% 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 ...: % 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} 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 $ + \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. % 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} Representative WMELs can be found in Figures [xxxxxx]. % \section{Types of spectroscopy} % ================================================================ 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. % \subsection{Linear vs multidimensional} % -------------------------------------------------------- Most familiar spectroscopic experiments are linear. % That is to say, they have just one frequency axis, and they interrogate just one resonance condition. % These are workhorse experiments, like absorbance, reflectance, FTIR, UV-Vis, and common old ordinary Raman spectroscopy (COORS). % These experiments are incredibly robust, and are typically performed using easy to use commercial desktop instruments. % There are now even handheld Raman spectrometers for use in industrial settings. [CITE] % Multidimensional spectroscopy contains a lot more information about the material under investigation. % In this work, by ``multidimensional'' I mean higher-order spectroscopy. % I ignore ``correlation spectroscopy'' [CITE], which tracks linear spectral features against non-spectral dimensions like lab time, pressure, and temperature. % So, in the context of this dissertation, multidimensional spectroscopy is synonymous with nonlinear spectroscopy. % 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? This means that nonlinear spectroscopy is typically very weak. % Still, nonlinear signals are fairly easy to isolate and measure using modern instrumentation, as this dissertation describes. % The most obvious advantage of multidimensional spectroscopy comes directly from the dimensionality itself. % Multidimensional spectroscopy can \emph{decongest} spectra with overlapping peaks by isolating peaks in a multidimensional resonance landscape. % Figure \ref{spc:fig:decongestion} shows... \begin{figure} \caption[Dimensionality and decongestion.]{ CAPTION TODO. } \label{spc:fig:decongestion} \end{figure} \subsection{Frequency vs time domain} % ---------------------------------------------------------- Broadly, there are two ways to collect nonlinear spectroscopic signals: frequency and time domain. % Both techniques involve exciting a sample with multiple pulses of light and measuring the output signal. % The techniques differ in how they resolve the multiple frequency axes of interest. % Frequency domain is probably the more intuitive strategy: frequency axes are resolved directly by iteratively tuning the frequency of excitation pulses against each-other. % 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. % 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. % 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. % 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. % 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] \item Pulse distortions become important. [CITE JONAS] \end{ditemize} %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}. % %See Paul's dissertation Time domain experiments require a phase-locked, independently controlled local oscillator in order to collect the interferogram at the heart of such techniques. % This local oscillator enhances the information-gathering power of time domain because it allows the experiment to explicitly collect nonlinear spectra with full phase information. % At the same time, the local oscillator requirement limits the flexability of the time-domain because it essentially requires that the output frequency must be the same as one of the inputs. % Novel, often fully coherent, experiments cannot be accomplished under this limitation. % %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{Homodyne vs heterodyne} % ------------------------------------------------------------ Within frequency domain multidimensional spectroscopy, one is free to use or forgo a local oscillator. % That is to say, frequency domain spectroscopy can be collected in a heterodyne or homodyne technique. % As discussed in the previous section, use of a local oscillator means that more useful phase information can be extracted from the spectrum. % At the same time, generation of a phase locked, controllable local oscillator can be cumbersome, limiting the flexibility of possible experiments. % Note that heterodyne techniques may be self heterodyned (as in transient absorption) or ``explicitly'' heterodyned with a local oscillator. % Besides the aforementioned phase information, probably the biggest difference between heterodyne and homodyne-detected experiments is their scaling with oscillator number density, $N$. % In all heterodyne spectroscopies, signal goes linearly, as $N$. % If the number of oscillators is doubled, the signal doubles. % In all homodyne spectroscopies, signal goes as $N^2$. % If the number of oscillators is doubled, the signal goes up by four times. % This is what we mean when we say that homodyne signals are ``intensity level'' and heterodyne signals are ``amplitude level''. % Recently we have been taking to representing homodyne-detected multidimensional experiments on the ``amplitude level'' by plotting the square root of the collected signal. % Many of the figures in this dissertation are plotted in this way. % In my opinion, this strategy makes interpretation of spectra easier. % Certainly it eases comparison with other experiments, like absorbance and COORS, which go as $N$. % One easy-to-miss consequence of homodyne collected experiments is the behavior of signals in delay space. % Since signal goes as $N^2$, signal decays much faster in homodyne-collected experiments. % If signal decays as a single exponential, the extracted decay is twice as fast for homodyne vs heterodyne-detected data. % [CITE DARIEN CORRECTION] \section{Instrumentation} % ====================================================================== In this section I introduce the key components of the MR-CMDS instrument. % \subsection{LASER} % ----------------------------------------------------------------------------- % TODO: add reference to MaimanTheodore.000 (ruby laser) \subsection{Optical parametric amplifiers} % ----------------------------------------------------- \subsection{Delay stages} % ---------------------------------------------------------------------- \subsection{Spectrometers} % ---------------------------------------------------------------------