## 4 Dipole Magnet Current Constraints  
To simplify our first commissioning with an assumption, the currents (\(D_{1},D_{3}\)) of the chicane dipole magnets CHICANE.D1 and CHICANE.D3 are set to equal magnitude, and the same goes to the currents (\(D_{2},D_{4}\)) of the chicane dipole magnets CHICANE.D2 and CHICANE.D4, which are written as  
\[D_{1}=-D_{3},\ \ \text{and}\ \ D_{4}=-D_{2}. \tag{1}\]  
The assumption is that the electron beam experiences a fringe field of CHICANE.D1 and CHICANE.D3 in the similar beam path as the beam straight enters them, and the same goes for CHICANE.D2 and CHICANE.D4 as the beam enters them with the same opening angle of 19 degrees. Figure 3 shows the evolution of absolute value of offset momentum in the \(y\)-axis of a reference particle throughout each chicane dipole magnet, which is simulated by program IMPACT-T. The simulation is implemented by the 1D magnetic field distribution (Figure 4) along the beamline longitudinal axis [2], simulated by CST Studio Suite [4], as the particle enters and leaves at a beamline longitudinal distance of \(\mp 0.645\) m offset from the center of each chicane dipole magnet, respectively. Note that the constant field region (longitudinal length) of each chicane dipole magnet of approximately 0.2 m is implemented in the program IMPACT-T. In order to achieve the bending angle of 19 deg at all chicane dipole magnets, current \(D_{1}=-D_{3}\) is set to \(1.155\) A, while \(D_{4}=-D_{2}\) is set to \(1.220\) A for the beam momentum of 17 MeV/c.  
Furthermore, beam transport throughout the chicane can be manipulated such that the beam follows the beam reference path by a simulation of beam tracking in program ASTRA, which is implemented by the 2D magnetic field distribution. Note that the simulation is performed without space charge effects. For instance, the chicane dipole magnet currents with the constraints in Equation (1) can be found to achieve both minimum momentum dispersion and \(y\)-position \(\Delta y=0\) of 17-MeV/c beam at the screen HIGH2.SCR3 (Figure 2) located downstream of the chicane in the simulation results shown in Figure 5. Figure 5 displays both the contour line of the \(y\)-position (or offset position \(\Delta y\)) and density plot of momentum dispersion as a function of its chicane dipole magnet currents with the constraints in Equation (1), which determine at least one set of the chicane dipole magnet  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\Delta D_{1}\) **(deg.)** & \(\Delta D_{2}\) **(deg.)** & \(\Delta D_{3}\) **(deg.)** & \(\Delta D_{4}\) **(deg.)** & **Energy Dispersion (mm)** \\ \hline
0.5 & 0 & 0 & 0 & 40.0 \\
0 & 0.5 & 0 & 0 & \(-\)49.3 \\
0.5 & 0 & 0.5 & 0 & 23.1 \\ \(-\)0.5 & 0 & \(-\)0.5 & 0 & \(-\)22.9 \\
0 & 0.5 & 0 & 0.5 & \(-\)23.1 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Energy dispersion for different cases with bending angle offsets (errors) \(\{\Delta D_{1},\Delta D_{2},\Delta D_{3},\Delta D_{4}\}\) (unit: deg.) of chicane dipole magnet CHICANE.D1, CHICANE.D2, CHICANE.D3, and CHICANE.D4, respectively.  
currents to achieve both the zero offset position and zero dispersion. In other words, the constraints in Equation (1) can be applied to the commissioning.  
Figure 4: Normalized \(x\)-magnetic field (\(B_{x}/B_{x,0}\)) profile at \(y=0\), simulated via CST Studio Suite, where \(B_{x,0}\) is the maximum dipole field in \(x\)-axis.  
Figure 3: Evolution of absolute value of offset \(y\)-momentum of a 17 MeV/c reference particle throughout each chicane dipole magnet including its fringe field, simulated by program IMPACT-T. The chicane dipole magnet current \(D_{1}=-D_{3}\) is set to 1.155 A, while \(D_{4}=-D_{2}\) is set to 1.220 A.