# Guided Self-Accelerating Airy Beams--A Mini-Review
## 2 Results and Discussion
### Dynamic Linear Potential Guidance  
As mentioned in the introduction, the AAF beams are radially symmetric beams that possess autofocusing property. It has been shown that the propagation trajectory as well as the positions of autofocusing points of the AAF beams can be controlled by potentials [41]. In our investigation [80], we theoretically analyzed the propagation and autofocusing effect of the AAF beams in a dynamic  
Figure 7: Propagation of circular Airy beams. From left to right: Intensity distributions at \(z=0\), \(z=\mathcal{D}/4\), \(z=\mathcal{D}/2\), and \(z=3\mathcal{D}/4\). Intervals inside give the relative measure of the beam size. (**a–d**) Outward abruptly autofocusing (AAF) beam; (**e–h**) Inward AAF beam; (**i**) Iso-surface plot of the propagation of the inward AAF beam. Parameters: \(\alpha=0.5\), \(a=0.1\) and \(r_{0}=10\). Reproduced with permission from [79], Copyright the Optical Society of America, 2015.  
linear potential. We found that the linear potential may weaken (even eliminate) or strengthen the autofocusing effect of the AAF beams, depending on the form of the linear potential. In this case, the governing equation is written as [41; 81; 82]:  
\[i\frac{d\psi}{dz}+\frac{1}{2}\left(\frac{\partial^{2}\psi}{\partial r^{2}}+ \frac{1}{r}\frac{\partial\psi}{\partial r}\right)-\frac{d(z)}{2}r\psi=0. \tag{40}\]  
Here, the external potential is linear in \(r\), with the scaling factor \(d\) that depends on the longitudinal coordinate; this is the so-called dynamical linear potential [81; 82]. Again, it is hard to find an analytical solution of Equation (40), so we resort to an approximate analysis, by introducing an azimuthal modulation of the inward AAF beam:  
\[\psi_{\mathrm{az}}(x,y)=\mathrm{Ai}(r_{0}-r)\exp[a(r_{0}-r)]\exp\left(-\frac{ \left(\theta-\theta_{0}\right)^{2}}{w_{0}^{2}}\right), \tag{41}\]  
with \(w_{0}\) being the width of the modulation, \(\theta=\arctan(y/x)\) being the azimuthal angle, \(\theta_{0}\) representing the modulation direction. If the value of \(w_{0}\) is small enough, the azimuthal modulation will result in a very narrow structure, so with \(w_{0}\to 0\) the result will be quite similar to the one-dimensional finite-energy Airy beam. In this way, the AAF beam can be transferred into the 1D finite-energy Airy beam [81; 83], as:  
\[\psi(x,y)=\sum_{\theta_{0}=-\pi}^{+\pi}\mathrm{Ai}(r_{0}-x_{p})\exp[a(r_{0}-x_ {p})]. \tag{42}\]  
Thus, by an analogy with the analytical result in [81], we describe the autofocusing effect and the propagation of the AAF beam as follows:  
\[ \begin{split}\psi(x,y,z)=C&\sum_{\theta_{0}=-\pi}^{ +\pi}\mathrm{Ai}\left[iaz+\left(\frac{1}{2}f_{1}-\frac{1}{4}z^{2}+(r_{0}-x_{p} )\right)\right]\exp\left[a\left(\frac{1}{2}f_{1}-\frac{1}{2}z^{2}+(r_{0}-x_{p} )\right)\right]\times\\ &\exp\left[i\left(\frac{1}{2}a^{2}z+\frac{1}{4}zf_{1}-\frac{1}{8 }f_{2}-\frac{1}{12}z^{3}-\frac{1}{2}(r_{0}-x_{p})g+\frac{1}{2}(r_{0}-x_{p})z \right)\right],\end{split}  \tag{43}\]  
where, \(g(z)=\int_{0}^{z}d(t)dt\), \(f_{1}(z)=f_{0}+\int_{0}^{z}g(t)dt\) and \(f_{2}(z)=\int_{0}^{z}g^{2}(t)dt\). The trajectory of each component in a linear dynamic potential is  
\[x_{p}=r_{0}+\frac{1}{2}f_{1}-\frac{1}{4}z^{2}. \tag{44}\]  
Clearly, the AAF beam can be effectively manipulated by the linear potential. If there is no autofocusing during propagation, Equation (43) can be reduced to the following form:  
\[ \begin{split}\psi(x,y,z)=&\mathrm{Ai}\left[iaz+ \left(\frac{1}{2}f_{1}-\frac{1}{4}z^{2}+(r_{0}-r)\right)\right]\exp\left[a \left(\frac{1}{2}f_{1}-\frac{1}{2}z^{2}+(r_{0}-r)\right)\right]\times\\ &\exp\left[i\left(\frac{1}{2}a^{2}z+\frac{1}{4}zf_{1}-\frac{1}{8} f_{2}-\frac{1}{12}z^{3}-\frac{1}{2}(r_{0}-r)g+\frac{1}{2}(r_{0}-r)z\right) \right],\end{split}  \tag{45}\]  
which becomes invalid when autofocusing happens. In this way, using Equations (43) and (45), the propagation of an AAF beam manipulated by a dynamic linear potential can now be reduced to a simple 1D case.