# Vector Optical Bullets in Dielectric Media: Polarization Structures and Group-Velocity Effects
## 1 Introduction  
The engineering of structured light is on the rise presently due to the wide range of possible applications and benefits not only within optics but also within physics in general [1]. This interest is caused by a large degree of controllable phenomena such as optical traction [2, 3], optical spinning [4], self-healing [5], imaging [6] and nondiffractive nondispersive propagation [7]. The diffraction of light beams is one of the obstacles to laser communication, information transfer, and long-distance laser interactions. As the beam propagates, its spatial components experience dephasing due to diffraction, so the spatial profile is distorted: the size increases and the intensity drops [8, 9, 10]. The nondiffracting beam is a beam that circumvents diffraction spreading through constructive interference in a zone called the Bessel zone. The best-known example of such a beam is the Bessel beam [11, 12]. However, other beams are also known--the elliptical Mathieu [13], the parabolic Weber [14], or the Airy beam [15]. These examples propagate infinitely large distances and possess infinite energy; however, giving them a Gaussian envelope results in finite-energy optical beams resisting diffractive spreading over finite distances [16]. Moving to pulsed beams [17], yet another obstacle appears for long-distance laser applications. Due to the dispersion of the material, the phase velocities of different temporal components with different wavelengths are not the same [8]. For this reason, they propagate differently [18], so the pulse experiences dispersive broadening and other shape distortions [19]. This was found to be overcome using the nonmonochromatic superposition of nondiffracting beams with a predefined spatiotemporal dispersion within the pulse [20, 21]. Joint dispersion and diffraction resistance of such pulsed fields have allowed us to call them optical bullets. Among such examples of pulsed nondiffracting and nondispersive beams are pulsed Bessel beams [22], focus wave mode [23], and X-waves [24, 25, 26]and spatiotemporal light sheets [27; 28], which have achieved their extension to elliptical and parabolical X-waves [29] and to nonlinear optics [30; 31]. Another degree of freedom, which becomes more noticeable within the field of structured light, is the polarization of the electromagnetic field. This advance is constantly increasing since the realization that inhomogeneous polarization enables a sharper focus of light [32; 33], which can be conveniently understood using a multipolar description of light [34; 35], valid also for pulsed fields [36]. As the numerical aperture increases, the scalar description of the optical field becomes invalid [37; 38; 39; 40]. Even for linearly polarized light, a noticeable longitudinal component appears, and even detectable cross-polarization is observed [41]. The longitudinal component is especially noticeable in radially and azimuthally polarized light [42], which is created using polarization-controllable optical elements [43; 44]. Anisotropic crystals have another interesting characteristic--birefringence. Birefringence materials have different refractive indices for different polarization directions. For even more control over birefringence, we could use liquid crystals, which can be controlled by external fields. By controlling polarization and amplitude spatially, we can enhance optical encryption and fine-tune vectorial holography [45]. Birefringence adds to the complexity of the problem, especially since anisotropic medium leads to polarization mixing between TE and TM polarizations [46] due to the Imbert-Fedorov shift [47; 48]. The concept of an inhomogeneously polarized nondiffracting beam was successfully introduced by [PERSON] [49], although some insight into the topic was already present in the book by [PERSON] [50] and the method of construction of such polarized structures is discussed in great detail by [PERSON] [51].