Riemann-Silberstein geometric phase in 4D polarization space

Abstract

Geometric phase is a far-reaching concept in quantum and classical physics. The first discovered geometric phase, the Pancharatnam-Berry (PB) phase, has profoundly shaped nanophotonics through metasurfaces. However, the PB phase arises from SU(2) polarization evolution and is constrained to a 2D polarization space, failing to capture the full polarization degrees of freedom. We generalize geometric phase to the 4D Riemann-Silberstein (RS) space that simultaneously describes electric, magnetic, and hybrid electric-magnetic polarizations. We show that SU(4) polarization evolution can generate a new geometric phase, the RS phase, alongside the PB phase. Unlike the PB phase that typically manifests in circularly polarized light, the RS phase can emerge in arbitrarily polarized light. Together, they enable a high-dimensional geometric framework for light propagation across general interfaces. We reveal that the phase shifts governed by Fresnel equations are direct manifestations of the RS-space geometric phases, integrating a century-old wave theory into this paradigm. We experimentally validate the framework using metasurfaces and achieve high-dimensional wavefront manipulation. Our work offers fundamental insights into the geometric nature of light-matter interactions, with implications for topological and non-Abelian physics in classical wave systems.

Figures (5)

• Figure 1: Complete 4D polarization of electromagnetic fields. (a) Poincaré hypersphere representation of 4D electromagnetic polarization, where the points “c-h” represent the polarizations of the plane waves in the panels (c-h), respectively. (b) RS geometric phase induced by the rotation of the local constitutive frame. Electric field (in red), magnetic field (in blue), and hybrid RS field (in black) of (c) $+z$-propagating linearly polarized plane wave, (d) $-z$-propagating linearly polarized plane wave, (e) $+z$-propagating RCP plane wave, (f) $-z$-propagating RCP plane wave, (g) $+z$-propagating LCP plane wave, and (h) $-z$-propagating LCP plane wave. The electric spin density $\mathbf{s_\mathrm{e}}$, magnetic spin density $\mathbf{s_\mathrm{m}}$, and RS spin density $\mathbf{s_\mathrm{rs}}$ are denoted by the red, blue, and black arrows, respectively.
• Figure 2: RS geometric phase at interfaces. (a) A plane wave with linearly polarized electric and magnetic fields impinges on an interface of isotropic media, where the RS polarization (blue arrows) undergoes evolutions but electric polarization (red arrows) remains unchanged. (b) Representation of the RS polarization evolution on the RS-sphere for the wave transmission and reflection. We choose the point on ${S}_1$ axis as the reference polarization. (c) Transmission and reflection phases for different orientation angles of the transmitted RS polarization. The circles denote the RS geometric phase. (d) A plane wave with circularly polarized electric and magnetic fields impinges on an interface of anisotropic media, where both the electric polarization (red arrows) and the RS polarization (blue arrows) undergo evolutions. (e) Representation of the 4D polarization evolution on the Poincaré hypersphere for the wave transmission and reflection. (f) Transmission and reflection phases for different orientation angle of the electric polarization. The circles denote the total geometric phase.
• Figure 3: RS geometric phase at a metasurface. (a) Schematic of the RS metasurface under the normally incident plane wave with linearly polarized electric field. The inset shows the structure of the meta-atom. (b) RS polarization evolution on the RS-sphere induced by the metasurface. (c) Theoretical RS geometric phase (symbols), simulated phase (solid lines), and simulated amplitude for the forward and backward scattering fields as a function of the orientation angle $\alpha_\mathrm{rs}$ of the RS dipole. (d) Simulated electric field scattered by the RS metasurface. (e) Experiment setup and the fabricated metasurface prototype. (f) Simulated and experimentally measured far-field intensity pattern. The shaded area marks the measurement blind zone [-15 deg, 15 deg] due to the source antenna obstruction.
• Figure 4: Complementary RS and PB phases at a metasurface. (a) Schematic of the quadruplex RS meta-deflector under the normal incidence of a plane wave with LCP electric field. (b) 4D polarization evolution on the Poincaré hypersphere induced by the higher-order RS metasurface. (c) Theoretical geometric phase (symbols), simulated phase (solid lines), and simulated amplitude in four polarization evolution channels as a function of the orientation angle of RS dipole in the meta-atom. (d) Simulated electric field distribution of four output waves. (e) Fabricated metasurface prototype. (f) Simulated and experimentally measured far-field intensity pattern under orthogonal incidence. The shaded area marks the measurement blind zone [-15 deg, 15 deg] due to the source antenna obstruction.
• Figure 5: Reconfigurable wave deflection by the general RS metasurface. (a) Schematic of the multiplexed beam forming with twelve distinct output wavefronts. (b) Simulated far-field intensity patterns under the normal incidence of plane waves with orthogonal 4D polarizations. The lobes are labelled in accordance with (a). (c) Incident 4D polarizations (labelled as “i-iv") for achieving reconfigurable far-field intensity patterns. (d) Far-field intensity patterns with different number of lobes induced by the incident polarizations “i-iv” in (c).