From ee6453ad4a44984bd354ff5220c32336a4df3c3a Mon Sep 17 00:00:00 2001 From: Blaise Thompson Date: Sat, 31 Mar 2018 17:56:55 -0500 Subject: 2018-03-31 17:56 --- spectroscopy/chapter.tex | 228 +++++++++++++++++++++++++++++++---------------- 1 file changed, 151 insertions(+), 77 deletions(-) (limited to 'spectroscopy/chapter.tex') diff --git a/spectroscopy/chapter.tex b/spectroscopy/chapter.tex index fe10f60..9062e89 100644 --- a/spectroscopy/chapter.tex +++ b/spectroscopy/chapter.tex @@ -10,7 +10,7 @@ % TODO: https://pubs.acs.org/doi/abs/10.1021/acs.jpcb.7b02693 % TODO: http://journals.sagepub.com/doi/10.1177/0003702816669730 -\chapter{Spectroscopy} +\chapter{Spectroscopy} \label{cha:spc} \begin{dquote} A hundred years ago, Auguste Comte, … a great philosopher, said that humans will never be able to @@ -26,48 +26,113 @@ \clearpage -In this chapter I lay out the foundations of spectroscopy. +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. % -\section{Light} +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} -% TODO: add reference to HuygensChristiaan1913.000 +For simplicity, we consider a single transition dipole, $\mu$. % -% TODO: add reference to MaimanTheodore.000 +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} -\section{Light-Matter Interaction} +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... -Spectroscopic experiments all derive from the interaction of light and matter. Many material -properties can be deduced by measuring the nature of this interaction. % +Where $\Omega$ is the \emph{Rabi frequency}: % +\begin{equation} + \Omega \equiv \frac{\mu E^\circ}{\hbar} +\end{equation} -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? +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 $ + + \left +\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. % -% TODO: Discuss dephasing induced resonance. Example: florescence +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}. % -\subsection{Representations} +% 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. % -\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{LeeDuckhwan1985a}. % \gls{WMEL} diagrams are drawn using the following rules. % -\begin{enumerate} +\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 @@ -77,15 +142,13 @@ spectroscopy for Wright group members. % \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} +\end{denumerate} -% TODO: Discuss Mukamel diagrams from a historical perspective +% TODO: representative WMEL? -\section{Linear Spectroscopy} +\section{Types of spectroscopy} % ================================================================ -\subsection{Reflectivity} +\subsection{Linear vs multidimensional} % -------------------------------------------------------- This derivation adapted from \textit{Optical Processes in Semiconductors} by Jacques I. Pankove \cite{PankoveJacques1975a}. % @@ -99,30 +162,14 @@ Further derivation adapted from \cite{KumarNardeep2013a}. % 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} +% TODO: spectral congestion figure -\subsection{Six Wave} - -\gls{multiple population-period transient spectroscopy} (\Gls{MUPPETS}) - -\section{Strategies for CMDS} +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 Detection} +\subsection{Homodyne vs heterodyne} % ------------------------------------------------------------ Two kinds of spectroscopies: 1) \gls{heterodyne} 2) \gls{homodyne}. Heterodyne techniques may be \gls{self heterodyne} or explicitly heterodyned with a local @@ -134,9 +181,7 @@ This literally means that homodyne signals go as the square of heterodyne signal 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} +\subsection{Frequency vs time domain} % ---------------------------------------------------------- Time domain techniques become more and more difficult when large frequency bandwidths are needed. % @@ -157,7 +202,7 @@ Since time-domain pulses in-fact possess all colors in them they cannot be trust perturbative fluence. % See that paper that Natalia presented... % -\subsection{Triply Electronically Enhanced Spectroscopy} +\subsection{Transient grating} % ----------------------------------------------------------------- Triply Electronically Enhanced (TrEE) spectroscopy has become the workhorse homodyne-detected 4WM experiment in the Wright Group. % @@ -166,7 +211,34 @@ experiment in the Wright Group. % % TODO: Discussion of old and current delay space -\subsection{Transient Absorbance Spectroscopy} +% 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} % -------------------------------------------------------------- \Gls{transient absorption} (\gls{TA}) @@ -247,13 +319,23 @@ expression for $\Delta A$ that only depends on my eight measurables. % \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 CMDS-probe} % ------------------------------------------------------------------- + +\clearpage +\section{Instrumentation} % ====================================================================== + -\subsection{Pump-TrEE-Probe} +\subsection{LASER} % ----------------------------------------------------------------------------- -\Gls{pump TrEE probe} (\gls{PTP}). +% TODO: add reference to MaimanTheodore.000 (ruby laser) -\section{Instrumental Response Function} +\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 @@ -261,7 +343,7 @@ attempt. % It is particularly useful to define bandwidth. -\subsection{Time Domain} +\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. % @@ -310,23 +392,15 @@ Finally, since we measure $\sigma_P$ and wish to extract $\sigma$: Again, all of these widths are on the \textit{intensity} level. -\subsection{Frequency Domain} +\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. % -\subsection{Time-Bandwidth Product} +\subsubsection{Time-Bandwidth Product} For a Gaussian, approximately 0.441 % TODO: find reference -% TODO: number defined on INTENSITY level! - - - - - - - - +% TODO: number defined on INTENSITY level! \ No newline at end of file -- cgit v1.2.3