# Vector Optical Bullets in Dielectric Media: Polarization Structures and Group-Velocity Effects
## 2 Materials and Methods
Vectorial solutions of the Maxwell equations are constructed from solutions of scalar wave equations using the method given in References [50; 51]  
\[\mathbf{M}(\mathbf{r},t)=\
abla\times\mathbf{a}E(\mathbf{r},t),\quad\mathbf{ N}(\mathbf{r},t)=\frac{1}{k}\
abla\times\mathbf{M}(\mathbf{r},t), \tag{4}\]  
where \(\
abla\) is a nabla operator and \(\mathbf{a}\) is a predefined vector [51]. Applying this operation to Equation (3) leads to the following expressions for transverse electric (TE) optical bullets:  
\[\mathbf{M}(\mathbf{r},t)=\mathrm{e}^{\mathrm{i}\gamma z}\int_{0}^{\infty}S( \omega)\mathbf{M}_{0}(\mathbf{r};\omega)\mathrm{e}^{-\mathrm{i}\omega\tau}d\omega, \tag{5}\]  
and transverse magnetic (TM) optical bullets:  
\[\mathbf{N}(\mathbf{r},t)=\mathrm{e}^{\mathrm{i}\gamma z}\int_{0}^{\infty}S( \omega)\mathbf{N}_{0}(\mathbf{r};\omega)\mathrm{e}^{-\mathrm{i}\omega\tau}d\omega. \tag{6}\]  
Vector functions \(\mathbf{M}_{0}(\mathbf{r},\omega)\) and \(\mathbf{N}_{0}(\mathbf{r},\omega)\) are basis vector functions of cylindrical coordinate systems obtained from Equation (4). Their exact expressions and physical interpretations depend on the choice of the vector \(\mathbf{a}\). We note here that, for anisotropic material, one must determine basis vector functions separately. Given the birefringence, it has been reported on the deformations of Bessel beams traveling in anisotropic materials [66], so the question of whether the basis functions will still be described using cylindrical functions is open. When \(\mathbf{a}=\mathbf{e}_{z}\), the expressions resulting for \(\mathbf{M}_{0}(\mathbf{r},\omega)\) and \(\mathbf{N}_{0}(\mathbf{r},\omega)\) are  
\[\mathbf{M}_{0}(\mathbf{r},\omega)=\bigg{[}\mathrm{i}\frac{m}{\rho}J_{m}\big{(} k_{\rho}\rho\big{)}\hat{e}_{\rho}-J_{m}^{\prime}\big{(}k_{\rho}\rho\big{)}\hat{e}_{ \varphi}\bigg{]}\mathrm{e}^{\mathrm{i}k_{z}z+\mathrm{i}m\varphi}, \tag{7}\]  
where \(J_{m}^{\prime}\) is the first derivative of the Bessel function of the \(m\)-th order. This expression describes an azimuthally polarized Bessel vortex beam. \[\mathbf{N}_{0}(\mathbf{r},\omega)=\bigg{[}\mathrm{i}\frac{k_{z}}{k}J_{m}^{ \prime}\big{(}k_{\rho}\rho\big{)}\hat{e}_{\rho}-\frac{mk_{z}}{k\rho}J_{m}\big{(} k_{\rho}\rho\big{)}\hat{e}_{\varphi}+\frac{k_{\rho}^{2}}{k}J_{m}\big{(}k_{\rho} \rho\big{)}\hat{e}_{z}\bigg{]}\mathrm{e}^{\mathrm{i}k_{z}z+\mathrm{i}m\varphi} \tag{8}\]  
This expression describes a radially polarized Bessel vortex beam, compared with Ref. [49]. In the context of inhomogeneously polarized vector beams, the index \(m\) in Equations (7) and (8) can be assigned to the so-called polarization order \(p\), see Reference [67]. When \(\mathbf{a}=\mathbf{e}_{x}\), the resulting expressions for \(\mathbf{M}_{0}(\mathbf{r},\omega)\) and \(\mathbf{N}_{0}(\mathbf{r},\omega)\) are  
\[\mathbf{M}_{0}(\mathbf{r},\omega)=\bigg{\{}\mathrm{i}k_{z}J_{m}\big{(}k_{\rho }\rho\big{)}\hat{e}_{y}-\bigg{[}\sin\varphi J_{m}^{\prime}\big{(}k_{\rho}\rho \big{)}+\mathrm{i}m\cos\varphi\frac{J_{m}\big{(}k_{\rho}\rho\big{)}}{\rho} \bigg{]}\hat{e}_{z}\bigg{\}}\mathrm{e}^{\mathrm{i}k_{z}z+\mathrm{i}m\varphi} \tag{9}\]  
where \(\mathbf{e}_{y}\) is the unit vector of the Cartesian coordinate system. This expression describes a linearly polarized (in the \(y\)-direction) Bessel vortex beam. \[