Conical diffraction, pseudospin, and nonlinear wave dynamics in photonic Lieb lattices

This paper explores wave dynamics near a singularity which occurs in the Lieb lattice built with three square sublattices. The Lieb lattice band singularity consists of three intersecting bands - two with a conical shape, and a third flat band in between. This intersection is associated with pseudo-spin 1, rather than the pseudospin 1/2 of the honeycomb lattice. In analogy with earlier results in the honeycomb lattice, it is found that the angular momentum of waves propagating near the singularity is naturally divided into macroscopic (on the scale of many lattice periods) orbital angular momentum, and a microscopic (on the scale of a lattice period), pseudospin component. Their sum is a conserved quantity for waves near the singularity. Consequently, the pseudospin plays an important role in wave dynamics, which is demonstrated by studying the diffraction of different pseudospin eigenstates. The diffraction of pseudo-spin 0 waves closely resembles the conical diffraction that occurs in honeycomb lattices. In contrast, waves with pseudospin ±1 also excite the flat band, resulting in a nondiffracting central spot in addition to conically diffracting rings. Finally, the effect of nonlinearity on wave dynamics near the singularity is looked into by considering a photonic lattice with Kerr-type nonlinearity. Nonlinear effective field equation for the pseudo-spin 1 waves is derived and its dynamics is compared with numerical solutions to the full nonlinear Schrödinger equation, finding excellent agreement. The nonlinearity transforms the diffracting rings of conical diffraction into squares, whose orientation depends on the sign of the nonlinearity. This is explained by the nonlinearity-induced mixing of waves between the three intersecting bands.