# Topological Photonic Media and the Possibility of Toroidal Electromagnetic Wavepackets
## 6 Propagation Characteristics of Toroidal Electromagnetic Wavepacket
To simplify the discussion, we shall omit the mode dependence of trial functions introduced above, and pick up only linearly-polarized wavepackets here. Figure 10 provides a sample electromagnetic wavepacket constructed from the procedure above, and which propagates at a positive \(z\)-direction. The wavepacket has an energy density distributed in a hollow torus-shape, and the wavepacket has a ring-shaped vortex along the internal hollow part of the torus, in addition to a line-shaped vortex associated with the orbital angular momentum directed to positive \(z\)-direction, which penetrates through the central hole of the torus. Figure 10a represents the \(xy\)-cross-section of the wavepacket, where we can find the eddy structure of energy flux density corresponding to the orbital angular momentum. On the other hand, Figure 10b represents the \(yz\)-cross-section of the wavepacket, where we can find another vortex structure whose core corresponds to the hollow part inside the torus. Let us take a closer look at a trial scalar wavepacket with toroidal-type vortex structure, \(f_{\text{TWP}}(\mathbf{r})\). This function contains seven types of parameters, namely \(m_{\text{line}}\): vorticity of line vortex; \(m_{\text{ring}}\): vorticity of ring vortex; \(\mathbf{k}_{\text{c}}\): central wavevector; \(\ell_{\text{R}}\): radius of the central ring inside the hollow region; \(\ell_{\text{r}}\): radius of torus-type tube; \(\ell_{\Delta}\): thickness of surface layer of hollow torus and; \(\ell_{\text{v}}\): size of vortex core, where we set the core sizes of two kinds of vortices to be the same. By introducing a right-handed basis set \(\{\mathbf{e}_{1},\mathbf{e}_{2},\mathbf{e}_{3}\}\) with the condition \(\mathbf{e}_{3}=\mathbf{k}_{\rm c}/|\mathbf{k}_{\rm c}|\), we can give an example such as  
\[f_{\rm TWP}\left(\mathbf{r}\right) = \mathcal{N}\left(\frac{z_{\rm line}}{|z_{\rm line}|}\right)^{m_{ \rm line}}\left(\frac{z_{\rm ring}}{|z_{\rm ring}|}\right)^{m_{\rm ring}}\tanh \left(\frac{|z_{\rm line}|}{\ell_{\rm v}}\right)\tanh\left(\frac{|z_{\rm ring} |}{\ell_{\rm v}}\right) \tag{15}\] \[\times\exp\left\{-\frac{1}{2\ell_{\Delta}^{2}}\left(|z_{\rm ring }|-\ell_{\rm r}\right)^{2}+i\mathbf{k}_{\rm c}\cdot\mathbf{r}\right\},\] \[z_{\rm line} = (\mathbf{e}_{1}+i\mathbf{e}_{2})\cdot\mathbf{r},\quad z_{\rm ring}=(|z_{\rm line }|-\ell_{\rm R})+i\mathbf{e}_{3}\cdot\mathbf{r}, \tag{16}\]  
where \(\mathcal{N}\) is a normalization factor. Figure 10 corresponds to the case where \(m_{\rm line}=1\), \(m_{\rm ring}=-1\), \(\lambda=2\pi/|\mathbf{k}_{\rm c}|=0.445\ell_{0}\), \(\ell_{\rm R}=6\ell_{0}\), \(\ell_{\rm r}=3\ell_{0}\), \(\ell_{\Delta}=\ell_{0}\), and \(\ell_{\rm v}=0.25\ell_{0}\) where \(\ell_{0}\) is a unit of length scale. Herein, we shall consider only this set of parameters for toroidal wavepackets, as our focus is limited on the topological properties of toroidal wavepackets, and does not extend to details of their shape deformations. For better understanding by way of comparison, we present trial scalar functions \(f_{\rm GWP}(\mathbf{r})\) and \(f_{\rm LGWP}(\mathbf{r})\) for Gaussian and Laguerre-Gaussian wavepackets, respectively, defined by  
\[f_{\rm GWP}\left(\mathbf{r}\right) = \mathcal{N}\exp\left(-\frac{r^{2}}{2\ell_{\rm R}^{2}}+i\mathbf{k}_{ \rm c}\cdot\mathbf{r}\right), \tag{17}\] \[f_{\rm LGWP}\left(\mathbf{r}\right) = \mathcal{N}\left(\frac{z_{\rm line}}{|z_{\rm line}|}\right)^{m_{ \rm line}}\tanh\left(\frac{|z_{\rm line}|}{\ell_{\rm v}}\right)\exp\left(- \frac{|z_{\rm ring}|^{2}}{2\ell_{\rm r}^{2}}+i\mathbf{k}_{\rm c}\cdot\mathbf{r}\right), \tag{18}\]  
The Gaussian wavepacket in Figure 1 and the Laguerre-Gaussian wavepacket in Figure 9 correspond to the cases with \(\ell_{\rm R}=6\ell_{0}\) and with \(m_{\rm line}=1\), \(\ell_{\rm R}=6\ell_{0}\), \(\ell_{\rm r}=2\ell_{0}\), \(\ell_{\rm v}=0.25\ell_{0}\), respectively. In both cases, the wavelength is set at \(\lambda=2\pi/|\mathbf{k}_{\rm c}|=0.445\ell_{0}\).