Photonic slide rule with metasurfaces.

As an elementary particle, a photon that carries information in frequency, polarization, phase, and amplitude, plays a crucial role in modern science and technology. However, how to retrieve the full information of unknown photons in an ultracompact manner over broad bandwidth remains a challenging task with growing importance. Here, we demonstrate a versatile photonic slide rule based on an all-silicon metasurface that enables us to reconstruct incident photons' frequency and polarization state. The underlying mechanism relies on the coherent interactions of frequency-driven phase diagrams which rotate at various angular velocities within broad bandwidth. The rotation direction and speed are determined by the topological charge and phase dispersion. Specifically, our metasurface leverages both achromatically focusing and azimuthally evolving phases with topological charges +1 and -1 to ensure the confocal annular intensity distributions. The combination of geometric phase and interference holography allows the joint manipulations of two distinct group delay coverages to realize angle-resolved in-pair spots in a transverse manner- a behavior that would disperse along longitudinal direction in conventional implementations. The spin-orbital coupling between the incident photons and vortex phases provides routing for the simultaneous identification of the photons' frequency and circular polarization state through recognizing the spots' locations. Our work provides an analog of the conventional slide rule to flexibly characterize the photons in an ultracompact and multifunctional way and may find applications in integrated optical circuits or pocketable devices.