Light deflection by light: Effect of incidence angle and inhomogeneity

We study the angular deflection of the circularly polarized components of a linearly polarized probe field in a weakly birefringent atomic system in tripod configuration. A spatially inhomogeneous control field incident obliquely onto an atomic vapor cell facilitates a large angular divergence between circular components. We show that the angular resolution can be dynamically controlled by optimally choosing the angle of incidence and the transverse profile of the control beam. For instance, by employing a Laguerre-Gaussian profile of the control field, one can impart a large angular divergence to the circular components close to the entry face of the atomic vapor cell. We further demonstrate how such a medium causes the focusing and refocusing of the probe field, thereby acting as a lens with multiple foci. The absorption in the medium remains negligible at resonance due to electromagnetically induced transparency.

Figure 1

(a) Energy-level configuration. The ${\sigma }_{\pm }$ components of the probe field drive the transitions $|1\rangle \leftrightarrow |3\rangle$ and $|2\rangle \leftrightarrow |3\rangle$, respectively. A control field incident at an angle ${\theta }_c$ to the entry face excites all the transitions of the tripod system. The degeneracy of the ground levels is removed by applying a feeble axial magnetic field. ${\gamma }_{j3}\left(j=0,1,2\right)$ are the spontaneous decay rates from $|3\rangle$ to $|j\rangle .$ (b) Schematic for the deflection of the circular components of the probe field.

Figure 2

Variation of angular divergence $\left(\varphi \right)$ (solid black line) between the circular components and the respective transmission of the circular components (solid blue line for ${\sigma }_{+}$ and dashed red line for ${\sigma }_{-}$) with the propagation distance $L$ (cm) for a Gaussian profile of the control field with (a) ${\theta }_c=\pi /10$, (b) ${\theta }_c=\pi /6$, (c) ${\theta }_c=\pi /4$, and (d) ${\theta }_c=\pi /3$. The corresponding transmission is also shown in the inflated graphs (e)–(h) for the respective ${\theta }_c$'s. We have used the parameters for ${}^{39}\mathrm{K}$ vapor with $A=2\pi \times 6.079$ MHz, $\gamma =\frac{A}{12}$, $\lambda =769.9$ nm, and $N=5\times {10}^{12}{\mathrm{cm}}^{-3}$. Other parameters are ${\mathrm{\Delta }}_z=0.01\gamma$ (32 kHz), $\delta =0$, $\mathrm{\Delta }=0$, ${\gamma }_{13}={\gamma }_{23}={\gamma }_{03}=\gamma$, and ${\gamma }_{\mathrm{coll}}=0$, and the transverse width $\left(\sigma \right)$ of the Gaussian profile is taken to be $\sqrt{2}$ mm.

Figure 3

Variation of angular divergence ($\varphi$) (solid black line) between the circular components of the probe field and the respective transmission of the circular components (solid blue line for ${\sigma }_{+}$ and dashed red line for ${\sigma }_{-}$) with the propagation distance $L$ (cm) for a Laguerre-Gaussian profile (with $m=3$) of the control field with (a) ${\theta }_c=\pi /10$, (b) ${\theta }_c=\pi /6$, (c) ${\theta }_c=\pi /4$, and (d) ${\theta }_c=\pi /3$. The corresponding transmission is also shown in the inflated graphs (e)–(h) for the respective ${\theta }_c$'s. The parameters used are beam waist ($w_0)=120\mu \mathrm{m}$, Rayleigh length $\left(z_R\right)=5.7$ cm, and the other parameters are the same as in Fig. 2.

Figure 4

The deflection of ${\sigma }_{\pm }$ components of the probe field with the propagation length $L$ (cm) for (a) the Gaussian and (b) the ${\mathrm{LG}}_3$ profile of the control field. We have chosen ${\theta }_c=\pi /6$ and the rest of the parameters are the same as in Fig. 2.