# Partially Coherent Flat-Topped Beam Generated by an Axicon
## 3 Numerical Simulation  
In this section, we will analyze the focused intensity of a MSSM beam shaped by an axicon based on Equation (15). The beam parameter \(\lambda\), the refractive index \(n_{d}\), and the apex angle \(\alpha\) of the axicon are set as \(\lambda=0.633\)\(\upmu\)m, \(\sigma=2.5\)\(\upmu\)m, \(n_{d}=1.51630\) and \(\alpha=170^{\circ}\). In order to clarify the axicon's effect on a MSSM beam, as seen in Figures 3 and 4, we first calculated the normalized intensity distributions of a completely coherent Gaussian beam with \(\delta=\infty,M=1\) and a partially coherent Gaussian Schell-model (GSM) beam with \(\delta=2\sigma,M=1\). One can see in Figure 3 that the focused field of a completely coherent Gaussian beam displays an optical needle and the transverse intensity profile remains unchanged and demonstrates a non-diffraction property. Figure 4 shows that an optical needle is also formed as the coherence length of the incident beam decreases while the transverse profile of the intensity at different propagation distances varies due to beam diffraction. To give a quantitative evaluation to the effect of the coherence length on the diffraction of focused beams, we calculate the full width at half maximum (FWHM) of the intensity along z for different values of \(\delta\), as shown in Figure 5. One sees that the FWHM along the z axis remains unchanged for a fully coherent beam, which is consistent with the transverse profiles of the intensity shown in Figure 3. For \(\delta=2\sigma\), the FWHM increases slightly with the increase of z, as shown in Figure 5. For the smaller value of \(\delta\), the focused field of a GSM beam shaped by an axicon diffracts more quickly. As a result, the length of coherence should not be too small to keep the non-diffraction property. Figure 4: Normalized intensity distribution of a focused partially coherent Gaussian Schell-model beam shaped by an axicon at different propagation distances z with \(\delta=2\sigma\): (**a**) 3-D shaded surface plot of the normalized intensity distribution (**b**) 2-D density plot of the normalized intensity distribution (**c**) Cross-section of the normalized intensity distribution. Figure 3: Normalized intensity distribution of a focused fully coherent Gaussian beam (GSM) shaped by an axicon at different propagation distances z: (**a**) 3-D shaded surface plot of the normalized intensity distribution (**b**) 2-D density plot of the normalized intensity distribution (**c**) Cross-section of the normalized intensity distribution. Next, we concentrated our attention on the focused field of a MGSM beam passing through an axicon. Figure 6 shows the normalized intensity generated by a MGSM beam with \(M=4\) and \(\delta=0.2\sigma\). One can see that the focused field of a MGSM beam diffracts more rapidly than that of a GSM beam. The flat-topped profile emerges at z = 18 mm and from z = 18 mm to z = 22 mm the beam profile remains flat-topped. The axicon lens plays the role of extending the longitudinal range of the flat-topped spatial profile. As we increase the beam order \(M\) to 12 in Figure 7, the focused field experiences a more complicated variance. The spatial profile varies from a concave top at z = 14 mm to a flat-top at z = 18 mm and then to a cusp top at z = 20 mm. It can be concluded that the focused field of a partially coherent beam is closely related to the characteristic parameters of a correlation function. The FWHM of different beam orders \(M\) with \(\delta=0.2\sigma\) in Figure 8 clearly demonstrates that as the beam order increases, the focused field diffracts more rapidly. Figure 5: Full width at half maximum (FWHM) of a focused GSM beam shaped by an axicon at different propagation distances z for different \(\delta\).