# Topological Photonic Media and the Possibility of Toroidal Electromagnetic Wavepackets
## 2 Geometric Phases and Angular Momenta of Electromagnetic Vortices
On the contrary, \((z_{\boldsymbol{k}}|\boldsymbol{\Omega}_{\boldsymbol{k}}|z_{\boldsymbol{k}})\) can be expressed as \((|z_{\boldsymbol{k}\boldsymbol{k}}|^{2}-|z_{\boldsymbol{k}\boldsymbol{k} }|^{2})/k^{2}\boldsymbol{e}_{k}\) using the corresponding representation \([z_{\boldsymbol{k}\boldsymbol{k}},z_{\boldsymbol{k}\boldsymbol{k}}]^{\top}\) of \(|z_{\boldsymbol{k}}\rangle\). Since the expected value of the spin angular momentum per photon \(\langle\boldsymbol{s}_{\boldsymbol{k}}\rangle\) is evaluated as \(\langle\boldsymbol{s}_{\boldsymbol{k}}\rangle=(|z_{\boldsymbol{k}\boldsymbol{k }}|^{2}-|z_{\boldsymbol{k}\boldsymbol{k}}|^{2})\boldsymbol{k}/k\), a close relation, \(\langle\boldsymbol{s}_{\boldsymbol{k}}\rangle=k^{2}(z_{\boldsymbol{k}}| \boldsymbol{\Omega}_{\boldsymbol{k}}|z_{\boldsymbol{k}})\), exists between the two quantities. We can extend the above discussion to a case accompanied by orbital angular momentum. A close relation between the internal orbital angular momentum and Berry curvature is derived in the same way as above, while its discussion gets a little bit complex. Herein, we consider a beam of a central wavevector \(\boldsymbol{k}_{\mathrm{c}}\) and introduce the extension  
Figure 1: (**a**) _xy_- and (**b**) _yz_-cross-sections of a right circularly polarized Gaussian wavepacket. \[\mathbf{e}_{k\alpha}\rightarrow\exp\left(il_{\alpha}\varphi_{k;\mathbf{k}_{c}}\right)\mathbf{e}_{ k\alpha},\quad\varphi_{k;\mathbf{k}_{c}}=\arctan\frac{\mathbf{e}_{\phi_{\text{c}}} \cdot\mathbf{k}}{\mathbf{e}_{\theta_{\text{c}}}\cdot\mathbf{k}}, \tag{2}\]  
where \(l_{\alpha}\) corresponds to the vorticity of the \(\alpha\)-polarized component and is an integer number, i.e., \(l_{\alpha}\in\mathbb{Z}\). For simplicity, we consider only a class of beams that are a superposition of a given polarization component with the vorticity \(l\) and its orthogonal component with \(l^{\prime}\). In other words, we restrict ourselves to a finite subspace of infinite whole space of states. On the basis of right and left circular polarizations, the Berry connection and curvature of this subspace are represented as,  
\[\mathbf{\Lambda}_{\mathbf{k}}=-\frac{\cos\theta}{k\sin\theta}(l_{\text{OAM}}+\sigma_{3 })\mathbf{e}_{\phi},\quad\mathbf{\Omega}_{\mathbf{k}}=\frac{1}{k^{2}}(l_{\text{OAM}}+ \sigma_{3})\mathbf{e}_{k}, \tag{3}\]  
where \(l_{\text{OAM}}\) is a \(2\times 2\) Hermitian matrix with a pair of integer eigenvalues \((l,l^{\prime})\). On the other hand, the expected value of the total angular momentum per photon \(\langle j_{\mathbf{k}}\rangle\) is evaluated as \(\langle j_{\mathbf{k}}\rangle=(z_{\mathbf{k}}|l_{\text{OAM}}+\sigma_{3}|z_{\mathbf{k}}) \mathbf{e}_{k}\). We can find a close relation \(\langle j_{\mathbf{k}}\rangle=k^{2}(z_{\mathbf{k}}|\mathbf{\Omega}_{\mathbf{k}}|z_{\mathbf{k}})\) again. Next, we consider the meaning of the form of the Berry curvature, \(\mathbf{k}/k^{3}\sigma_{3}\), on the basis of circular polarizations which we shall call helicity basis hereafter. As we shall see, this form reflects the photon characteristics of a spin-1 gauge symmetric massless boson. To this end, we introduced a degenerate two-band model of a spin-1/2 fermion system with conical dispersions, similar to relativistic electrons. The Hamiltonian of this model and its projection to the subspace of definite wavevector \(\mathbf{k}\) are given by  
\[H=-iv\mathbf{\
abla}_{\mathbf{r}}\cdot\mathbf{\alpha}+\Delta\,\beta\to H_{\mathbf{k}}=v\mathbf{k} \cdot\mathbf{\alpha}+\Delta\,\beta, \tag{4}\]  
where \(\mathbf{\alpha}\) and \(\beta\) are Dirac matrices and \(v\) and \(\Delta\) are parameters of effective phase velocity and band gap, respectively. The band gap of this model plays a similar role as the mass gaps in relativistic theories; hence, we shall refer to the limit \(\Delta\to 0\) as massless limit. The eigenvalue problem of \(H_{\mathbf{k}}\) can be easily solved. The eigenvalues of the upper and lower bands are obtained as \(\pm\sqrt{v^{2}k^{2}+\Delta^{2}}\); only the upper bands will be considered hereafter.