# Topological Photonic Media and the Possibility of Toroidal Electromagnetic Wavepackets
## 4 Topological Photonic Media
A wider range of topological materials, including superconductors, have been systematically classified based on symmetry and dimensionality [44; 45]. Typical physical conditions under which topological insulators emerge are as follows: (1) two pairs of bands with different parities opposite to each pair, which are energetically close to each other, hybridize with each other through a strong spin-orbit interaction and; (2) the resultant level repulsion forms an enough sized bulk gap [46]. By contrast, for photon systems in periodic structures, what kind of DOF should be regarded as spin DOF remains unclear. Nevertheless, if the difference between certain degenerate modes is approximately regarded as pseudo-spin DOF and the coupling between electric and magnetic fields introduced by an artificial chiral medium are regarded as effective spin-orbit interaction of photon, then a similar mechanism could be applied to photon systems in periodic structures. Photonic versions of topological crystalline insulator using metamaterials as artificial chiral media have been proposed [41; 42]. After such proposals, it has been pointed out that those topological photonic media could be realized even by photonic crystals composed of only ordinary dielectrics [43], whereas such systems needed to introduce some complication to the unit cell of the crystal. Therefore, we should regard the chiral medium as an example of effective spin-orbit interaction implementation, and not an item of necessity. Figure 4a,b show examples closely related to the all-dielectric topological photonic crystals in Ref. [43]. The structure of Figure 4a is an inversion asymmetric deformation of a topological photonic crystal; hence, we shall call it a quasi-topological photonic crystal for convenience. Compared to the case of Figure 2a, the degree of the symmetry breaking is so weak that it is not easy to distinguish two kinds of rods colored by dark gray and black. The bulk bands are almost unchanged from the symmetric case as depicted in Figure 4c. However, as we shall see below, this symmetry breaking clearly resolves the degeneracy of edge modes and opens a recognizable gap in edge bands. The crystal comes into the topological phase when the inversion symmetry is restored by setting the same values of relative permittivity, i.e., 10, in the dark gray and black regions. Contrastingly, the crystal of Figure 4b is in non-topological phase. Figure 4c is the band diagram of the TM modes of the quasi-topological photonic crystal in Figure 4a. We can find sufficiently-sized band gap at approximately 0.5 in the unit \(\omega a/(2\pi c)\). Figure 5a displays the unit cell of the superlattice composed of the crystals of Figure 4a,b. Figure 5b is a closeup of the projected band diagram of TM modes in the superlattice. The edge modes are emphasized in red. The red-dotted lines are the edge modes when the inversion symmetry of the middle part is restored. In this superlattice, the structure around the edge part also breaks the inversion symmetry of the whole system, as evidenced by a small gap in the edge band modes even when the middle part is in the topological phase. The explicit breaking of the inversion symmetry in the middle part increases the size of this gap as well as resolves the degeneracy of the edge modes. Figure 5c gives the energy flux density of an edge mode belonging to the lowest branch. We can see that the mode is well confined around the boundary and accompanies some eddies. The propagation characteristics of the edge modes were demonstrated in a Y-shaped waveguide in Figure 6a, where the Y-shaped part is composed of the crystal in Figure 4a, whereas the block layer is composed of the crystal in Figure 4b with a sufficiently large band gap to cover the relevant frequency  
Figure 4: **(a)** A sample inversion asymmetric 2D quasi-topological photonic crystal for relative permittivity of 1, 9, and 11 in white, dark gray, and black regions, respectively. When the inversion symmetry is restored, the crystal can be in the topological phase.