# Guided Self-Accelerating Airy Beams--A Mini-Review
## 2 Results and Discussion
### Harmonic Potential Guidance
#### 2.2.2 Two-Dimensional Case  
Naturally, the harmonic oscillator model in Equation (14) is easily extended to two, three or even four dimensions [64]. In 2D, it has the form:  
\[i\frac{\partial\psi}{\partial z}+\frac{1}{2}\left(\frac{\partial^{2}\psi}{ \partial x^{2}}+\frac{\partial^{2}\psi}{\partial y^{2}}\right)-\frac{1}{2} \alpha^{2}\left(x^{2}+y^{2}\right)\psi=0, \tag{36}\]  
with the initial beam being:  
\[\psi(x,y,z=0)=\mathrm{Ai}(x)\mathrm{Ai}(y)\exp[a(x+y)]. \tag{37}\]  
By the separation of variables, the 2D problem can be reduced to two 1D cases [64; 77]. The result is displayed in Figure 6. Similar to the 1D case, the 2D Airy beam displays inversion and phase transition (the gaps represent the single-peak regions) during propagation. From Figure 6c,d, one can clearly see that the beam at \(z=\pi/(4\alpha)\) is still an SF beam, just like in the 1D case. Thus, the wave function is a product of two finite-energy Airy beams: one along \(x\) and the other along the \(y\) direction. In a 2D parabolic potential, the wave exhibits all the properties of 1D Airy beams: periodic inversion, phase transition, and anharmonic oscillation.  
Figure 5: (Color online) Comparison of intensities of an Airy beam at \(z=\pi/4\) in real space and inverse space, corresponding to Figure 3a. Intensities in real and frequency spaces refer to the left and right \(y\) scales, respectively. Reproduced with permission from [63], Copyright Elsevier, 2015.  
Figure 6: Propagation of a 2D finite energy Airy beam \(\psi(x,y)=\exp(ax)\mathrm{Ai}(x)\exp(ay)\mathrm{Ai}(y)\) in a harmonic potential. (**a**) Iso-surface plot; (**b**) Intensity in the cross section \(x-y=0\); (**c**) Intensity of the beam at \(\pi/(4\alpha)\) in the real spaces; (**d**) The corresponding intensity at \(\pi/(4\alpha)\) in the inverse space. The parameters are \(a=0.1\) and \(\alpha=0.5\). Reproduced with permission from [64], Copyright the Optical Society of America, 2015.  
For the cases that cannot be treated with the variable separation method, e.g., when the initial beam is a superposition of AAF beams carrying orbital angular momentum [18; 41; 42; 45; 78], in the radially symmetric case one can switch to the polar coordinates. Then, the input can be written as:  
\[\psi(r,\theta)=\text{Ai}[\pm(r_{0}-r)]\exp[\pm a(r_{0}-r)]\sum_{n=1}^{4}\exp( in\theta), \tag{38}\]  
where \(\pm\) represents the inward and outward AAF beams, \(r_{0}\) determines the location of the main ring, and \(\theta\) represents the spatial frequency in polar coordinates. An analytical propagating solution of Equation (38) is hard to obtain. However, using a fairly accurate approximation method developed in [18; 79], the AAF beam propagation can be described as a superposition of Bessel beams:  
\[\psi(r,\theta,z)\approx-A_{0}f(r,\theta,z)\exp\left(ibr_{0}^{2}\right)\sum_{n =1}^{4}i^{1-n}\exp(in\theta)J_{n}(r_{0}r), \tag{39}\]  
where \(A_{0}\approx(1-a^{2}/r_{0})\exp(a^{3}/3)\). The results are depicted in Figure 7. The first row of panels presents the intensity of an outward AAF beam; the second row of panels presents the intensity of an inward AAF beam.  
In the 3D plot, one can see that the oscillation is more continuous, and there are no phase transition points.