diff options
| -rw-r--r-- | BiVO4/chapter.tex (renamed from BiVO4.tex) | 0 | ||||
| -rw-r--r-- | MX2/chapter.tex (renamed from MX2.tex) | 0 | ||||
| -rw-r--r-- | PbSe/chapter.tex (renamed from PbSe.tex) | 0 | ||||
| -rw-r--r-- | colophon/chapter.tex (renamed from readme.tex) | 2 | ||||
| -rw-r--r-- | dissertation.pdf | bin | 9664415 -> 9667202 bytes | |||
| -rw-r--r-- | dissertation.tex | 68 | ||||
| -rw-r--r-- | hardware.tex | 37 | ||||
| -rw-r--r-- | hardware/auto/chapter.el | 7 | ||||
| -rw-r--r-- | hardware/chapter.tex | 72 | ||||
| -rw-r--r-- | instrument/chapter.tex (renamed from instrument.tex) | 0 | ||||
| -rw-r--r-- | introduction/chapter.tex (renamed from introduction.tex) | 0 | ||||
| -rw-r--r-- | materials/chapter.tex (renamed from materials.tex) | 0 | ||||
| -rw-r--r-- | procedures/chapter.tex (renamed from procedures.tex) | 0 | ||||
| -rw-r--r-- | public/chapter.tex (renamed from public.tex) | 0 | ||||
| -rw-r--r-- | software/chapter.tex (renamed from software.tex) | 0 | ||||
| -rw-r--r-- | spectroscopy/auto/chapter.el | 9 | ||||
| -rw-r--r-- | spectroscopy/chapter.tex (renamed from spectroscopy.tex) | 106 |
17 files changed, 207 insertions, 94 deletions
diff --git a/BiVO4.tex b/BiVO4/chapter.tex index 604a75b..604a75b 100644 --- a/BiVO4.tex +++ b/BiVO4/chapter.tex diff --git a/MX2.tex b/MX2/chapter.tex index 2914e65..2914e65 100644 --- a/MX2.tex +++ b/MX2/chapter.tex diff --git a/PbSe.tex b/PbSe/chapter.tex index 6d59c72..6d59c72 100644 --- a/PbSe.tex +++ b/PbSe/chapter.tex diff --git a/readme.tex b/colophon/chapter.tex index 3e76fc7..e0ace85 100644 --- a/readme.tex +++ b/colophon/chapter.tex @@ -1,4 +1,4 @@ -\chapter{README} +\chapter{Colophon} This chapter lays out the technical aspects of this dissertation as a software and data product, including instructions for obtaining the source and regeneration of figures and documents. %
\ No newline at end of file diff --git a/dissertation.pdf b/dissertation.pdf Binary files differindex 1b27c74..1011949 100644 --- a/dissertation.pdf +++ b/dissertation.pdf diff --git a/dissertation.tex b/dissertation.tex index 1282485..52f040a 100644 --- a/dissertation.tex +++ b/dissertation.tex @@ -34,6 +34,7 @@ \setlength\parindent{0pt}
\setlength{\parskip}{1em}
\usepackage{enumitem}
+\setlist{noitemsep, topsep=0pt, parsep=0pt, partopsep=0pt}
\renewcommand{\familydefault}{\sfdefault}
\newcommand{\RomanNumeral}[1]{\textrm{\uppercase\expandafter{\romannumeral #1\relax}}}
\usepackage{etoolbox}
@@ -95,13 +96,23 @@ \makeglossaries
\include{glossary}
+
+\usepackage{tocloft}
+
+\setlength\cftparskip{0pt}
+\setlength\cftbeforechapskip{-5pt}
+\setlength\cftbeforesecskip{-7pt}
+\setlength\cftbeforesubsecskip{-10pt}
+
\begin{document}
+% pre ---------------------------------------------------------------------------------------------
+
\begin{centering}
\thispagestyle{empty}
% TITLE PAGE
-Strategies for Coherent Multidimensional Spectroscopy of Semiconductors \\
+\textbf{Spectroscopy and Such (Working Title)} \\
\vspace{80 pt}
By \\
Blaise Jonathan Thompson \\
@@ -136,8 +147,6 @@ This dissertation is approved by the following members of the Final Oral Committ \listoffigures
\listoftables
-\doublespacing % double spacing required for body of paper
-
% ACKNOWLEDGEMENTS
\cleardoublepage
\chapter*{Acknowledgments}
@@ -152,46 +161,55 @@ This dissertation is approved by the following members of the Final Oral Committ \pagenumbering{gobble}
\cleardoublepage
-\begin{singlespace}
-
\setlength{\parskip}{\baselineskip}
\vspace*{2 cm}
-\noindent \emph{The explanatory stories that people find compelling are simple; are concrete rather than abstract; assign a larger role to talent, stupidity and intentions than to luck; and focus on a few striking events that happened rather than on the countless events that failed to happen.}
+\noindent \emph{The explanatory stories that people find compelling are simple; are concrete rather
+ than abstract; assign a larger role to talent, stupidity and intentions than to luck; and focus
+ on a few striking events that happened rather than on the countless events that failed to
+ happen.}
-\noindent \emph{The ultimate test of an explanation is whether it would have made the event predictable in advance.}
+\noindent \emph{The ultimate test of an explanation is whether it would have made the event
+ predictable in advance.}
-\noindent \emph{Paradoxically, it is easier to construct a coherent story when you know little, when there are fewer pieces to fit into the puzzle. Our comforting conviction that the world makes sense rests on a secure foundation: our almost unlimited ability to ignore our ignorance.}
+\noindent \emph{Paradoxically, it is easier to construct a coherent story when you know little,
+ when there are fewer pieces to fit into the puzzle. Our comforting conviction that the world
+ makes sense rests on a secure foundation: our almost unlimited ability to ignore our ignorance.}
\hfill -- Daniel Kahneman \cite{KahnemanDaniel2013.000}
\cleardoublepage
-\end{singlespace}
-
\pagebreak
+\doublespacing % double spacing required for body of paper
\pagenumbering{arabic}
-%chapters
-\include{introduction}
-\include{spectroscopy}
-\include{materials}
-\include{software}
-\include{instrument}
-\include{PbSe}
-\include{MX2}
-\include{BiVO4}
-
-%appendix
+% chapters ----------------------------------------------------------------------------------------
+
+\include{introduction/chapter}
+\include{spectroscopy/chapter}
+\include{materials/chapter}
+\include{software/chapter}
+\include{instrument/chapter}
+\include{PbSe/chapter}
+\include{MX2/chapter}
+\include{BiVO4/chapter}
+
+% appendix -----------------------------------------------------------------------------------------
+
\begin{appendix}
-\include{public}
-\include{procedures}
-\include{hardware}
-\include{readme}
+\include{public/chapter}
+\include{procedures/chapter}
+\include{hardware/chapter}
+\include{colophon/chapter}
\end{appendix}
+% post --------------------------------------------------------------------------------------------
+
+\pagenumbering{gobble}
+
\singlespacing
\renewcommand{\arraystretch}{2} % there is probably a better way...
\printglossaries
diff --git a/hardware.tex b/hardware.tex deleted file mode 100644 index b5a02d7..0000000 --- a/hardware.tex +++ /dev/null @@ -1,37 +0,0 @@ -\chapter{Hardware} % -----------------------------------------------------------------------------
-
-
-\section{Adjustable periscopes} % ----------------------------------------------------------------
-
-
-Our light sources take on horizontal or vertical polarizations according to which tuning process is
-used. %
-Our experiments are opinionated about polarization, so some strategy for aligning polarization is
-necessary. %
-Desire fully reflective, easy to switch without changing path length (delay) etc... For several
-years, we used brewster-angle polarization unifiers... %
-These worked by... %
-But these were very difficult to align, and they were too lossy for some of the weaker tuning
-processes. %
-As an alternative, we designed a more traditional periscope with adjustability for our unique
-needs. %
-
-\begin{figure}[htp!]
- \centering
- \includegraphics[scale=0.1]{"hardware/periscope"}
- \label{f:periscope}
- \caption{CAPTION TODO}
-\end{figure}
-
-\begin{enumerate}
- \item in flipped polarization:
- \begin{itemize}
- \item stage near
- \item upper mirror far
- \end{itemize}
- \item in kept polarization:
- \begin{itemize}
- \item stage x and upper mirror height near
- \item lower mirror far
- \end{itemize}
-\end{enumerate}
diff --git a/hardware/auto/chapter.el b/hardware/auto/chapter.el new file mode 100644 index 0000000..bff5926 --- /dev/null +++ b/hardware/auto/chapter.el @@ -0,0 +1,7 @@ +(TeX-add-style-hook + "chapter" + (lambda () + (LaTeX-add-labels + "f:periscope")) + :latex) + diff --git a/hardware/chapter.tex b/hardware/chapter.tex new file mode 100644 index 0000000..c1b505d --- /dev/null +++ b/hardware/chapter.tex @@ -0,0 +1,72 @@ +\chapter{Hardware} % -----------------------------------------------------------------------------
+
+In this chapter I collect some of the specific hardware contribution details that do not belong in
+the body of the dissertation. %
+
+\section{Adjustable periscopes} % ----------------------------------------------------------------
+
+OPAs output horizontal or vertical polarizations according to which tuning process is used. %
+Our experiments are opinionated about polarization, so some strategy for aligning polarization is
+necessary. % TODO: cite opinionated about polarization
+In addition, it is useful to bring all excitation beams to the same height. %
+To this end, I designed and constructed two adjustable periscopes. %
+Each periscope is designed to bring OPA output to table height standard (5 inches) while either
+keeping or switching polarization. %
+Both polarization configurations take the same path length, so source polarization can be switched
+without large changes to zero delay. %
+All of this is done with just two (switched polarization) or three (kept polarzation)
+reflections. %
+A picture of these periscopes is shown in \ref{f:periscope}. %
+
+\begin{figure}[htp!]
+ \centering
+ \includegraphics[width=\textwidth]{"hardware/periscope"}
+ \label{f:periscope}
+ \caption{CAPTION TODO}
+\end{figure}
+
+While these periscopes are easy to align, their unique design means that it is not necessarily
+obvious what the correct strategy is. %
+The following strategy will always converge:
+\begin{enumerate}
+ \item use two ``magic'' apertures along the output beamline
+ \item in flipped polarization (two mirror configuration):
+ \begin{itemize}
+ \item use the stage (green X, Y) to align near aperture
+ \item use the upper mirror (yellow TA, TB) to align far aperture
+ \item iterate above
+ \end{itemize}
+ \item in kept polarization (three mirror configuration):
+ \begin{itemize}
+ \item use stage X (green X) and upper mirror height (yellow TC) to align near aperture
+ \item use lower mirror (pink SA, SB) to align far aperture
+ \item iterate above
+ \end{itemize}
+\end{enumerate}
+The kept polarization alignment is derivative of the fixed polarization alignment. %
+One must ensure that the fixed polarization is correctly aligned at all times. %
+
+Mirror B (aqua) is magnetically mounted to switch between polarization conditions. %
+Ensure that the lower turning mirror (pink) does not bump into mirror B (aqua) in polarization
+swtiching configuration. %
+The lower turning mirror is on a rail (pink SC). %
+This rail is a rough adjust for the same degree of freedom as pink SA. %
+Adjust the rail only to ensure that the beam is roughly centered on the free aperture of the
+turning mirror. %
+
+The first reflection is often accomplished using a wedge, as OPA output may be strong enough to
+damage downstream optics. %
+This optic can and should be replaced if more of the OPA output is desired on the table (keeping
+damage thresholds in mind). %
+
+\subsection{Wedge polarization preference}
+
+TODO: wedges will be more efficent at reflecting horizontal / vertical at 45 degrees
+
+\section{Automated transmissive filters} % -------------------------------------------------------
+
+TODO
+
+\section{Electronics} % --------------------------------------------------------------------------
+
+TODO
\ No newline at end of file diff --git a/instrument.tex b/instrument/chapter.tex index 6eee6e8..6eee6e8 100644 --- a/instrument.tex +++ b/instrument/chapter.tex diff --git a/introduction.tex b/introduction/chapter.tex index 210b1e6..210b1e6 100644 --- a/introduction.tex +++ b/introduction/chapter.tex diff --git a/materials.tex b/materials/chapter.tex index e868421..e868421 100644 --- a/materials.tex +++ b/materials/chapter.tex diff --git a/procedures.tex b/procedures/chapter.tex index 3cb42bf..3cb42bf 100644 --- a/procedures.tex +++ b/procedures/chapter.tex diff --git a/public.tex b/public/chapter.tex index 9c859e7..9c859e7 100644 --- a/public.tex +++ b/public/chapter.tex diff --git a/software.tex b/software/chapter.tex index e2c9652..e2c9652 100644 --- a/software.tex +++ b/software/chapter.tex diff --git a/spectroscopy/auto/chapter.el b/spectroscopy/auto/chapter.el new file mode 100644 index 0000000..ae8a477 --- /dev/null +++ b/spectroscopy/auto/chapter.el @@ -0,0 +1,9 @@ +(TeX-add-style-hook + "chapter" + (lambda () + (LaTeX-add-labels + "fig:ta_and_tr_setup" + "eq:ta_complete" + "eq:generic")) + :latex) + diff --git a/spectroscopy.tex b/spectroscopy/chapter.tex index 4e3546e..27d763b 100644 --- a/spectroscopy.tex +++ b/spectroscopy/chapter.tex @@ -14,15 +14,18 @@ In this chapter I lay out the foundations of spectroscopy. \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.
+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?
+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.
+Many strategies have been introduced for diagrammatically representing the interaction of multiple
+electric fields in an experiment. %
\subsubsection{Circle Diagrams}
@@ -36,11 +39,19 @@ Many strategies have been introduced for diagrammatically representing the inter \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.
+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 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.
@@ -54,13 +65,16 @@ So-called wave mixing energy level (\gls{WMEL}) diagrams are the most familiar w \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
+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
+Further derivation adapted from \cite{KumarNardeep2013.000}. %
+To extend reflectivity to a differential measurement
% TODO: finish derivation
\section{Coherent Multidimensional Spectroscopy}
@@ -88,30 +102,43 @@ Raman \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.
+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.
+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:
-
+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}.
+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}.
+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.
+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.
+Triply Electronically Enhanced (TrEE) spectroscopy has become the workhorse homodyne-detected 4WM
+experiment in the Wright Group. %
% TODO: On and off-diagonal TrEE pathways
@@ -123,7 +150,9 @@ Triply Electronically Enhanced (TrEE) spectroscopy has become the workhorse homo \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.
+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
@@ -132,9 +161,11 @@ Transient absorbance (TA) spectroscopy is a self-heterodyned technique. Through \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
+\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
+Typically one attempts to calculate the change in absorbance $\Delta A$ \dots %
\begin{eqnarray}
\Delta A &=& A_{\mathrm{on}} - A_{\mathrm{off}} \\
@@ -171,9 +202,13 @@ Define two new proportionality constants... 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.
+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 $I_0$.
+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}}} \\
@@ -184,7 +219,8 @@ 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.
+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)
@@ -198,15 +234,18 @@ Wonderfully, the $I_0$ cancels when plugged back in to \autoref{eq:ta_complete}, \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.
+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).
-
-
+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}:
@@ -214,8 +253,9 @@ The driven polarization, $P$, goes as the product of my input pulse \textit{inte 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
-
+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}}.
@@ -231,7 +271,9 @@ The width of the polarization (across $T$) is therefore % 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
+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}} \\
@@ -249,7 +291,9 @@ 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.
+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}
|