# Investigation into the Uniformization of Proton Beams for FLASH Therapy
## 3 The Physical Principle of Beam Uniformization Using Nonlinear Magnet
Notably, for optimal beam uniformization, specific designs are required for the placement of nonlinear magnets. These designs can be categorized into two aspects: horizontal and vertical requirements. Firstly, to minimize beam coupling in the horizontal and vertical directions, the beam cross-section at the nonlinear magnet should be \"flattened\". This means that the horizontal size-to-vertical size ratio of the beam should be greater than 4 [21]. This ensures that coupling effects between the two directions do not adversely affect uniformization. The mentioned transverse size ratio refers to the ratio of the maximum beam width in the x-direction to that in the y-direction. At the first nonlinear magnet, this ratio is defined as the maximum value in the x-direction divided by the maximum value in the y-direction. At the second nonlinear magnet, it is defined as the maximum value in the y-direction divided by the maximum value in the x-direction. Secondly, the position of the nonlinear magnet should have a phase shift with the target position as close to \(\pi\) or \(2\pi\) as possible. This is necessary to achieve specific rotation angles \begin{table}
\begin{tabular}{c c c c} \hline
**Element** & **Length (mm)** & **Radius (mm)** & \multicolumn{2}{c}{**Magnetic Gradient (T/m)/Radius**} \\  & & & **Curvature of the Central Trajectory (mm)** \\ \hline QP1 & 200 & 36 & 17.12 \\ \hline QP2 & 200 & 36 & \(-\)16.3 \\ \hline DIP1 & 200 & 20 & 1500 \\ \hline QP3 & 200 & 36 & 24.2 \\ \hline QP4 & 200 & 60 & \(-\)24.75 \\ \hline DIP2 & 200 & 20 & 1500 \\ \hline QP5 & 200 & 80 & \(-\)12.98 \\ \hline QP6 & 200 & 80 & 13.4 \\ \hline QP7 & 200 & 80 & 5.92 \\ \hline QP8 & 300 & 90 & \(-\)18.85 \\ \hline QP9 & 300 & 90 & 25.8 \\ \hline QP10 & 400 & 90 & \(-\)28.07 \\ \hline \end{tabular}
\end{table} 
Table 3: The design parameters for dipole magnets and quadrupole magnets in the transport line. Figure 7: Flow chart of LATTICE parameter modulation. in phase space, allowing distorted edge particles to overlay in the central region of the beam, thereby improving the uniformity of particle distribution [23]. The action of the quadrupole magnet before the first nonlinear magnet is to adjust the ratio of the \(\beta\) functions in the x and y directions, making the \(\beta\)x value larger and the \(\beta\)y value smaller. This ensures a suitable ratio at the nonlinear magnet, as shown in Figure 8, with the ratio at the first nonlinear magnet in the case being \(\beta\)x/\(\beta\)y = 66/3. Similarly, after passing through the first nonlinear magnet, the beam undergoes magnetic focusing through the FODO structure [29], increasing the \(\beta\)y value and decreasing the \(\beta\)x value. In this case, the ratio at the second nonlinear magnet is \(\beta\)y/\(\beta\)x = 70/0.12. Due to a significant increase in the horizontal \(\varepsilon_{\text{rms}}\) after the first nonlinear magnet, lowering the \(\beta\)x value is necessary to match the transverse size ratio of the beam at the second nonlinear magnet. After passing through the second nonlinear magnet, the beam undergoes vertical compression and horizontal expansion using three quadrupole magnets. This process ensures that, at the exit of the final quadrupole magnet, the horizontal and vertical beta functions are maintained, and \(\varepsilon_{\text{rms}}\) remains consistent. Subsequently, a long drift section is employed to enlarge the beam spot. According to Formula (5), the width of the beam spot, denoted as \(2\tau_{\text{r}}\), is dependent on the beta functions at the target location. Therefore, the length of the drift section can be adjusted to alter the size of the beam spot. In order to compare the uniformity at the target when different nonlinear magnet elements are inserted, two pairs of nonlinear magnets were inserted at the same position (m1 and m2), and the cross-section of the particles was intercepted in the same range.