IPAC 2012

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\begin{document} \title{TRANSVERSE BEAM EMITTANCE MEASUREMENTS OF\\A 16 MeV LINAC AT THE IDAHO ACCELERATOR CENTER (IAC)} %\title{TRANSVERSE BEAM EMITTANCE MEASUREMENTS OF A 16 MeV LINAC AT THE IAC\thanks{ Work supported by ...}}

\author{S. Setiniyaz, K. Chouffani, T. Forest, Y. Kim\\

move the names C. O'Neill, C. F. Eckman, and D. Wells to an acknowledgement section

Idaho State University, Pocatello, ID, 83209, USA\\ A. Freyberger, Jefferson Lab, Newport News, Virginia, 23606, USA}

\maketitle

\begin{abstract} A beam emittance measurement of a 16 MeV S-band High Repetition Rate Linac (HRRL) was performed at Idaho State University's Idaho Accelerator Center (IAC). The HRRL is one of several low energy accelerators operating at the IAC. Originally, the linac structure of the HRRL was similar to that used by a typical medical linac. Its RF system was upgraded to facilitate a maximum repetition rate of 1 kHz. The transverse beam emittance of the HRRL is currently underway to optimize the production of positrons using an intense electron beam on a tungsten converter. In this paper we describe a beam imaging system based on an OTR screen and a digital CCD camera, a MATLAB tool to extract beamsize and emittance, detailed measurement procedures, and the results of measured transverse emittances for an arbitrary beam energy. The HRRL beam line is being reconfigured into an achromat to be used as a positron source. \end{abstract}

\section{Introduction}

The HRRL is an S-band electron linac located in the Beam Lab of the Physics Department at Idaho State University (ISU). It is one of the fifteen low energy linacs operated by the IAC. The HRRL accelerate electrons to energies between 3~MeV and 16~MeV with a maximum repetition rate of 1 kHz. The HRRL beamline had recently been reconfigured to generate and collect positrons. More detailed operational parameters on HRRL are summarized in Table~\ref{tab:hrrl}.

Originally a Yttrium Aluminum Garnet (YAG ) screen was used to measure emittance. It was observed that the electron beam image was large due to the high sensitivity of YAG~\cite{otr-yag}. It was decided to use OTR which is better for for high current beam. When a relativistic electron beam crosses through the boundary of two mediums with different dielectric constants, Optical Transition Radiation (OTR) is emitted. Photons are emitted both in the forward direction, along the particle motion, and backward direction~\cite{OTR}. When beam is incident with 45${^\circ}$, backward radiation emitted with a 90${^\circ}$ angle to the incident beam direction~\cite{OTR2}. Backward-emitted photons are observed in a camera for beam diagnostics. Shape and intensity of the beam image reflects its size and distribution.

Emittance is an important parameter in accelerator physics. It is the a parameter to determine the quality of electron beam. Emittance measurement can be done in a few ways~\cite{emit-ways, sole-scan-Kim}. Quadrupole scanning method~\cite{quad-scan} was used to measure emittance, Twiss parameters, and energy of the beam.

\begin{table}[hbt]

  \centering
  \caption{Operational Parameters of HRRL Linac.}
  \begin{tabular}{lccc}
      \toprule
      Parameter     & Unit   & Value \\ 
      \midrule
       maximum electron beam energy $E$   &  MeV     &  16   \\
       \midrule
       electron beam peak current $I_{\textnormal{peak}}$ &  mA      &  80     \\
       \midrule
       macro-pulse repetition rate                   &   Hz       &  1000  \\
       \midrule
       macro-pulse pulse length (FWHM)          &   ns       &  250    \\
       \midrule
       rms energy spread                                &  \%      &   4.23   \\
 \bottomrule

\end{tabular} \label{tab:hrrl} \end{table}

\section{The experiment}

\subsection{Theory of Quadrupole Scanning Method}

As shown in Fig.~\ref{q-scan-layout}, in quadrupole scanning, the quadrupole next to the exit of the linac was used for focusing/de-focusing the beam to the screen located downstream. The quadrupole and the screen are located far away to minimize chromatic effects and for better thin lens approximation.

\begin{figure}[htb] \centering

 \includegraphics[scale=0.40]{quad_scan_setup.eps}

\caption{Layout for quadrupole scanning method.} \label{q-scan-layout} \end{figure}

Assuming that thin lens approximation, $\sqrt{k_1}L << 1$, is satisfied, the transfer matrix of a quadrupole magnet is given by % thin lens approximation (sqrt{k1}*L << 1). In our case sqrt{k1}*L =0.07 \begin{equation} \label{quad-trans-matrix} \mathrm{\mathbf{Q}}=\Bigl(\begin{array}{cc} 1 & 0\\ -k_{1}L & 1 \end{array}\Bigr)=\Bigl(\begin{array}{cc} 1 & 0\\ -\frac{1}{f} & 1 \end{array}\Bigr), \end{equation} where $k_{1}$ is the quadrupole strength, $L$ is the length of quadrupole, and $f$ is the focal length. The matrix of drift space between quadrupole and screen is given by \begin{equation} \label{drift-trans-matrix} \mathbf{\mathbf{S}}=\Bigl(\begin{array}{cc} 1 & l\\ 0 & 1 \end{array}\Bigr), \end{equation} where $l$ is the distance between the scanning quadrupole and the screen. The transfer matrix of the scanning region is given by \begin{equation} \mathbf{\mathbf{M}}=\mathrm{\mathbf{SQ}} =\Bigl(\begin{array}{cc} m_{11} & m_{12}\\ m_{21} & m_{22} \end{array}\Bigr). \end{equation} For horizontal plane, beam matrix at the screen $\mathbf{\sigma_{s}}$ is related to the beam matrix of the quadrupole $\mathbf{\sigma_{q}}$ through means of a similarity transformation \begin{equation} \mathbf{\mathbf{\sigma_{s}=M\mathrm{\mathbf{\mathbf{\sigma_{q}}}}}M}^{\mathrm{T}} \end{equation} where the $\mathbf{\sigma_{s}}$ is defined as~\cite{SYLee}\\

\begin{equation} \mathbf{\mathbf{\sigma_{s,\mathnormal{x}}=}}\Bigl(\begin{array}{cc} \sigma_{\textnormal{s},x}^{2} & \sigma_{\textnormal{s},xx'}\\ \sigma_{\textnormal{s},xx'} & \sigma_{\textnormal{s},x'}^{2} \end{array}\Bigr), \end{equation}

\begin{equation} \mathbf{\mathbf{\sigma_{q,\mathnormal{x}}}}=\Bigl(\begin{array}{cc} \sigma_{\textnormal{q},x}^{2} & \sigma_{\textnormal{q},xx'}\\ \sigma_{\textnormal{q},xx'} & \sigma_{\textnormal{q},x'}^{2} \end{array}\Bigr). \end{equation}

\noindent By defining new parameters~\cite{quad-scan}, \begin{equation} A \equiv \sigma_{11},~B \equiv \frac{\sigma_{12}}{\sigma_{11}},~C \equiv \frac{\epsilon_{x}^{2}}{\sigma_{11}} \end{equation} the matrix element describes square of beam size at the screen. $\sigma_{\textnormal{s},x}^{2}$, becomes a parabolic function of the product of $k_1$ and $L$ \begin{equation} \sigma_{\textnormal{s},x}^{2}=A(k_{1}L)^{2}-2AB(k_{1}L)+(C+AB^{2}) \label{q_scan_layout} \end{equation}

When the quadrupole current, $k_{1}L$, is changed, the beam image at screen changes accordingly. The beam image was projected to the $x$, $y$ axes and fitted with Gaussian function to extract rms value, which is $\sigma_{s,x}$ in Eq.~(\ref{q_scan_layout}). With rms values, plot $\sigma_{x}^2$ $vs$ $k_{1}L$ was created and fitted with parabolic function to find the constants $A$, $B$, and $C$. After that, with the relation given in Eq.~(\ref{emit-relation}), emittance and Twiss parameters can be found. \begin{equation} \epsilon=\sqrt{AC},~\beta=\sqrt{\frac{A}{C}},~\alpha=-B\sqrt{\frac{A}{C}} \label{emit-relation} \end{equation}

\subsection{Imaging System Setup} The OTR target is 10 $\mu$m thick aluminum foil with 1.25 inches of diameter. OTR is emitted in a cone shape with the maximum intensity at an angle $1/\gamma$ with respect to the reflecting angle of the electron beam~\cite{OTR}. To avoid optical distortion, when OTR is performed at lower energy regions, three lenses with 2 inches in diameter are used for the imaging system. Focal lengths and position of the lenses are shown in Fig.~\ref{image_sys}.

\begin{figure} \begin{tabular}{lr} {\scalebox{0.25} [0.25]{\includegraphics{image_sys.eps}}} {\scalebox{0.3} [0.30]{\includegraphics{imaging_sys}}} \end{tabular} \caption{Square of rms and parabolic fitting for x-projection.} \label{image_sys} \end{figure}

\subsection{Quadrupole Scanning} At the beginning of the scanning, beam was steered by the quadrupole due to the mis-alignment of the beam. To align the beam, the dipole, quadrupoles, solenoids and steering magnets on the linac were turned off. Beam current was observed by a Faraday cup located at the end of beamline. This Current was maximized by first steering sets near to the electron gun. Then the first solenoid near the gun was turned to focus the beam at the OTR screen. When the beam was steered by the solenoid, steering magnets were reset. These process was repeated until beam was not steered by the solenoid. Then beam was focused even more by the second solenoid near the exit of the linac, and round and minimum size beam was observed on the screen. The second set of steering magnets near the exit of the linac then used to steer the beam for the maximum current the Faraday cup. After these proses, the steering of the beam by quadrupole was minimized.

Scanning was performed at 14~MeV beam energy and 40~mA macro pulse peak current with the first quadrupole after the exit of the linac, and all the other quadrupoles and dipoles were turned off. The quadrupole current was increased from -5~A to 5~A with an increment of 0.2~A. Seven scans were performed to take average along with background. To see the dark current, backgrounds were taken when the RF was on and the gun was off. Background image and beam images before and after background subtraction were shown in Fig.~\ref{bg}. Dark currents were visible from images. It was generated when electrons on the wall of the cavity were ``polled off" and accelerated. Background images were taken while the RF was on and electron gun was off.

\begin{figure} \begin{tabular}{ccc} \centerline{\scalebox{0.3} [0.25]{\includegraphics{sg_no_bg_subtraction_0Amp.eps}}} \\ \centerline{\scalebox{0.3} [0.25]{\includegraphics{Background.eps}}}\\ \centerline{\scalebox{0.3} [0.25]{\includegraphics{bg_subtracted_0Amp.eps}}} \end{tabular} \caption{Background subtracted to minimize impact of dark current; (top) a beam with the dark current and background noises, (middle) a background image, (bottom) a beam image when dark background was subtracted.} \label{bg} \end{figure}

After scanning, the current was measured with the Faraday cup, for calculation of a single bunch charge. The dipole was turned on to bend beam by 45 degree for energy measurement. The beam current were observed at another second Faraday cup located at the end of the 45 degree bend. The strongest signal observed while dipole was set for the 14~MeV electron beam. %${^\circ}$ \subsection{Data Analysis and Results} Images need to be converted from camera pixels to physical length. Diameter of the OTR screen is 31.75~mm. The scaling factor can be obtained by dividing this length with the pixels numbers in the image. Horizontal scaling factor is 0.04327$\pm$0.00016~mm/pixel, and vertical scaling factor is 0.04204$\pm$0.00018~mm/pixel.

In the data analysis, the beam image was projected to a single axis and fitted it with Gaussian function. The beam projection is sharper than Gaussian distribution, it is more like Lorentzian. However, the rms of the Lorentzian function is not defined. Fitting it with the super Gaussian function seems to be best option~\cite{sup-Gau}, since rms values can be extracted.

Two plots were generated with square of rms ($\sigma^2$) $vs$ $k_1L$ for $x$ (horizontal) and $y$ (vertical) projections of beam profiles, as shown in Fig.~\ref{par-fit} along with parabolic fits. Emittances and Twiss parameters from fits summarized in Table.~\ref{results}. All these were done using MATLAB. For details, please look at~\cite{emit-mat}.

%\subsection{Measured Results} %Parabolic fits are plotted in Fig.~\ref{par-fit}.

%Parabolic fits for x and y projections are given by Eq.~(\ref{eq:x-fit-eq}) and ~(\ref{eq:y-fit-eq}), which are plotted in Fig.~\ref{par-fit-x} and Fig.~\ref{par-fit-y}. %\begin{equation} %\sigma_x^2 = (3.68 \pm 0.02) + (-4.2 \pm 0.2)k_{1}L + (5.6 \pm 0.4)(k_{1}L)^2 %\label{eq:x-fit-eq} %\end{equation} %\begin{equation} %\sigma_y^2 = (2.84 \pm 0.04) + (1.0 \pm 0.5)k_{1}L + (3.8 \pm 1.2)(k_{1}L)^2 %\label{eq:y-fit-eq} %\end{equation}

\begin{figure} \begin{tabular}{cc} {\scalebox{0.21} [0.20]{\includegraphics{par_fit_x.eps}}} {\scalebox{0.21} [0.20]{\includegraphics{par_fit_y.eps}}} \end{tabular} \caption{Square of rms values and parabolic fittings.} \label{par-fit} \end{figure}

%\begin{figure} %\begin{tabular}{cc} %\centerline{\scalebox{0.20} [0.20]{\includegraphics{par_fit_x.eps}}} \\ %\centerline{\scalebox{0.20} [0.20]{\includegraphics{par_fit_y.eps}}} %\end{tabular} %\caption{Square of rms values and parabolic fittings.} %\label{par-fit} %\end{figure}

%Projected emittance $\epsilon$, normalized emittance $\epsilon_n$, and Twiss parameters are shown in Table~\ref{results}. %\begin{table}[hbt] % \centering % \caption{Emittance Measurement Results.} % \begin{tabular}{llll} % \toprule % \textbf{} & \textbf Unit & Horizontal Plane & Vertical Plane\\ % \midrule % $\epsilon$ &$\mu$m & $0.369 \pm 0.019$ & $0.294 \pm 0.038$ \\ % $\epsilon_n$ &$\mu$m & $10.10 \pm 0.51$ & $8.06 \pm 1.1 $ \\ % $\beta$ &m & $1.40 \pm 0.06$ & $1.17 \pm 0.13$ m \\ % $\alpha$ &rad & $0.97 \pm 0.06$ & $0.24 \pm 0.07$ rad \\ % \bottomrule % \end{tabular} % \label{results} %\end{table}

\begin{table}[hbt]

  \centering
  \caption{Emittance Measurement Results.}
  \begin{tabular}{lcc}
      \toprule
       {Parameter}         & {Unit}     &    {Value}    \\ 
      \midrule
        projected emittance $\epsilon_x$        &   $\mu$m    &    $0.37 \pm 0.02$     \\
         projected emittance $\epsilon_y$            &   $\mu$m    &    $0.30 \pm 0.04$     \\

normalized emittance $\epsilon_{n,x}$ & $\mu$m & $10.10 \pm 0.51$ \\ normalized emittance $\epsilon_{n,y}$ & $\mu$m & $8.06 \pm 1.1$ \\

        $\beta_x$-function                            &  m                           &   $1.40  \pm  0.06$          \\
        $\beta_y$-function                                &  m                           &   $1.17   \pm 0.13$         \\

$\alpha_x$-function & rad & $0.97 \pm 0.06$ \\ $\alpha_y$-function & rad & $0.24 \pm 0.07$ \\ single bunch charge & pC & 11 \\ energy of the beam $E$ & MeV & 14 \\

 \bottomrule
  \end{tabular}
  \label{results}

\end{table}

\section{Conclusions} Diagnostic tools to measure the beam emittance at the HRRL was established. Electron beam profiles from HRRL are not Gaussian, but rather super Gaussian or Lorentzian. When beam was projected to a single plane, due to more pixel numbers of horizontal plane, projection to vertical plane has bigger signal to noise ratio. Thus rms beam sizes measured in vertical plane have bigger error bars, which also lead to bigger estimated errors on the results of vertical beam profile.

%\section{ACKNOWLEDGMENT} %Thanks to A, B, and C.

%\begin{figure} %\begin{tabular}{cc} %\centerline{\includegraphics[width=60mm]{par_fit_x.eps}} \\ %\centerline{\includegraphics[width=60mm]{par_fit_y.eps}} %\end{tabular} %\caption{Square of rms and parabolic fitting for x-projection.} %\end{figure} % %\begin{figure}[htb] %\centerline %\scalebox{0.21} [0.3]{{\includegraphics{sg_no_bg_subtraction_0Amp.png}}}\\ %\scalebox{0.21} [0.3]{{\includegraphics{Background.png}}}\\ %\scalebox{0.21} [0.3]{{\includegraphics{bg_subtracted_0Amp.png}}} %\caption{\small Background subtraction.} %\label{par-fit} %\end{figure} % %\begin{figure}[htb] %\centerline{ %%\scalebox{0.21} [0.28]{{\includegraphics{par_fit_x.png}}}\\ %%\scalebox{0.21} [0.28]{\includegraphics{par_fit_y.png}}} %\scalebox{0.21} [0.3]{{\includegraphics{par_fit_x.eps}}}\\ %\scalebox{0.21} [0.3]{\includegraphics{par_fit_y.eps}}} %\caption{\small $\sigma^2$ from super Gaussian and parabolic fittings.} %\label{par-fit} %\end{figure} %\begin{figure}[htb] % \centering % \includegraphics*[width=80mm]{image_sys} % \caption{Imaging system.} % \label{image_sys} %\end{figure} % %\begin{figure}[htb] % \centering % \includegraphics*[width=40mm]{imaging_sys} % \caption{Imaging system.} % \label{imaging_sys} %\end{figure}

%\begin{figure} % \centering % \subfloat[X-projection.]{\label{par-fit-x}\includegraphics[width=0.25\textwidth]{par_fit_x.eps}} % ~ %add desired spacing between images, e. g. ~, \quad, \qquad etc. (or a blank line to force the subfig onto a new line) % \subfloat[Y-projection.]{\label{par-fit-y}\includegraphics[width=0.25\textwidth]{par_fit_y.eps}} % ~ %add desired spacing between images, e. g. ~, \quad, \qquad etc. (or a blank line to force the subfig onto a new line) % \caption{Square of rms and parabolic fittings.} % \label{par-fit} % \label{fig:animals} %\end{figure}

%\begin{figure}[htb] % \centering % \includegraphics[width=60mm]{par_fit_x.eps} % \caption{Square of rms and parabolic fitting for x-projection.} % \label{par-fit-x} %\end{figure} %\begin{figure}[htb] % \centering % \includegraphics[width=60mm]{par_fit_y.eps} % \caption{Square of rms and parabolic fitting for y-projection.} % \label{par-fit-y} %\end{figure}

%\begin{figure} %\centering %\begin{tabular}{cc} %\epsfig{file=par_fit_x.eps,width=0.5\linewidth,clip=a} & %\epsfig{file=par_fit_y.eps,width=0.5\linewidth,clip=b} \\ %\epsfig{file=par_fit_x.eps,width=0.5\linewidth,clip=} & %\epsfig{file=par_fit_y.eps,width=0.5\linewidth,clip=} %\end{tabular} %\end{figure}

%\begin{table}[hbt] % \centering % \caption{HRRL Parameters.} % \begin{tabular}{lccc} % \toprule % \textbf{En} & {$\textbf I_{peak}$} & \textbf{Rep Rate} & \textbf{Pulse Length}\\ % \midrule % 16 MeV & 80 mA & 1 kHz & 250 ns (FWHM) \\ % \bottomrule %\end{tabular} %\label{tab:hrrl} %\end{table}

%\begin{figure*}[htb] % \centering % %\includegraphics*[width=168mm]{HRRL_BeamLine} % \includegraphics*[width=160mm]{HRRL_Pos_and_Ele_Go.pdf} % \caption{HRRL beamline for positron production. } % \label{hrrl-beamline} %\end{figure*}

%\bibliographystyle{aipproc} %\bibliography{bibtex} % %\IfFileExists{\jobname.bbl}{} % {\typeout{} % \typeout{******************************************} % \typeout{** Please run "bibtex \jobname" to optain} % \typeout{** the bibliography and then re-run LaTeX} % \typeout{** twice to fix the references!} % \typeout{******************************************} % \typeout{} % }

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