# Vector Optical Bullets in Dielectric Media: Polarization Structures and Group-Velocity Effects
## 2 Materials and Methods  
For the investigation of the optical bullets (focus wave modes), we have selected one of the most common materials used for laser applications--the BK7 glass. The dielectric constants of BK7 glass are well known. The Sellmeier equations are easily accessible and well researched. The feasibility of using BK7 glass for experiments is high due to its widespread use, availability, excellent optical properties, and cost-effectiveness. Angular dispersion curves of BK7 glass (see Figure 1) allow us to represent the O- and X-shaped focus wave modes within the optical wavelength range, with available Light Conversion, PHAROS laser (wavelength of 1030 nm) and harmonics available at 515 nm, 343 nm, 258 nm, and 206 nm. We should take into account the maximum angle at the interface between the air and the glass, which limits the maximum feasible angle to 0.73 radians. At higher angles, we see the required incident angles become complex, which shows that evanescent waves would be required. Nondiffracting and nondispersive wave packets are usually introduced within the frame of the cylindrical coordinate system and are perceived as a coherent superposition of monochromatic solutions of the respective cylindrical coordinate system. For such a coherent superposition to have nondiffracting and nondispersive properties, the wave vectors and frequencies within the wave packet must correlate with each other. This requirement results in a wide range of different wave packets that are resistant to diffraction and dispersion. In this work, our aim is to investigate the so-called optical bullets, which are characterized by the following relation [21]:  
\[k_{z}(\omega)=\frac{\omega}{c}n(\omega)\cos\theta=\frac{\omega}{V}+\gamma, \tag{1}\]  
where \(k_{z}\) is the \(z\) component of the wave vector \(\mathbf{k}=(k_{p},0,k_{z})\), \(k_{p}\) is the transverse component of the wave vector, \(\omega\) is the wave frequency, \(c\) is the speed of light, \(n\) is the refractive index of the material, \(\theta\) is the angle between the \(z\) axis and the \(\mathbf{k}\) vector, \(V\) is the group velocity of the focus wave mode, and \(\gamma\) is the propagation constant [21]. Sometimes, it can be attributed to the integration constant [23], which represents the phase period of a resulting plane wave traveling at zero propagation angle (\(\theta_{0}=0\)). This dependency (1) introduces the spatial-temporal relationship between the wave vector and the wave frequency. It can be visually perceived as the angular dispersion law for different frequencies:  
\[\theta(\omega)=\arccos\bigg{[}\frac{c}{Vn(\omega)}+\frac{\gamma c}{\omega n( \omega)}\bigg{]}. \tag{2}\]  
Figure 1: (**a**) Angular dispersion of the optical bullet inside the BK7 glass, when \(\gamma=1.25\) μm\({}^{-1}\), and \(V/c\): 0.7 (1), 0.75 (2), 0.85 (3), 1 (4), 1.4 (5). **(b**) Angular dispersion of the optical bullet within the BK7 glass, when \(\gamma=-0.628\) μm\({}^{-1}\), and \(V/c\): 0.629 (1), 0.63685 (2), 0.645 (3), 0.65495 (4), 0.685 (5), **(c**) Angular dispersion of the optical bullet within the BK7 glass, when \(V/c=-1.2\), and \(\gamma\): \(6\pi\) μm\({}^{-1}\) (1), \(7\pi\) μm\({}^{-1}\) (2), \(8\pi\) μm\({}^{-1}\) (3), \(9\pi\) μm\({}^{-1}\) (4), \(10\pi\) μm\({}^{-1}\) (5). The frequency is normalized to the value of \(\omega_{0}=1.7716\) fs\({}^{-1}\). Substituting the relation between the vector \(k\) and the frequency of the wave into the general expression of the superposition of monochromatic waves results in the following integral expression:  
\[E(\rho,\varphi,z,t)=\mathrm{e}^{\mathrm{i}\gamma z}\int_{0}^{\infty}S(\omega)J_{m }\big{(}k_{\rho}\rho\big{)}\mathrm{e}^{\mathrm{i}m\varphi}\mathrm{e}^{-\mathrm{ i}\omega\tau}d\omega \tag{3}\]  
where \(\tau=t-z/V\) and \(m\) is the topological charge of the focus wave mode, \((\rho,\varphi,z)\) are cylindrical coordinates. The scalar focus wave modes are well known and have been studied. Our aim here is to construct their vector counterparts.