Fig. 1: Working principle of a nonreciprocal space–time phase modulated metasurface.

a An illustration showing the concept of a space–time phase modulated metasurface consisting of resonating dielectric nanoantennas operating in reflection mode. A travelling phase modulation in sinusoidal form is superposed on the designed phase gradient along the x direction. Light impinging on the metasurface with frequency ω is converted to a reflecting beam with frequency ω – Δω due to the parametric process arising from dynamic phase modulation, while the back-propagating beam with frequency ω – Δω is converted to ω – 2Δω instead of ω, resulting in a nonreciprocal effect. b, c and d, e Comparison between a regular space modulated metasurface (b and c) and a space–time phase modulated metasurface (d and e). A regular space-modulated metasurface supports only symmetric forward (b) and backward (c) reflections, as shown in the dispersion diagrams. There is no frequency conversion (i.e., no transition in the energy space). The process is reciprocal, and the forward and backward beams share the same trajectory. In contrast, a space–time phase modulated metasurface supports asymmetric forward (b) and backward (c) reflections. It not only offers additional momentum along the x direction to the reflected light but also converts its frequency. In either the forward or backward case, the upward transition is forbidden because the resulting wavevector will be too large to be supported in free space, resulting in unidirectional photonic transitions in both the energy and momentum spaces. Therefore, the trajectories of beams differ and reveal nonreciprocity effects. k_s is the linear momentum introduced by the spatial phase modulation, and k_M is the additional linear momentum introduced by the temporal phase modulation.