Optical Spin Hall Effect

A remarkable analogy is established between the well-known spin Hall effect and the polarization dependence of Rayleigh scattering of light in microcavities. This dependence results from the strong spin effect in elastic scattering of exciton polaritons: if the initial polariton state has a zero spin and is characterized by some linear polarization, the scattered polaritons become strongly spin polarized. The polarization in the scattered state can be positive or negative dependent on the orientation of the linear polarization of the initial state and on the direction of scattering. Very surprisingly, spin polarizations of the polaritons scattered clockwise and anticlockwise have different signs. The optical spin Hall effect is possible due to strong longitudinal-transverse splitting and finite lifetime of exciton polaritons in microcavities.

Figure 1

(a) Scheme of the considered experimental configuration: linearly polarized light is incident on a semiconductor microcavity under oblique angle. Polarization of the scattered light is analyzed. (b) The red arrows show the distribution of the effective magnetic field induced by the TE-TM splitting on an elastic circle in reciprocal space. (c) The green circle sketches the polariton state resonantly excited by a pulse having an in-plane wave vector along the $x$ axis. This initial polariton state is TM polarized and has its pseudospin parallel to the $x$ direction (green arrow). The blue circles and blue arrows show the short-time distribution of particles and their pseudospin orientation. The red arrows show the distribution of the effective magnetic field induced by the TE-TM splitting.

Figure 2

The black arrows show the polariton pseudospin orientation in the reciprocal space at a time $t=\frac{\pi }{2\Omega }$ after arrival of the TM-polarized excitation pulse. In the first and third quarter (emission angles [0°–90°] and [180°–270°], respectively) polaritons have their pseudospins parallel to the $z$ direction, which corresponds to the right-circularly polarized emission (${\sigma }^{+}$), shown in red. In the second and fourth quarter (emission angles [90°–180°] and [270°–360°], respectively) the polariton pseudospins are antiparallel to the $z$ axis, which corresponds to the left-circularly polarized emission (${\sigma }^{-}$), shown in blue.

Figure 3

(a) Dispersion of the LPB in the model microcavity. The bare exciton energy is 1450 meV, and the bare photon energy at $k=0$ is 1439 meV. (b) Wave-vector dependence of the energy splitting $\Omega$ between TE- and TM-polarized polariton modes. (c) Wave-vector dependence of the product $\Omega \tau$, where $\tau$ is the polariton radiative lifetime.

Figure 4

(a) Time dependence of the circular polarization degree of light emitted by the states whose in-plane wave vector makes an angle $\theta$ of $+45°$ (solid line) and $-45°$ (dashed line) with respect to the wave vector of the incident light. The inset shows the time dependence of the populations of the corresponding states. (b) Integrated circular polarization degree versus the scattering angle $\theta$ for two different in-plane wave vectors of the incident light, namely, $2.6\times {10}^6{\mathrm{m}}^{-1}$ (solid line) and $1.3\times {10}^6{\mathrm{m}}^{-1}$ (dashed line).