% 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.

\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!