# Localized Vector Optical Nondiffracting Subcycle Pulses
## 3 Vector Subcycle Pulses
The FWHM sizes are \(\Delta z_{TE}=0.1899\) fs and \(\Delta z_{TM}=0.1899\) fs in longitudinal directions. spikes with alternating phases; see the corresponding inset in Figure 4. The \(z\) component is weakest in this case compared to all the polarizations we consider here. The FWHM sizes are \(\Delta x_{TE}=0.2312\)\(\upmu\)m, \(\Delta y_{TE}=0.3518\)\(\upmu\)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--it has only two transverse components; see Figure 4, third column. However, the radially polarized backward propagating bullet demonstrates a rather interesting feature in the subcycle mode--its z component is the strongest due to tight focusing conditions; see Figure 4, last column. The FWHM sizes are \(\Delta x_{azi}=0.3518\)\(\upmu\)m, \(\Delta y_{azi}=0.3518\)\(\upmu\)m and \(\Delta x_{rad}=0.1709\)\(\upmu\)m, \(\Delta y_{rad}=0.1709\)\(\upmu\)m in transverse directions. We now proceed to discussing the spatial structure of the vector subcycle optical bullets that propagate backward. For the sake of comparison, we investigated the same particular case of the longitudinal plane; see Figure 5.  
We plotted the total intensity distributions as well as the individual ones in a specific longitudinal plane. Firstly, we observed a distinct difference in the shapes of linearly polarized TE and TM optical bullets, as shown in Figure 5. The cross-shaped peripheral part is very pronounced for the TE mode, as seen in the first and second columns. For the TM polarization, the profile is less pronounced, and the overall shape appears to be more concentrated. In particular, axial secondary maxima appear before and after the central part, giving the general shape a resemblance to a rotated Cyrillic letter 'K', whereas the  
\[\lim_{\Delta\omega\to\infty}\frac{\Delta\omega}{\omega_{c}}\to 2,\]0 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 \(V/c=3.45,\gamma=0\)\(\upmu\)m\({}^{-1}\). The insets represent the phase distribution for each individual component. TE mode still resembles the well-known X-shape. Due to the choice of the longitudinal plane, the TE mode has a single non-zero component. The phase of this component reveals a conical wave-front structure. The phase gradient indicates the direction of energy flow, which, as seen in the phase distribution, includes parts moving inward and outward, as previously noted in the literature [57; 73]. Similarly, linear TM polarization exhibits two electric field components in this longitudinal plane, a dominant \(x\) component and a very weak \(z\) component. The intensity profile of the \(x\) component resembles a rotated Cyrillic letter 'X', while the \(z\) component maintains a general X-shape with an axial void due to the singularity. The phase distributions are typical for this type of conical beam, and the energy flow for each component is as expected. The FWHM sizes are \(\Delta z_{TE}=1.6583\) fs and \(\Delta z_{TM}=1.4171\) fs in longitudinal directions. Azimuthal and radial polarizations are shown in the third and fourth columns of \[\lim_{\Delta\omega\to\infty}\frac{\Delta\omega}{\omega_{c}}\to 2,\]1 The azimuthal polarization exhibits a very pronounced peripheral part of the beam. This pulsed beam has only one component in this plane. The phase distribution shows conical energy flow with a phase jump on the \(z\) axis due to the polarization singularity [39; 40]. The radially polarized mode has a dominant \(z\) component and an almost unnoticeable \(x\) component. The FWHM sizes are \(\Delta z_{azi}=1.6583\) fs, and \(\Delta z_{rad}=1.5377\) fs in longitudinal directions.