From 92b0eaa59ac85b9fa7b4bbb61ce84949e96e286b Mon Sep 17 00:00:00 2001
From: Blaise Thompson <blaise@untzag.com>
Date: Fri, 13 Apr 2018 22:39:33 -0500
Subject: 2018-04-13 22:39

---
 spectroscopy/chapter.tex | 109 +++++++++++++++++++++++++----------------------
 1 file changed, 58 insertions(+), 51 deletions(-)

(limited to 'spectroscopy/chapter.tex')

diff --git a/spectroscopy/chapter.tex b/spectroscopy/chapter.tex
index cdd3955..819fe94 100644
--- a/spectroscopy/chapter.tex
+++ b/spectroscopy/chapter.tex
@@ -14,9 +14,9 @@
   
 \clearpage
 
-In this chapter I lay out the foundations of spectroscopy as relevant to this dissertation.  %
-Spectroscopy is the study of the interaction of light (electromagnetic radiation) and matter
-(molecules, crystals, solids, liquids etc).  %
+%In this chapter I lay out the foundations of spectroscopy as relevant to this dissertation.  %
+%Spectroscopy is the study of the interaction of light (electromagnetic radiation) and matter
+%(molecules, crystals, solids, liquids etc).  %
 
 \section{Light-matter interaction}  % =============================================================
 
@@ -36,7 +36,12 @@ many photons.  %
 
 % TODO: language from 'how a photon is created or destroyed'
 
-For simplicity, consider a two state system: ``a'' and ``b''.  %
+The basics of light matter interaction have been covered in many texts.  %
+For a beginners introduction I recommend ``How a Photon is Created or Absorbed'' by
+\textref{HendersonGiles1994a}.  %
+Here I present a very minimal overview.  %
+
+Consider a two state system: ``a'' and ``b''.  %
 These two states might be the inital and final states in a transition.  %
 The wavefunction for this system can be written as a sum of the stationary states (eigenstates)
 with appropriate scaling coefficients:
@@ -47,42 +52,41 @@ The time dependence lies in the $c_a$ and $c_b$ coefficients, and the spatial de
 the $\psi_a$ and $\psi_b$ eigienstates.  %
 
 Now we will expose this two-state system to an electric field:
-\begin{equation}
-  E = E^{\circ}\left[ \me^{i(kz-\omega t)} + \me^{-i(kz-\omega t)} \right]
+\begin{equation} \label{spc:eqn:E}
+  E = E^{0}\left[ \me^{i(kz-\omega t)} + \me^{-i(kz-\omega t)} \right]
 \end{equation}
 
 For simplicity, we consider a single transition dipole, $\mu$.  %
 
 The Hamiltonian which controls the coupling of or simple system to the electric field described in
-...:
-% jcw- ISN'T IT JUST MU DOT E WHERE E IS A VECTOR THAT IS TIME DEPENDENT, NOT A TIME DERIVATIVE 
-\begin{equation}
-  H = H_{\circ} - \mu \dot E
-\end{equation}
+\autoref{spc:eqn:E} can be written.  %
+\begin{eqnarray}
+  H &=& H_{0} - \mu \cdot E \\
+  &=& H_{0} - \mu \cdot \frac{E^0}{2}\left[ \me^{i(kz-\omega t)} + \me^{-i(kz-\omega t)} \right]
+\end{eqnarray}
 
 Solving for the time-dependent coefficients, then:
 \begin{eqnarray}
   c_a(t) &=& \cos{\frac{\Omega t}{2}} \me^{-i\omega_at} \\
   c_b(5) &=& \sin{\frac{\Omega t}{2}} \me^{-i\omega_bt}
 \end{eqnarray}
-Fast and slow parts...
-Bohr and Rabi freuencies...
-
-Where $\Omega$ is the \emph{Rabi frequency}:  %
+Where $\omega_a$ and $\omega_b$ are the fast (and familiar) Bohr frequencies and $\Omega$ is the
+\emph{Rabi frequency}:  %
 \begin{equation}
   \Omega \equiv \frac{\mu E^\circ}{\hbar}
 \end{equation}
 
-In Dirac notation \cite{DiracPaulAdrienMaurice1939a}., an observable (such as $\mu(t)$) can be written simply:  %
+In Dirac notation \cite{DiracPaulAdrienMaurice1939a}, an observable (such as $\mu(t)$) can be
+written simply:  %
 \begin{equation}
-  \mu(t) = \left< c_aa + c_bb \left| \hat{\mu} \right| c_aa + c_bb \right>
+  \mu(t) = \left< c_aa + c_bb \left| \hat{H} \right| c_aa + c_bb \right>
 \end{equation}
-The complex wavefunction is called a \emph{ket}, represented $|b>$.  %
-The complex conjugate is called a \emph{bra}, represented $<a|$.  %
-When expanded,  % JCW- MU IS NOT THE OPERATOR. THE OPERATOR IS THE TIME DEPENDENT HAMILTONIAN. MU MULIPLIES ca and cb
+The complex wavefunction is called a \emph{ket}, represented $\left|b\right>$.  %
+The complex conjugate is called a \emph{bra}, represented $\left<a\right|$.  %
+When expanded,
 \begin{equation}
-  \mu(t) = c_a^2\mu_a + c_b^2\mu_b + \left< c_aa \left| \hat{mu} \right| c_bb \right> +
-  \left<c_bb \left| \hat{mu} \right| c_aa \right>
+  \mu(t) = c_a^2\mu_a + c_b^2\mu_b + \left< c_aa \left| \hat{H} \right| c_bb \right> +
+  \left<c_bb \left| \hat{H} \right| c_aa \right>
 \end{equation}
 The first two terms are populations and the final two terms are coherences.  %
 The coherent terms will evolve with the rapid Bohr oscillations, coupling the dipole observable
@@ -91,18 +95,9 @@ with the time-dependent electric field.  %
 We commonly represent quantum mechanical systems using density matrices, where diagonal elements
 are populations and off-diagonal elements are coherences.  %
 Each density matrix element has the form $\rho_{kb}$, where $k$ is the ket and $b$ is the bra.  %
-% TODO: 4 member density matrix representing system above
 A more complete discussion of the formalism we use to describe light-matter interaction is
 presented in \autoref{cha:mix}.  %
 
-% TODO: homogeneous line-width
-
-Spectroscopic experiments are typically performed on an ensemble of states.  %
-In such circumstances, inhomogeneous broadening becomes relevant.  %
-Inhomogeneous broadening arises from permanent differences between different oscillators in the
-ensemble.  %
-% TODO: why is inhomogeneous broadening important?
-
 Many strategies have been introduced for diagrammatically representing the interaction of multiple
 electric fields in an experiment.  %
 Spectroscopists have used diagrams to represent nonlinear optical phenomena since 1965.
@@ -139,9 +134,8 @@ WMELs can be found throughout this dissertation.  %
 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 detail regarding a few experiments that are particularly relevant to this
-dissertation.  %
+I take a compare-and-contrast approach, rather than getting too caught up in the infinite diversity
+of possible spectroscopic strategies.  %
 
 \subsection{Linear vs multidimensional}  % --------------------------------------------------------
 
@@ -172,6 +166,8 @@ The most obvious advantage of multidimensional spectroscopy comes directly from
 itself.  %
 Multidimensional spectroscopy can \emph{decongest} spectra with overlapping peaks by isolating
 peaks in a multidimensional resonance landscape.  %
+This decongestion arises directly from the multiple resonances the multidimensional spectroscopy
+demands of a material.  %
 
 \subsection{Frequency vs time domain}  % ----------------------------------------------------------
 
@@ -188,30 +184,32 @@ 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.  %
+The delay (phase) 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.  %
+This dissertation focuses on less-popular 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.  %
+Not so for multidimensional spectroscopy.  %
 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.  %
+more optomechanical engineering---more moving parts.  %
 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{ChengJixin2001a}
+	\item Non-resonant signal becomes brighter relative to resonant signal (the resonance advantage
+    is lost). \cite{ChengJixin2001a}
 	\item Pulse distortions become essentially unavoidable. \cite{SpencerAustinP2015a}
 \end{ditemize}
 
@@ -268,13 +266,14 @@ heterodyne-detected data.  %
 \section{Instrumentation}  % ======================================================================
 
 In this section I introduce the key components of the MR-CMDS instrument.  %
+This also serves to introduce the reader to the particular components used in my research.  %
 
 \subsection{LASER}  % -----------------------------------------------------------------------------
 
 Light Amplified by Stimulated Emission of Radiation (LASER) light sources are absolutely crucial
 components of the modern MR-CMDS instrument.  %
 The first laser was built in 1960 by \textcite{MaimanTheodore1960a}, and pulsed lasers were
-invented soon after [CITE].  %
+invented soon after.  %
 Today, ultrafast light sources are relatively cheap and reliable.  %
 Our SpectraPhysics ``Tsunami'' oscillator uses passive Kerr-lens mode-locking to generate $\sim$35
 fs seed pulses at $\sim$80 MHz (one pulse every 12.5 nanoseconds). \cite{Tsunami}  %
@@ -298,23 +297,31 @@ idler the lower energy photon.  %
 Signal and idler are then either used directly, or amplified through sum or difference frequency
 processes to provide broadband tuneability.  %
 All available optical processes for the TOPAS-C OPAs \cite{TOPAS-C} used on the femtosecond table
-are shown in \autoref{spc:fig:power_curves}.  %
+are shown in \autoref{spc:fig:ranges}.  %
+Amazingly, these OPAs offer tunability from the mid-infrared to the ultraviolet.  %
+Currently we do not have proper automated filters to make it possible to continuously scan between
+regions, but such things are possible.  %
 
 On the picosecond table we have three separate kinds of OPAs, including one TOPAS-800
 \cite{TOPAS-800} and two OPA-800 models that have been modified with precision micro control
 \cite{PMC} servo motors to provide automated tunability.  %
 
-\begin{figure}
-  \caption[TOPAS-C optical processes]{
-    CAPTION TODO.
-  }
-  \label{spc:fig:power_curves}
-\end{figure}
-
 \subsection{Delay stages}  % ----------------------------------------------------------------------
 
-Delay stages are simple, one-motor devices which...
+Delay stages are simple, one-motor devices which are used to control the relative arrival time of
+pulses at the sample.  %
+The Wright Group currently owns four models of delay stage: 1. Newport MFA-CC [CITE], 2. Aerotech
+... [CITE], 3. Thorlabs LTS300 [CITE].  %
+The fourth ``model'' are actually homemade stages that are driven using PMC motors.  %
 
 \subsection{Spectrometers}  % ---------------------------------------------------------------------
 
-Spectrometers...
\ No newline at end of file
+Spectrometers...
+
+\begin{figure}
+  \includegraphics[width=\linewidth]{spectroscopy/ranges}
+  \caption{
+    CAPTION TODO
+  }
+  \label{spc:fig:ranges}
+\end{figure}
\ No newline at end of file
-- 
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