# Vector Optical Bullets in Dielectric Media: Polarization Structures and Group-Velocity Effects
## 3 Vector Optical Bullets in Dielectric Material
An example of the azimuthally polarized optical bullet for this particular case is given in Figure 3.0 The longitudinal intensity distribution is shown in Figure 8a. As expected, no \(z\)-component is present in the beam; see Equation (7). For this reason, the optical pulse has a void-like intensity structure with no axial intensity. In particular, the general shape of the transverse intensity distribution is a distinct rhombus-like shape, resembling the capital Latin letter O, compared to Figure 4a. The transverse intensity distribution now represents many concentric rings with oscillating directions of clockwise and counterclockwise electric field  
Figure 3.1 Intensity distributions in the longitudinal (**a**) and transverse (**b**) planes of transverse electric (TE) linearly 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. The frequency \(\omega_{c}=1.6\) fs\({}^{-1}\). Topological charge \(m=0\), pulse duration \(dt=100\) fs, \(V/c=0.63685\), \(\gamma=-0.628\) μm\({}^{-1}\). rotation; see Figure 8b. As the change in the central frequency has increased spatial angles of the individual Bessel beams, creating an optical bullet, the focus wave mode became more spatially confined, see Figure 8c,d. The individual electric field components are spatially oriented in the same way as in the previous case; see Figure 4c,d, but a finer structure with a larger number of rings has appeared. As expected, no \(z\) components are present; see Figure 8e. The situation depicting the radially polarized optical bullet is given in Figure 3.2 The longitudinal intensity distribution, given in Figure 9a, resembles the intensity distribution of azimuthally polarized FWM; see Figure 8a, with a noticeable difference. The axial intensity is no longer zero and is on a level comparable to the general pattern of the beam. This can be especially noticed in the transverse intensity distribution of the electric field shown in Figure 9b. The central part of the beam is not hollow anymore, though we observe a polarization singularity with the star-like flow of the electric field streamlines. Due to the phase jumps in the adjacent concentric rings, the orientation of the electric field changes from inwards to outwards; see Figure 9b. The individual components of the electric field are shown in Figure 9c-e. As expected, the two strongest field components are the \(x\) and \(y\) components; see Figure 9c,d and compared to the previous case, in Figure 5c,d. The longitudinal component became stronger when compared to the previous case; please note the change in the color bar values; see Figures 5e and 9e. We have also investigated cases of higher polarization order, with a topological charge of \(m=1\) (translating to polarization order \(p=2\)[67]), see Figures 10 and \(m=2\) (translating to the polarization order \(p=3\)[67]), see Figure 3.3 We can see that in the case of Figure 10, there is a minimum intensity in the center of the beam, except for the \(z\) component, which has a considerable intensity in the center of the beam. The \(x\) and \(y\) components are the same in intensity distributions. In Figure 11, we notice a total intensity zero in the center of the beam (\(m=2\)). The distributions of the components \(x\) and \(y\) are exactly the same. There is an intensity minimum at the center of the \(z\) component, unlike in the previous case. Figure 3.4 Intensity distributions in the longitudinal (**a**) and transverse (**b**) planes of transverse magnetic (TM) linearly 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. The frequency \(\omega_{c}=1.6\) fs\({}^{-1}\). Topological charge \(m=0\), pulse duration \(dt=100\) fs, \(V/c=0.63685\), \(\gamma=-0.628\) μm\({}^{-1}\).