# Beam Propagation Factor of a Cosh-Airy Beam
## 2 Analytical Expression of the Beam Propagation Factor
By giving the decay factor \(a\), the square of the beam waist in the \(x\)-direction of the cosh-Airy beam as a function of the cosh parameter \(\gamma\) is shown in \[E(s,0)=\frac{1}{2}\Bigg{[}Ai\Bigg{(}\frac{s}{w_{0}}\Bigg{)}\exp\Bigg{(}\frac{ a_{+}s}{w_{0}}\Bigg{)}+Ai\Bigg{(}\frac{s}{w_{0}}\Bigg{)}\exp\Bigg{(}\frac{a_{-}s}{w_{0}} \Bigg{)}\Bigg{]} \tag{3}\]0 When the decay factor \(a\) was greater than 1, the square of the beam waist in the \(x\)-direction first increased, then decreased, and finally increased again with the increase of cosh parameter \(\gamma\). As the decay factor \(a\) decreased, the first extremum position of the square of the beam waist versus the parameter \(\gamma\) increased. When the decay factor \(a\) was less than 1, the square of the beam waist in the \(x\)-direction increased with the increase of cosh parameter \(\gamma\). By giving the cosh parameter \(\gamma\), the square of the beam waist in the \(x\)-direction of the cosh-Airy beam as a function of the decay factor \(a\) is shown in \[E(s,0)=\frac{1}{2}\Bigg{[}Ai\Bigg{(}\frac{s}{w_{0}}\Bigg{)}\exp\Bigg{(}\frac{ a_{+}s}{w_{0}}\Bigg{)}+Ai\Bigg{(}\frac{s}{w_{0}}\Bigg{)}\exp\Bigg{(}\frac{a_{-}s}{w_{0}} \Bigg{)}\Bigg{]} \tag{3}\]1  
When the cosh parameter \(\gamma\) was relatively large, such as 0.6, the square of the beam waist in the \(x\)-direction first decreased and then increased with the increase of decay factor \(a\). In this case, the square of the beam waist in the \(x\)-direction had a minimum value. Moreover, the position where the minimum value appeared increased with the increase of the cosh parameter \(\gamma\). When the cosh parameter \(\gamma\) was relatively small, such as 0.1, the square of the beam waist in the \(x\)-direction decreased and finally tended to the minimum value with the increase of decay factor \(a\). By giving the decay factor \(a\), the square of the divergence in the \(x\)-direction of the cosh-Airy beam as a function of the cosh parameter \(\gamma\) is shown in \[E(s,0)=\frac{1}{2}\Bigg{[}Ai\Bigg{(}\frac{s}{w_{0}}\Bigg{)}\exp\Bigg{(}\frac{ a_{+}s}{w_{0}}\Bigg{)}+Ai\Bigg{(}\frac{s}{w_{0}}\Bigg{)}\exp\Bigg{(}\frac{a_{-}s}{w_{0}} \Bigg{)}\Bigg{]} \tag{3}\]2 When the decay factor \(a\) was greater than 1, the square of the divergence in the \(x\)-direction decreased with the increase of the cosh parameter \(\gamma\). However, there was a concavity in the curve of the square of the divergence versus the cosh parameter \(\gamma\). The concavity was more obvious for the small decay factor \(a\). When the decay factor \(a\) was less than 1, the square of the divergence in the \(x\)-direction increased with the increase of the cosh parameter \(\gamma\). Figure 7 shows the square of the divergence in the \(x\)-direction of the cosh-Airy beam as a function of the decay factor \(a\)  
\[E(s,0)=\frac{1}{2}\Bigg{[}Ai\Bigg{(}\frac{s}{w_{0}}\Bigg{)}\exp\Bigg{(}\frac{ a_{+}s}{w_{0}}\Bigg{)}+Ai\Bigg{(}\frac{s}{w_{0}}\Bigg{)}\exp\Bigg{(}\frac{a_{-}s}{w_{0}} \Bigg{)}\Bigg{]} \tag{3}\]3 By giving the decay factor \(a\), the square of the beam waist in the \(x\)-direction of the cosh-Airy beam as a function of the cosh parameter \(\gamma\) is shown. \[E(s,0)=\frac{1}{2}\Bigg{[}Ai\Bigg{(}\frac{s}{w_{0}}\Bigg{)}\exp\Bigg{(}\frac{ a_{+}s}{w_{0}}\Bigg{)}+Ai\Bigg{(}\frac{s}{w_{0}}\Bigg{)}\exp\Bigg{(}\frac{a_{-}s}{w_{0}} \Bigg{)}\Bigg{]} \tag{3}\]4 By giving the cosh parameter \(\gamma\), the square of the beam waist in the \(x\)-direction of the cosh-Airy beam as a function of the decay factor \(a\) is shown. with the determined \(\cosh\) parameter \(\gamma\). As the decay factor \(a\) increased, the square of the divergence in the \(x\)-direction decreased. A \(\cosh\)-Airy beam can be obtained by a superposition of two Airy beams. The decay factors of these two Airy beams are \(a+\gamma\) and \(a-\gamma\), respectively. According to Equation (20), the \({M_{x}}^{2}\) factor of the Airy beam is small for a large decay factor.