# Localized Vector Optical Nondiffracting Subcycle Pulses
## 3 Vector Subcycle Pulses  
Subcycle duration pulses are common nowadays [23]; however, the main question of this study is whether the subcycle duration can be combined with the nondiffracting and nondispersing properties of the focus wave modes described by Equation (1). In this study, we introduce the rectangular spectral envelope \(S(\omega)=\Pi(\omega_{c}-\omega/\Delta\omega)\), where  
\[\Pi\Big{(}\frac{\omega}{\Delta\omega}\Big{)}= \begin{cases}0,&\text{if }| \omega|>\frac{\Delta\omega}{2},\\ \frac{1}{2},&\text{if }|\omega|=\frac{\Delta\omega}{2},\\ 1,&\text{if }|\omega|<\frac{\Delta\omega}{2},\end{cases}  \tag{7}\]  
and where \(\Delta\omega\) is the spectral width of the rectangular spectral envelope, and \(\omega_{c}\) is the central frequency of the wave packet. For the rectangular spectral envelope, the spectral width \(d\omega\) can be related to the temporal duration \(dt\) using the expression \(d\omega=5.56/dt\)[72]. To quantify the resulting pulses as the subcycle we introduce the following parameter:  
\[\frac{\Delta\omega}{\omega_{c}}=\frac{T_{c}}{\Delta t},\] \[\lim_{\Delta\omega\to\infty}\frac{\Delta\omega}{\omega_{c}}\to 2,\] \[\lim_{\Delta\omega\to\infty}\frac{\Delta t}{T_{c}}\to\frac{1}{2}, \tag{8}\]  
where \(T_{c}\) is the cycle duration of the central frequency \(\omega_{c}\). Obviously, the limit of the shortest pulse is half the cycle of the central frequency, leading to the spectral width being twice the frequency. We now combine the spatiotemporal dispersion of the optical bullets with this definition and investigate the dependency of the parameter \(\Delta\omega/\omega_{c}\) on the group velocity \(V/c\) and the parameter \(\gamma\). The results are depicted in Figure 1a for vacuum conditions. In general, we observe values of \(\Delta\omega/\omega_{c}\) very close to 2. Most strikingly, we observe the possibility of creating subcycle optical bullets that propagate backward. On the other hand, for positive group velocities, we note that the spatiotemporal dispersion restricts the shortest possible duration of the nondiffracting pulsed beam; see Figure 1a. Here we note that the duration of the subcycle optical bullet in a dispersive medium is larger than in vacuum; see Figure 1b. The cause of this is the presence of absorption lines which restrict the resulting bandwidth, showing Equation (1) is valid in those regions. In further numerical simulations, we select three distinct examples of spatiotemporal dispersion; see Figure 1c and numerically integrate Equations (5) and (6) using the expression mentioned above for the spectral envelope. We start our investigation by selecting point (**1**) from Figure 1a. Here, the group velocity of the focus wave mode is \(V/c=3.45\) and the parameter \(\gamma=0\)\(\upmu\)m\({}^{-1}\), which is a particular case that is referred to as X-waves in the literature. We investigated the following  
Figure 1: **(a)** Dependency of the parameter \(\Delta\omega/\omega_{c}\) for focus wave modes on the parameters \(\gamma\) and \(V/c\) in vacuum; (**b**) dependency of the parameter \(\Delta\omega/\omega_{c}\) for focus wave modes on the parameters \(\gamma\) and \(V/c\) in BK7 glass. **(c)** Angular dispersion curves of focus wave modes in vacuum for parameters: (**1**) blue curve \(V/c=3.45,\gamma=0\)\(\upmu\)m\({}^{-1}\); (**2**) red curve \(V/c=-5.65,\gamma=4.35\)\(\upmu\)m\({}^{-1}\); (**3**) orange curve \(V/c=1,\gamma=-2.45\)\(\upmu\)m\({}^{-1}\). four polarization cases: linear TE (transverse electric), linear TM (transverse magnetic), radial, and azimuthal polarizations; see Equations (5) and (6). First we plotted the intensity distributions of these subcycle pulsed beams in the transverse plane; see Figure 2, first row. The linearly polarized TE mode is elongated along the polarization direction, as expected from the literature, and this occurs due to the strong longitudinal component; see Figure 2, last row.