# Vector Optical Bullets in Dielectric Media: Polarization Structures and Group-Velocity Effects
## 3 Vector Optical Bullets in Dielectric Material
The same applies to the radially and azimuthally polarized beams, though they are larger than the linearly polarized optical bullets. For this particular case, the spatial angles \(\theta\) are rather modest, so the difference between the azimuthal and radial polarizations is vanishingly small, as is the case between the linearly polarized TM and TE optical bullets. We also investigate the beam sizes of the backward-propagating focus wave modes, where \(V/c=-1.2\) and \(\gamma=6\pi\)\(\upmu\)m\({}^{-1}\), see Figure 16a, blue curves. In our previous investigation, we observed the most notable differences between polarizations in this case. As a result, we observe that the FWHM beamwidth dependencies are separate for all different cases of vector FWMs. Most notably, the radial polarization becomes comparable in the beam sizes to the linearly polarized TE FWM as the frequency increases. This is an expected outcome because it is well known that radial polarization could even result in a tighter focal spot than linear polarization [32]. Radial polarization has beam sizes smaller than those of the linearly polarized TM mode; see Figure 16a. The largest beam sizes are observed for the azimuthally polarized FWHM. Figure 3:7 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**)). 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, \(V/c=-1.2\), \(\gamma=6\pi\)\(\upmu\)m\({}^{-1}\). The next metric is the definition of the second moment of the beam width. This metric usually has the characteristic of artificially increasing the importance of peripheral parts of the beam. It represents the radial distance weighted intensity; therefore, if the intensity drops moving away from the center of the beam slower than the square of the radial distance, it might result in larger beam sizes than those obtained using the FWHM definition. The result of the numerical estimate of this beam width for the same set of parameters is given in Figure 16b. We immediately notice differences in the general trend of the beamwidth dependencies of the carrier frequency. Although the FWHM beam width was monotonically decreasing for four cases out of eight, the second-moment beam width depends on the carrier frequency non-monotonically. For small carrier frequencies, it starts to increase to some particular value and then drops. In this case, the linear polarization has the largest beam width. These subtle differences that occur at smaller carrier frequencies \(\omega_{c}\) are observable for the four cases considered here. Moving to the case of backward-propagating FWM--see Figure 16b, blue curves--we note yet another tendency: the beam widths defined using the second-order moment of the beam behave differently than those defined using the FWHM definition; compare to Figure 16a. This means that even though the beam is above the 50 percent level is larger, the total intensity in the beam drops a lot faster than the radial coordinate for smaller carrier frequencies. The fact that the second-moment beamwidth increases with increasing central frequency indicates that significant side lobes become present in the vector FWMs. The longitudinal beam size has also been investigated. The result did not show any change in that size across all investigated frequencies, and for the sake of brevity, figures covering this claim are not provided. In the case of \(V/c=0.63685\) and \(\gamma=-0.628\)\(\upmu\)m\({}^{-1}\), the longitudinal beam size remains constant at 26.01 \(\upmu\)m using the FWHM definition for all polarizations investigated. For the case of \(V/c=-1.2\) and \(\gamma=6\pi\)\(\upmu\)m\({}^{-1}\), the longitudinal beam size remains constant at 49.05 \(\upmu\)m for all the investigated scenarios.