Here, I will introduce a technique to control beam profile from Gaussian beam to uniform distribution [1]. First, I calculated a beamline by excel file and initially assumed Twiss parameters as \(\beta_x=\beta_y=\)11m, \(\alpha_x=\alpha_y=\)5, and beam energy of 10MeV for electrons at the end of an accelerator. Then, the beam is widely expanded (\(\beta_x=\beta_y=\)100m) in the target to reduce the thermal loading. This is a simple calculation.

In the next step, I calculate particle tracking by GPT.
As you see in the excel file (Sheet Beta or Alpha), the Twiss parameters are matched with the GPT simulation result of Courant-Snyder parameters (CSalphax, CSalphay, CSbetax, CSbetay). In the GPT input file, we might write as my script.
I set octupole magnets in the script and manipulated the Gaussian distribution to uniform distribution, as shown in Fig.1.

Figure 1: The left figure shows the Gaussian distribution, and the right figure shows the uniform distribution by using octupoles.

Thanks to the formulation to estimate the required octupole strength \(K_{OCT}\) as follow;
\(\displaystyle{K_{OCT}=\frac{1}{\varepsilon\beta^2_0tan\phi}}\)
where \(\varepsilon\) is the root-mean-square (RMS) emittance of the beam, \(\beta_0\) is an average beta function at the octupole, and \(\phi\) is the betatron phase advance from the octupole magnet to the target. This estimation works well in my calculation.

Let me explain my batch files below;

set INPUT=HEBT_FLASH.in
set OUTPUT=Result.gdf
set EMIT=1
set Q1=-0.21
set Q2=0.21
set Q3=-0.22
set Q4=0.1
set O1=1800
set O2=-2500
set dE=0
set ndir=Q1_%Q1%Q2_%Q2%_Q3_%Q3%_Q4_%Q4%O1_%O1%O2_%O2%_dE_%dE%_10k
mkdir %ndir%
gpt -o %OUTPUT% %INPUT% emit=%EMIT%  quad1=%Q1% quad2=%Q2% quad3=%Q3% quad4=%Q4% oct1=%O1% oct2=%O2% enespread=%dE%
gdfa -o %ndir%\emittance.gdf %OUTPUT% position numpar CSalphax CSalphay CSbetax CSbetay
gdf2a -w 15 -o %ndir%\emit.txt %ndir%\emittance.gdf position numpar CSalphax CSalphay CSbetax CSbetay
sed -e "s/\s\+/ /g" -e "s/^  *//" -e "s/  *$//" %ndir%\emit.txt > %ndir%\ana.txt 
gdf2a -w 15 %OUTPUT% x y z Bx By Bz G t ID | perl gdf2a_CutData.pl position 6.635 6.64 > %ndir%\acc_part.dat

$EMIT means RMS emittance with the dimension of [mm-mrad].
$Q1~4 is the gradient of the quadrupoles with the dimension of [T/m].
$O1~2 is the gradient of the octupoles with the dimension of [T/m\(^3\)].
$dE is the energy spread of the beam with the dimension of [keV].




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References