Fig. 1: Tilted moiré lattices.
From: Two-dimensional Thouless pumping of light in photonic moiré lattices
a Schematics of the tilted moiré lattice based on a reference square lattice V(r) with primitive translation vectors ai and aj, where a is the lattice constant (here set to 1) (i). One of the constitutive sublattices is obtained by twisting the reference lattice by the angle θ around the z-axis, yielding the potential V(R(θ)r) (the red-colored lattice in (ii)). The second sublattice is obtained by rotating the reference lattice by an angle α around the x-axis (the olive-colored lattice in (iii)), thus producing the potential V(r − αzj). Superimposing V(r − αzj) and V(R(θ)r) results in the moiré potential U shown in (iii). The tilted moiré lattice has a period Z along the direction of the incident beam (shown by a dark-red arrow). b Cross-sections of the optically induced moiré lattice U(r, z) with p = 0.3 and E0 = 7 (upper row) and the instantaneous bandgap structures (lower row) at distances z = 0, Z/4, Z/2, and 3Z/4. The red arrows indicate the primitive translation vectors e1 and e2 of the tilted moiré lattice. Blue and red lines in lower row of (b) show the first and second bands in the instantaneous lattice spectrum for the same distances. c Blue and red lines show the evolution of the bottom edge of the first band and top edge of the second band with distance z on one lattice period Z. In (b) and (c), α = 0.015.
