Spin-orbit interaction of light in three-dimensional microcavities

We investigate the spin-orbit coupling of light in three-dimensional cylindrical and tubelike whispering gallery mode resonators. We show that its origin is the transverse confinement of light in the resonator walls, even in the absence of inhomogeneities or anisotropies. The spin-orbit interaction results in elliptical far-field polarization (spin) states and causes spatial separation of polarization handedness in the far field. The ellipticity and spatial separation are enhanced for whispering gallery modes with higher excitation numbers along the resonator height. We analyze the asymmetry of the ellipticity and the tilt of the polarization orientation in the far field of conelike microcavities. Furthermore, we find a direct relationship between the tilt of the polarization orientation in the far field and the local inclination of the resonator wall. Our findings are based on finite-difference time-domain simulations and are supported by three-dimensional diffraction theory.

Figure 1

Far-field polarization properties. (a) Characterization of the far-field polarization state by $\mathrm{\Psi }$, the angle of orientation; $\chi$, the ellipticity angle; and $\sigma$, the handedness. (b) Plot of $\mathrm{\Psi }$ and $\chi$ of a 3D-ring WGM far field derived from the Kirchhoff diffraction formula for the fundamental axial mode ($q=1$). See the text for details.

Figure 2

Geometry and WGM. (a) Illustration of a cylindrical ring of thickness $d$ and corresponding WGMs, and spectrum of the modes with different vertical excitations. The parameters are the azimuthal number $m=24$, mean radius $R=2\mu \text{m}$, height $h=R=2\mu \text{m}$, wall thickness $d=0.2\mu \text{m}$, and refractive index $n=1.5$. (b) Shown on the left is the field distribution of $E_{\text{z}}$ and $E_{\varphi }$ in the exterior space ($r>R$) of the ring and on the right $E_{\text{z}}$ and $E_{\varphi }$ at $r=R$ as a function of $z$. See the text for details.

Figure 3

Far fields and far-field polarization states. (a) Polar plots of far-field intensities of different vertical excitation numbers $q$. The frames show the polarization ellipses along the (maximum) far-field direction $\theta$ indicated above each ellipse. (b) Poincaré spheres of polarization displaying observable polarization states when scanning the far field within the given $\theta$ range. See the text for details.

Figure 4

Polarization states in conelike 3D-ring cavities. (a) Polar plots of far-field intensity and polarization evolution for $q=2$ modes. (b) Polarization states at intensity maxima of the upper and lower lobes for different inclination angles $\gamma =0^{\circ },2^{\circ },4^{\circ },5^{\circ }.6^{\circ },8^{\circ },{10}^{\circ },{12}^{\circ },{15}^{\circ }$. (c) Examples for electric field distributions inside ring resonators with inclination angles $\gamma =0^{\circ },5^{\circ },{10}^{\circ },{15}^{\circ }$; the left panels are for $\gamma ={15}^{\circ }$.

Figure 5

Geometry and polarization states of conelike tube cavities. (a) Illustration of the geometry of the tube cavity and different confinement profiles of the confining region. (b) Polarization states at a maximum far-field intensity of $q=2$ modes of different confinement profiles. The parameters displayed in the legend are given in microns. (c) Comparison of field distributions $E_{\varphi }$ at different profile configurations and inclination angles. See the text for details.

Figure 6

Illustration of the geometry. The blue cone surface is parametrized by ($u^{\prime },{\varphi }^{\prime }$), $\theta$ and $\phi$ are the far-field angles, and $\gamma$ is the angle between the cone surface and the $z$ axis.