Engineering nonreciprocal responses in traveling-wave spacetime crystals via Clausius-Mossotti homogenization

Here, we investigate the effective response of three-dimensional spacetime crystals formed by spherical scatterers under a traveling-wave modulation. We develop an analytical formalism to homogenize the spacetime crystals that extends the renowned Clausius-Mossotti formula to time-varying platforms. Our formalism shows that traveling-wave spacetime crystals can be used to engineer a wide range of classes of nonreciprocal bianisotropic couplings in the long-wavelength limit. In particular, our theory reveals the possibility of realizing a purely isotropic Tellegen response in crystals formed by interlaced sublattices of scatterers subjected to different modulation velocities. Furthermore, we introduce a class of generalized Minkowskian crystals that display invariance under arbitrary Lorentz boosts aligned with a fixed spatial direction. We prove that such systems are formed by pseudouniaxial materials with the principal axis aligned parallel to the modulation velocity. The electromagnetic response of such generalized Minkowskian crystals is indistinguishable from that of moving photonic crystals.