# Topological Photonic Media and the Possibility of Toroidal Electromagnetic Wavepackets
## 6 Propagation Characteristics of Toroidal Electromagnetic Wavepacket  
The electromagnetic vortices shown in Figure 9 can be implemented into a quantum digit for information communication [13] and mode division multiplexing in telecommunication technology [14; 15]. Figure 10: (a) _xy_- and (b) _yz_-cross-sections, and (c) isosurface of the energy density of a linearly polarized toroidal wavepacket with orbital angular momentum directed to the positive \(z\)-axis. Figure 9: (a) _xy_- and (b) _yz_-cross-sections of a linearly polarized Laguerre-Gaussian wavepacket with orbital angular momentum directed to the positive \(z\)-axis. These applications use the fact that there are multiple quasi-orthogonal modes in a narrow frequency band. Hence, it is a meaningful task to devise various extensions of such electromagnetic vortices. This section brings up the toroidal vortex structure in Figure 10 as one of those extensions. Moreover, it aims to answer questions such as whether the electromagnetic wavepacket with a toroidal vortex can exist as a solution to [PERSON]'s equations and how stable it is when it can be present. To answer the initial question, we shall present a procedure to construct wavepacket solutions with generic vortex structures. In the process, we assume that we already have the information of Fourier components \(\mathbf{e}_{\mathbf{k}\alpha}\) of plane wave solutions of mode \(\alpha\) with eigenfrequency \(\omega_{\mathbf{k}\alpha}\), and that the set \(\{\mathbf{e}_{\mathbf{k}\alpha}\}\) (\(\alpha=1,2,\cdots\)) constitutes a perfect orthonormal system, at least at a practical approximation level. The construction procedure consists of four steps as follows:  
1. Construct normalized scalar wavepackets \(\{f_{\alpha}(\mathbf{r})\}\) with trial vortex structures for mode \(\alpha\). 2. Calculate the Fourier transform \(\{\tilde{f}_{\mathbf{k}\alpha}\}\) of \(\{f_{\alpha}(\mathbf{r})\}\). 3. Construct the solution of electromagnetic field \(\tilde{\mathbf{E}}(\mathbf{k},t)\) in \(\mathbf{k}\)-space by \[\tilde{\mathbf{E}}(\mathbf{k},t)=\sum_{\alpha}\tilde{f}_{\mathbf{k}\alpha}z_{\mathbf{k}\alpha} \mathbf{e}_{\mathbf{k}\alpha}\exp\left(i\omega_{\mathbf{k}\alpha}t\right),\] (14) where the set of parameters \(\{z_{\mathbf{k}\alpha}\}\) reduces to Jones vector in simple cases. 4. Calculate the inverse Fourier transform of \(\tilde{\mathbf{E}}(\mathbf{k},t)\), and take its real part as the solution \(\mathbf{E}(\mathbf{r},t)\). Generally, the above procedure can contain numerical calculations and is inevitably accompanied by approximation due to discretization of both real and wavevector spaces. Nonetheless, in principle the obtained solution converges to an exact solution in the continuous limit. Electromagnetic wavepackets with any vortex structure can actually exist, whereas the stability remains uncertain. In other words, this procedure is applicable as long as the quasi-complete set of \(\{\mathbf{e}_{\mathbf{k}\alpha}\}\) is obtained by either an analytical or numerical method. For instance, a typical case for the former may include the reflection and refraction of linearly polarized plane waves at a flat interface between two different media of homogeneous isotropic permittivity \(\epsilon\) and permeability \(\mu\). Analytical expressions of \(\{\mathbf{e}_{\mathbf{k}\alpha}\}\) are given by [PERSON]'s equations, where the index \(\alpha\) stands for either P- or S-polarization, while the index \(\mathbf{k}\) can represent the wavevector of an incident plane wave. As for the latter, we can consider an extension to periodic systems by replacing momentum \(\mathbf{k}\) by crystal momentum and making mode index \(\alpha\) include a band index, along with a degenerate mode index, as in \(\alpha\to n\lambda\) (\(n\): band index and \(\lambda\): degenerate mode index). Finally, note that the center of a wavepacket can be easily shifted by \(\mathbf{r}_{0}\) through the replacement \(\{f_{\alpha}(\mathbf{r})\}\rightarrow\{f_{\alpha}(\mathbf{r}-\mathbf{r}_{0})\}\).