Solitons on Patterned Metasurfaces: Glide, Sector and Seam

Abstract

A patterned metasurface is a lattice of resonators, and its symmetries decide where a Kerr soliton can exist, whether it radiates, and how its polarization survives a closure. This article follows these questions in the sector language of the Möbius–Klein Transform (MKT) canon. A glide, a half-period shift with a mirror, makes the bands stick on the zone edge, and a glide-odd defect opens the gap linearly. In one dimension the gap soliton then has, at a fixed position in the gap, a width proportional to the inverse defect and a power independent of it. In two and three dimensions the full gap opens only above a threshold set by the transverse couplings; gap solitons exist above it, and near it the three-dimensional soliton widens across the stack. The glide-odd defect acts as a mass: a sign change binds a state near mid-gap, a glide-patterned Möbius strip forces such a wall, and near mid-gap the Kerr soliton of the wall needs no threshold power. A flux turns a mirror into a glide in momentum space. There the real gap soliton is made of two valleys and its rows repeat with period 4; its valley phase locks, with a power ratio of 2 between the bond-centred and the row-centred family, and the Bloch period on a time-reversal-invariant line of the zone (a TRIM line) is halved. In the dark channels the n-th harmonic of a mode of rotation label l radiates at normal incidence only if nl1, so on a crystalline metasurface with a rotation axis no harmonic of a -point bound state in the continuum (BIC) protected by that axis radiates at normal incidence. A soliton on a BIC band decays algebraically, turns exponential below a crossover set by a mirror defect, and stays dark in the zero order until its peak cell power is close to the dimer value. A lossless, time-reversal-symmetric Kerr metasurface has no nonlinear optical activity; a circular soliton flips its helicity at every circuit of a Möbius closure, and a linearly polarized one beats after the seam of a planar-chiral lattice. Design rules and refutable predictions follow. The glide threshold, the forced mass wall of a glide pattern, the locked valley phase, the half Bloch period, the dark harmonics, the lattice form of the decay law and its defect crossover, and the behaviour at the seam were not found in the literature checked. All results are model computations.