Steering far-field spin-dependent splitting of light by inhomogeneous anisotropic media

An inhomogeneous anisotropic medium with specific structure geometry can apply the tunable spin-dependent geometrical phase to the light passing through the medium, and thus can be used to steer the spin-dependent splitting (SDS) of light. In this paper, we exemplify this inference by the $q$ plate, an inhomogeneous anisotropic medium. It is demonstrated that when a linearly polarized light beam normally passes through a $q$ plate, $k$-space SDS first occurs, and then the real-space SDS in the far-field focal plane of a converging lens is distinguishable. Interestingly, the SDS, described by the normalized Stokes parameter $S_3$ shows a multilobe and rotatable splitting pattern with rotational symmetry. Further, by tailoring the structure geometry of the $q$ plate and/or the incident polarization angle of light, the lobe number and the rotation angle both are tunable. Our result suggests that the $q$ plate can serve as a potential device for manipulating the photon spin states and enable applications such as in nano-optics and quantum information.

Figure 1

Examples of the $q$-plate geometries for $q=1$. The tangent to the lines shown indicates the local optical axis orientation (fast axis). (a)–(d) represent the geometries for ${\alpha }_0$ $=$ 0, $\pi /6$, $\pi /3$, and $\pi /2$, respectively.

Figure 2

Schematic illustrating the SDS produced by a $q$ plate. A linearly polarized light beam normally passes through the $q$ plate and is focused by a lens. ${\sigma }_{+}$ and ${\sigma }_{-}$ represent the left and right circular polarization components, respectively. The two curves with arrows indicate that the splitting patterns will rotate when rotating the incident polarization direction ($\theta$). The additional devices for measuring the Stokes parameter $S_3$ are not shown.

Figure 3

Normalized Stokes parameter $S_3$ in the focal plane under the irradiation of a linearly $x$-polarized light for (a) $q=1$, (b) $q=2$, and (c) $q=3$, respectively. Here ${\alpha }_0=0$. In the calculations, we set $\Phi =\pi /2$, $\lambda =1$ $\mu$m, $w_0=1$ mm, and $f_0=0.5$ m.

Figure 4

Normalized Stokes parameter $S_3$ in the focal plane under the irradiation of a linearly $x$-polarized light for (a) ${\alpha }_0=0$, (b) ${\alpha }_0=\pi /6$, (c) ${\alpha }_0=\pi /4$, and (d) ${\alpha }_0=\pi /2$, respectively. Other parameters are the same as in Fig. 3. The white arrows show the clockwise rotation of $S_3$ with the increase of ${\alpha }_0$.

Figure 5

Normalized Stokes parameter $S_3$ in the focal plane under the irradiation of a linear polarization light with polarization angle (a) $\theta =0$, (b) $\theta =\pi /6$, (c) $\theta =\pi /4$, and (d) $\theta =\pi /2$, respectively. Here ${\alpha }_0=0$. Other parameters are the same as in Fig. 3. The white arrows show the counterclockwise rotation of $S_3$ with the increase of $\theta$.