% TODO: discuss and cite CerulloGiulio2003.000
% TODO: discuss and cite BrownEmilyJ1999.000
% TODO: cite and discuss Sheik-Bahae 1990 (first z-scan)
% Modeling of Transient Absorption Spectra in Exciton–Charge-Transfer Systems 10.1021/acs.jpcb.6b09858
% TODO: Multidimensional Spectral Fingerprints of a New Family of Coherent Analytical Spectroscopies
% TODO: https://www.nature.com/articles/nature21425
% TODO: https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.76.4793
% TODO: https://www.nature.com/articles/ncomms2405
% TODO: https://www.nature.com/articles/ncomms2405
% TODO: https://pubs.acs.org/doi/abs/10.1021/acs.jpcb.7b02693
% TODO: http://journals.sagepub.com/doi/10.1177/0003702816669730

\chapter{Spectroscopy} \label{cha:spc}

\begin{dquote}
  A hundred years ago, Auguste Comte, … a great philosopher, said that humans will never be able to
  visit the stars, that we will never know what stars are made out of, that that's the one thing
  that science will never ever understand, because they're so far away.  %
  And then, just a few years later, scientists took starlight, ran it through a prism, looked at
  the rainbow coming from the starlight, and said: ``Hydrogen!''  %
  Just a few years after this very rational, very reasonable, very scientific prediction was made,
  that we'll never know what stars are made of.  %

  \dsignature{Michio Kaku}
\end{dquote}
  
\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).  %

\section{Light-matter interaction}  % =============================================================

As scientists, light is perhaps the most useful tool we have for interrogating materials.  %
Light is relatively easy to create and control, and light-matter interaction tells us a lot about
the microscopic physics of the material under investigation.  %
Spectroscopists use light-matter interaction as an analytical tool.  %
For the purposes of this document, light can be treated as a classical electromagnetic wave and
matter can be treated in the quantum mechanical density matrix formalism.  %
More complete treatments which also take the quantum-mechanical nature of light into account are
possible (see: ``quantum optics'', ``quantum electrodynamics''), but beyond the scope of this
dissertation.  %
This classical treatment still captures the full richness of the wave-nature of light, including
interference effects.  \cite{HuygensChristiaan1913a}  %
It merely ignores the quantitization of the electric field---a valid assumption in the limit of
many photons.  %

% TODO: language from 'how a photon is created or destroyed'

For simplicity, 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:
\begin{equation}
  \Psi(r, t) = c_a(t)\psi_a(r) + c_b(t)\psi_b(r)
\end{equation}
The time dependence lies in the $c_a$ and $c_b$ coefficients, and the spatial dependence lies in
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]
\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
...:
\begin{equation}
  H = H_{\circ} - \mu \dot E
\end{equation}

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}:  %
\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:  %
\begin{equation}
  \mu(t) = \left< c_aa + c_bb \left| \hat{\mu} \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,
\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>
\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
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.
\cite{WardJF1965a}  %
Several competing strategies have been defined over the years.  %
In 1978, \textcite{YeeTK1978a} defined the ``circle diagram'' convention.  %
Since then, the more popular  ``closed-time path-loop'' \cite{MarxChristophA2008a,
  RoslyakOleksiy2009a} and ``double-sided Feynman'' diagrams \cite{MukamelShaul1995a} (also known
as Mukamel diagrams) were introduced.  %
\textcite{BiggsJasonD2012a} have written a paper which does an excellent job defining and comparing
these two strategies.  %
In their seminal 1985 work, \emph{A Unified View of Raman, Resonance Raman, and Fluorescence
  Spectroscopy}, \textcite{LeeDuckhwan1985a} defined the conventions for a ``wave-mixing energy
level'' (WMEL) diagram.  %
Today, double-sided Feynman diagrams are probably most popular, but WMELs will be used in this
document due to author preference.  %

WMEL diagrams are drawn using the following rules.  %
\begin{denumerate}
	\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{denumerate}

% TODO: representative WMEL?

\section{Types of spectroscopy}  % ================================================================

\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

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?

\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.

\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...  %

\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}


\begin{figure}
  \includegraphics[scale=1]{"spectroscopy/wmels/trive_off_diagonal"}
  \caption[CAPTION TODO]{
    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}
\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{Pump CMDS-probe}  % -------------------------------------------------------------------

\clearpage
\section{Instrumentation}  % ======================================================================


\subsection{LASER}  % -----------------------------------------------------------------------------

% TODO: add reference to MaimanTheodore.000 (ruby laser)

\subsection{Optical parametric amplifiers}  % -----------------------------------------------------

\subsection{Delay stages}  % ----------------------------------------------------------------------

\subsection{Spectrometers}  % ---------------------------------------------------------------------

\subsection{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.
 
\subsubsection{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.

\subsubsection{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.  %

\subsubsection{Time-Bandwidth Product}

For a Gaussian, approximately 0.441

% TODO: find reference
% TODO: number defined on INTENSITY level!