# Topological Photonic Media and the Possibility of Toroidal Electromagnetic Wavepackets
## 2 Geometric Phases and Angular Momenta of Electromagnetic Vortices  
In this section, we first review the relation that exists between the spin angular momentum and geometric phase of an electromagnetic wavepacket, following those between polarization state and spin angular momentum and between polarization vector and geometric phase. Figure 1 describes the relation between polarization state and angular momentum [17] of a right circularly polarized wavepacket propagating in the \(z\)-direction (upward in the drawing). The arrows represent the deviation of the wavepacket energy flux density from the product of the energy density and the averaged velocity vector, whereas the hue represents the energy density (cold color \(<\) warm color). For simplicity, we considered the situation wherein the wavepacket spread is sufficiently large with respect to its central wavelength so that its deformation can be ignored. In Figure 1, we can find a clockwise vortex in the view facing the propagation direction (\(z\)-direction) of the wavepacket, indicating that the right circularly polarized wavepacket has a spin angular momentum in the propagation direction. By contrast, we could not find such structure of energy flux density in a similar plot of a linearly polarized wavepacket in the same scale as Figure 1.  
Furthermore, once the well-known indications, i.e., the geometric phase appearing due to the change in polarization state [9] and the geometric phase due to orbital change of polarized beam [18], are accepted, the relation between spin and geometric phase looms into view. Herein, by specifically looking at the relation between polarization vector and geometric phase, we confirm the relation between the three concepts more explicitly. For that purpose, we introduce two quantities, the Berry connection and curvature, defined as  
\[\left[\mathbf{\Lambda}_{k}\right]_{\alpha\beta}=\left( \begin{array}{c}-i \boldsymbol{e}_{k\alpha}^{\dagger}\cdot\frac{\partial}{\partial k^{\dagger}} \boldsymbol{e}_{k\beta}\\ -i\boldsymbol{e}_{k\alpha}^{\dagger}\cdot\frac{\partial}{\partial k^{\dagger} }\boldsymbol{e}_{k\beta}\\ -i\boldsymbol{e}_{k\alpha}^{\dagger}\cdot\frac{\partial}{\partial k^{\dagger} }\boldsymbol{e}_{k\beta}\end{array} \right)=-i\boldsymbol{e}_{k\alpha}^{ \dagger}\cdot\boldsymbol{\
abla}_{k}\boldsymbol{e}_{k\beta},\quad\boldsymbol {\Omega}_{k}=\boldsymbol{\
abla}_{k}\times\boldsymbol{\Lambda}_{k}+i\boldsymbol {\Lambda}_{k}\times\boldsymbol{\Lambda}_{k}, \tag{1}\]  
where \(\{\boldsymbol{e}_{k\alpha}\}\) is an orthonormal basis of polarization vectors normal to the wavevector \(\boldsymbol{k}\) (\(\boldsymbol{e}_{k\alpha}^{\dagger}\cdot\boldsymbol{e}_{k\beta}=\delta_{\alpha \beta}\), and \(\boldsymbol{e}_{k\alpha}^{\dagger}\cdot\boldsymbol{e}_{k}=0\)); the symbol \(\alpha\) is an indicator of polarization state; and \(\boldsymbol{e}_{k\alpha}^{\dagger}\) is the complex conjugate transpose (the Hermitian conjugate) of \(\boldsymbol{e}_{k\alpha}\). Herein, we also introduce the Jones vector corresponding to this orthonormal basis as \(|z_{\boldsymbol{k}}\rangle\). With \((z_{\boldsymbol{k}}|\) as the Hermitian conjugate of \(|z_{\boldsymbol{k}}\rangle\), they satisfy \((z_{\boldsymbol{k}}|z_{\boldsymbol{k}})=\sum_{\boldsymbol{\alpha}}|z_{k\alpha }|^{2}=1\). Although representations of \(\boldsymbol{\Lambda}_{k}\) and \(\boldsymbol{\Omega}_{k}\) depend on how the basis is selected, \((z_{\boldsymbol{k}}|\boldsymbol{\Omega}_{k}|z_{\boldsymbol{k}})\) is uniquely determined once a state is given. On the basis of right and left circular polarizations, the Berry curvature is represented as \(\boldsymbol{k}/k^{3}\sigma_{3}\), where \(\sigma_{3}\) is the third component of Pauli matrices \(\boldsymbol{\sigma}=(\sigma_{1},\sigma_{2},\sigma_{3})^{\top}\) and is diagonal in the standard representation.