# Generating Bessel-Gaussian Beams with Controlled Axial Intensity Distribution
## 1 Introduction  
Research on the methods of focusing light beams associated with the names of such scientists as [PERSON], [PERSON], and [PERSON] have a long history and remain relevant to this day. Tight focusing of laser radiation, which allows one to obtain high power densities and directivity, is in demand in many problems of photonics and laser physics [1; 2; 3; 4; 5; 6].  
In practice, Gaussian laser beams are widely used. When they are focused using an optical system with spherical (or parabolic) surfaces, it is possible to concentrate laser radiation in the region corresponding to the beam waist diameter. However, in Gaussian beams, the wavefront is flat only in the waist region, the length of which is equal to the confocal parameter. With a decrease in the waist diameter of a Gaussian beam, its confocal parameter decreases [7; 8; 9].  
By choosing the type and shape of the surfaces of the laser optical system elements, it is possible to correct the geometry of the wavefront. Correction methods can also be applied to increase the focusing length of the formed beam. In the end, when the lens takes the form of a cone, the phase front of the transmitted wave becomes conical. Such a conical lens is called an axicon [10]. The axicon forms a beam with a radial distribution of field intensity, which is described by the Bessel function of the first kind of the zero-order. Such beams are often used for scientific and applied problems [11; 12], in which they have advantages over Gaussian beams [13; 14]. The main ones are: (i) the transverse distribution of field intensity remains practically unchanged over a significant portion along the optical axis, (ii) the distribution of the field is restored behind obstacles, and (iii) excellent depth of field.  
Axicon is a reliable optical element for the formation of a zero-order Bessel beam, including schemes with high-power laser radiation. However, it has disadvantages such as the dependence of theparameters of the generated beam on a specific element, the dependence of the intensity distribution of the output beam on the quality of element manufacturing (in particular, on the quality of the apex), and uneven distribution of axial intensity [15].  
There is a huge number of different tools used for the formation of Bessel beams [16; 17] which allow controlling axial intensity distribution. Diffraction [18; 19] and holographic methods [20] are outside of the scope of this study due to the features of their implementation. Methods capable of realizing the required Bessel beam intensity profiles are widely used in microscopy [21], including biological imaging [22; 23; 24], as well as in applications of ultrashort laser pulses [25; 26]. In almost all of the abovementioned applications, a Bessel beam with a controlled quasi-uniform axial intensity distribution is formed.  
The only way to change the axial intensity is to apodize the beam incident on the element (axicon or lens). In research, phase masks with a gradient, step, or both dependences of the phase shift value on the radius vector in combination with an axicon or a lens are actively used [27; 28]. There are also sophisticated ways to achieve the same goal [29].  
Research in the field of parametrization of the quasi-uniform axial intensity distribution of the Bessel beam is relevant. The purpose of this work was to develop a laser optical system for forming a Bessel beam with a controlled quasi-uniform axial distribution of field intensity on a standard element base with a minimum number of optical elements without significant restrictions on their radiation resistance. Research is being carried out on the possibility of realizing such a laser optical system by apodizing the input aperture of the axicon.