Generalized Pancharatnam-Berry Phase in Rotationally Symmetric Meta-Atoms

Pancharatnam-Berry geometric phase has attracted enormous interest in subwavelength optics and electromagnetics during the past several decades. Traditional theory predicts that the geometric phase is equal to twice the rotation angle of anisotropic elements. Here, we show that high-order geometric phases equal to multiple times the rotation angle could be achieved by meta-atoms with highfold rotational symmetries. As a proof of concept, the broadband angular spin Hall effect of light and optical vortices is experimentally demonstrated by using plasmonic metasurfaces consisting of space-variant nanoapertures with C2, C3, and C5 rotational symmetries. The results provide a fundamentally new understanding of the geometric phase as well as light-matter interaction in nanophotonics.

Figure 1

(a) Illustration of meta-atoms consisting of rectangular, regular triangular, and regular pentagonal apertures. LCP illumination and RCP transmission are plotted. The apertures can rotate in the $x\text{−}y$ plane with an orientation angle of $\phi$. (b),(c) Simulated phase shift of the transmitted cross-polarized light as a function of the orientation angle $\phi$ of the regular (b) triangular and (c) pentagonal nanoapertures under LCP (solid lines, left axis) and RCP (dotted lines, right axis) illuminations. The insets illustrate the geometry of the designed meta-atoms, where 120-nm-thick gold films are deposited on ${\mathrm{SiO}}_2$ substrate. The periods and length are $p_x=p_y=200\mathrm{nm}$ and $l=160\mathrm{nm}$ for the triangular meta-atoms, and $p_x=p_y=400\mathrm{nm}$ and $l=200\mathrm{nm}$ for the pentagonal meta-atoms.

Figure 2

Rotation dependence of the principal axis and meta-atoms with (a) C1, (b) C2, (c) C3, (d) C6, (e) C5, and (f) C10 rotational symmetry. The double-headed arrows indicate the orientations of the principal array axis, which correspond to the evolution of the polarization states. (g) Schematic of polarization evolution on the Poincaré sphere for the triangular meta-atom. (h) Absolute value of the coefficient ($|\mathrm{\Phi }/\phi |$) of geometric phase for different rotational symmetries.

Figure 3

(a) Scanning electron microscopy (SEM) images of the fabricated samples. Scale bars, 500 nm. All the metasurfaces have a total area of $14.4\mu \mathrm{m}\times 14.4\mu \mathrm{m}$. (b) Simulated $E_{\text{cross}}$ field distributions of the three metasurfaces for the incident wavelength of 633 nm. (c) Simulated far-field intensity distributions for different incident wavelengths. The first, second, and third rows correspond to the incident wavelengths of 633, 650, and 670 nm, respectively. (d) Measured diffraction patterns from the three samples under LCP illumination (left column) and RCP illumination (right column).

Figure 4

(a),(b) SEM images of the fabricated vortex plates consisting of nanoapertures with C2 (first row), C3 (second row), and C5 (third row) rotational symmetry. All the samples have a radius $R=8\mu \mathrm{m}$. Scale bars, (a) $2\mu \mathrm{m}$, (b) 500 nm. (c) Measured interference patterns of the vortex beams under LCP illumination with a wavelength of 633 nm.