# Vector Optical Bullets in Dielectric Media: Polarization Structures and Group-Velocity Effects
## 3 Vector Optical Bullets in Dielectric Material
Topological charge \(m=0\), \(dt=100\) fs, \(V/c=-1.2\), \(\gamma=6\pi\) μm\({}^{-1}\). The solution to this riddle is presented in Figure 15c-e, where the individual electric field components of the optical pulse are given. The \(x\) and \(y\) components are comparable to each other and rotated by 90 degrees in respect to one another; see Figure 15c,d. However, in reverse to the previous cases, the strongest electric field component is the longitudinal one; see Figure 15e. This behavior is expected due to the choice of the central frequency in the optical pulse, causing the appearance of high spatial angles \(\theta\). Under these conditions, the radial polarization becomes strongly focused, which in return causes the appearance of very tight small spots with longitudinal polarization [32]. We summarize the investigation of these three particular cases by investigating the effects that the choice of group velocities \(V\) and the propagation constant \(\gamma\) have on the spatiotemporal parameters of the optical bullets, such as effective beam width in transverse and longitudinal directions. In the case of uniform beam shapes, such as a Gaussian beam or similar modes, the question of the beam width definition is pretty straightforward. The full width at half maximum of the intensity is the most commonly used within the field. However, in our case, the complex vectorial nature, together with a rich ring-like structure observed in the variously polarized pulses, suggests looking at alternative definitions for the beam width. One of the alternatives that is very efficient at accounting for peripheral parts of the complex optical beams is the so-called second-moment definition. Figure 3:5  
where \(\rho\) is the radial coordinate in the plane under consideration, and \(I(\rho)\) is the total intensity of the electric field. As we have concluded from the three cases investigated in the previous part, the spatiotemporal dependence between frequencies of individual cone angles (see Figure 1) strongly affects intensity distributions both of individual electric field components and total electric field for vector optical bullets. Therefore, it is natural to ask ourselves whether these effects can be qualified using standard metrics, such as beam length. In this numerical  
Figure 3:6 Intensity distributions in the longitudinal (**a**) and transverse (**b**) planes of azimuthally 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\({}^{-1}\). experiment, we keep the same duration (\(dt=100\) fs) for all cases and change the central frequency \(\omega_{c}\) of the pulse. We consider linearly polarized TE and TM optical bullets along with radially and azimuthally polarized ones. The transverse beam width is evaluated together with the longitudinal extent of the optical pulse using two different definitions: FWHM and second moment. The beam widths of the optical pulses using the FWHM definition are given in Figure 16a. First, we note that for particular parameters \(V/c=0.63685\) and \(\gamma=-0.628\)\(\upmu\)m\({}^{-1}\) the FWHM beam width behaves very distinctly: the beam width first decreases, up to central frequencies of \(\omega_{c}/\omega_{0}\approx 1\), then increases to the maximum value of \(100\)\(\upmu\)m at frequency \(\omega_{c}/\omega_{0}\approx 2.4\). A further increase in the carrier frequency \(\omega_{c}\) results in a monotonically decreasing beam size. This behavior can be understood by looking at Figure 1b, curve (2). The spike in FWHM dependency occurs at the same frequency when the spatial angles are rather small. When the spatial angles increase, we obtain optical pulses with small transverse sizes. Most notably, the linearly polarized modes have, for this case, indistinguishable beam width; see Figure 16a, red curves.