# Propagation of Cosh-Airy and Cos-Airy Beams in Parabolic Potential
## 2 Propagation of Cosh-Airy and Cos-Airy Beams in Parabolic Potential  
In the Cartesian coordinate system, the \(z\)-axis is taken to be the propagation axis. For simplicity, the cosh-Airy and cos-Airy beams considered here are one-dimensional. The propagation of one-dimensional cosh-Airy and cos-Airy beams obeys the normalized dimensionless linear parabolic equation as follows:  
\[\frac{1}{2}\frac{\partial^{2}\psi(x,z)}{\partial x^{2}}+i\frac{\partial\psi( x,z)}{\partial z}-V(x)\psi(x,z)=0, \tag{1}\]  
where \(V(x)=L^{2}x^{2}/2\) is the parabolic potential, and the parameter \(L\) describes the depth of the potential. The variables \(x\) and \(z\) are the normalized transverse coordinate and the propagation distance, respectively; \(\psi(x,z)\) is the envelope of the cosh-Airy and cos-Airy beams. It is known that the propagation of a beam in a parabolic potential is equivalent to the propagation in a quadratic-index waveguide [42; 43] as well as in a multilens system [44] and can be described by using the fractional Fourier transform [45; 46]. A reformulation of the \"separation of variables\" scheme in the parabolic equation has been presented [47]. The solution of Equation (1) can be written as [48; 49]:  
\[\psi(x,z)=f(x,z)\int_{-\infty}^{+\infty}\left[\widetilde{\psi}(\xi) \widetilde{G}(\xi)\right]\exp(-iK\xi)d\xi, \tag{2}\]  
with the functions \(f(x,z)\) and \(\widetilde{G}(\xi)\) as well as the parameter \(K\) being given by:  
\[f(x,z)=\sqrt{\frac{K}{2\pi i\dot{x}}}\exp(ibx^{2}), \tag{3}\]  
\[\widetilde{G}(\xi)=\sqrt{\frac{i\pi}{b}}\exp(-\frac{i\dot{x}^{2}}{4b}), \tag{4}\]  
\[K=Lx\csc(Lz). \tag{5}\]  
The parameter \(b\) is defined by:  
\[b=\frac{L}{2}\cot(Lz), \tag{6}\]  
where \(\widetilde{\psi}(\xi)\) is the Fourier transform of \(\psi(x,0)\), and \(\psi(x,0)\) is the input cosh-Airy or cos-Airy beam. \(\widetilde{G}(\xi)\) is the Fourier transform of \(\exp(ibx^{2})\). First, the input beam is a cosh-Airy beam, which has the form:  
\[\psi(x,0)=Ai(x-x_{0})\exp[a(x-x_{0})]\cosh[\gamma(x-x_{0})], \tag{7}\]where \(Ai(\cdot)\) is the Airy function, \(a\) is the decay factor, \(\gamma\) is the cosh factor, and \(x_{0}\) is the transverse displacement. The cosh-Airy beam is the result of the superposition of two Airy beams with different decay factors \(a_{+}\) and \(a_{-}\). Here, \(a_{+}\) and \(a_{-}\) are defined by \(a_{\pm}=a\pm\gamma\). Therefore, \(\gamma\) should be smaller than \(a\). The corresponding Fourier transform of \(\psi(x,0)\) is found to be [50]:  
\[\overline{\psi}(\xi)=\frac{1}{2}\exp\!\left(-ix_{0}\xi+\frac{i\xi^{3}}{3} \right)\!\left[\exp(-a_{+}\xi^{2})\exp\!\left(\frac{a_{+}^{3}}{3}-ia_{+}^{2} \xi\right)+\exp(-a_{-}\xi^{2})\exp\!\left(\frac{a_{-}^{3}}{3}-ia_{-}^{2}\xi \right)\right]. \tag{8}\]  
The point \(z\
eq\left(m+\frac{1}{2}\right)\frac{\pi}{L}\) is referred to as the general point, where \(m\) is an arbitrary integer, and the point \(z=\left(m+\frac{1}{2}\right)\frac{\pi}{L}\) is called the phase transition point [26]. When \(z=\left(m+\frac{1}{2}\right)\frac{\pi}{L}\), \(\cos(Lz)=0\) and \(b=0\). When \(z\
eq\left(m+\frac{1}{2}\right)\frac{\pi}{L}\), the general solution of the one-dimensional cosh-Airy beam in the parabolic potential is expressed as:  
\[ \begin{array}{rl}\psi(x,z)=&-\frac{1}{2}\sqrt{\frac{K}{2 Nz}}\exp(ibx^{2}) \left\{\exp\!\left(\frac{a_{+}^{3}}{3}\right)Ai\!\left(\frac{K}{2D}-\frac{1}{16 b^{2}}-x_{0}+\frac{ia_{+}}{2D}\right)\right.\\ &\times\exp\!\left[\left(a_{+}+\frac{i}{2b}\right)\!\left(\frac{K}{2b}-\frac{1 }{16b^{2}}-x_{0}+\frac{ia_{+}}{2b}\right)\right]\exp\!\left[-\frac{iK^{2}}{4b} -\frac{1}{3}\!\left(a_{+}+\frac{i}{4b}\right)^{3}\right]\\ &+\exp\!\left(\frac{a_{-}^{3}}{3}\right)Ai\!\left(\frac{K}{2D}-\frac{1}{16b^{2 }}-x_{0}+\frac{ia_{-}}{2b}\right)\exp\!\left[\left(a_{-}+\frac{i}{4b}\right) \!\left(\frac{K}{2D}-\frac{1}{16b^{2}}-x_{0}+\frac{ia_{-}}{2b}\right)\right]\\ &\times\exp\!\left[-\frac{iK^{2}}{4b}-\frac{1}{3}\!\left(a_{-}+\frac{i}{4b} \right)^{3}\right]\!.\end{array}