## 3 Design and Construction
### B-Field Collimation  
The required magnet dimensions were initially studied by Biot-Savart current coil approximation simulations [9], exploring axial B-field strength along the magnet assembly z-axis as the gap size is varied, as seen in Figure 6a. Code for all models are provided at [[https://github.com/garnet/CircularLaserMagnet](https://github.com/garnet/CircularLaserMagnet)]([https://github.com/garnet/CircularLaserMagnet](https://github.com/garnet/CircularLaserMagnet)) (accessed on 16 September 2021). Each magnet is modeled as a pair of inner and outer coaxial, z-centered, and counter-rotating current loops. Currents more uniformly distributed over the magnet surfaces were simulated and found unnecessary for this initial analysis. The summation of infinitesimal magnetization current loops across the magnet bulk yields the current loops model [9]. From the Biot-Savart law:  
\[d\overset{\rightarrow}{B}=kl\frac{\left(d\overset{\rightarrow}{l}\times \overset{\rightarrow}{r}\right)}{|\overset{\rightarrow}{r}|^{3}} \tag{4}\]  
where \(k\) is the magnetic constant, \(l\) is the current, and \(r\) is the point of interest position. The well-known analytical solution of the axial field for a circular coil is [10]:  
\[B_{z,axial}=\frac{\mu_{0}I}{2}\frac{r^{2}}{\left(z^{2}+r^{2}\right)^{\frac{3}{ 2}}} \tag{5}\]  
Therefore, the assembly axial field \(B_{z}(z)\) is a summation over all the coils/magnets:  
\[B_{z}(z)=\frac{\mu_{0}I}{2}\left\{ \begin{array}{c}\sum_{\begin{array}{c}i= 2,3\\ j=1,2\end{array}}\frac{(-1)^{j}r_{j}^{2}}{\left[(z-z_{i})^{2}+r_{j}\right]^{\frac {3}{2}}}-\sum_{\begin{array}{c}i=1,4\\ j=in,out\end{array}}\frac{(-1)^{j}r_{j}^{2}}{[(z-z_{i})^{2}+r_{j}]^{\frac{3}{2} }}\\ j=in,out\end{array} \right\} \tag{6}\]  
The magnets are numbered left to right, 1 to 4, and are indexed as \(i\). The inner and outer coils are represented by \(j\) and are numbered 2 and 1, respectively. Antiparallel magnetization directions sum depending on gap size. Axial (z-axis) field strength is also influenced by the ratio of inner radius (R\({}_{\text{i}}\)), outer radius (R\({}_{\text{o}}\)), and thickness of each magnet. In Figure 6c, solenoid simulation identifies the optimum thickness and R\({}_{\text{i}}\)/R\({}_{\text{o}}\) values. However, thickness and outer diameter (OD) selection were constrained by commercially available options: 1/8\({}^{\prime\prime}\) or 1/16\({}^{\prime\prime}\) and OD (2R\({}_{\text{o}}\)) desired to 3/8\({}^{\prime\prime}\) (mentioned  
Figure 5: Simulated transverse B\({}_{\text{z}}\) and B\({}_{\text{x}}\) on z = 0 plane for: (**a**) single; (**b**) inner pair only; and (**c**) full stack (no space between inner and outer pairs). earlier), respectively. Conceptually, a smaller thickness would be preferable when the annular magnet is modeled as inner and outer solenoids. Thicker magnets relative to outer diameter tend to the infinite coil solution where the axial fields sum vanishes. Axial fields of the inner and outer coils are equal and opposite. [PERSON]'s [9] (p. 225) solenoid-faces-subtending-angles solution shows this behavior, see Figure 7.  
In other words, the outer loop's negative impact on the sum axial (B\({}_{\text{z}}\)) field is reduced by the largest possible OD. Experiment apparatus 3/8\({}^{\prime\prime}\) available space, therefore, sets the OD.3/8\({}^{\prime\prime}\) OD constraint and commercial unavailability of 3/8\({}^{\prime\prime}\)\(\times\) 1/4\({}^{\prime\prime}\)\(\times\) 1/16\({}^{\prime\prime}\) forced customization of 3/8\({}^{\prime\prime}\)\(\times\) 1/8\({}^{\prime\prime}\)\(\times\) 1/16\({}^{\prime\prime}\) via Dremel-lathe working to widen the 1/8\({}^{\prime\prime}\) inner diameter (ID) to 1/4\({}^{\prime\prime}\) to allow sufficient atom transit time as discussed shortly. Commercial annular magnets are sintered powders encased in a thin metallic jacket; in this case, Ni-Cu-Ni. Usual cutting tools result in fracturing and disintegration. Grinding proved to be a viable option.