# Investigation into the Uniformization of Proton Beams for FLASH Therapy
## 2 Materials and Methods
### The Physical Principle of Beam Uniformization Using Nonlinear Magnets
#### 2.1.2 Step-like Magnet  
The principle of a step-like magnet is similar to that of a multipole magnet, and they can both be called nonlinear magnets. From the images generated in Figure 4 for the step-like field and the field produced by multiple poles, it is evident that their magnetic  
Figure 1: Variation principle of beam uniformization of step field in X-X\({}^{\prime}\) phase space, from left to right, is the image of the phase space before, when and after the nonlinear magnet.  
Figure 3: Cross-section view of multipole magnet. Each pole contains a pole head and a winding coil, and the magnetic field within the circle is affected by the current of the coil and the material properties of the pole head.  
Figure 2: Compares the impact on particle distribution at the target when incorporating dodecapole magnet components. The left side depicts the octupole component with a gradient of 2978 T/m\({}^{3}\), while the right side shows the octupole and dodecapole gradient components at 2978 T/m\({}^{3}\) and \(-\)150,000 T/m\({}^{5}\), respectively.  
fields are significantly higher at the transverse edges, generating a field much larger than the central field uniformization.  
The fitting curve of the step-like magnet is [19; 24]:  
\[B(x)=\frac{F_{s}/L}{1+e^{b(x_{0}-x)}} \tag{6}\]  
The fitting curve in the other direction can be approximated by calculation as:  
\[B_{Y}(X,Y)=B_{X}(1-\frac{B_{X}}{B_{0}})y \tag{7}\]  
The magnetic field strength of the step field, denoted as \(B_{0}\), is defined as \(B_{0}=F_{s}/L\), where \(F_{s}\) represents the step field strength, \(L\) is the step width, \(b\) represents the step steepness, and \(x_{0}\) denotes the position of the step's rise or fall [19]. Figure 4 shows the difference between the magnetic field map of the step-like field and octupole field.  
Referring to Formula (5), with certain adjustments, the density between the peaks is enhanced, while the peaks at the edges of the beam spot are diminished. The design parameters of the two pairs of step-like magnets are shown in Table 2[20; 25]:  
The schematic figure of the step-like magnet designed by OPERA 2021 modeling software is shown in the Figure 5; a step-like magnetic field can be achieved by a magnet with a nested structure of two pairs of yokes and two pairs of coils. The flat-top field is used to fold transverse particles at the magnet into the beam core at the target. The outer yoke and coil are also called the main core and main coil, while the inner yoke and coil are called the auxiliary core and coil.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **B (T)** & **L (m)** & **Step Location (mm)** & **b (1/mm)** & **Gap (mm)** \\ \hline SFMX1-1 & 0.84 & 0.2 & 10 & 0.22 & 100 \\ SFMX1-2 & 0.84 & 0.2 & 50 & 0.22 & 100 \\ SFMY2-1 & 0.2 & 0.2 & 10 & 0.22 & 100 \\ SFMY2-2 & 0.2 & 0.2 & 60 & 0.22 & 100 \\ \hline \hline \end{tabular}
\end{table}
Table 2: The design parameters of step-like magnet field.  
Figure 4: (**a**) Magnetic field map of step-like field; (**b**) magnetic field map of octupole field.