# Topological Photonic Media and the Possibility of Toroidal Electromagnetic Wavepackets
## 6 Propagation Characteristics of Toroidal Electromagnetic Wavepacket
For the second question, we shall consider the stability of the toroidal wavepacket against reflection and refraction on flat interfaces between different kinds of homogeneous isotropic media. The dielectric constants on the lower and upper sides are represented by the symbols \(\epsilon_{1}\) and \(\epsilon_{2}\), respectively. Figures 11-13 show the time lapses for cases with \(\epsilon_{2}/\epsilon_{1}=0.40\), \(0.75\), and \(2.50\), respectively. The incident angle is set at \(45^{\circ}\). The time is measured in units of \(\ell_{0}\sqrt{\epsilon_{1}\mu_{0}}\). The dimensionless time \(\tau\) of each frame is \(\tau=-16\), \(-8\), \(0\), \(+8\), \(+16\) from left to right. Figure 14 shows the incident-angle dependence for \(\epsilon_{2}/\epsilon_{1}=2.50\). The incident angle \(\theta\) of each frame is \(\theta=0^{\circ},15^{\circ},30^{\circ},45^{\circ},60^{\circ}\) from left to right, and the dimensionless time \(\tau\) is \(\tau=+16\) in every frame. In all cases, the magnetic permeability is set at \(\mu=\mu_{0}\) everywhere, and only the \(xz\)-cross-sections are depicted. (\(x\)- and \(z\)-axes correspond to horizontal and vertical directions, respectively.) We adopt quasi-P-type configuration for the polarization state of every incident wavepacket. (The mean magnetic field of every incident wavepacket is parallel to the interface and normal to the quasi-incident-plane.) From the result in Figure 11, we can presume that the toroidal vortex is stable against reflection. On the other hand, two cases for refraction emerge. First, as the refractive index at the transmission side (Figure 12) decreases, the wavepacket shape stretches in a similar manner to its central wavelength, leading to an unstable ring vortex. Second, as the refractive index at the transmission side (Figure 13) increases, the wavepacket compresses in a similar manner to its central wavelength, resulting in a stable ring vortex at least up to \(\theta=60^{\circ}\) (Figure 14). Finally, we would like to mention the transformation laws of the wavepacket topological properties. The vorticity \(m_{\rm line}\) of the linear vortex corresponding to the orbital angular momentum changes as \(m_{\rm line}\to-m_{\rm line}\) in reflection, while it does not in refraction, suggested by our analogy with the Laguerre-Gaussian beam with an orbital angular momentum. By contrast, the vorticity \(m_{\rm ring}\) of the ring vortex does not change in both reflection and refraction. In general, recognizing a wavepacket as a particle-like object may give  
media as the photonic version of the electronic topological insulator and further interpreted it as an extension of a class of photonic crystals where various types of electromagnetic vortex modes emerge with their time-reversal partners. Furthermore, we referred to the fact that modern information transmission technology via electromagnetic waves started to pay attention to photonic orbital angular momentum in a unique way. For instance, optically-based communication technologies have demonstrated the use of photonic orbital angular momentum in the realization of quantum digits for single-photon communication and in the development of a new scheme for multiplexing signals in telecommunications. A key concept common in both examples is the presence of multiple nearly-orthogonal modes within a narrow range of frequencies. From this point of view, we stressed the meaningful benefits of investigating the extensions of this concept, and proposed the toroidal electromagnetic wavepacket as a fusional application with the exotic surface modes of topological photonic media. The electromagnetic wavepacket with toroidal-type dual vortices is an extension of the Laguerre-Gaussian wavepacket whose line vortex corresponds to the photonic orbital angular momentum. Herein, we presented the procedure to construct the solutions of [PERSON]'s equations with multiple types of vortices.