Figure 2

Optimal waist of the Gaussian beam \(\bar{w}^{\max}_{0}\) divided by the width of the atomic ensemble \(\sqrt{2} \sigma _{\perp }\) (upper panels a) and the corresponding maxima of the geometric factor \(\mathcal{G} \equiv 6|\xi |^{2}/(\bar{w}^{\max}_{0})^{2}\) for \(N=10^{3}\) atoms (lower panels b) versus the dimensionless width and length \(\bar{\sigma }_{\perp ,z} = k_{e} \sigma _{\perp ,z}\) of the atomic cloud, for the cases of the spatial phase \(\phi (\boldsymbol{r})\) of the atomic spin wave given by: Eq. (16), uniform phase, (a1), (b1); Eq. (20), Gouy phase compensation, (a2), (b2); and Eq. (23), full spatial phase compensation, (a3), (b3). For the values of \(\bar{\sigma }_{\perp ,z}\) where \(\bar{w}^{\max}_{0} \simeq \sqrt{2}\bar{\sigma }_{\perp }\) we also have \(|\xi |^{2} \simeq 1/4\) and thus \(\mathcal{G} \simeq 3/4 \bar{\sigma }_{\perp }^{2}\)