# Generating Bessel-Gaussian Beams with Controlled Axial Intensity Distribution
## 2 Modeling
Using the integral Rayleigh-Sommerfeld transform of the first type, we find the amplitude-phase distribution of the beam field in the plane of analysis after the axicon [34; 35]:  
\[u(\rho,\theta,z_{\rm AP})=\frac{z_{\rm AP}}{2\pi}\int\limits_{0}^{\infty}\int \limits_{0}^{2\pi}u^{\prime}(r)\tau(r)\frac{\exp\!\left(-ikL\right)}{L}\!\left( ik+\frac{1}{L}\right)\!drd\varphi \tag{4}\]  
where \(u^{\prime}\) is distribution of the complex field amplitude before the axicon, \(L^{2}=\rho^{2}+r^{2}+z_{\rm AP}^{2}-2\rho r\cos(\varphi-\theta)\) denotes oblique length equal to the distance between the point at which the initial distribution of the complex amplitude of the field \(u^{\prime}\) is specified and a point in the analysis plane after the axicon. To calculate the axial intensity of the output field, we use the following expression for \(L\): \(L^{2}=r^{2}+z_{\rm AP}^{2}\). Figure 1: Schemes for the formation of a zero-order Bessel beam with a quasi-rectangular dependence of axial intensity: (**a**) the Bessel zone starts right after the axicon; (**b**) the Bessel zone is moved away from the axicon. \(2h_{\rm w}\)—waist diameter of the Gaussian beam, \(2\alpha_{0}\)—apex angle axicon, \(n\)—refractive index axicon, \(z_{AP}\)—analysis plane after the axicon, \(z_{B}\)—length of the Bessel zone, D—clear aperture of the binary mask, d—diameter of the inner aperture of the mask, \(\tau_{A}\)—amplitude transmittance. When calculating amplitude-phase field distributions using the integral Rayleigh-Sommerfeld transform within the nonparaxial scalar model, it is necessary to consider the efficiency and error of numerical methods when working with rapidly oscillating functions. One of such practical methods is [PERSON]'s method [36], which allows one to calculate integrals with complex integrands of strongly oscillating functions. According to [PERSON]'s method, the one-dimensional oscillating integral is transformed into an ordinary differential equation which can then be solved by the collocation method [37]. The parameters of the axicon directly determine the distribution of the beam field in the analysis plane after the axicon, but do not affect the general nature of the effect under study. The presence of a binary mask in front of the axicon leads to the fact that the longitudinal dependence of axial intensity of the Bessel beam does not obey Equation (3). The distribution of the axial intensity of the Bessel zone is modified in such a way that the width of the central core and concentric rings remains unchanged in a certain interval, while the length of the Bessel zone decreases. With different parameters of the binary mask transmission function, it is possible to achieve a sharp change in the edge of the axial intensity distribution of the Bessel zone (Figure 2a-d). The analysis of the axial intensity distributions of the Bessel zone was carried out using the gradient of the scalar function, which shows the direction and rate of its most rapid increase/decrease [38; 39]. In Figure 2, orange plots represent the value of the function gradient for each point in the analysis plane after the \(z_{\text{AP}}\) axicon. Based on the data, it follows that there was an abrupt change in the speed and direction of the gradient of the function. The position of the beam focusing plane after the axicon can be calculated using Equation (2) for the length of the Bessel zone \(z_{B}\), in which the input beam waist is substituted for beam height. The results of this calculation are also compared in Figure 2, where the dotted vertical lines show the positions of the focusing plane according to the ray model. The calculations based on the wave  
Figure 2: Axial intensity distributions of a Bessel beam and the gradient from the longitudinal coordinate \(z_{\text{AP}}\) for various parameters of the mask before the axicon: (**a**) without binary mask, \(d=0\) mm, \(D\rightarrow\infty\); (**b**) with binary mask, \(d=4\) mm, \(D\rightarrow\infty\); (**c**) with binary mask, \(d=0\), \(D=7\) mm; (**d**) with binary mask, \(d=4\), \(D=7\) mm.