# Topological Photonic Media and the Possibility of Toroidal Electromagnetic Wavepackets
## 3 Electromagnetic Vortices in Periodic Systems  
The spin DOF of photon is no longer well-defined in the presence of periodic structure, while we shall need an alternative concept of photonic spin to discuss the possibility of toroidal electromagnetic wavepackets in Section 5, based on an electron dynamics around the interface of a topological insulator where the concept of electronic spin still works well. (The photonic version of topological insulator will be introduced in Section 4.) One possibility of the alternative is the internal rotational motion of electromagnetic Bloch modes with vortices. In this section, we step further into electromagnetic vortices in periodic structures such as photonic crystals. After looking back briefly on the relation between the Berry curvature and real-space vortex structure in periodic systems, we present a numerical simulation on propagation modes around a standing vortex mode in a photonic crystal, indicating the effect of the internal rotation in the propagation process. Useful functionalities of a photonic crystal, such as light confinement and waveguide, are realized by adjusting energy (frequency) dispersion relations and forbidden bands by the periodic structure and symmetry design of the system. The required design procedures stem from the band theory common to wave phenomena in periodic structures. Moreover, based on the studies of geometric Hall effect in electron systems [21; 22], we can also use such design to control the Berry curvature of each band [23; 24]. For instance, by employing a two-dimensional (2D) periodic system in the \(xy\) direction, we can consider a situation where two bands are immediately prior to touching each other at a point \(\mathbf{k}_{0}\) in a wavevector space. Given an approximate description, \(f_{\mathbf{k}}\pm\sqrt{v^{2}|k-\mathbf{k}_{0}|^{2}+\Delta^{2}}\) (\(f_{\mathbf{k}}\): a function of wavevector \(\mathbf{k}\)) for the local energy dispersion of each, the \(z\)-component of the Berry curvature of each band is estimated as \(\Omega_{\mathbf{k}\pm}^{z}=\pm v^{2}\Delta/\left\{2(v^{2}|\mathbf{k}-\mathbf{k}_{0}|^{2}+ \Delta^{2})\right\}^{\frac{3}{2}}\). In other words, we can control the Berry curvature by adjusting the level repulsion \(\Delta\). We confirmed this mechanism in a more strict theory exactly treating the periodic structure, as well as the relation among the Berry curvature, the angular momentum [19; 20] and the real-space vortex structure [16; 25]. The electromagnetic vortex can be also controlled through the adjustment of \(\Delta\). Figure 2 shows an example of a set of such periodic structure, band diagram, and electromagnetic vortex. As for the 2D photonic crystals, we considered only photonic modes of two distinct polarizations, i.e., transverse-electric (TE) and transverse-magnetic (TM), which propagate strictly parallel to the plane with a 2D periodicity. Herein, we adopted this definition: TE modes have magnetic fields normal to the plane and electric fields in the plane; conversely, TM modes have electric fields normal to the plane and magnetic fields in the plane. The electromagnetic vortex shown in Figure 2c corresponds to a standing wave mode of a zero group velocity. On the other hand, modes around it have finite group velocities in addition to the vortex structure, and can propagate through the crystal with rotational motion. Next, we describe the propagation characteristics of such modes in a Y-shaped waveguide in Figure 3a composed of the crystal in Figure 2a and a block layer with a sufficiently large band gap covering the relevant frequency range. Since it was not easy to excite a specific electromagnetic vortex mode in a real-time-and-space simulation, we adopted an excitation using a linear source with a line width of a few percent around the central frequency of the targeted vortex modes. Moreover, the linear source was set at the left end of the left branch of the waveguide and the vortex modes were excited by electric field oscillations along the source.