# Vector Optical Bullets in Dielectric Media: Polarization Structures and Group-Velocity Effects
## 4 Discussion and Conclusions  
We have introduced vector formalism to the concept of nondiffracting and nondispersive optical bullets within a dielectric material. Vector optical bullets for different polarizations were introduced: linear polarization, azimuthal polarization, and radial polarization. Two different propagation velocities \(V\) were chosen to showcase the wide variety of spatiotemporal dispersions of vector optical bullets, one positive and one negative (i.e., backward-propagating). The first value of the group velocity \(V\) was chosen because it enforces a particularly interesting dispersion, with cone angles that extend from small paraxial (i.e., scalar description is valid) to larger values, which now require a proper vector treatment of the electric field. For this particular velocity, we selected two different values of central frequencies, showcasing two possible situations: one giving the so-called X spatiotemporal profile and another giving the so-called O spatiotemporal profile. We can conclude that the spatiotemporal shape of the vector optical bullet experiences the same changes as reported for scalar optical pulses, with differences arising from nonzero longitudinal and cross-polarization components. Expectedly, high cone angles result in a vector description of the vector FWM, also demonstrating distinct distances between polarizations. For the second case, when the forward-propagating superposition of Bessel beams creates within the Bessel zone a backward-propagating pulse of negative velocity \(V\), we  
Figure 16: Dependencies of FWHM (**a**) and second-moment (**b**) pulse widths in BK7 glass for different values of frequency \(\omega_{c}\) for linear, azimuthal and radial polarizations. The red color represents \(V/c=0.63685\)\(\gamma=-0.628\)\(\upmu\)m\({}^{-1}\), and the blue color \(V/c=-1.2\)\(\gamma=6\pi\)\(\upmu\)m\({}^{-1}\). Topological charge \(m=0\), \(dt=100\) fs. Figure 17: Normalized intensities of individual components of FWM pulses (\(E_{x}\)—blue, \(E_{y}\)—red, \(E_{z}\)—orange). Topological charge \(m=0\), pulse duration \(dt=100\) fs. For the cases: (**a**) \(V/c=0.63685\), \(\gamma=-0.628\)\(\upmu\)m\({}^{-1}\), (**b**) \(V/c=-1.2\), \(\gamma=6\pi\)\(\upmu\)m\({}^{-1}\). The solid line represents the linear polarization of TE, the dashed line represents the linear polarization of TM, the dotted line represents the azimuthal polarization, and the dashed-dotted line represents the radial polarization. observed the most notable differences in the properties of vector optical bullets within the dielectric material. Due to the highly nonparaxial nature of such exotic waves, the longitudinal component is quite noticeable in the TE and TM polarized optical bullets. Most notable is the appearance of a relatively large cross-polarized component in the linearly polarized TM mode. As expected, no differences did appear in the azimuthally polarized optical bullet, but the radially polarized wave was most affected--the longitudinal electric field component dominates the intensity pattern. Lastly, we have studied how polarization and spatiotemporal dispersion influence the beam sizes of the optical bullets in the focal plane. For this, we have used two distinct metrics--the full-width half maximum and the second-order moment of the beam. As expected, the FWHM beam size gradually increases, where the central frequency and the group velocity allow the angular dispersion curve to touch the frequency axis. This has been shown using two different methods to calculate the beam size: the full width at half maximum and the method of the second moment. For optical bullets with negative group velocity, the FWHM beam sizes monotonically increase as we move along the dispersion curve to higher cone angles of the spatial components. The second metric is well known for its sensitivity to the side lobe energy, so this parameter has revealed complex dynamics within the optical bullets, showcasing a competition between the main spike of the signal and the side lobes.