# Beam Propagation Factor of a Cosh-Airy Beam
## 2 Analytical Expression of the Beam Propagation Factor  
In the Cartesian coordinate system, the \(z\)-axis is taken to be the propagation axis. The cosh-Airy beam in the source plane \(z=0\) takes the form of [25]  
\[E(x_{0},y_{0},0)=E(x_{0},0)E(y_{0},0) \tag{1}\]  
with \(E(x_{0},0)\) and \(E(y_{0},0)\) being given by  
\[E(s,0)=Ai\Bigg{(}\frac{s}{w_{0}}\Bigg{)}\exp\Bigg{(}\frac{as}{w_{0}}\Bigg{)} \cosh\Bigg{(}\frac{bs}{w_{1}}\Bigg{)} \tag{2}\]  
where \(s=x_{0}\) or \(y_{0}\), \(Ai(\cdot)\) is the Airy function, \(\cosh(. )\) is the cosh function, \(a\) and \(w_{0}\) are the decay factor and the transverse scale factor of the Airy part, respectively, \(b\) is the parameter associated with the cosh function, and \(w_{1}\) is the transverse scale associated with the cosh function. Equation (2) can be written in another form:  
\[E(s,0)=\frac{1}{2}\Bigg{[}Ai\Bigg{(}\frac{s}{w_{0}}\Bigg{)}\exp\Bigg{(}\frac{ a_{+}s}{w_{0}}\Bigg{)}+Ai\Bigg{(}\frac{s}{w_{0}}\Bigg{)}\exp\Bigg{(}\frac{a_{-}s}{w_{0}} \Bigg{)}\Bigg{]} \tag{3}\]  
where \(a_{\pm}=a\pm\gamma\) and \(\gamma=bw_{0}/w_{1}\). Therefore, a cosh-Airy beam can be considered as a superposition of two Airy beams with different decay factors. As the beam propagation factor is an invariant, we calculated it in the source plane. The method we used to calculate the beam propagation factor was the second-order moments [29; 30; 31; 32; 33]. According to the standard definition, the first-order moment of the cosh-Airy beam in the \(x\)-direction is found to be  
\[<x_{0}>=\frac{\int_{-\infty}^{\infty}x_{0}\big{|}E(x_{0},0)\big{|}^{2}dx_{0} }{\int_{-\infty}^{\infty}\big{|}E(x_{0},0)\big{|}^{2}dx_{0}}=\frac{\int_{- \infty}^{\infty}x\big{|}f_{1}(a_{+},x)+f_{1}(a_{-},x)\big{|}^{2}dx}{\int_{- \infty}^{\infty}\big{|}f_{1}(a_{+},x)+f_{1}(a_{-},x)\big{|}^{2}dx}=\frac{T_{1 }(a_{+},a_{-})}{T_{0}(a_{+},a_{-})} \tag{4}\]  
where \(f_{1}(a_{+},x)\) and \(f_{1}(a_{-},x)\) are given by  
\[f_{1}(a_{\pm},x)=Ai\Bigg{(}\frac{x}{w_{0}}\Bigg{)}\exp\Bigg{(}\frac{a_{\pm}x} {w_{0}}\Bigg{)}. \tag{5}\]\(<x_{0}>\) is just the center of gravity in the \(x\)-direction. The Fourier transforms of \(f_{1}(a_{\pm},x)\) and \(f_{1}\)'\((a_{\pm},x)\) turn out to be [34]  
\[F_{1}(a_{\pm},\xi)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}A\Bigg{(}\frac{x} {w_{0}}\Bigg{)}\exp\Bigg{(}\frac{a_{\pm}x}{w_{0}}\Bigg{)}\exp(i\xi x)dx=\frac{w_ {0}}{2\pi}\exp\Bigg{[}\frac{(a_{\pm}+iw_{0}\xi)}{3}^{3}\Bigg{]} \tag{6}\]  
where the asterisk denotes the complex conjugation. The Fourier transform of \(xf_{1}(a_{\pm},x)\) yields  
\[F_{2}(a_{\pm},\xi)=-i\frac{dF_{1}(a_{\pm},\xi)}{d\xi}=\frac{w_{0}^{2}}{2\pi}(a_ {\pm}+iw_{0}\xi)^{2}\exp\Bigg{[}\frac{(a_{\pm}+iw_{0}\xi)}{3}^{3}\Bigg{]}. \tag{7}\]  
Based on the property of the Fourier transform [34], one can obtain the analytical expressions of \(T_{0}(a_{+},a_{-})\) and \(T_{1}(a_{+},a_{-})\). As the analytical expressions of \(T_{0}(a_{+},a_{-})\) and \(T_{1}(a_{+},a_{-})\) are lengthy, they are listed in the Appendix A. The second-order moment of the cosh-Airy beam in the \(x\)-direction of the spatial domain reads as  
\[<x_{0}^{2}>=\frac{\int_{-\infty}^{\infty}{(x_{0}-cx_{0}>)}^{\infty}[E(x_{0},0 )]^{2}dx_{0}}{\int_{-\infty}^{\infty}{|E(x_{0},0)|}^{2}dx_{0}}=\frac{\int_{- \infty}^{\infty}{x_{0}^{2}|E(x_{0},0)|}^{2}dx_{0}}{\int_{-\infty}^{\infty}{|E( x_{0},0)|}^{2}dx_{0}}-<x_{0}>^{2}=\frac{T_{2}(a_{+},a_{-})}{T_{0}(a_{+},a_{-})}-<x_{0}>^{ 2}. \tag{8}\]  
The analytical expression of \(T_{2}(a_{+},a_{-})\) is also listed in the Appendix A. \(<x_{0}^{2}>\) is the square of the beam waist in the \(x\)-direction. The second-order moment of the cosh-Airy beam in the spatial-frequency domain is found to be [29]  
\[<\theta_{x}^{2}>=\frac{1}{k^{2}\!\!\int_{-\infty}^{\infty}{\big{|}E(x_{0},0) \big{|}}^{2}dx_{0}}\!\int_{-\infty}^{\infty}\!\!\left|\frac{\partial E(x_{0},0 )}{\partial x_{0}}\right|^{2}dx_{0}=\frac{T_{3}(a_{+},a_{-})}{k^{2}T_{0}(a_{+ },a_{-})} \tag{9}\]  
where \(k=2\pi/\lambda\), with \(\lambda\) being the optical wavelength.