From 4ddc0bcecdd172e6fbed0df2e80dfc7663b6ab73 Mon Sep 17 00:00:00 2001 From: Blaise Thompson Date: Sun, 12 Nov 2017 18:51:13 -0600 Subject: structure --- spectroscopy/chapter.tex | 311 +++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 311 insertions(+) create mode 100644 spectroscopy/chapter.tex (limited to 'spectroscopy/chapter.tex') diff --git a/spectroscopy/chapter.tex b/spectroscopy/chapter.tex new file mode 100644 index 0000000..27d763b --- /dev/null +++ b/spectroscopy/chapter.tex @@ -0,0 +1,311 @@ +% 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! + + + + + + + + -- cgit v1.2.3