# Vector Optical Bullets in Dielectric Media: Polarization Structures and Group-Velocity Effects
## 3 Vector Optical Bullets in Dielectric Material
For the rectangular spectral envelope, the spectral width \(d\omega\) can be related to the temporal duration \(dt\) using the expression \(d\omega=5.56/dt\)[70]. In further numerical simulations, we numerically integrate Equations (5) and (6) using the aforementioned expression for the spectral envelope. For our convenience, we select one particular value of the pulse duration \(dt=100\) fs and two particular values of the central frequency, one giving us an example of the so-called X spatiotemporal profile and another giving us an example of the so-called O spatiotemporal profile. We start our simulations by choosing \(\mathbf{a}=\mathbf{e}_{x}\), which gives us two linearly polarized optical bullets. The case of the transverse electric optical bullet(the main component is the \(y\) component). We present the results in Figure 2. The longitudinal intensity profile of the vector optical bullet is depicted in Figure 2a. We note a distinct cross profile, which is the cause of the most used name for the optical bullets, and a clear resemblance to the capital X letter is observed. In Figure 2b, we depict the total intensity profile of the transverse electric optical bullet in the \(\tau=0\) plane. The streamlines depict the orientation of the electric field. In Figure 2c-e, we demonstrate individual electric field components of the vector optical bullet. The component \(x\) is absent due to the transversality of the electric field with respect to \(\mathbf{a}=\mathbf{e}_{x}\). The main dominant component is the \(y\) component, and the main effect of vectorization of the scalar optical bullet is the appearance of the \(z\) component, see Figure 2e. In the next numerical simulation, we turn our attention to the transverse magnetic focus wave mode; see Figure 3. The intensity distribution of the transverse magnetic optical bullet in the longitudinal plane is presented in Figure 3a. Due to the selection of the central carrier frequency \(\omega_{c}\), in both cases, we end up with relatively moderate angles of the angular dispersion curve. For this reason, the electric intensity distribution in Figure 3a is similar to the one in Figure 2a. The main difference is revealed in Figure 3b, where the transverse intensity distribution is depicted together with the streamlines with the electric field. As one can see, the transverse magnetic optical bullet is polarized in the \(x\) direction. A closer look at the individual electric field components, see Figure 3c-e, shows that the dominating electric field component is the \(x\) component, the \(z\) component is present as in the previous case, but it is rotated by 90 degrees. The most notable change is the presence of the cross-polarized (\(y\) component), which is diminishingly small due to the rather modest angles of the angular dispersion curve. Figure 3: 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\({}^{-1}\). Topological charge \(m=0\), pulse duration \(dt=100\) fs, \(V/c=0.63685\), \(\gamma=-0.628\) μm\({}^{-1}\). We start our investigation of inhomogeneously polarized optical bullets (\(\textbf{a}=\textbf{e}_{z}\)) by analyzing the case of the azimuthally polarized optical pulse; for this purpose, we use Equation (7), and obtain individual electric field components by integration of Equation (5). Figure 2: Intensity distributions in the longitudinal (**a**) and transverse (**b**) planes of transverse electric (TE) linearly polarized optical bullets and their 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. The frequency \(\omega_{c}=4\) fs\({}^{-1}\).