# Partially Coherent Flat-Topped Beam Generated by an Axicon
## 1 Introduction  
Over the past years, axicon lenses have been studied widely due to their unique properties of producing focal lines of narrow transverse width over extended distances [1; 2]. The axicon lens and its combination with other optical elements have been applied in many fields such as laser machining, corneal surgery, and atom alignment and trapping [2; 3; 4; 5]. Since 1986, when [PERSON] first put forward a non-diffracting beam named a Bessel beam [6], a great deal of attention has been concentrated on the axicons lens as it provides the most energy efficient method of realizing generally diffraction-free beams [7; 8; 9]. Later, the research was extended from completely coherent beams to partially coherent beams. For a spatially partially coherent light, [PERSON] et al. proposed a method of designing annular-aperture axicons or apodized annular logarithmic axicons to generate a uniform intensity axial line image [10; 11]. [PERSON] et al. designed diffractive axicons to produce uniform line images in a twisted Gaussian Schell-model (TGSM) illumination [12; 13]. [PERSON] et al. generated adjustable partially coherent bottle beams by focusing partially coherent light with an axicon-lens system [14; 15]. [PERSON] et al. examined the focal depth of partially coherent fields without aperture [16]. To date, the research on focusing partially coherent fields by an axicon lens has been restricted to Gaussian Schell-model beams. The results of the above-mentioned research show that the coherence length of partially coherent light has an intimate relationship with the axial intensity distribution of the focused field formed by an axicon. In an experiment, we can easily obtain different values of the coherencewidth by varying the focused beam spot on a rotating ground glass disk (RGGD) through adjusting the distance between the focus lens and RGGD [17].  
With the rapid development of partially coherent light and its wide applications [17], nonconventional partially coherent light, the correlation function of which does not satisfy a Gaussian distribution, has been proposed [18; 19]. It has been confirmed both theoretically and experimentally that the correlation function provides a new effective way to shape the intensity profile [20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30]. What is the effect of the nonconventional correlation function on the intensity distribution of beams focused by an axicon lens? Among a variety of nonconventional partially coherent beams, we are more interested in beams with a flat-topped spatial profile, which have shown great value in lithography, integrated circuit trimming, laser welding, and biomedical engineering. In 2012, a multi-Gaussian Schell-model (MGSM) beam [31], the degree of coherence (DOC) of which was modeled by multi-Gaussian distribution, was proposed by [PERSON] et al. The MGSM beam can form flat square or rectangular intensity distributions in the far field [31; 32; 33]. Later, generalized GMSM beam and elliptical MGSM beam were proposed, which can produce prescribed flat-topped spatial profiles [34; 35]. For these nonconventional partially coherent beams, extra parameters of a correlation function can be used to control the flat-topped spatial profiles. In this paper, we will investigate the intensity distribution of a MGSM beam focused by an axicon and explore the effect of a nonconventional correlation function on the focusing properties of a MGSM beam.