# Topological Photonic Media and the Possibility of Toroidal Electromagnetic Wavepackets  
[PERSON]  
Graduate School of Engineering Science, Akita University, 1-1 Tegatagakuen-machi, Akita 010-8502, Japan; [EMAIL_ADDRESS]  
November 6, 2  
clear-cut recognition was reapplied to an electromagnetic system, i.e., a photon system, and its validity became clear in a photon system than in electron systems [11]. On the contrary, the vortex structure of an electromagnetic wave became widely recognized to closely relate with the orbital angular momentum of photons [12], a view currently being implemented in the optical and quantum information communication technology [13; 14; 15]. Moreover, electromagnetic vortices can appear in periodic structures, such as photonic crystals [16]. This suggests a new kind of internal orbital angular momenta of photon in such systems. These internal orbital angular momenta may be interpreted as quasi-spin DOF and potentially cause a variety of geometric phase effects. Specifically, an electromagnetic wavepacket composed from wave modes with such vortices can have orbital angular momentum perpendicular to its propagation direction. This relation between angular momentum and propagation direction for such a wavepacket is similar to that for an atmospheric tornado which shows unexpected exotic motions.  
Herein, we theoretically investigate a possible electromagnetic wavepacket with toroidal-type dual vortices, i.e., having a ring vortex inside the wavepacket and a line vortex along its propagation direction. The line vortex resembles that of a Laguerre-Gaussian beam and suggests a finite orbital angular momentum of the wavepacket. This paper is organized as follows. In Section 2, we review the relation between geometric phases and angular momenta, followed by the discussion on electromagnetic vortices in periodic structures in Section 3, which further demonstrates the propagation characteristics of such electromagnetic vortices by conducting numerical simulations on Y-shaped waveguides. In Section 4, we introduce the topological photonic media, while reviewing topological phenomena of electron systems in solids, such as quantum Hall systems and topological insulators. Herein, a class of topological photonic media is interpreted as a photonic version of topological insulator and can be realized as an extension of photonic crystals accompanying the electromagnetic vortices. Moreover, we present another simulation of waveguide propagation via exotic boundary modes of such a medium. In Section 5, referring to the dynamics of an electronic wavepacket around the boundary between a topological insulator and conductor, we consider the possibility of toroidal electromagnetic wavepackets with an argument on a correspondence relation between electronic helical surface states of topological insulator and photonic helical surface modes of topological photonic media. In Section 6, we present an algorithm for constructing wavepacket solutions of Maxwell's equations with multiple types of vortices. Next, we numerically investigate the stability of the toroidal electromagnetic wavepacket in reflection and refraction at interfaces between homogeneous isotropic dielectrics, and reveal the transformation laws of topological charges of line and ring vortices.  
In the next section and beyond, we adopt the natural system of units as \(\hbar=1\) (\(\hbar\): Dirac constant or reduced Planck constant) and \(c=1\) (\(c\): speed of light), unless those symbols are explicitly stated. We will not distinguish between the wavevector \(\mathbf{k}\) of a plane wave and the momentum \(\hbar\mathbf{k}\) of a quantum particle derived from second quantization of the wave. Likewise, the frequency \(\omega\) of a harmonic wave and the energy \(\hbar\omega\) of a corresponding quantum particle will not be distinguished. For convenience on later discussions, we introduce a spherical basis \(\{\mathbf{e}_{k},\mathbf{e}_{\theta},\mathbf{e}_{\phi}\}\) (\(\mathbf{e}_{k}=\mathbf{k}/k\)) in wavevector space.