

+The instrumental response function (IRF) is a classic concept in analytical science.  %

+Defining IRF becomes complex with instruments as complex as these, but it is still useful to

+attempt.  %

+

+It is particularly useful to define bandwidth.

+ 

+\subsection{Time Domain}

+

+I will use four wave mixing to extract the time-domain pulse-width.   %

+I use a driven signal \textit{e.g.} near infrared carbon tetrachloride response.  %

+I'll homodyne-detect the output.  %

+In my experiment I'm moving pulse 1 against pulses 2 and 3 (which are coincident).  %

+

+The driven polarization, $P$, goes as the product of my input pulse \textit{intensities}:

+

+\begin{equation}

+P(T) = I_1(t-T) \times I_2(t) \times I_3(t)

+\end{equation}

+

+In our experiment we are convolving $I_1$ with $I_2 \times I_3$.  %

+Each pulse has an \textit{intensity-level} width, $\sigma_1$, $\sigma_2$, and $\sigma_3$. $I_2

+\times I_3$ is itself a Gaussian, and  

+\begin{eqnarray}

+\sigma_{I_2I_3} &=& \dots \\

+&=& \sqrt{\frac{\sigma_2^2\sigma_3^2}{\sigma_2^2 + \sigma_3^2}}.

+\end{eqnarray}

+

+The width of the polarization (across $T$) is therefore

+

+\begin{eqnarray}

+\sigma_P &=& \sqrt{\sigma_1^2 + \sigma_{I_2I_3}^2} \\

+&=& \dots \\ 

+&=& \sqrt{\frac{\sigma_1^2 + \sigma_2^2\sigma_3^2}{\sigma_1^2 + \sigma_2^2}}. \label{eq:generic}

+\end{eqnarray}

+

+% TODO: determine effect of intensity-level measurement here

+

+I assume that all of the pulses have the same width.   %

+$I_1$, $I_2$, and $I_3$ are identical Gaussian functions with FWHM $\sigma$. In this case,

+\autoref{eq:generic} simplifies to  

+

+\begin{eqnarray}

+\sigma_P &=& \sqrt{\frac{\sigma^2 + \sigma^2\sigma^2}{\sigma^2 + \sigma^2}} \\

+&=& \dots \\

+&=& \sigma \sqrt{\frac{3}{2}}

+\end{eqnarray}

+

+Finally, since we measure $\sigma_P$ and wish to extract $\sigma$:

+

+\begin{equation}

+\sigma = \sigma_P \sqrt{\frac{2}{3}}

+\end{equation}

+

+Again, all of these widths are on the \textit{intensity} level.

+

+\subsection{Frequency Domain}

+

+We can directly measure $\sigma$ (the width on the intensity-level) in the frequency domain using a

+spectrometer.  %

+A tune test contains this information.  %

+

+\subsection{Time-Bandwidth Product}

+

+For a Gaussian, approximately 0.441

+

+% TODO: find reference

+% TODO: number defined on INTENSITY level!

+

+

+

+

+

+

+

+