Topological Floquet bound states in the continuum.

A honeycomb array of helical waveguides with zigzag-zigzag edges and a refractive index gradient orthogonal to the edges may support Floquet bound states in the continuum (BICs). The gradient of the refractive index leads to strong asymmetry of the Floquet-Bloch spectrum. The mechanism of creation of such Floquet BICs is understood as emergence of crossings and avoided crossings of the branches supported by spatially limited stripe array. The whole spectrum of a finite array is split into the bulk branches being a continuation of the edge states in the extended zone revealing multiple self-crossings and bulk modes disconnected from the gap states by avoided crossings. Nearly all states in the system are localized due to the gradient, but topological edge states manifest much stronger localization than other states. Such strongly localized Floquet BICs coexist with localized Wannier-Stark-like bulk modes. Robustness of the edge Floquet states is confirmed by their passage through a localized edge defect in the form of a missing waveguide.