# Localized Vector Optical Nondiffracting Subcycle Pulses
## 1 Introduction  
The development of structured light is progressing rapidly due to its broad range of potential applications and advantages, not only in optics but also in various branches of physics as a whole [1]. This surge of interest is driven by the ability to control a wide range of phenomena, including optical traction [2; 3], optical spinning [4], self-healing [5], imaging [6], nondiffractive and nondispersive propagation [7], ultrafast near-subcycle pulses [8; 9], and the generation of ultra-high-power laser radiation [10; 11]. The diffraction of light beams presents a significant challenge for laser communication, information transfer, and long-distance laser interactions. Spatial components of the beam during diffraction undergo dephasing, leading to the distortion of the spatial profile; the beam size and intensity change [12; 13; 14]. A nondiffracting beam avoids diffractive spread thanks to constructive interference within a region known as the Bessel zone. The Bessel beam is the most well-known example of a nondiffracting beam [15; 16]. An elliptical Mathieu [17] beam, a parabolic Weber beam [18], and an Airy beam [19] are also members of the family of nondiffracting beams. These beams can propagate over infinitely long distances and possess infinite energy; however, when given a Gaussian envelope, they form finite-energy optical beams that resist diffractive spreading over limited distances [20]. When shifting to pulsed beams [21], the most intriguing aspect is the shortest possible duration of pulsed beams, known as the subcycle [22; 23], which was experimentally achieved for various frequencies [24; 25]. Dispersion is a new challenge that arises in long-distance laser applications. Due to material dispersion, the phase velocities of different temporal components with varying wavelengths differ [12], leading to different propagation behaviors. Consequently, the pulse undergoes dispersive broadening and other distortions in its shape [26]. However, this issue can be mitigated by employing a non-monochromatic superposition of nondiffracting beams with a predefined spatiotemporaldispersion within the pulse [27; 28]. The joint resistance of these pulsed beams to both dispersion and diffraction has led to their designation as optical bullets. Examples of such pulsed nondiffracting and nondispersive beams include pulsed Bessel beams [29], focus wave modes [30], X waves [31; 32; 33], and spatiotemporal light sheets [34; 35], which have been extended to elliptical and parabolic X-waves [36] and further into nonlinear optics [37; 38]. The polarization of electromagnetic fields is a third critical degree of freedom which is becoming increasingly significant in the field of structured light. Advances in this area have increased following the realization that inhomogeneous polarization can enable the sharper focusing of light [39; 40], which is effectively described using an electromagnetic multipole model [41; 42], applicable even to pulsed fields [43]. As the numerical aperture increases, the scalar description of the optical fields becomes insufficient [44; 45; 46; 47]. Even linearly polarized light has a significant longitudinal component and detectable cross-polarization [48]. This longitudinal component is especially prominent in radially and azimuthally polarized light [49], which is generated using geometrical phase elements [50; 51]. In addition, vector beams can have topological structures such as skyrmions, merons, or other topological quasiparticles in their polarization or other derived forms of the electromagnetic field. These configurations of the electromagnetic field could be robust to perturbations and have been observed in Laguerre-Gaussian [52] or Bessel [53] beams. The concept of an inhomogeneously polarized nondiffracting beam was first introduced by [PERSON] [54], although earlier insights into this topic can be found in [PERSON]'s work [55], with [PERSON] providing a detailed discussion on constructing such polarized structures [56].