# Guided Self-Accelerating Airy Beams--A Mini-Review
## 2 Results and Discussion
### Harmonic Potential Guidance
#### 2.2.1 One-Dimensional Airy Beams
Here, \(z\
eq(2m+1)\mathcal{D}/4\), where \(m\) is a positive integer. This trajectory is ideal, because it indicates that the Airy beam can accelerate all the way to infinity \(x\rightarrow\pm\infty\), when \(z\rightarrow(2m+1)D/4\). However, upon exponential apodization of the Airy beam, such an acceleration will stop when \(z\) is close to the points mentioned. At these points, the beam will turn around, accelerate in the opposite direction, and change the shape. We call these points the phase transition points, for the reasons explained below. So, several issues concerning Equation (21) must be addressed:  
(i) When \(z=m\mathcal{D}\), we have \(\psi(x,\,z)=\psi(x,\,0)\)--an initial beam recurrence. (ii) When \(z=(2m+1)\mathcal{D}/2\), we have \(\psi(x,\,z)=\psi(-x,\,0)\)--an inversion of the initial beam. 3. When \(z=(2m+1)\mathcal{D}/4\), by directly solving Equation (18) for the FT of the initial beam, we have \[i\frac{\partial\hat{\psi}}{\partial z}+\frac{1}{2}\alpha^{2}\frac{\partial^{2 }\hat{\psi}}{\partial k^{2}}-\frac{1}{2}k^{2}\hat{\psi}=0. \tag{15}\]4 (23) where \(s=1\) if \(m\) is even and \(s=-1\) if \(m\) is odd. This field is unrelated to the initial Airy beam--that is, it represents a new \"phase\" of the propagating beam. Equation (23) displays a Gaussian intensity profile, which is completely different from the intensity profiles elsewhere during propagation. It is similar to the propagating Gaussian pulse as it bounces off the harmonic potential wall--but the Gaussian beam remains Gaussian in propagation, whereas this pulse inverts and becomes an inverse Airy beam. On the other hand, it is different from a free Gaussian wave packet hitting an infinite potential wall--there, during the bounce, the packet becomes a rapidly oscillating multi-peaked structure, owing to the interference between the incoming and the reflected beam. Since the inversion introduces a discontinuity in the velocity and a singularity in the acceleration, for lack of a better word, we refer to the phenomenon as the phase transition of the finite energy Airy beam, due to the harmonic potential. Correspondingly, \(z=(2m+1)\mathcal{D}/4\) are the phase transition points. When we introduce a transverse displacement \(x_{0}\) of the beam, the initial beam is \(\psi(x,0)=Ai(x-x_{0})\exp[a(x-x_{0})]\), and the solution can be written as:  
\[i\frac{\partial\hat{\psi}}{\partial z}+\frac{1}{2}\alpha^{2}\frac{\partial^{2 }\hat{\psi}}{\partial k^{2}}-\frac{1}{2}k^{2}\hat{\psi}=0. \tag{15}\]5 \[i\frac{\partial\hat{\psi}}{\partial z}+\frac{1}{2}\alpha^{2}\frac{\partial^{2 }\hat{\psi}}{\partial k^{2}}-\frac{1}{2}k^{2}\hat{\psi}=0. \tag{15}\]6  
The corresponding trajectory is:  
\[i\frac{\partial\hat{\psi}}{\partial z}+\frac{1}{2}\alpha^{2}\frac{\partial^{2 }\hat{\psi}}{\partial k^{2}}-\frac{1}{2}k^{2}\hat{\psi}=0. \tag{15}\]7  
At the transition points, we have:  
\[i\frac{\partial\hat{\psi}}{\partial z}+\frac{1}{2}\alpha^{2}\frac{\partial^{2 }\hat{\psi}}{\partial k^{2}}-\frac{1}{2}k^{2}\hat{\psi}=0. \tag{15}\]8  
Comparing with the former case, the transverse displacement introduces a linear chirp at the transition points. There is no change of the period and phase transition points, so one can predict that the beam executes the same motion as before, but it is transversely stretched. One can also predict that with an increasing transverse displacement \(|x_{0}|\), the beam will accelerate along an ever more elongated cosine curve. To explore the phase transition region more clearly, we show the numerical simulations of trajectories, velocities, and accelerations of the beam during propagation in Figure 4, for different cases. One can observe that the accelerating trajectories are modulated by the transverse displacement; the beam acceleration with \(x_{0}<0\) being opposite to the case with \(x_{0}>0\).