From 9c6afe7dc748bf26e4ab6f6df0c70d57e74b610f Mon Sep 17 00:00:00 2001
From: Blaise Thompson <blaise@untzag.com>
Date: Sat, 7 Apr 2018 14:56:52 -0500
Subject: 2018-04-07 14:56

---
 spectroscopy/chapter.tex | 285 ++++++++++++++++++++---------------------------
 1 file changed, 121 insertions(+), 164 deletions(-)

(limited to 'spectroscopy/chapter.tex')

diff --git a/spectroscopy/chapter.tex b/spectroscopy/chapter.tex
index 2cf088b..a70cd69 100644
--- a/spectroscopy/chapter.tex
+++ b/spectroscopy/chapter.tex
@@ -147,193 +147,150 @@ WMEL diagrams are drawn using the following rules.  %
 
 Representative WMELs can be found in Figures [xxxxxx].  %
 
-% TODO: representative WMEL?
-
 \section{Types of spectroscopy}  % ================================================================
 
 Scientists have come up with many ways of exploiting light-matter interaction for measurement
 purposes.  %
 This section discusses several of these strategies.  %
 I start broadly, by comparing and contrasting differences across categories of spectroscopies.  %
-I then go into relevant detail regarding a few experiments that are particularly relevant in this
+I then go into detail regarding a few experiments that are particularly relevant to this
 dissertation.  %
 
 \subsection{Linear vs multidimensional}  % --------------------------------------------------------
 
-This derivation adapted from \textit{Optical Processes in Semiconductors} by Jacques I. Pankove
-\cite{PankoveJacques1975a}.  %
-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{KumarNardeep2013a}.  %
-To extend reflectivity to a differential measurement
-% TODO: finish derivation
-
-% TODO: (maybe) include discussion of photon echo famously discovered in 1979 in Groningen
-
-% TODO: spectral congestion figure
+Most familiar spectroscopic experiments are linear.  %
+That is to say, they have just one frequency axis, and they interrogate just one resonance
+condition.  %
+These are workhorse experiments, like absorbance, reflectance, FTIR, UV-Vis, and common old
+ordinary Raman spectroscopy (COORS).  %
+These experiments are incredibly robust, and are typically performed using easy to use commercial
+desktop instruments.  %
+There are now even handheld Raman spectrometers for use in industrial settings. [CITE]  %
+
+Multidimensional spectroscopy contains a lot more information about the material under
+investigation.  %
+In this work, by ``multidimensional'' I mean higher-order spectroscopy.  %
+I ignore ``correlation spectroscopy'' [CITE], which tracks linear spectral features against
+non-spectral dimensions like lab time, pressure, and temperature.  %
+So, in the context of this dissertation, multidimensional spectroscopy is synonymous with nonlinear
+spectroscopy.  %
 
 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?
+This means that nonlinear spectroscopy is typically very weak.  %
+Still, nonlinear signals are fairly easy to isolate and measure using modern instrumentation, as
+this dissertation describes.  %
 
-TODO: Basic ``advantage of dimensionality'' figure.
-
-\subsection{Homodyne vs heterodyne}  % ------------------------------------------------------------
-
-Two kinds of spectroscopies: 1) heterodyne 2) homodyne.
-Heterodyne techniques may be self heterodyne or explicitly heterodyned with a local
-oscillator.
-
-In all heterodyne spectroscopies, signal goes as $N$.  %
-In all homodyne spectroscopies, signal goes as $N^2$.  %
-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.
-
-Homodyne dynamics go faster: cite Darien correction
-
-\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{ChengJixin2001a}.  %
-
-An excellent discussion of pulse distortion phenomena in broadband time-domain experiments was
-published by \textcite{SpencerAustinP2015a}.  %
-
-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...  %
-
-See Paul's dissertation
-
-\subsection{Transient grating}  % -----------------------------------------------------------------
-
-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
-
-% TODO: discuss current delay space physical conventions (see inbox)
-
-\begin{figure}
-  \includegraphics[scale=1]{"spectroscopy/wmels/trive_on_diagonal"}
-  \caption[CAPTION TODO]{
-    CAPTION TODO
-  }
-  \label{spc:fig:trive_on_diagonal}
-\end{figure}
-
+The most obvious advantage of multidimensional spectroscopy comes directly from the dimensionality
+itself.  %
+Multidimensional spectroscopy can \emph{decongest} spectra with overlapping peaks by isolating
+peaks in a multidimensional resonance landscape.  %
+Figure \ref{spc:fig:decongestion} shows...
 
 \begin{figure}
-  \includegraphics[scale=1]{"spectroscopy/wmels/trive_off_diagonal"}
-  \caption[CAPTION TODO]{
-    CAPTION TODO
+  \caption[Dimensionality and decongestion.]{
+    CAPTION TODO.
   }
-  \label{spc:fig:trive_off_diagonal}
-\end{figure}
-
-\begin{figure}
-  \includegraphics[scale=1]{"spectroscopy/wmels/trive_population_transfer"}
-  \caption[CAPTION TODO]{
-    CAPTION TODO
-  }
-  \label{spc:fig:trive_population_transfer}
-\end{figure}
-
-\subsection{Transient absorbance}  % --------------------------------------------------------------
-
-\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}
-	\includegraphics[width=\textwidth]{"spectroscopy/TA setup"}
-	\label{fig:ta_and_tr_setup}
-	\caption{CAPTION TODO}
+  \label{spc:fig:decongestion}
 \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.  %
+\subsection{Frequency vs time domain}  % ----------------------------------------------------------
 
-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 $I_0$.  %
+Broadly, there are two ways to collect nonlinear spectroscopic signals: frequency and time
+domain.  %
+Both techniques involve exciting a sample with multiple pulses of light and measuring the output
+signal.  %
+The techniques differ in how they resolve the multiple frequency axes of interest.  %
+
+Frequency domain is probably the more intuitive strategy: frequency axes are resolved directly by
+iteratively tuning the frequency of excitation pulses against each-other.  %
+This relies on pulsed light sources with tunable frequencies.  %
+
+Time domain experiments use an interferometric technique to resolve frequency axes.  %
+Broadband excitation pulses which contain all of the necessary frequencies are used to excite the
+sample.  %
+The delay (time) between pulses is scanned, and the resonances along that axis are resolved through
+Fourier transform of the resulting interferogram.  %
+In modern experiments, pulse shapers are used to control the delay between pulses in a very
+precise, fast, and reproducible way.  %
+The time domain strategy is by-far the most popular technique in multidimensional spectroscopy
+because these technologies allow for rapid, robust data collection.  %
+
+This dissertation focuses on frequency domain strategies, so some discussion of the advantages of
+frequency domain when compared to time domain are warranted.  %
+
+One of the biggest instrumental limitations of multidimensional spectroscopy is bandwidth.  %
+It is easy to get absorbance spectra over the entire visible spectrum, and even into the
+ultraviolet and near infrared.  %
+Multidimensional spectroscopy is limited by the bandwidth of our (tunable) light sources.  %
+For frequency domain techniques, this limitation is incidental: sources with greater tunability
+will be easy to incorporate into these instruments, and creating such sources is only a matter of
+more optomechanical engineering.  %
+Time domain techniques, on the other hand, have a more fundamental issue with bandwidth.  %
+Time domain requires that all of the desired frequencies be present within the single excitation
+pulse, and pulses with very large frequency bandwidth (very short in time) become very hard to use
+and control.   %
+With short, broad pulses:
+\begin{ditemize}
+	\item Non-resonant signal becomes brighter relative to resonant signal. [CITE]
+	\item Pulse distortions become important. [CITE JONAS]
+\end{ditemize}
+%This epi-CARS paper might have some useful discussion of non-resonant vs resonant for shorter and
+%shorter pulses \cite{ChengJixin2001a}.  %
+%An excellent discussion of pulse distortion phenomena in broadband time-domain experiments was
+%published by \textcite{SpencerAustinP2015a}.  %
+%See Paul's dissertation
+
+Time domain experiments require a phase-locked, independently controlled local oscillator in order
+to collect the interferogram at the heart of such techniques.  %
+This local oscillator enhances the information-gathering power of time domain because it allows the
+experiment to explicitly collect nonlinear spectra with full phase information.  %
+At the same time, the local oscillator requirement limits the flexability of the time-domain
+because it essentially requires that the output frequency must be the same as one of the inputs.  %
+Novel, often fully coherent, experiments cannot be accomplished under this limitation.  %
+
+%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...  %
 
-\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.  %
+\subsection{Homodyne vs heterodyne}  % ------------------------------------------------------------
 
-\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}
+Within frequency domain multidimensional spectroscopy, one is free to use or forgo a local
+oscillator.  %
+That is to say, frequency domain spectroscopy can be collected in a heterodyne or homodyne
+technique.  %
+As discussed in the previous section, use of a local oscillator means that more useful phase
+information can be extracted from the spectrum.  %
+At the same time, generation of a phase locked, controllable local oscillator can be cumbersome,
+limiting the flexibility of possible experiments.  %
+
+Note that heterodyne techniques may be self heterodyned (as in transient absorption) or
+``explicitly'' heterodyned with a local oscillator.  %
+
+Besides the aforementioned phase information, probably the biggest difference between heterodyne
+and homodyne-detected experiments is their scaling with oscillator number density, $N$.  %
+In all heterodyne spectroscopies, signal goes linearly, as $N$.  %
+If the number of oscillators is doubled, the signal doubles.  %
+In all homodyne spectroscopies, signal goes as $N^2$.  %
+If the number of oscillators is doubled, the signal goes up by four times.  %
+This is what we mean when we say that homodyne signals are ``intensity level'' and heterodyne
+signals are ``amplitude level''.  %
+
+Recently we have been taking to representing homodyne-detected multidimensional experiments on the
+``amplitude level'' by plotting the square root of the collected signal.  %
+Many of the figures in this dissertation are plotted in this way.  %
+In my opinion, this strategy makes interpretation of spectra easier.  %
+Certainly it eases comparison with other experiments, like absorbance and COORS, which go as
+$N$.  %
+
+One easy-to-miss consequence of homodyne collected experiments is the behavior of signals in delay
+space.  %
+Since signal goes as $N^2$, signal decays much faster in homodyne-collected experiments.  %
+If signal decays as a single exponential, the extracted decay is twice as fast for homodyne vs
+heterodyne-detected data.  %
+[CITE DARIEN CORRECTION]
 
-\clearpage
 \section{Instrumentation}  % ======================================================================
 
 In this section I introduce the key components of the MR-CMDS instrument.  %
-- 
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