# Beam Propagation Factor of a Cosh-Airy Beam
## 4 Conclusions  
Based on the second-order moments, the analytical expression of the beam propagation factor of a cosh-Airy beam has been derived. The beam propagation factor was determined by the decay factor \(a\) and the \(\cosh\) parameter \(\gamma\). Since the beam propagation factors in the \(x\)- and \(y\)-directions of the \(\cosh\)-Airy beam have the same form, only the \({M_{\mathrm{x}}}^{2}\) factor was selected here for numerical calculation. The effects of the decay factor and the \(\cosh\) parameter on the \({M_{\mathrm{x}}}^{2}\) factor were investigated. When the decay factor was greater than 1, the \({M_{\mathrm{x}}}^{2}\) factor first increased, then decreased, and finally tended to the minimum value with the increase of the \(\cosh\) parameter. When the decay factor was less than 1, the \({M_{\mathrm{x}}}^{2}\) factor always increased with the increase of the \(\cosh\) parameter. As the decay factor increased, the \({M_{\mathrm{x}}}^{2}\) factor decreased and finally tended to the minimum value. The effects of the decay factor and the \(\cosh\) parameter on the squares of the beam waist and the divergence were further investigated. When the decay factor was greater than 1, the square of the beam waist first increased, then decreased, and finally increased again with the increase of the \(\cosh\) parameter. However, the square of the divergence always decreased with the increase of the \(\cosh\) parameter. When the decay factor was less than 1, the squares of the \(\cosh\) waist and the divergence increased with the increase of the \(\cosh\) parameter. When the \(\cosh\) parameter was relatively large, the square of the beam waist first decreased and then increased with the increase of the decay factor. When the \(\cosh\) parameter was relatively small, the square of the beam waist decreased with the increase of the decay factor and finally tended to the minimum value. As the decay factor increased, the square of the divergence decreased. The research results can be roughly summarized as follows. The greater the decay factor is than the \(\cosh\) parameter, the smaller the beam propagation factor of the \(\cosh\)-Airy beam is. The results obtained here are useful for the further understanding and application of \(\cosh\)-Airy beams. When one wants to obtain a \(\cosh\)-Airy beam by means of the superposition of different Airy beams, the present results provide a basis by which one can select the beam parameters to control the beam quality of the expected \(\cosh\)-Airy beam. Let us use a numerical example to illustrate this problem. For example, the first Airy beam in the source plane is characterized by \(E_{1}(s,0)=Ai(s/w_{0})\exp(s/w_{0})\). How does one choose the second Airy beam? According to Figure 2, one can choose the second Airy beam that has the form of \(E_{2}(s,0)=Ai(s/w_{0})\exp(3s/w_{0})\). The superposition of these two Airy beams results in a \(\cosh\)-Airy beam with \(a=(3+1)/2=2\) and \(\gamma=(3-1)/2=1\). The beam propagation factor of the obtained \(\cosh\)-Airy beam is close to 1.  
[PERSON] and [PERSON] conceived of and wrote the paper; [PERSON] and [PERSON] performed the numerical simulations; [PERSON] and [PERSON] analyzed and evaluated the results. This research was supported by National Natural Science Foundation of China under Grant No. 11574272, and Zhejiang Provincial Natural Science Foundation of China under Grant No. LY16A040014. The authors declare no conflict of interests.  
Figure 8: Normalized intensity in the \(x\)-direction of the \(\cosh\)-Airy beam in the source plane \(z=0\) (**a**) and the observation plane \(z=5z_{0}\) (**b**).