# Vector Optical Bullets in Dielectric Media: Polarization Structures and Group-Velocity Effects
## 3 Vector Optical Bullets in Dielectric Material
The linearly polarized transverse magnetic optical bullet for this particular case is evaluated and presented in Figure 3:0 The most noticeable difference between the backward-propagating TM mode compared to the TE mode can be seen in Figure 13a. The transverse beam size is considerably larger in this case, but the longitudinal dimension is the same. The changes are even more pronounced in the transverse pattern of the optical bullet at its center; see Figure 13b. The streamlines representing the electric field are not oriented along the \(x\)-axis anymore, compared to the previous cases; see Figures 3b and 7b. The intensity distribution is no longer symmetrical with respect to the beam center; instead, it is elongated along the \(x\)-axis. This behavior is a direct manifestation of high spatial  
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**)). The white arrows in (**b**) represent the orientation of the electric field. Frequency \(\omega_{c}=4\) fs\({}^{-1}\). Topological charge \(m=0\), \(dt=100\) fs, \(V/c=-1.2\), \(\gamma=6\pi\) μm\({}^{-1}\). frequencies comprising the backward-propagating FWHM. In a similar fashion, linearly polarized light is elongated along the direction of polarization in high numerical aperture systems. Moreover, from the flow of the electric field lines, we conclude that the notable cross-polarized component is present; see Figure 13b. The individual components of the electric field are given in Figure 13c-e. The \(x\)-polarized component is no longer symmetric, though the concentric ring system is observable in the beam profile. For this particular case, we observe the strongest cross-polarized \(y\) component from all the considered cases; see Figure 13d. Its maximum value is even comparable to that longitudinally polarized in Figure 13e. The general shapes of these individual components are in line with expectations. We see that the longitudinal component is elongated along the direction of polarization and that the cross-polarized component has two perpendicular splits in its concentric ring pattern. We consider the inhomogeneously polarized backward-propagating optical bullets with the case of azimuthal polarization; see Figure 3:2 The longitudinal intensity distribution is given in Figure 14a. Compared to previously investigated cases, the most distinct effect caused by spatiotemporal dispersion is the change in the transverse sizes and observation of the general shape of O type; see Figure 14a. In the transverse plane, we see many concentric rings with clockwise and counterclockwise directions of the electric field flow; see Figure 14b. The longitudinal field component for azimuthal polarization is zero, and the \(x\) and \(y\) components are of the same strength and intensity distributions with one notable difference being a 90-degree rotation with respect to each other; see Figure 14c-e. The radially polarized backward-propagating optical bullet is depicted in Figure 3:3 Here, we observe a distinct difference to the azimuthally polarized optical bullet and compare Figures 14a and 15a. Although the central part of the azimuthally polarized optical bullet is hollow because of the presence of azimuthal phase singularity, the central part of the radially polarized optical bullet is the brightest one. The transverse intensity distribution depicts the same trend--the central part still contains the star-like radial polarization singularity, but the intensity is still highest at its very core; see Figure 15b. 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**)). The white arrows in (**b**) represent the orientation of the electric field. Frequency \(\omega_{c}=4\) fs\({}^{-1}\).