# Vector Optical Bullets in Dielectric Media: Polarization Structures and Group-Velocity Effects
## 3 Vector Optical Bullets in Dielectric Material
The linearly polarized transverse electric optical bullet is presented in Figure 6. We immediately notice the changes in the longitudinal intensity distribution; see Figure 6a. Long-range peripheral arms similar to the X shape are no longer present, compared to Figure 2a. Although the spatial extent of the central part is smaller now, and we observe many needle-like structures when we move away from the center, we can deduce a rhombus-like area with approximately the same levels of intensity; see Figure 6a. This becomes especially obvious when we turn our attention to the intensity levels of approximately 20% of the normalized value. Although this rhombus shape has sharp corners, it can be seen as a distorted O-shaped. Thus, we can conclude that the spatiotemporal shape of the vector optical bullet experiences the same changes as reported for scalar optical pulses [21; 24]. The transverse intensity distribution of the linearly polarized optical bullet is given for this case in Figure 6b. The electric field lines in this plane are expectedly oriented along the \(y\) direction. However, the intensity distribution of the total electric field has some signs of nonuniformity, with the tendency to have higher intensities in the direction of the electric field. In the expected manner for the transverse electric optical bullet, no \(x\) component is present; see Figure 6c. The dominant electric field component is the \(y\) component; see Figure 6d. When we plot the longitudinal electric field component, see Figure 6e. We recognize the cause of the nonuniformity in the total electric field. The maximum value of the \(z\) component reaches 10% of the intensity of the \(y\) component. This increase in the  
Figure 5: Intensity distributions in the longitudinal (**a**) and transverse (**b**) planes of radially 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\({}^{-1}\). relative strength of the longitudinal component is mainly caused by changes in the central frequency of the optical pulse, compared to Figure 2e. In the next set of simulations, we move on to investigate the transverse magnetic linearly polarized optical bullet; see Figure 7. The longitudinal intensity distribution is given in Figure 7a. As in the case of the transverse electric optical bullet, in this case, we observe a disappearance of the pronounced X shape in the intensity profile, compared to Figure 3a. This behavior is expected because of the change in the central frequency of the optical pulse. The transverse intensity distribution is depicted in Figure 7b together with the streamlines depicting the orientation of the electric field. We note the appearance of some inhomogeneity in the flow of the electric field, compared to Figure 6b. There are some particular points in the regions of vanishingly low intensity when the electric field is not oriented along the \(x\)-axis anymore. This behavior is expected when one notices the presence of the component \(y\) in Equation (10). The intensity distributions of the individual electric field components are given in Figure 7c-e. As expected, the \(x\) component of the electric field is the strongest; see Figure 7c. Most notably, we observe the appearance of a vanishingly small \(y\) component; see Figure 7d. Its maximum value reaches the single percent digits compared to the dominant \(x\) component. The longitudinal component reaches 10% of the value of the \(x\) component. This behavior is comparable to the case of the transverse electric optical bullet; see Figure 6e. We investigate radially and azimuthally polarized optical bullets of the O type using the same set of parameters as we did in the case of linearly polarized FWM.