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    \begin{tabular}{c}
        \huge
        Validation of a Model of a Resonant\\
        \huge 
        Optothermoacoustic Trace Gas Sensor\medskip \\
        \Large N.~Petra$^1$, J.~Zweck$^1$, S.~E.~Minkoff$^1$, 
A.~A.~Kosterev$^2$, and J.~H.~Doty~III$^2$\smallskip \\
\large $^1$Mathematics and Statistics,
         University of Maryland Baltimore County and
$^2$Electrical and Computer Engineering,
        Rice University
    \end{tabular}
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\boxit[box/header height = {12ex}]{col = 0, at top, name=box1}{
{\bf \color{\cred}R}esonant 
{\bf \color{\cred}O}pto{\bf \color{\cred}T}hermo{\bf \color{\cred}A}coustic 
{\bf \color{\cred}DE}tection\\ ({\bf \color{\cred} ROTADE})
}
{
\ctext  

ROTADE uses a quartz tuning fork (QTF) to detect diffusion waves
generated when optical radiation interacts with a gas

\begin{center}
     \includegraphics[width=0.8\textwidth]{PAS}
\\[1.5ex]
{\bf \color{\cgreen}
\large Detection}
  \end{center}

\begin{itemize}
\item Gas molecules diffuse and collide with QTF surface
\item Excited molecules transfer vibrational energy to QTF
\item Heating induces a stress and displacement of the QTF
\item A resonance is induced by modulating the laser
\item A current is generated via piezoelectricity of quartz
\end{itemize}

\begin{center}
{\bf \color{\cred}
Current is proportional to gas concentration.  
}
\end{center}


}

\boxit{col = 0, below of=box1, name=box2}
{ROTADE Characteristics}
{\ctext

  \begin{itemize}
    \item Effective at low ambient pressures ($\lesssim 50$~Torr)
    \item High wavelength selectivity
    \item Allows for analysis of small gas samples ($< 1$~mm$^3$)
    \item Offers immunity to environmental acoustic noise
  \end{itemize}
  }

\boxit{col = 1, at top, name=box3}
{Geometry of QTF}
{\ctext

\begin{center}
     \includegraphics[width=0.5\textwidth]{TFCh4}
\\[1ex]
Laser beam axis is parallel to $y$-axis through $(0,0,z)$
  \end{center}
}


\boxit{col = 1, below of=box3, name=box4}
{Helmholtz Equation for Diffusion Wave}
{\ctext

{\Large \color{\cred}
\begin{equation*}
 - \nabla \cdot ( k \nabla T) \,\,+\,\, i\omega_L  T \,\,=\,\, S
\end{equation*}
}

\begin{tabular}{ll}
$T$ = temperature & $\omega_L = $ laser modulation frequency \\
$k = $ thermal diffusivity & $S = $ laser source amplitude 
\end{tabular}
}

\boxit{col = 1, below of=box4, name=box5}
{Helmholtz Equation\\ for Thermoelastic Deformation}
{\ctext

{\Large \color{\cred}
\begin{equation*}
\nabla\cdot C[\nabla \mathbf{u}] \,\,+\,\,
(\rho \omega_{\text{TF}}^2 - ib\omega_{\text{TF}})\,\mathbf{u} \,\,=\,\,
 \nabla \cdot C[\alpha T]
\end{equation*}
}

\begin{tabular}{ll}
$\mathbf{u}  = $ displacement field  &  $C = $ elasticity tensor\\
$\rho =     $ density & $\omega_{\text TF} = $ QTF resonance freq. 
= $\omega_L$ \\
$b = $ damping constant & $\alpha = $ thermal expansion tensor
\end{tabular}

\begin{center}
 \includegraphics[width=0.7\textwidth]{Displacement_Comsol}
\\[1ex]
32.8~kHz eigenmode of vibration of QTF
\end{center}

}

\boxit{col = 2, at top, name=box6}
{Experimental Validation}
{\ctext

\begin{center}
     \includegraphics[width=0.45\textwidth]{ampthVSexpN2_semilog}\hskip2pt
     \includegraphics[width=0.45\textwidth]{phasethVSexpN2}
\\[1ex]
Amplitude (left) and phase (right) of signal as function of vertical
position, $z$, of source for 0.5\%  C$_2$H$_2$ in N$_2$ at 20~Torr
\end{center}

\begin{itemize}
\item Discrepancy for $z>0.25$~mm is due to QEPAS signal 
\item Signal is largest when laser is near base of QTF
\item Molecular dynamics of C$_2$H$_2$ do not influence signal
\end{itemize}

}

\boxit{col = 2, below of=box6, name=box7}
{Design Optimization of QTF}
{\ctext

\begin{center}
Numerical optimization was performed using 
the mesh-adaptive direct search (MADS) algorithm 
\\[1ex]  
     \includegraphics[height=0.3\textwidth]{STRESSmap_3kHzTF}\hskip5pt
     \includegraphics[height=0.3\textwidth]{STRESSmap_32p8kHzTF}
\\[1ex]
First fundamental stress of a 3~kHz QTF (left) whose
response is $24\times$ that of standard $32.8$~kHz QTF (right)
\end{center}
}

\boxit{col = 2, below of=box7, name=box8}
{Further Information}
{\ctext

\begin{center}
{\bf \color{\cred}
 N. Petra, \emph{et al.},
{\it SIAM J. Applied Mathematics}, {\bf 71} (1), 2011}
\\ [1ex]
Funded by the U.S. NSF MIRTHE-ERC program\\ [1ex]
\textsf{http://www.mirthecenter.org}
\end{center}
}

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