# Vector Optical Bullets in Dielectric Media: Polarization Structures and Group-Velocity Effects
## 3 Vector Optical Bullets in Dielectric Material
Topological charge \(m=0\), pulse duration \(dt=100\) fs, \(V/c=0.63685\), \(\gamma=-0.628\) μm\({}^{-1}\). As in the previous case, we chose the same parameters to obtain an azimuthally polarized optical pulse with the X letter in the longitudinal plane; see Figure 4.  
First, we note a hollow center of the optical pulse (see Figure 4a), which is caused by the presence of a polarization singularity on the optical axis of the optical bullet. In this polarization, the singularity is visualized in Figure 4b and is common for azimuthally polarized beams. The presence of this structure results in the \(x\) and \(y\) components being rotated 90 degrees with respect to each other; see Figure 4c,d. The intensity structure of the azimuthally polarized optical bullet contains several concentric rings with the clockwise and counterclockwise pace of the electric field rotation; see Figure 4b. Changes in the orientation of the azimuthally polarized light are caused by different phases on concentric rings when moving away from the center of the beam. As expected, the longitudinal component is absent in this case; see Figure 4e. The next case that we study is the radially polarized optical bullet. We obtain individual electric field components using Equation (8), which we use to integrate Equation (6). For consistency, the set of parameters is the same in this integral as in previous cases. The numerical results are given in Figure 5. The longitudinal intensity distribution of the radially polarized optical bullet largely resembles the previous case--we observe the presence of the on-axis intensity minima, though the intensity never reaches zero here, see Figure 5a. The intensity distribution in the transverse plane, together with the depiction of the polarization state, is given in Figure 5b. We observe here the presence of the polarization singularity: the electric field lines are not defined in the very center of the beam and are oriented either away or towards the center of the beam. Oscillations in the orientations are caused by phase changes in the concentric intensity structure of the radially polarized optical bullet. This behavior is well known and expected and is caused by the intensity distributions of the \(x\) and \(y\) components; see Figure 5c,d. The \(x\) component of the radially polarized optical pulse is rotated 90 degrees compared to the \(x\) component of the azimuthally polarized pulse; see Figure 4c. The \(y\) component also rotates 90 degrees; see Figure 5d. Most notably, we observe the appearance of the longitudinally polarized \(z\) component; see Figure 5e. The presence of the nonzero \(z\) component causes the  
Figure 4: Intensity distributions in the longitudinal (**a**) and transverse (**b**) planes of azimuthally polarized optical bullets and its individual components (\(E_{x}\), (**c**), \(E_{y}\), (**d**), \(E_{z}\), (**e**)) in the transverse plane. The white arrows in (**b**) represent the orientation of the electric field. Frequency \(\omega_{c}=4\) fs\({}^{-1}\). Topological charge \(m=0\), pulse duration \(dt=100\) fs, \(V/c=0.63685\), \(\gamma=-0.628\) μm\({}^{-1}\). region of the on-axis polarization singularity to become somewhat less pronounced. As the maximum value of the \(z\) component reaches 10 % of the maximum intensity on the axis. Our next aim is to investigate the set of parameters for which the longitudinal shape of the optical bullet is expected to change from the capital Latin letter X to the capital Latin letter O. For ease of comparison, the dispersion curve, which we investigate, is the same as previously, see Figure 1b, curve (2), but the central frequency has now decreased. This change will lead to larger spatial angles in the optical bullet, and thus, we expect vectorial properties to become more pronounced. We start the verification of this claim by repeating numerical simulations of the linearly polarized optical bullet using Equations (5) and (6).