From 4ddc0bcecdd172e6fbed0df2e80dfc7663b6ab73 Mon Sep 17 00:00:00 2001
From: Blaise Thompson <blaise@untzag.com>
Date: Sun, 12 Nov 2017 18:51:13 -0600
Subject: structure

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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!
+
+
+
+
+
+
+
+
-- 
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