# Localized Vector Optical Nondiffracting Subcycle Pulses
## 2 Materials and Methods  
Nondiffracting and nondispersive pulsed beams are typically introduced within a cylindrical symmetry, where they are understood as a coherent superposition of monochromatic solutions of the cylindrical coordinate framework. To achieve nondiffracting and nondispersive characteristics, the wave vectors \(k\) and frequencies \(\omega\) within the pulsed beam must be related. This condition leads to a variety of pulsed beams that resist both diffraction and dispersion. In this study, we focus on the investigation of the so-called optical bullets, which are defined by the following relation [28]:  
\[k_{z}(\omega)=\frac{\omega}{c}n(\omega)\cos\theta=\frac{\omega}{V}+\gamma. \tag{1}\]  
Here, \(k_{z}\) represents the \(z\) component of the wave vector \(k=(k_{\rho},0,k_{z})\), with \(k_{\rho}\) being the transverse component, \(\omega\) the wave frequency, \(c\) the speed of light, \(n(\omega)\) the refractive index of the material, \(\theta\) the angle between the \(z\) axis and the \(k\) vector, \(V\) the group velocity of the focus wave mode, and \(\gamma\) the propagation constant [28]. Occasionally, \(\gamma\) is considered the integration constant [30], which corresponds to the phase period of a plane wave traveling at a zero propagation angle (\(\theta_{0}=0\)). The relationship given by (1) establishes the spatiotemporal connection between the wave vector and frequency, which can be interpreted as the angular dispersion law for different frequencies as follows:  
\[\theta(\omega)=\arccos\biggl{[}\frac{c}{Vn(\omega)}+\frac{\gamma c}{\omega n( \omega)}\biggr{]}. \tag{2}\]  
By substituting the relationship between the wave vector \(k\) and the wave frequency into the general formula for the superposition of monochromatic waves, we obtain the following integral expression:  
\[E(\rho,\varphi,z,t)=\mathrm{e}^{\mathrm{i}\gamma z}\int_{0}^{\infty}S(\omega )J_{m}\bigl{(}k_{\rho}\rho\bigr{)}\mathrm{e}^{\mathrm{i}m\varphi}\mathrm{e}^{ -\mathrm{i}\omega\tau}d\omega \tag{3}\]  
where \(\tau=t-z/V\) and \(m\) is the topological charge of the pulsed beam; \((\rho,\varphi,z)\) are cylindrical coordinates.  
The scalar focus wave modes are well known, and here we construct their vector counterparts. Vectorial solutions of the Maxwell equations are constructed from solutions of scalar wave equations using the method given in References [55; 56], as follows:  
\[\mathbf{M}(\mathbf{r},t)=\mathbf{\
abla}\times\mathbf{a}E(\mathbf{r},t),\quad\mathbf{N}(\mathbf{r},t)= \frac{1}{k}\mathbf{\
abla}\times\mathbf{M}(\mathbf{r},t), \tag{4}\]  
where \(\mathbf{\
abla}\) is a nabla operator and \(\mathbf{a}\) is a predefined vector [56]. Applying this operation to Equation (3) leads to the following expressions for transverse electric (TE) optical bullets:  
\[\mathbf{M}(\mathbf{r},t)=\mathrm{e}^{\mathrm{i}\gamma z}\int_{0}^{\infty}S(\omega) \mathbf{M}_{0}(\mathbf{r};\omega)\mathrm{e}^{-\mathrm{i}\omega\tau}d\omega, \tag{5}\]  
and transverse magnetic (TM) optical bullets:  
\[\mathbf{N}(\mathbf{r},t)=\mathrm{e}^{\mathrm{i}\gamma z}\int_{0}^{\infty}S(\omega) \mathbf{N}_{0}(\mathbf{r};\omega)\mathrm{e}^{-\mathrm{i}\omega\tau}d\omega. \tag{6}\]  
Vector functions \(\mathbf{M}_{0}(\mathbf{r},\omega)\) and \(\mathbf{N}_{0}(\mathbf{r},\omega)\) are the basis vector functions of the cylindrical coordinate systems obtained from Equation (4). Their exact expressions and physical interpretations depend on the choice of vector \(\mathbf{a}\). In this work, we select \(\mathbf{a}=\mathbf{e}_{z}\) to obtain azimuthally and radially polarized modes and \(\mathbf{a}=\mathbf{e}_{x}\) to obtain linearly polarized ones. The exact expressions for vector functions \(\mathbf{M}_{0}(\mathbf{r},\omega)\) and \(\mathbf{N}_{0}(\mathbf{r},\omega)\) in these two cases are provided in Reference [68], so we omit them here for the sake of brevity. For the integration of Equations (5) and (6) we use the Matlab 2023a procedure 'quadgk', with both relative and absolute tolerances set to \(10^{-8}\).