# Guided Self-Accelerating Airy Beams--A Mini-Review
## 2 Results and Discussion
### Harmonic Potential Guidance
#### 2.2.1 One-Dimensional Airy Beams
In addition, the beam inversion produces a discontinuity in the velocity and a singularity in the acceleration, which demonstrates nicely that the motion is not harmonic and that there exist two phase regions: the Airy phase and the single-peak phase. According to Equations (24) and (26), the single-peak structure only occurs at the phase transition points; before and after these points, the beam still exhibits multi-peak structure, having to reconnect the accelerating motion before the point with the decelerating motion after the point. The length of the single-peak phase is determined by the size of the decay parameter, as displayed in Figure 4b; the smaller \(a\), the smaller the length, and the harder it is for the beam to make a sudden inversion, so the region in Figure 4c is narrower than that in Figure 4d. However, \(x_{0}\) has no effect on the width of the single-peak phase region when \(a\) is fixed. We next consider the initial beam with a linear chirp:  
\[i\frac{\partial\hat{\psi}}{\partial z}+\frac{1}{2}\alpha^{2}\frac{\partial^{2 }\hat{\psi}}{\partial k^{2}}-\frac{1}{2}k^{2}\hat{\psi}=0. \tag{15}\]9  
with \(\beta\) being the constant wavenumber. Thus, the solution is written as:  
\[ \begin{split}\psi(x,\,z)=&-f(x,\,z)\sqrt{i\frac{ \pi}{b}}\exp\left(\frac{a^{3}}{3}\right)\text{Ai}\left(\frac{K^{\prime}}{2b}- \frac{1}{16b^{2}}+i\frac{a}{2b}-x_{0}\right)\\ &\times\exp\left[\left(a+\frac{i}{4b}\right)\left(\frac{K^{\prime }}{2b}-\frac{1}{16b^{2}}+i\frac{a}{2b}-x_{0}\right)\right]\exp\left[-i\frac{K^ {\prime 2}}{4b}-\frac{1}{3}\left(a+\frac{i}{4b}\right)^{3}\right],\end{split}  \tag{28}\]  
with \(K^{\prime}=K-\beta\). Clearly, the period \(\mathcal{D}\) does not change and the phase transition point is still an odd integer multiple of the quarters of the period. Mathematically, the trajectory is given by  
\[x=\frac{1}{4\alpha^{2}}\frac{\sin^{2}(\alpha z)}{\cos(\alpha z)}+x_{0}\cos( \alpha z)+\frac{\beta}{\alpha}\sin(\alpha z),\]  
and is modulated greatly by the linear chirp. At the phase transition points, we have:  
\[ \begin{split}\psi\left(x,\,z=\frac{2m+1}{4}\mathcal{D}\right)=& \sqrt{-i\frac{s\alpha}{2\pi}}\exp[-ix_{0}(\alpha x-\beta)]\exp \left[-a(\alpha x-\beta)^{2}\right]\\ &\times\exp\left\{\frac{a^{3}}{3}+i\frac{s}{3}\left[(\alpha x- \beta)^{3}-3a^{2}(\alpha x-\beta)\right]\right\}.\end{split}  \tag{29}\]  
In this case, the beam is equivalent to an obliquely incident beam, but without the ballistic properties due to the harmonic potential [3, 73]. Figure 4: (**a**) Numerical trajectory (**a1**), velocity (**a2**) and acceleration (**a3**) of the Airy beam during propagation in a harmonic potential. Red, black, and blue curves correspond to the transversely displaced beams, with displacements \(x_{0}=-10\), \(0\), and \(10\), respectively, and with \(a=0.1\); (**b**) The width of the single-peak phase region versus the decay parameter \(a\). (**c**,**d**) Corresponding to the green dots in (**b**). In the left panel, \(a=0.01\); in the right, \(a=0.05\). Other parameters: \(\alpha=0.5\). Reproduced with permission from [64], Copyright the Optical Society of America, 2015. If the initial finite energy Airy beam carries a quadratic chirp,  
\[\psi(x,\,0)=\text{Ai}(x-x_{0})\exp\left[a(x-x_{0})\right]\exp\left(i\beta x^{2} \right), \tag{30}\]  
the analytical solution will be  
\[\psi(x,\,z)= -f(x,\,z)\sqrt{i\frac{\pi}{b^{\prime}}}\exp\left(\frac{a^{3}}{3} \right)\text{Ai}\left(\frac{K}{2b^{\prime}}-\frac{1}{16b^{\prime 2}}+i\frac{a}{2b^{ \prime}}-x_{0}\right) \tag{31}\] \[\times\exp\left[\left(a+\frac{i}{4b^{\prime}}\right)\left(\frac{K }{2b^{\prime}}-\frac{1}{16b^{\prime 2}}+i\frac{a}{2b^{\prime}}-x_{0}\right) \right]\exp\left[-i\frac{K^{2}}{4b^{\prime}}-\frac{1}{3}\left(a+\frac{i}{4b^{ \prime}}\right)^{3}\right].\]  
Here, \(b^{{}^{\prime}}=b+\beta\), but the period in this case is still \(\mathcal{D}\).