# Localized Vector Optical Nondiffracting Subcycle Pulses
## 4 Discussion and Conclusions  
We introduced vector formalism to the concept of nondiffracting and nondispersive optical bullets, exploring the following different polarizations: linear, azimuthal, and radial. To showcase the variety of spatiotemporal dispersions, we selected three propagation velocities, two positive and one negative (backward propagating). The first two group velocities \(V\) were chosen for their intriguing dispersion, one being the dispersion of the X-waves and the second of the FWM with cone angles ranging from small paraxial (valid for scalar descriptions) to larger values that require a vector treatment of the electric field. For these velocities, we examined such frequencies, resulting in the following two scenarios: one with X-wave spatiotemporal dispersion and profile and another with focus wave mode dispersion. We found that the spatiotemporal shape of the vector-subcycle optical bullet (FWM) mirrors the scalar optical X-pulses, with differences due to the presence of small non-zero longitudinal and cross-polarization components. Polarization-dependent features for these situations are negligible, thus enabling nearly scalar subcycle nondiffract  
Figure 7: 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 \(V/c=1,\gamma=-2.45\)\(\mu\)m\({}^{-1}\). The insets represent the phase distribution. ing fields. Nevertheless, for the case of the X-wave, high cone angles can necessitate a vector description of the vector FWM, highlighting the distinct differences between polarizations. In the second case, a forward-propagating superposition of Bessel beams within the Bessel zone creates a backward-propagating pulse with negative velocity \(V\). This scenario revealed significant differences in the properties of the optical vector bullets within the dielectric material. The highly nonparaxial nature of these waves creates the longitudinal component prominent in TE and TM polarized bullets. Notably, the linearly polarized TM mode exhibits a large cross-polarized component. The azimuthally polarized bullet shows no differences, but the radially polarized wave is greatly affected, with the longitudinal electric field component dominating the intensity pattern. A quick look into the research field gives the following hints regarding practical realizability: a high-energy subcycle pulse synthesizer based on a mid-infrared optical parametric amplifier was reported [74], subcycle-isolated attosecond pulses were generated by pumping ionizing gating [75], phase-stable subcycle mid-infrared pulses were generated from filamentation in nitrogen [76], etc. A harder task for experimentally realizing nondiffracting and nondispersing optical bullets is organizing a proper spatiotemporal dispersion. A solution to this problem might be the usage of metasurfaces [77]. These results could help improve ultrafast optics, imaging, and material processing by giving us a better understanding of the behavior and characteristics of such subcycle pulses. Conceptualization, S.O. ; methodology, S.O. ; software, K.L. ; validation, K.L. ; formal analysis, S.O. and K.L. ; investigation, K.L. ; resources, S.O. ; data curation, K.L. ; writing--original draft preparation, S.O. and K.L. ; writing--review and editing, S.O. and K.L. ; visualization, K.L. ; supervision, S.O. ; project administration, S.O. ; funding acquisition, S.O. All authors have read and agreed to the published version of the manuscript. Not applicable. The raw data supporting the conclusions of this article will be made available by the authors on request. This work is dedicated to the memory of [PERSON] (1942-2022), a mentor, colleague, pioneer, and leader in the field of optics. The authors declare no conflicts of interest.