# Generating Bessel-Gaussian Beams with Controlled Axial Intensity Distribution
## 2 Modeling  
The fundamental TEM\({}_{00}\) mode formed by laser resonators of a stable configuration corresponds to a Gaussian beam with a radially symmetric field distribution [30]:  
\[ \begin{array}{l}u_{00}(r,z)=A_{\rm w}\frac{h_{\rm w}}{h(z)}\exp\left[-\frac{ r^{2}}{h^{2}(z)}\right]\exp\left(-i\frac{\pi}{\Lambda}\frac{r^{2}}{R_{0}(z)}- \arctan\frac{z}{z_{\rm c}}+kz\right]\right),\\ h(z)=h_{\rm w}\sqrt{1+\left(z/z_{\rm c}\right)^{2}},\ R_{\Phi}(z)=z\Big{(}1+ z_{\rm c}^{2}/z^{2}\Big{)}.\end{array}  \tag{1}\]  
where \(u_{00}\) is the distribution of the complex amplitude of the Gaussian beam in the section with the \(z\) coordinate from the waist; \(r=\sqrt{x^{2}+y^{2}}\) denotes the distance from the observation point in the beam to the \(z\) axis; \(x,y\) are the transverse coordinates; \(A_{\rm w}\) denotes the amplitude constant or the field amplitude on the axis (\(r=0\)) in the section of the beam waist (\(z=0\)); \(h(z)\) and \(R_{\Phi}(z)\) are the dependences of the caustic and radius of wavefront curvature of the beam in section \(z\); \(2h_{\rm w}\) denotes the waist diameter of the Gaussian beam; \(z_{\rm c}=\frac{\pi ln_{\rm w}^{2}}{M\Lambda}\) is Rayleigh length of the Gaussian beam; \(\lambda\) is laser wavelength; \(M^{2}\) is the parameter quality of a Gaussian beam. Let us consider the transformation of a Gaussian beam with field distribution Equation (1) by a binary mask and a refractive axicon with a refractive index \(n\) and apex angle \(2\alpha_{0}\) (Figure 1). In wave optics, the effect of an axicon is described by the complex amplitude transmittance [31]: \(\tau(r)=\tau_{A}(r)\tau_{\Phi}(r)\), where \(\tau_{A}(r)\) is amplitude transmittance, \(\tau_{\Phi}(r)=\exp(ikpr)\) is phase transmittance, \(p=\sin\beta\) is the numerical aperture of the axicon, \(\beta=\arcsin(n\cos\alpha_{0})+\alpha_{0}-\frac{\pi}{2}\) is the angle of wavefront inclination to the optical axis after passing through the axicon in the case of the plane homogeneous wave. If the effect of the axicon has only a phase nature, then the generated Bessel beam of the zero-order is characterized by the length of the Bessel zone \(z_{B}\) and the diameter of the core \(D_{B}\) (diameter of the central maximum) of the field intensity distribution of the Bessel beam. They are defined by the following expressions [32; 33]:  
\[z_{B}=\frac{h_{\rm w}}{\tan\beta},\ D_{B}=\frac{2.405\lambda}{2\pi\sin\beta}, \tag{2}\]Such a Bessel beam also has an uneven distribution of axial field intensity [33; 34]:  
\[I(0,z_{\rm AP})\sim\frac{8\pi P\sin\beta}{\lambda h_{\rm w}}\frac{z_{\rm AP}}{z_{B }}\exp\!\left(-\frac{2z_{\rm AP}^{2}}{z_{B}^{2}}\right) \tag{3}\]  
where \(z_{\rm AP}\) is the analysis plane after the axicon and \(P\) denotes the power of laser radiation. Let us investigate the influence of binary mask parameters and the input Gaussian beam on forming a quasi-rectangular axial intensity distribution of the zero-order Bessel beam. Amplitude transmittance \(\tau_{A}(r)\) is determined by the finite axicon aperture and the nature of its transmission within the clear aperture. The calculation of the transverse distribution of field intensity in the analysis plane \(z_{\rm AP}\) after the axicon is performed using a certain transfer operator. The choice of the transfer operator depends on the distance between the original plane and the analysis plane. The nonparaxial scalar model based on Rayleigh-Sommerfeld theory allows to obtain correct results at sufficiently close distances from the aperture of the optical element at which the diffraction occurs.