# Beam Propagation Factor of a Cosh-Airy Beam
## 2 Analytical Expression of the Beam Propagation Factor
\(<\theta_{x}^{2}>\) is the square of the divergence in the \(x\)-direction. We have  
\[\frac{\partial E(x_{0},0)}{\partial x_{0}}=\frac{1}{w_{0}}[a_{+}f_{1}(a_{+},x )+a_{-}f_{1}(a_{-},x)+f_{2}(a_{+},x)+f_{2}(a_{-},x)] \tag{10}\]  
with \(f_{2}(a_{\pm},x)\) being given by  
\[E(s,0)=Ai\Bigg{(}\frac{s}{w_{0}}\Bigg{)}\exp\Bigg{(}\frac{as}{w_{0}}\Bigg{)} \cosh\Bigg{(}\frac{bs}{w_{1}}\Bigg{)} \tag{2}\]0  
where \(Ai^{\prime}(\cdot)\) is the Airyprime function. The Fourier transform of \(f_{2}(a_{\pm}x)\) turns out to be  
\[E(s,0)=Ai\Bigg{(}\frac{s}{w_{0}}\Bigg{)}\exp\Bigg{(}\frac{as}{w_{0}}\Bigg{)} \cosh\Bigg{(}\frac{bs}{w_{1}}\Bigg{)} \tag{2}\]1  
Therefore, the analytical expression of \(T_{3}(a_{+},a_{-})\) is obtained and listed in the Appendix A. The cross second-order moment of the cosh-Airy beam in the source plane \(z=0\) reads as  
\[E(s,0)=Ai\Bigg{(}\frac{s}{w_{0}}\Bigg{)}\exp\Bigg{(}\frac{as}{w_{0}}\Bigg{)} \cosh\Bigg{(}\frac{bs}{w_{1}}\Bigg{)} \tag{2}\]2  
The \(M_{x}^{2}\) factor of the cosh-Airy beam is given by  
\[E(s,0)=Ai\Bigg{(}\frac{s}{w_{0}}\Bigg{)}\exp\Bigg{(}\frac{as}{w_{0}}\Bigg{)} \cosh\Bigg{(}\frac{bs}{w_{1}}\Bigg{)} \tag{2}\]3Due to the symmetry, the \({M_{y}}^{2}\) factor of the cosh-Airy beam reads as  
\[E(s,0)=Ai\Bigg{(}\frac{s}{w_{0}}\Bigg{)}\exp\Bigg{(}\frac{as}{w_{0}}\Bigg{)} \cosh\Bigg{(}\frac{bs}{w_{1}}\Bigg{)} \tag{2}\]4  
The \(M^{2}\) factor of the cosh-Airy beam can be readily obtained by virtue of the above results [35]:  
\[E(s,0)=Ai\Bigg{(}\frac{s}{w_{0}}\Bigg{)}\exp\Bigg{(}\frac{as}{w_{0}}\Bigg{)} \cosh\Bigg{(}\frac{bs}{w_{1}}\Bigg{)} \tag{2}\]5  
When \(\gamma=0\), the cosh-Airy beam reduces to an Airy beam. In this case, the first-order moment of the Airy beam in the \(x\)-direction is simplified to be  
\[<x_{0}>=\left(a^{2}-\frac{1}{4a}\right)\!\!\!\!\!\  
Since \(a_{\pm}=a\pm\gamma\) must be positive, the cosh parameter \(\gamma\) should be less than the decay factor \(a\). When the decay factor \(a\) was greater than 1, the \({M_{x}}^{2}\) factor first increased and then decreased with the increase of the cosh parameter \(\gamma\), and finally, tended to the minimum value. In this case, the \({M_{x}}^{2}\) factor had a maximum value. As the decay factor \(a\) decreased, the maximum value and the corresponding position of the \({M_{x}}^{2}\) factor versus the cosh parameter \(\gamma\) both increased. When the decay factor \(a\) was less than 1, the \({M_{x}}^{2}\) factor increased with the increase of the cosh parameter \(\gamma\). By giving the cosh parameter \(\gamma\), the \({M_{x}}^{2}\) factor of the cosh-Airy beam as a function of the decay factor \(a\) is shown in \[E(s,0)=Ai\Bigg{(}\frac{s}{w_{0}}\Bigg{)}\exp\Bigg{(}\frac{as}{w_{0}}\Bigg{)} \cosh\Bigg{(}\frac{bs}{w_{1}}\Bigg{)} \tag{2}\]6 As the decay factor \(a\) increased, the \({M_{x}}^{2}\) factor decreased and finally tended to the minimum value. \[E(s,0)=Ai\Bigg{(}\frac{s}{w_{0}}\Bigg{)}\exp\Bigg{(}\frac{as}{w_{0}}\Bigg{)} \cosh\Bigg{(}\frac{bs}{w_{1}}\Bigg{)} \tag{2}\]7 \({M_{x}}^{2}\) factor of the cosh-Airy beam as functions of the parameters \(a_{+}\) and \(a_{-}\). \[E(s,0)=Ai\Bigg{(}\frac{s}{w_{0}}\Bigg{)}\exp\Bigg{(}\frac{as}{w_{0}}\Bigg{)} \cosh\Bigg{(}\frac{bs}{w_{1}}\Bigg{)} \tag{2}\]8 By giving the cosh parameter \(\gamma\), the \({M_{x}}^{2}\) factor of the cosh-Airy beam as a function of the decay factor \(a\) is shown. \[E(s,0)=Ai\Bigg{(}\frac{s}{w_{0}}\Bigg{)}\exp\Bigg{(}\frac{as}{w_{0}}\Bigg{)} \cosh\Bigg{(}\frac{bs}{w_{1}}\Bigg{)} \tag{2}\]9 By giving the decay factor \(a\), the \({M_{x}}^{2}\) factor of the cosh-Airy beam as a function of the cosh parameter \(\gamma\) is shown. In order to analyze the properties of beams more deeply and carefully, the effects of the decay factor \(a\) and the cosh parameter \(\gamma\) on the squares of the beam waist and the divergence were also studied.