## 5 Collimation Performance  
Sensitivity to z-axis atomic beam offset was investigated via simulations. Findings are shown in Figure 14. Atomic beam z-position offsets as large as 30 mils from z = 0 can be tolerated for x < 0.08\({}^{\prime\prime}\), based on our 1/10 or 80 G B\({}_{\text{x}}\)-threshold requirement as seen in (b). Better than 0.005\({}^{\prime\prime}\) (5 mils or 0.13 mm) of positioning precision is typical for the apparatus. However, for x > 0.08\({}^{\prime\prime}\), even a 1 mil z-offset (purple curve) shows a significant increase, reaching ~50 G at x = 0.125 (ID edge). In practice, exposure to B\({}_{\text{x}}\) greater than 80 G is likely beyond x = 80 mils. Polarization purity would be adversely affected by this field on the atomic beam exit side of the preparation area. The figure of merit for He\({}^{*}\) preparation is the survival fraction of 2\({}^{3}\)S\({}_{1,\ m_{j}=0}\) remaining from the He\({}^{*}\) mixture produced in the electron-bombardment source. The relevant radiative excitation and relaxation paths for \(\sigma^{+}\) (\(\Delta\text{m}_{\text{s}}=1\)) are depicted in Figure 15. \(\sigma^{+}\) and \(\sigma^{-}\) excitations are symmetric about m\({}_{\text{s}}=0\). The following rate equation quantifies the population behavior at level i, N\({}_{\text{i}}\), based on normalized electric dipole matrix elements D\({}_{\text{ik}}\) for transitions between levels i and k as dictated by the laser polarization, \(\sigma\), e.g., \(\sigma^{+}\) polarization [14; 15] (p. 518). Equal laser power is assumed across all energy levels. \[\frac{dN_{i}}{dt}=\sum_{k=\{\sigma\}}\text{D}_{ik}N_{k}-\text{D}_{ik}N_{i} \tag{15}\]  
Figure 16 shows the survival ratio with increasing laser power. The preparation laser is verified to >1/1000 circular polarization purity by the crossed-polarizer method. Two features are apparent in (a): a high clearing and our required optical pumping efficiency are achieved, but the model predicts sharper population decline than observed. The  
Figure 13: Far-field asymptotic behavior, 1/\(r^{5}\) (blue broken line), and 1/\(r^{3}\) (green broken line). best ratio is \(\sim\)99.3%, with the other extreme spin state below the noise level of \(\sim\)0.1%. The Bz fringe effect presented in Figure 14b is theorized as the cause for higher survival. Approximate exponential decline in signal channel population with laser power is determined by the atom-photon cross-section [16] as assumed in Equation (15). Transitions are occurring for many of the available transitions between magnetic field-dependent energy levels that are reachable by the \(\sim\)2 GHz bandwidth and \(\sim\)2 mm waist laser, as shown in Figure 15. Reaching 99.3% clearing is a validation of B-field 1/10 collimation nonetheless. Otherwise, significant parallel polarization (\(\Delta\)m\({}_{\text{j}}\) = 0) transitions would wash out clearing by returning atoms to the signal channel even at low power. Transit time constraint is known to be power-independent from previous work [4]. Figure 16b adds detail to the high power or infinite power asymptote survival ratio. A pure \(\Delta\)m\({}_{\text{j}}\) = +1 model (Model) predicts 0.01% survival while we measure 0.7%. A modeled 0.07 rad misalignment of \(\overset{\rightarrow}{k}\) and \(\overset{\rightarrow}{B}\) (Model 0.07) matches the measured survival asymptote casting further interest on the gap fringe field mentioned earlier. Modeling results are the same for 0 and 810 G within experimental uncertainty. In addition, low to intermediate power survival ratio fluctuations in (a) are now clearly visible in (b). The Yb-doped fiber preparation laser is known to hop between 3 MHz separated modes (50 m cavity) therefore stimulating the 1.6 MHz FWHM transitions in fluctuating fashion. Figure 14: B\({}_{\text{x}}\)(x) at varying longitudinal offset (z-axis, in mils): (**a**) wide x range; (**b**) small x range. Simulation uses no inner to outer pair spacing.