Difference between revisions of "IPAC 2012"
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\begin{abstract} | \begin{abstract} | ||
| − | A beam emittance measurement of | + | A beam emittance measurement of the 16~MeV S-band High Repetition Rate Linac (HRRL) was performed at Idaho State University's Idaho Accelerator Center (IAC). The HRRL linac structure was upgraded beyond the capabilities of a typical medical linac so it can achieve a repetition rate of 1~kHz. Measurements of the HRRL transverse beam emittance are underway that will be used to optimize the production of positrons using HRRL's intense electron beam on a tungsten converter. In this paper, we describe a beam imaging system using on an OTR screen and a digital CCD camera, a MATLAB tool to extract beamsize and emittance, detailed measurement procedures, and the measured transverse emittances for an arbitrary beam energy of 15~MeV. |
\end{abstract} | \end{abstract} | ||
\section{Introduction} | \section{Introduction} | ||
| − | The HRRL is an S-band electron linac located in the beam lab of the Physics Department at Idaho State University (ISU). The HRRL accelerates electrons to energies between 3 and 16~MeV with a maximum repetition rate of 1~kHz. The HRRL beamline has recently been reconfigured to generate | + | The HRRL is an S-band electron linac located in the beam lab of the Physics Department at Idaho State University (ISU). The HRRL accelerates electrons to energies between 3 and 16~MeV with a maximum repetition rate of 1~kHz. The HRRL beamline has recently been reconfigured to generate positrons to be used as a secondary beam. |
%The electron beam characteristics of the HRRL are summarized in Table~\ref{tab:hrrl}. | %The electron beam characteristics of the HRRL are summarized in Table~\ref{tab:hrrl}. | ||
| − | An Optical Transition Radiation (OTR) based viewer was installed to allow measurements at the high electron currents available using the HRRL. The visible light from the OTR based viewer is produced when a relativistic electron beam crosses the boundary of two mediums with different dielectric constants. | + | An Optical Transition Radiation (OTR) based viewer was installed to allow measurements at the high electron currents available using the HRRL. The visible light from the OTR based viewer is produced when a relativistic electron beam crosses the boundary of two mediums with different dielectric constants. Visible radiation is emitted at an angle of 90${^\circ}$ with respect to the incident beam direction~\cite{OTR} when the electron beam intersects the OTR target at a 45${^\circ}$ angle. These backward-emitted photons are observed using a digital camera and can be used to measure the shape and intensity of the electron beam based on the OTR distribution. |
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| + | Emittance is a key parameter in accelerator physics that is used to quantify the quality of an electron beam produced by an accelerator. An emittance measurement can be performed in a several ways~\cite{emit-ways, sole-scan-Kim}. This work used the Quadrupole scanning method~\cite{quad-scan} to measure the emittance, Twiss parameters, and beam energy. | ||
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\section{The Experiment} | \section{The Experiment} | ||
\subsection{Quadrupole Scanning Method} | \subsection{Quadrupole Scanning Method} | ||
| − | + | Fig.~\ref{q-scan-layout} illustrates the apparatus used to measure the emittance using the quadrupole scanning method. A quadrupole is positioned at the exit of the linac to focus or de-focus the beam as observed on a downstream view screen. The 3.1~m distance between the quadrupole and the screen was chosen in order to minimize chromatic effects and to satisfy the thin lens approximation. | |
%The quadrupole and the screen are located far away to minimize chromatic effects and to increase the veracity of the thin lens approximation used to calculate beam optics. | %The quadrupole and the screen are located far away to minimize chromatic effects and to increase the veracity of the thin lens approximation used to calculate beam optics. | ||
\begin{figure}[htb] | \begin{figure}[htb] | ||
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\end{array}\Bigr), | \end{array}\Bigr), | ||
\end{equation} | \end{equation} | ||
| − | where $k_{1}$ is the quadrupole strength, $L$ is the length of quadrupole, and $f$ is the focal length. A matrix representing the drift space between quadrupole and screen is given by | + | where $k_{1}$ is the quadrupole strength, $L$ is the length of quadrupole, and $f$ is the focal length. A matrix representing the drift space between the quadrupole and screen is given by |
\begin{equation} | \begin{equation} | ||
\label{drift-trans-matrix} | \label{drift-trans-matrix} | ||
| Line 106: | Line 90: | ||
\end{array}\Bigr), | \end{array}\Bigr), | ||
\end{equation} | \end{equation} | ||
| − | where $l$ is the distance between the scanning quadrupole and the screen. The transfer matrix of the scanning region is given by the matrix product $\mathbf{ | + | where $l$ is the distance between the scanning quadrupole and the screen. The transfer matrix of the scanning region is given by the matrix product $\mathbf{SQ}$. |
In the horizontal plane, the beam matrix at the screen ($\mathbf{\sigma_{s}}$) is related to the beam matrix of the quadrupole ($\mathbf{\sigma_{q}}$) using the similarity transformation | In the horizontal plane, the beam matrix at the screen ($\mathbf{\sigma_{s}}$) is related to the beam matrix of the quadrupole ($\mathbf{\sigma_{q}}$) using the similarity transformation | ||
\begin{equation} | \begin{equation} | ||
| Line 129: | Line 113: | ||
A \equiv l^2\sigma_{\textnormal{q},x}^{2},~B \equiv \frac{1}{l} + \frac{\sigma_{\textnormal{q},xx'}}{\sigma_{\textnormal{q},x}^{2}},~C \equiv l^2\frac{\epsilon_{x}^{2}}{\sigma_{\textnormal{q},x}^{2}}. | A \equiv l^2\sigma_{\textnormal{q},x}^{2},~B \equiv \frac{1}{l} + \frac{\sigma_{\textnormal{q},xx'}}{\sigma_{\textnormal{q},x}^{2}},~C \equiv l^2\frac{\epsilon_{x}^{2}}{\sigma_{\textnormal{q},x}^{2}}. | ||
\end{equation} | \end{equation} | ||
| − | the matrix element | + | the matrix element $\sigma_{\textnormal{s},x}^{2}$, the square of the rms beam size at the screen, may be expressed as a parabolic function of the product of $k_1$ and $L$ |
\begin{equation} | \begin{equation} | ||
\sigma_{\textnormal{s},x}^{2}=A(k_{1}L)^{2}-2AB(k_{1}L)+(C+AB^{2}). | \sigma_{\textnormal{s},x}^{2}=A(k_{1}L)^{2}-2AB(k_{1}L)+(C+AB^{2}). | ||
\label{par_fit} | \label{par_fit} | ||
\end{equation} | \end{equation} | ||
| − | The emittance measurement was performed by changing the quadrupole current, which changes $k_{1}L$, and measuring the corresponding beam image on the view screen. The measured two-dimensional beam image was projected along the | + | The emittance measurement was performed by changing the quadrupole current, which changes $k_{1}L$, and measuring the corresponding beam image on the view screen. The measured two-dimensional beam image was projected along the image's abscissa and ordinate axes. A Gaussian fitting function is used on each projection to determine the rms value, $\sigma_\textnormal{s}$ in Eq.~(\ref{par_fit}). Measurements of $\sigma_\textnormal{s}$ for several quadrupole currents ($k_{1}L$) is then fit using the parabolic function in Eq.~(\ref{par_fit}) to determine the constants $A$, $B$, and $C$. The emittance ($\epsilon$) and the Twiss parameters ($\alpha$ and $\beta$) can be found using Eq.~(\ref{emit-relation}). |
\begin{equation} | \begin{equation} | ||
\epsilon=\frac{\sqrt{AC}}{l^2},~\beta=\sqrt{\frac{A}{C}},~\alpha=\sqrt{\frac{A}{C}}(B+\frac{1}{l}). | \epsilon=\frac{\sqrt{AC}}{l^2},~\beta=\sqrt{\frac{A}{C}},~\alpha=\sqrt{\frac{A}{C}}(B+\frac{1}{l}). | ||
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\end{equation} | \end{equation} | ||
\subsection{The OTR Imaging System} | \subsection{The OTR Imaging System} | ||
| − | The OTR target is 10 $\mu$m thick aluminum foil with a 1.25 inch | + | The OTR target is 10 $\mu$m thick aluminum foil with a 1.25 inch diameter. The OTR is emitted in a cone shape with the maximum intensity at an angle of $1/\gamma$ with respect to the reflecting angle of the electron beam~\cite{OTR}. Three lenses, 2 inches in diameter, are used for the imaging system to avoid optical distortion at lower electron energies. The focal lengths and position of the lenses are shown in Fig.~\ref{image_sys}. The camera used was a JAI CV-A10GE digital camera with a 767 by 576 pixel area. The camera images were taken by triggering the camera synchronously with the electron gun. |
\begin{figure} | \begin{figure} | ||
\centering | \centering | ||
| − | {\scalebox{0. | + | {\scalebox{0.16} [0.16]{\includegraphics{image_sys.eps}}} {\scalebox{0.20} [0.20]{\includegraphics{imaging_sys}}} |
\caption{The OTR Imaging system.} | \caption{The OTR Imaging system.} | ||
\label{image_sys} | \label{image_sys} | ||
\end{figure} | \end{figure} | ||
\subsection{Quadrupole Scanning} | \subsection{Quadrupole Scanning} | ||
| − | The current | + | The quadrupole current is changed to alter the strength and direction of the quadrupole magnetic field such that a measurable change in the beam shape is seen by the OTR system. Initially, the beam was steered by the quadrupole indicating that the beam was not entering along the quadrupole's central axis. Several magnetic elements upstream of this quadrupole were adjusted to align the incident electron beam with the quadrupole's central axis. First, the beam current observed by a Faraday cup located at the end of beam line was maximized using upstream steering coils within the linac nearest the gun. Second, the first solenoid nearest the linac gun was used to focus the electron beam on the OTR screen. Steering coils were adjusted to maximize the beam current to the Faraday cup and minimize the deflection of the beam by the solenoid first then by the quadrupole. A second solenoid and the last steering magnet, both near the exit of the linac, were used in the final step to optimize the beam spot size on the OTR target and maximize the Faraday cup current. A configuration was found that minimized the electron beam deflection when the quadrupole current was altered during the emittance measurements. |
| − | The emittance measurement was performed using an electron beam energy of 15~MeV | + | The emittance measurement was performed using an electron beam energy of 15~MeV and a 200~ns long, 40~mA, macro pulse peak current. The current in the first quadrupole after the exit of the linac was changed from $-$~5~A to 5~A with an increment of 0.2~A. Seven measurements were taken at each current step in order to determine the average beam width and the variance. Background measurements were taken by turning the linac's electron gun off while keep the RF on. Background image and beam images before and after background subtraction are shown in Fig.~\ref{bg}. A small dark current is visible in Fig.~\ref{bg}b that is known to be generated when electrons are pulled off the cavity wall and accelerated. |
\begin{figure} | \begin{figure} | ||
| Line 164: | Line 148: | ||
\label{bg} | \label{bg} | ||
\end{figure} | \end{figure} | ||
| − | The electron beam energy was measured using a dipole magnet downstream of the quadrupole used for the emittance measurements. Prior to energizing the dipole, the electron micro-pulse bunch charge passing through the dipole was measured using a Faraday cup located approximately 50~cm downstream. The dipole current was adjusted until a maximum beam current was observed on another Faraday cup located just after the 45 degree exit port of the dipole. A magnetic field map of the dipole suggests that the electron beam energy was 15~MeV. Future emittance measurements are planned to cover the entire energy range of the linac. | + | The electron beam energy was measured using a dipole magnet downstream of the quadrupole used for the emittance measurements. Prior to energizing the dipole, the electron micro-pulse bunch charge passing through the dipole was measured using a Faraday cup located approximately 50~cm downstream of the OTR screen. The dipole current was adjusted until a maximum beam current was observed on another Faraday cup located just after the 45 degree exit port of the dipole. A magnetic field map of the dipole suggests that the electron beam energy was 15~$\pm$~1.5~MeV. Future emittance measurements are planned to cover the entire energy range of the linac. |
\subsection{Data Analysis and Results} | \subsection{Data Analysis and Results} | ||
| − | Images from the JAI camera were calibrated using the OTR target frame. An LED was used to illuminate the OTR aluminum frame that has a known inner diameter of 31.75~mm. Image processing software was used to inscribe a circle on the image to measure the circular OTR inner frame in units of pixels. The scaling factor can be obtained by dividing this length with the number of pixels observed. The result is a horizontal scaling factor | + | Images from the JAI camera were calibrated using the OTR target frame. An LED was used to illuminate the OTR aluminum frame that has a known inner diameter of 31.75~mm. Image processing software was used to inscribe a circle on the image to measure the circular OTR inner frame in units of pixels. The scaling factor can be obtained by dividing this length with the number of pixels observed. The result is a horizontal scaling factor of 0.04327~$\pm$~0.00016~mm/pixel and vertical scaling factor of 0.04204~$\pm$~0.00018~mm/pixel. |
Digital images from the JAI camera were extracted in a matrix format in order to take projections on both axes and perform a Gaussian fit. The observed image profiles were not well described by a single Gaussian distribution. The profiles may be described using a Lorentzian distribution, however, the rms of the Lorentzian function is not defined. The super Gaussian distribution seems to be the best option~\cite{sup-Gau}, because rms values may be directly extracted. | Digital images from the JAI camera were extracted in a matrix format in order to take projections on both axes and perform a Gaussian fit. The observed image profiles were not well described by a single Gaussian distribution. The profiles may be described using a Lorentzian distribution, however, the rms of the Lorentzian function is not defined. The super Gaussian distribution seems to be the best option~\cite{sup-Gau}, because rms values may be directly extracted. | ||
| − | Fig.~\ref{par-fit} shows the square of the rms ($\sigma^2_\textnormal{s}$) $vs$ $k_1L$ for $x$ (horizontal) and $y$ (vertical) beam projections along with the parabolic fits using Eq.~\ref{par_fit} . The emittances and Twiss parameters from these fits are summarized in Table.~\ref{results}. Further details | + | Fig.~\ref{par-fit} shows the square of the rms ($\sigma^2_\textnormal{s}$) $vs$ $k_1L$ for $x$ (horizontal) and $y$ (vertical) beam projections along with the parabolic fits using Eq.~\ref{par_fit} . The emittances and Twiss parameters from these fits are summarized in Table.~\ref{results}. Further details of the fitting procedures are described in reference~\cite{emit-mat}. |
\begin{figure} | \begin{figure} | ||
\begin{tabular}{cc} | \begin{tabular}{cc} | ||
| Line 187: | Line 171: | ||
projected emittance $\epsilon_x$ & $\mu$m & $0.37 \pm 0.02$ \\ | projected emittance $\epsilon_x$ & $\mu$m & $0.37 \pm 0.02$ \\ | ||
projected emittance $\epsilon_y$ & $\mu$m & $0.30 \pm 0.04$ \\ | projected emittance $\epsilon_y$ & $\mu$m & $0.30 \pm 0.04$ \\ | ||
| − | + | % normalized \footnote{normalization procedure assumes appropriate beam chromaticity.} emittance $\epsilon_{n,x}$ & $\mu$m & $10.10 \pm 0.51$ \\ | |
| − | normalized emittance $\epsilon_{n,y}$ & $\mu$m & $8.06 \pm 1.1$ \\ | + | %normalized emittance $\epsilon_{n,y}$ & $\mu$m & $8.06 \pm 1.1$ \\ |
$\beta_x$-function & m & $1.40 \pm 0.06$ \\ | $\beta_x$-function & m & $1.40 \pm 0.06$ \\ | ||
$\beta_y$-function & m & $1.17 \pm 0.13$ \\ | $\beta_y$-function & m & $1.17 \pm 0.13$ \\ | ||
| Line 195: | Line 179: | ||
micro-pulse charge & pC & 11 \\ | micro-pulse charge & pC & 11 \\ | ||
micro-pulse length & ps & 350.16 \\ | micro-pulse length & ps & 350.16 \\ | ||
| − | energy of the beam $E$ & MeV & 15 | + | energy of the beam $E$ & MeV & 15 $\pm$ 1.6 \\ |
relative energy spread $\Delta E/E$ & \% & 10.4 \\ | relative energy spread $\Delta E/E$ & \% & 10.4 \\ | ||
| Line 204: | Line 188: | ||
\section{Conclusions} | \section{Conclusions} | ||
| − | A diagnostic tool was developed and used to measure the beam emittance of the High Rep Rate Linac at the Idaho Accelerator Center. The tool relied on measuring the images generated by the optical transition radiation of the electron beam on a polished thin aluminum target. The electron beam profile was not described well using a single Gaussian distribution but rather a super Gaussian or Lorentzian distribution. The | + | A diagnostic tool was developed and used to measure the beam emittance of the High Rep Rate Linac at the Idaho Accelerator Center. The tool relied on measuring the images generated by the optical transition radiation of the electron beam on a polished thin aluminum target. The electron beam profile was not described well using a single Gaussian distribution but rather by a super Gaussian or Lorentzian distribution. The larger uncertainties observed for $\sigma^2_y$ are still under investigation. The projected emittance of the High Rep Rate Linac, similar to medical linacs, at ISU was measured to be less than 0.4~$\mu$m as measured by the OTR based tool described above when accelerating electrons to an energy of 15~MeV. The normalized emittance may be obtained by multiplying the projected emittance by the average relativistic factor $\gamma$ and $\beta$ of the electron beam. We plan to perform similar measurements over the energy range of the linac in the near future. |
\section{ACKNOWLEDGMENT} | \section{ACKNOWLEDGMENT} | ||
| Line 282: | Line 266: | ||
%} | %} | ||
\bibitem{quad-scan} | \bibitem{quad-scan} | ||
| − | D.F.G. Benedetti, $et$ $al$., Tech. Rep., DAFNE Tech. | + | D.F.G. Benedetti, $et$ $al$., Tech. Rep., DAFNE Tech. Not., Frascati, Italy (2005). |
| − | Not., Frascati, Italy (2005). | ||
%000000000000000000000000000000000000000000000000000000000000 | %000000000000000000000000000000000000000000000000000000000000 | ||
Latest revision as of 00:16, 17 May 2012
latex file: File:IPAC 2012 Sadiq.txt
Latex
\documentclass[acus]{JAC2003}
%% %% This file was updated in March 2011 by T. Satogata to be in line with Word templates. %% %% Use \documentclass[boxit]{JAC2003} %% to draw a frame with the correct margins on the output. %% %% Use \documentclass{JAC2003} %% for A4 paper layout %%
\usepackage{graphicx} \usepackage{booktabs} \usepackage{epstopdf} \usepackage{subfig} \usepackage{graphicx} \usepackage{amstext} %\usepackage{hyperref} %\usepackage[bottom]{footmisc} %\usepackage{tabularx } %\usepackage{footnote} %\usepackage{caption} %\usepackage{subcaption}
%\usepackage{epsfig} %% %% VARIABLE HEIGHT FOR THE TITLE BOX (default 35mm) %%
\setlength{\titleblockheight}{40mm}
\begin{document} \title{TRANSVERSE BEAM EMITTANCE MEASUREMENTS OF\\A 16 MeV LINAC AT THE IDAHO ACCELERATOR CENTER} %\title{TRANSVERSE BEAM EMITTANCE MEASUREMENTS OF A 16 MeV LINAC AT THE IAC\thanks{ Work supported by ...}}
\author{S. Setiniyaz\thanks{Email: sadik82@gmail.com}, K. Chouffani, T. Forest, and Y. Kim\\ Idaho State University, Pocatello, ID, 83209, USA\\ A. Freyberger, Jefferson Lab, Newport News, Virginia, 23606, USA}
\maketitle
\begin{abstract} A beam emittance measurement of the 16~MeV S-band High Repetition Rate Linac (HRRL) was performed at Idaho State University's Idaho Accelerator Center (IAC). The HRRL linac structure was upgraded beyond the capabilities of a typical medical linac so it can achieve a repetition rate of 1~kHz. Measurements of the HRRL transverse beam emittance are underway that will be used to optimize the production of positrons using HRRL's intense electron beam on a tungsten converter. In this paper, we describe a beam imaging system using on an OTR screen and a digital CCD camera, a MATLAB tool to extract beamsize and emittance, detailed measurement procedures, and the measured transverse emittances for an arbitrary beam energy of 15~MeV. \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). The HRRL accelerates electrons to energies between 3 and 16~MeV with a maximum repetition rate of 1~kHz. The HRRL beamline has recently been reconfigured to generate positrons to be used as a secondary beam. %The electron beam characteristics of the HRRL are summarized in Table~\ref{tab:hrrl}.
An Optical Transition Radiation (OTR) based viewer was installed to allow measurements at the high electron currents available using the HRRL. The visible light from the OTR based viewer is produced when a relativistic electron beam crosses the boundary of two mediums with different dielectric constants. Visible radiation is emitted at an angle of 90${^\circ}$ with respect to the incident beam direction~\cite{OTR} when the electron beam intersects the OTR target at a 45${^\circ}$ angle. These backward-emitted photons are observed using a digital camera and can be used to measure the shape and intensity of the electron beam based on the OTR distribution.
Emittance is a key parameter in accelerator physics that is used to quantify the quality of an electron beam produced by an accelerator. An emittance measurement can be performed in a several ways~\cite{emit-ways, sole-scan-Kim}. This work used the Quadrupole scanning method~\cite{quad-scan} to measure the emittance, Twiss parameters, and beam energy.
\section{The Experiment} \subsection{Quadrupole Scanning Method} Fig.~\ref{q-scan-layout} illustrates the apparatus used to measure the emittance using the quadrupole scanning method. A quadrupole is positioned at the exit of the linac to focus or de-focus the beam as observed on a downstream view screen. The 3.1~m distance between the quadrupole and the screen was chosen in order to minimize chromatic effects and to satisfy the thin lens approximation. %The quadrupole and the screen are located far away to minimize chromatic effects and to increase the veracity of the thin lens approximation used to calculate beam optics. \begin{figure}[htb] \centering
\includegraphics[scale=0.40]{quad_scan_setup.eps}
\caption{Apparatus used to measure the beam emittance.} \label{q-scan-layout} \end{figure} Assuming the thin lens approximation, $\sqrt{k_1}L << 1$, is satisfied, the transfer matrix of a quadrupole magnet may be expressed as % 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. A matrix representing the drift space between the 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 the matrix product $\mathbf{SQ}$. In the horizontal plane, the beam matrix at the screen ($\mathbf{\sigma_{s}}$) is related to the beam matrix of the quadrupole ($\mathbf{\sigma_{q}}$) using the similarity transformation \begin{equation} \mathbf{\mathbf{\sigma_{s}=M\mathrm{\mathbf{\mathbf{\sigma_{q}}}}}M}^{\mathrm{T}}. \end{equation} where the $\mathbf{\sigma_{s}}$ and $\mathbf{\sigma_{q}}$ are 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) ,\; \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 the new parameters~\cite{quad-scan}, $A \equiv \sigma_{11},~B \equiv \frac{\sigma_{12}}{\sigma_{11}},~C \equiv \frac{\epsilon_{x}^{2}}{\sigma_{11}}$ By defining the new parameters~\cite{quad-scan} \begin{equation} A \equiv l^2\sigma_{\textnormal{q},x}^{2},~B \equiv \frac{1}{l} + \frac{\sigma_{\textnormal{q},xx'}}{\sigma_{\textnormal{q},x}^{2}},~C \equiv l^2\frac{\epsilon_{x}^{2}}{\sigma_{\textnormal{q},x}^{2}}. \end{equation} the matrix element $\sigma_{\textnormal{s},x}^{2}$, the square of the rms beam size at the screen, may be expressed as 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{par_fit} \end{equation} The emittance measurement was performed by changing the quadrupole current, which changes $k_{1}L$, and measuring the corresponding beam image on the view screen. The measured two-dimensional beam image was projected along the image's abscissa and ordinate axes. A Gaussian fitting function is used on each projection to determine the rms value, $\sigma_\textnormal{s}$ in Eq.~(\ref{par_fit}). Measurements of $\sigma_\textnormal{s}$ for several quadrupole currents ($k_{1}L$) is then fit using the parabolic function in Eq.~(\ref{par_fit}) to determine the constants $A$, $B$, and $C$. The emittance ($\epsilon$) and the Twiss parameters ($\alpha$ and $\beta$) can be found using Eq.~(\ref{emit-relation}). \begin{equation} \epsilon=\frac{\sqrt{AC}}{l^2},~\beta=\sqrt{\frac{A}{C}},~\alpha=\sqrt{\frac{A}{C}}(B+\frac{1}{l}). \label{emit-relation} \end{equation} \subsection{The OTR Imaging System} The OTR target is 10 $\mu$m thick aluminum foil with a 1.25 inch diameter. The OTR is emitted in a cone shape with the maximum intensity at an angle of $1/\gamma$ with respect to the reflecting angle of the electron beam~\cite{OTR}. Three lenses, 2 inches in diameter, are used for the imaging system to avoid optical distortion at lower electron energies. The focal lengths and position of the lenses are shown in Fig.~\ref{image_sys}. The camera used was a JAI CV-A10GE digital camera with a 767 by 576 pixel area. The camera images were taken by triggering the camera synchronously with the electron gun. \begin{figure} \centering {\scalebox{0.16} [0.16]{\includegraphics{image_sys.eps}}} {\scalebox{0.20} [0.20]{\includegraphics{imaging_sys}}} \caption{The OTR Imaging system.} \label{image_sys} \end{figure} \subsection{Quadrupole Scanning} The quadrupole current is changed to alter the strength and direction of the quadrupole magnetic field such that a measurable change in the beam shape is seen by the OTR system. Initially, the beam was steered by the quadrupole indicating that the beam was not entering along the quadrupole's central axis. Several magnetic elements upstream of this quadrupole were adjusted to align the incident electron beam with the quadrupole's central axis. First, the beam current observed by a Faraday cup located at the end of beam line was maximized using upstream steering coils within the linac nearest the gun. Second, the first solenoid nearest the linac gun was used to focus the electron beam on the OTR screen. Steering coils were adjusted to maximize the beam current to the Faraday cup and minimize the deflection of the beam by the solenoid first then by the quadrupole. A second solenoid and the last steering magnet, both near the exit of the linac, were used in the final step to optimize the beam spot size on the OTR target and maximize the Faraday cup current. A configuration was found that minimized the electron beam deflection when the quadrupole current was altered during the emittance measurements.
The emittance measurement was performed using an electron beam energy of 15~MeV and a 200~ns long, 40~mA, macro pulse peak current. The current in the first quadrupole after the exit of the linac was changed from $-$~5~A to 5~A with an increment of 0.2~A. Seven measurements were taken at each current step in order to determine the average beam width and the variance. Background measurements were taken by turning the linac's electron gun off while keep the RF on. Background image and beam images before and after background subtraction are shown in Fig.~\ref{bg}. A small dark current is visible in Fig.~\ref{bg}b that is known to be generated when electrons are pulled off the cavity wall and accelerated.
\begin{figure} \begin{tabular}{ccc} \centerline{\scalebox{0.28} [0.22]{\includegraphics{sg_no_bg_subtraction_0Amp.eps}}} \\ (a)\\ \centerline{\scalebox{0.28} [0.22]{\includegraphics{Background.eps}}}\\ (b)\\ \centerline{\scalebox{0.28} [0.22]{\includegraphics{bg_subtracted_0Amp.eps}}}\\ (c) \end{tabular} \caption{Background subtracted to minimize impact of dark current; (a) a beam with the dark current and background noise, (b) a background image, (c) a beam image when dark background was subtracted.} \label{bg} \end{figure} The electron beam energy was measured using a dipole magnet downstream of the quadrupole used for the emittance measurements. Prior to energizing the dipole, the electron micro-pulse bunch charge passing through the dipole was measured using a Faraday cup located approximately 50~cm downstream of the OTR screen. The dipole current was adjusted until a maximum beam current was observed on another Faraday cup located just after the 45 degree exit port of the dipole. A magnetic field map of the dipole suggests that the electron beam energy was 15~$\pm$~1.5~MeV. Future emittance measurements are planned to cover the entire energy range of the linac. \subsection{Data Analysis and Results} Images from the JAI camera were calibrated using the OTR target frame. An LED was used to illuminate the OTR aluminum frame that has a known inner diameter of 31.75~mm. Image processing software was used to inscribe a circle on the image to measure the circular OTR inner frame in units of pixels. The scaling factor can be obtained by dividing this length with the number of pixels observed. The result is a horizontal scaling factor of 0.04327~$\pm$~0.00016~mm/pixel and vertical scaling factor of 0.04204~$\pm$~0.00018~mm/pixel. Digital images from the JAI camera were extracted in a matrix format in order to take projections on both axes and perform a Gaussian fit. The observed image profiles were not well described by a single Gaussian distribution. The profiles may be described using a Lorentzian distribution, however, the rms of the Lorentzian function is not defined. The super Gaussian distribution seems to be the best option~\cite{sup-Gau}, because rms values may be directly extracted.
Fig.~\ref{par-fit} shows the square of the rms ($\sigma^2_\textnormal{s}$) $vs$ $k_1L$ for $x$ (horizontal) and $y$ (vertical) beam projections along with the parabolic fits using Eq.~\ref{par_fit} . The emittances and Twiss parameters from these fits are summarized in Table.~\ref{results}. Further details of the fitting procedures are described in reference~\cite{emit-mat}. \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{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 \footnote{normalization procedure assumes appropriate beam chromaticity.} 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 & $0.24 \pm 0.07$ \\ micro-pulse charge & pC & 11 \\ micro-pulse length & ps & 350.16 \\ energy of the beam $E$ & MeV & 15 $\pm$ 1.6 \\ relative energy spread $\Delta E/E$ & \% & 10.4 \\
\bottomrule
\end{tabular}
\label{results}
\end{table}
\section{Conclusions} A diagnostic tool was developed and used to measure the beam emittance of the High Rep Rate Linac at the Idaho Accelerator Center. The tool relied on measuring the images generated by the optical transition radiation of the electron beam on a polished thin aluminum target. The electron beam profile was not described well using a single Gaussian distribution but rather by a super Gaussian or Lorentzian distribution. The larger uncertainties observed for $\sigma^2_y$ are still under investigation. The projected emittance of the High Rep Rate Linac, similar to medical linacs, at ISU was measured to be less than 0.4~$\mu$m as measured by the OTR based tool described above when accelerating electrons to an energy of 15~MeV. The normalized emittance may be obtained by multiplying the projected emittance by the average relativistic factor $\gamma$ and $\beta$ of the electron beam. We plan to perform similar measurements over the energy range of the linac in the near future.
\section{ACKNOWLEDGMENT} We would like to acknowledge the outstanding efforts of Idaho Accelerator Center. This work was supported by DOE award \# DE-SC0002600. %Thanks to C. F. Eckman, C. O'Neill, and Dr. D. Wells. \begin{thebibliography}{9} % Use for 1-9 references
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