This thesis in theoretical physics deals with the topological properties of 2D periodically driven systems in the context of photonics.
In the first part, topological regimes beyond that of band insulators are investigated. We have shown the existence of two new topological regimes : the first one is analogous to a semimetallic phase, while the other one exploits the winding of the spectrum thus having no static counterpart. Despite the existence of gapless edge states along with the gapless bulk states, in both the cases, their topological description strongly differs. In the winding regime, one can define a meaningful topological invariant by continuously deforming the bands to end up with a direct band gap. In this regime, there exists an interplay of two topological properties: one is the winding of the quasienergy bands, and the another is the presence of chiral edge states in a finite geometry. The former property manifests as Bloch oscillations of wavepackets, where stationary points in the oscillations are related to the winding number of the bands. This topological property can thus be probed directly in an experiment by the state-of-art technology. In the second regime (semimetallic), we see how degeneracies can be specifically manipulated at the quasienergy 0 or pi. Unlike previous regime, as a consequence of the absence of any kind of gap (direct or indirect) the topology can be captured by enclosing the degeneracies in parameter space and calculating the Berry flux piercing through the enclosed surface.
In the second part of our work, we explore how topological properties can be engineered in 1D photonics evanescently coupled waveguide arrays. This is made possible by the interplay between crystalline symmetries of the network and the other discrete symmetries responsible for topology like chiral symmetry. However, due to the generalities of the argument, these concepts can easily be extended to higher dimensions, as well. In these same 1D waveguide network, we identify the link between breaking bipartiteness of the structure and existence of a symmetry that has been overlooked before, namely the shifted-particle hole symmetry. We clearly point out in the same waveguide arrays its dissimilarity with respect to usual particle hole symmetry. Similar to particle hole symmetry, it is also responsible for giving rise to non-trivial topology.