From ff2e66aeef80259d60df46261305d387ad604baa Mon Sep 17 00:00:00 2001 From: Blaise Thompson Date: Thu, 5 Apr 2018 17:29:20 -0500 Subject: 2018-04-05 17:29 --- dissertation.tex | 1 + irf/chapter.tex | 69 ++++++++++++++++++++++++++++++++++ spectroscopy/auto/chapter.el | 3 +- spectroscopy/chapter.tex | 88 ++++++++------------------------------------ 4 files changed, 87 insertions(+), 74 deletions(-) create mode 100644 irf/chapter.tex diff --git a/dissertation.tex b/dissertation.tex index b1e219b..d8bf32a 100644 --- a/dissertation.tex +++ b/dissertation.tex @@ -89,6 +89,7 @@ This dissertation is approved by the following members of the Final Oral Committ %\include{procedures/chapter} %\include{hardware/chapter} % TODO: consider inserting WrightTools documentation as PDF +\include{irf/chapter} %\include{errata/chapter} %\include{colophon/chapter} \end{appendix} diff --git a/irf/chapter.tex b/irf/chapter.tex new file mode 100644 index 0000000..07cd561 --- /dev/null +++ b/irf/chapter.tex @@ -0,0 +1,69 @@ +\chapter{Instrumental response function} \label{cha:irf} + +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! \ No newline at end of file diff --git a/spectroscopy/auto/chapter.el b/spectroscopy/auto/chapter.el index f316ae6..b067440 100644 --- a/spectroscopy/auto/chapter.el +++ b/spectroscopy/auto/chapter.el @@ -7,7 +7,6 @@ "spc:fig:trive_off_diagonal" "spc:fig:trive_population_transfer" "fig:ta_and_tr_setup" - "eq:ta_complete" - "eq:generic")) + "eq:ta_complete")) :latex) diff --git a/spectroscopy/chapter.tex b/spectroscopy/chapter.tex index 030edd5..2cf088b 100644 --- a/spectroscopy/chapter.tex +++ b/spectroscopy/chapter.tex @@ -145,10 +145,19 @@ WMEL diagrams are drawn using the following rules. % \item Output is represented as a solid wavy line. \end{denumerate} +Representative WMELs can be found in Figures [xxxxxx]. % + % TODO: representative WMEL? \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 relevant detail regarding a few experiments that are particularly relevant in this +dissertation. % + \subsection{Linear vs multidimensional} % -------------------------------------------------------- This derivation adapted from \textit{Optical Processes in Semiconductors} by Jacques I. Pankove @@ -170,6 +179,8 @@ To extend reflectivity to a differential measurement 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: Basic ``advantage of dimensionality'' figure. + \subsection{Homodyne vs heterodyne} % ------------------------------------------------------------ Two kinds of spectroscopies: 1) heterodyne 2) homodyne. @@ -182,6 +193,8 @@ 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. +Homodyne dynamics go faster: cite Darien correction + \subsection{Frequency vs time domain} % ---------------------------------------------------------- Time domain techniques become more and more difficult when large frequency bandwidths are @@ -203,6 +216,8 @@ Since time-domain pulses in-fact possess all colors in them they cannot be trust perturbative fluence. % See that paper that Natalia presented... % +See Paul's dissertation + \subsection{Transient grating} % ----------------------------------------------------------------- Triply Electronically Enhanced (TrEE) spectroscopy has become the workhorse homodyne-detected 4WM @@ -318,11 +333,10 @@ 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{Pump CMDS-probe} % ------------------------------------------------------------------- - \clearpage \section{Instrumentation} % ====================================================================== +In this section I introduce the key components of the MR-CMDS instrument. % \subsection{LASER} % ----------------------------------------------------------------------------- @@ -333,73 +347,3 @@ expression for $\Delta A$ that only depends on my eight measurables. % \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! \ No newline at end of file -- cgit v1.2.3