# Topological Photonic Media and the Possibility of Toroidal Electromagnetic Wavepackets
## 5 Electronic State with Twisted Spin-polarization
The Hamiltonian of the sequence is given by  
\[\hat{H}_{\text{3 DTI}} = \sum_{\mathbf{r}}\sum_{\mu}\left(-1\right)^{\mathbf{r}}t_{\mu}\left[ \hat{c}_{\mathbf{r}+\mathbf{a}_{\mu}}^{\dagger}U_{\mu}^{\left(-1\right)^{\mathbf{r}}} \hat{c}_{\mathbf{r}}+\text{H.c.}\right] \tag{13}\] \[+\sum_{\mathbf{r}}\sum_{\mu<\
u}\left(-1\right)^{\mathbf{r}}t_{\mu\
u} \left[\left(\hat{c}_{\mathbf{r}+\mathbf{a}_{\mu}+\mathbf{a}_{\
u}}^{\dagger}\hat{c}_{\mathbf{ r}}+\hat{c}_{\mathbf{r}-\mathbf{a}_{\mu}+\mathbf{a}_{\
u}}^{\dagger}\hat{c}_{\mathbf{r}}\right)+ \text{H.c.}\right]+\sum_{\mathbf{r}}\left(-1\right)^{\mathbf{r}}v_{s}\hat{c}_{\mathbf{r}}^ {\dagger}\hat{c}_{\mathbf{r}}.\]  
This sequence is advantageous in that the edge states of a member with open boundary condition can be analytically investigated in some parameter regions, provided that \(\{U_{\mu}\}\) satisfies the conditions \(U_{\mu}^{\dagger}U_{\
u}+U_{\
u}^{\dagger}U_{\mu}=2\delta_{\mu\
u}v_{0}\) and \(\sigma_{2}\overline{U}_{\mu}\sigma_{2}=U_{\mu}\). Here, \(U_{\mu}^{\dagger}\) and \(\overline{U}_{\mu}\) are Hermitian and complex conjugates of \(U_{\mu}\), respectively. The symbol \(v_{0}\) stands for the \(2\times 2\) unit matrix in the spin space. Unfortunately, there remain unresolved issues in plausible modeling of the interface between a member of this sequence and conductor. The analysis also accompanies technical complications and will be given elsewhere. Besides, 3D versions of topological photonic crystals have also been proposed [51; 52]. Although the relation between a photon's pseudo-spin and actual angular momentum in a periodic structure remains ambiguous currently, examples of energy flux densities of photonic chiral edge modes in References [25; 33] and photonic helical edge modes in Figure 5 suggest that the former corresponds to a vortex structure stemming from the latter. A schematic of the ideal photonic helical surface modes is shown in Figure 8b. Here, each set of yellow and black arrows represent the propagation direction and local angular momentum density of a photonic helical surface mode, respectively, whereas the yellow circles containing arrows represent the local vortex structures of the modes. Let us look back to the electronic wavepacket with a twisted spin structure in Figure 7b and consider its 3D extension. Suppose a case exists where a wavepacket homogeneously polarized in the propagation direction is perpendicularly incident on the surface of an idealistic 3D topological insulator depicted in Figure 8a. Since in this case we can find the rotational symmetry around the incident axis, the reflected wavepacket is expected to accompany a toroidally-twisted spin texture derived by rotating  
Figure 8: Conceptual diagrams of (**a**) electronic helical surface states and (**b**) photonic helical surface modes. Figure 7b around the incident axis. Therefore, the question that arises is how do we extend the above discussion of electoronic wavepacket to its photonic version. Section 4 argues that various types of topological photonic media can be proposed based on the same idea in electron systems. As speculated in Figure 8b, the electronic spin would be replaced by a photonic vortex structure. It is reasonable to replace the incident electronic wavepacket homogeneously spin-polarized to the propagation direction by an electromagnetic wavepacket with photonic orbital angular momentum in the propagation direction, i.e., by that as depicted in Figure 9. Similarly, a simple thinking on the reflected wavepacket would suggest that a toroidally-twisted spin structure can be replaced by a toroidal vortex structure, as depicted in Figure 10. (The hue and the arrows in Figures 9 and 10 represent the energy density and the deviation of energy flux density, respectively, as in Figure 1.) This speculation appears to be extremely naive, because in general, the vortex structure of a photonic helical surface mode is complicated, as displayed in Figure 5c.