# Propagation of Cosh-Airy and Cos-Airy Beams in Parabolic Potential
## 4 Conclusions  
In summary, different analytical expressions of one-dimensional \(\cosh\)-Airy and \(\cos\)-Airy beams in the general and the phase transition points of the parabolic potential are derived. The optical fields of the \(\cosh\)-Airy and \(\cos\)-Airy beams in the phase transition point have a symmetric Gaussian intensity profile and are independent of the Airy function. The phase transition point \(z=(2m+1)T/4\) is also the inversion point of the intensity distribution. The intensity, the center of gravity, and the effective beam size of the \(\cosh\)-Airy and \(\cos\)-Airy beams in the parabolic potential are periodic and have the same period with \(T=2\pi/L\).  
The direction of self-acceleration of \(\cosh\)-Airy and \(\cos\)-Airy beams with the positive transverse displacement is opposite to those with the nonpositive transverse displacement. \(Z=(m+1)T/2\) is the inversion point of the direction of self-acceleration. Due to the positive transverse displacement, the \(\cosh\)-Airy and \(\cos\)-Airy beams in the first half-period accelerate towards the negative direction of the \(x\)-axis. In the latter half-period, however, they accelerate towards the positive direction of the \(x\)-axis.  
Figure 11: The effective beam size of \(\cos\)-Airy beams in the parabolic potential. (**a**) \(\beta=0.04\). (**b**) \(x_{0}=5\).  
Figure 10: The effective beam size of \(\cosh\)-Airy beams in the parabolic potential. (**a**) \(\gamma=0.04\). (**b**) \(x_{0}=5\).  
The evolution of the center of gravity in the case of the positive transverse displacements is opposite to that in the case of the nonpositive transverse displacement. When the transverse displacement is positive, the center of gravity in the first half-period decreases, while that in the latter half-period is opposite. The amplitude of the center of gravity in the case of the negative transverse displacement is much larger than that in the case of the contrary transverse displacement. As the cosh factor increases, the amplitude of the center of gravity of the cosh-Airy beam decreases. However, the amplitude of the center of gravity of the cos-Airy beam decreases slowly with the increase of cosine factor.  
The effective beam size in the first half-period decreases, while the situation in the latter half-period is just the opposite. The effective beam size of the cosh-Airy beam is sensitive to the transverse displacement. However, the effect of the transverse displacement on the effective beam size of the cos-Airy beam is too small to be detected. As the cosh factor increases, the number of the side lobes of the cosh-Airy beam increases, which results in the increase of the effective beam size. However, the number of the side lobes of the cos-Airy beam decreases with the increase of the cosine factor, whose outcome is the decrease of the effective beam size. Under the same conditions, the effective beam size of the cosh-Airy beam is much larger than that of the cos-Airy beam.  
The periodic behaviors of the normalized intensity, the center of gravity, and the effective beam size of the cosh-Airy and cos-Airy beams in the parabolic potential are abundant and well exhibited. The results obtained here may have potential applications in particle manipulation and signal processing, as well as in other fields.  
[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 the National Natural Science Foundation of China (Grant No. 11974313) and the Open Research Fund of Key Laboratory of Optical Engineering, Chinese Academy Sciences (Grant No. 2017 LBC). The authors declare no conflict of interest.