# Localized Vector Optical Nondiffracting Subcycle Pulses
## 3 Vector Subcycle Pulses
\[\lim_{\Delta\omega\to\infty}\frac{\Delta\omega}{\omega_{c}}\to 2,\]2 Intensity distributions of subcycle pulses in vacuum and their individual components \(E_{x}\) (second row), \(E_{y}\) (third row), \(E_{z}\) (fourth row) in the transverse plane for linear TE (first column), TM (second column), azimuthal (third column), and radial (fourth column) polarizations when \(V/c=-5.65\), \(\gamma=4.35\) μm\({}^{-1}\). The white arrows in first row represent the orientation of the electric field. The insets represent the phase distribution. Lastly, we consider point (**3**) in Figure 1a, where the group velocity of the focus wave mode is \(V/c=1\), and the parameter \(\gamma=-2.45\)\(\upmu\)m\({}^{-1}\). We compare the same polarizations as follows: linear TE (transverse electric), linear TM (transverse magnetic), radial, and azimuthal; see Equations (5) and (6). In the transverse plane, we illustrate the intensity distributions of these subcycle pulsed beams (refer to Figure 4, first row). The linearly polarized TE mode does not stretch along the polarization direction in this case, a behavior well documented in the literature and usually attributed to the dominant longitudinal component (see Figure 4, last row). This occurs due to the rather weak \(z\) component, which is caused by a rather modest spatiotemporal dispersion; see Figure 1b. The linearly polarized TM mode does not diverge from this pattern and looks identical to the TE mode. Furthermore, in addition to the emergence of the \(z\)-component, a non-zero cross-polarized \(y\)-component is absent in this case. The spikes in the \(x\) components share the same phases (see the inset in Figure 6), while the other two components exhibit spikes with alternating phases. Among all the polarizations considered, the \(z\) component is the weakest. The FWHM sizes are \(\Delta x_{TE}=0.1307\)\(\upmu\)m, \(\Delta y_{TE}=0.1307\)\(\upmu\)m, and \(\Delta x_{TM}=0.2714\)\(\upmu\)m, \(\Delta y_{TM}=0.1307\)\(\upmu\)m in transverse directions. An azimuthally polarized subcycle optical bullet behaves as expected, having only two transverse components (see Figure 6, third column). However, the radially polarized optical bullet exhibits a change in the subcycle mode when compared to the previous two cases; its \(z\) component is no longer dominant due to the focusing conditions (see  
\[\lim_{\Delta\omega\to\infty}\frac{\Delta\omega}{\omega_{c}}\to 2,\]3 Intensity distributions of subcycle pulses in vacuum and their individual components (\(E_{x}\) (second row), \(E_{y}\) (third row), \(E_{z}\) (fourth row)) in the longitudinal plane for linear TE (first column), TM (second column), azimuthal (third column), and radial (fourth column) polarizations when \(V/c=-5.65,\gamma=4.35\)\(\upmu\)m\({}^{-1}\). The insets represent the phase distribution. Figure 6, last column). The FWHM sizes are \(\Delta x_{azi}=0.2714\)\(\upmu\)m, \(\Delta y_{azi}=0.2714\)\(\upmu\)m and \(\Delta x_{rad}=0.2714\)\(\upmu\)m, \(\Delta y_{rad}=0.2714\)\(\upmu\)m in transverse directions. We now examine the spatial structure of the vector subcycle optical bullets that propagate forward. For comparison, we focus on the longitudinal plane, as illustrated in \[\lim_{\Delta\omega\to\infty}\frac{\Delta\omega}{\omega_{c}}\to 2,\]4  
We plotted the total and individual intensity distributions in this specific plane. In particular, there was no distinct difference between the shapes of linearly polarized TE and TM optical bullets. As shown in Figure 5, both the TE and the TM modes exhibit a pronounced cross-shaped peripheral part. The resulting profile resembles a rotated Cyrillic letter 'K', with axial intensity lines before and after the central part. In this plane, the TE mode has a single non-zero component, and its phase reveals a conical wave-front structure with energy flow directions indicated by the phase gradient. This includes both inward and outward movements.