# Localized Vector Optical Nondiffracting Subcycle Pulses
## 3 Vector Subcycle Pulses
In addition, we present the individual components of the pulsed beam together with their phases, which resemble the behavior of the tightly focused, linearly polarized, monochromatic beam [44]. However, the linearly polarized TM mode does not follow this trend, as it occupies a larger area in the direction perpendicular to the polarization, though the central spike is elongated along the direction of the polarization. A careful look at the individual components reveals the cause of that. The \(x\) component is no longer symmetric and has two additional spikes further away from the central part; see Figure 2, second column. Moreover, besides the appearance of the \(z\) component, we observe a cross-polarized non-zero \(y\) component. Individual spikes in the \(x\) components have the same phases; see the corresponding inset in Figure 2, while the remaining two individual components have spikes with alternating phases. The FWHM sizes are \(\Delta x_{TE}=0.0332\)\(\upmu\)m, \(\Delta y_{TE}=0.0533\)\(\upmu\)m, and \(\Delta x_{TM}=0.0392\)\(\upmu\)m, \(\Delta y_{TM}=0.0211\)\(\upmu\)m in the transverse plane. Figure 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 \(V/c=3.45,\gamma=0\)\(\upmu\)m\({}^{-1}\). The white arrows in first row represent the orientation of the electric field. The insets represent the phase distributions of the individual components. An azimuthally polarized subcycle X-wave behaves in an expected manner--it has only two transverse components, see Figure 2, third column. However, the radially polarized X-wave demonstrates a rather interesting feature in the subcycle mode--its \(z\) component is the strongest due to rather tight focusing conditions; see Figure 2, last column. The FWHM sizes are \(\Delta x_{azi}=0.0533~{}\upmu\)m, \(\Delta y_{azi}=0.0533~{}\upmu\)m and \(\Delta x_{rad}=0.0392~{}\upmu\)m, \(\Delta y_{rad}=0.0271~{}\upmu\)m in the transverse plane. After analyzing the spatial structure of the vector X-waves for this particular case in the transverse plane, we examined their behavior in the longitudinal plane, where the temporal duration influenced the longitudinal shape of the vector pulsed beams; see Figure 3. This figure shows the total and individual intensity distributions in a specific longitudinal plane. First, we noted the distinct difference in the shape of the linearly polarized TE and TM optical bullets; see Figure 3. The cross-shaped peripheral part is more pronounced for the TE mode; compare the first and second columns. For the TM polarization, the profile is less pronounced, but the overall shape looks smoother. The most important thing is the appearance of axial secondary maxima before and after the central part, giving the intensity profile the resemblance of a rotated Cyrillic letter '\(\mathcal{\Omega}\)K'. Due to the choice of the longitudinal plane, the TE mode has a single non-zero component. The phase of the component reveals that the structure of the wave front had a conical nature. The phase gradient is related to the direction of the energy flow. This can be seen from the phase distribution in the figures, which contains moving parts inward and outward, as previously observed in the literature [57; 73]. The situation is similar in the case of linear TM polarization, though it has two electric field components in this particular longitudinal plane, the \(x\) and \(z\) components. The intensity profile of the \(x\) component is identical to that of a rotated Cyrillic letter '\(\mathcal{\Omega}\)K'. However, the \(z\) component still resembles a general X-shape with an axial void due to the zero line. The phase distributions are typical for that type of conical beam, and the energy flow for each component is as expected.