# Vector Optical Bullets in Dielectric Media: Polarization Structures and Group-Velocity Effects
## 3 Vector Optical Bullets in Dielectric Material  
In this section, we systematically investigate and present a comparison of linearly polarized optical bullets with their radially/azimuthally polarized counterparts in the dielectric material. We selected a BK7 glass for this investigation, but note that these steps can be easily applied to any other material whose dispersion can be described using a Sellmeier formula [68]. The angular dispersion of the focus wave modes is described by Equation (2). The properties of such dispersion curves have already been extensively studied in the literature, see Reference [21], so we shall briefly recall them without diving into the problematics. An example of angular dispersion curves for BK7 glass is shown in Figure 1 for a few selections of group velocities \(V/c\) and parameters \(\gamma\). The central frequency of the beam \(\omega_{c}\) is normalized to the frequency \(\omega_{0}\), corresponding to zero group-velocity dispersion (GVD) in BK7 glass. In the first case, we select a positive value of \(\gamma=1.25\)\(\upmu\)m\({}^{-1}\) and change the group-velocity values \(V/c\). As a result, we observe the angular dispersion that increases monotonically with frequency. The main feature of these dependencies is that the range of frequencies where the optical bullet exists changes with velocity; see Figure 1a. In the second case, we select one particular negative value of \(\gamma=-0.628\)\(\upmu\)m\({}^{-1}\) to plot the angular dispersions for 5 different values of the group velocity, see Figure 1b. This case is especially interesting as a demonstration of the rich and complex dependencies of the angular dispersion. Moreover, for some values, we observe monotonically decreasing angles for single-wave components, which, after reaching the inflection point, start to increase. At a particular value of the group velocity \(V/c\), the inflection point touches the frequency axis. A further change in the group-velocity results in the dispersion curve splitting into two parts; see Figure 1b. In the last example, we selected a negative group velocity \(V/c=-1.2\) to show the possibility of creating a backward-propagating optical bullet inside the dielectric material; see Figure 1c and plot dispersion curves for a selection of parameters \(\gamma\). Before proceeding to the analysis of numerical simulations, we note that the situation depicted in Figure 1b is the most interesting and most studied in the literature. As for the scalar optical bullets, the rich dynamics of spatiotemporal profiles are observed. Usually, the optical bullets have a spatial profile that resembles the capital Latin letter X; for this reason, the optical bullets are very often called mistakenly X-waves or Bessel-X pulses, although the latter solutions are obtained as a particular case of the focus wave modes with parameter \(\gamma=0\)\(\upmu\)m\({}^{-1}\) and very specific and distinct expressions of the spectral envelope \(S_{\omega}\). Yet another example of the transformations in the spatiotemporal profile of the nondiffracting and nondispersive focus wave mode is the change in the capital Latin letter O, caused by specific angular dispersion due to which the peripheral intensities merge; see References [21; 24; 69]. Motivated by this particular case receiving great attention in the literature, we select one particular angular dispersion curve from Figure 1b, when \(\gamma=-0.628\)\(\upmu\)m\({}^{-1}\) and \(V/c=0.63685\), as this is the case with most changes in the spatiotemporal profile of the vector optical bullet. For this study, we introduce the rectangular shape of the spectral envelope \(S(\omega)=\Pi(\omega_{c}-\omega/d\omega)\), where  
\[\Pi\Big{(}\frac{\omega}{d\omega}\Big{)}= \begin{cases}0,&\text{if}\ |\omega|>\frac{d \omega}{2},\\ \frac{1}{2},&\text{if}\ |\omega|=\frac{d\omega}{2},\\ 1,&\text{if}\ |\omega|<\frac{d\omega}{2},\end{cases}  \tag{12}\]  
where the \(d\omega\) is the spectral width of the rectangular spectral envelope, and \(\omega_{c}\) is the central frequency of the wave packet.