# Guided Self-Accelerating Airy Beams--A Mini-Review
## 2 Results and Discussion  
In quantum mechanics, the Schrodinger equation (SE) for a quantum particle moving in free space is written as:  
\[i\hbar\frac{\partial\psi}{\partial t}+\frac{\hbar^{2}}{2m}\frac{\partial^{2} \psi}{\partial x^{2}}=0, \tag{1}\]where \(\hbar\) is the reduced Planck's constant and \(m\) is the particle mass. This equation describes the development of the wave function \(\psi\) of the particle, but in appropriate units it could also describe the development of an Airy wave packet.  
In optics, the propagation of a scalar wave packet obeys the Helmholtz equation:  
\[\left(\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial z^{2}} \right)\psi+k^{2}\psi=0, \tag{2}\]  
where \(x\) and \(z\) are the transverse and longitudinal coordinates, and \(k=2\pi n/\lambda_{0}\) is the wavenumber (\(n\) is the index of refraction and \(\lambda_{0}\) the wavelength in free space). Under the paraxial approximation \(|\partial_{z}^{2}\psi|\ll|2k\partial_{z}\psi|\), one obtains the paraxial wave equation,  
\[i\frac{\partial\psi}{\partial z}+\frac{1}{2}\frac{\partial^{2}\psi}{\partial x ^{2}}=0, \tag{3}\]  
where now the variables \(x\) and \(z\) are the normalized transverse coordinate and the propagation distance, scaled by some characteristic transverse width \(x_{0}\) and the corresponding Rayleigh range \(kx_{0}^{2}\). Obviously, Equation (3) has the same form as Equation (1) in scaled units--it is just the SE without potential--so one of the accelerating solutions of Equation (3) is the well-known Airy function with the characteristic infinite oscillatory tail,  
\[\psi(x,z)=\text{Ai}\left(x-\frac{z^{2}}{4}\right)\exp\left[\frac{i}{12}\left( 6 xz-z^{3}\right)\right]. \tag{4}\]  
From this solution, it is not hard to see that the trajectory is determined by the transverse accelerating term \(x-z^{2}/4\), so the beam propagates along a parabolic curve. Note that the intensity of an ideal Airy wave packet remains invariant during propagation, as displayed in Figure 1a. However, the ideal Airy beam does not exist in reality, due to its infinite energy. To make it realizable in an experiment, an exponentially tapered Airy beam is introduced [2; 9],  
\[\psi(x)=\text{Ai}(x)\exp(ax), \tag{5}\]  
where \(a\geq 0\) is an arbitrary real decay parameter. In the momentum space, the corresponding Fourier transform is  
\[\hat{\psi}(k)=\exp(-ak^{2})\exp\left[\frac{a^{3}}{3}+\frac{i}{3}\left(k^{3}-3a ^{2}k\right)\right], \tag{6}\]  
Figure 1: (**a**) Propagation of the ideal Airy beam according to Equation (3). The inset in the right corner represents the energy distribution of the beam at \(z=0\); (**b**) Same as (**a**), but for a truncated Airy beam, with \(a=0.1\); (**c**) Self-healing process of the truncated Airy beam. The white dashed line is the theoretical trajectory.  
which is of limited energy. So, according to the [PERSON]'s theorem, the energy of the attenuated Airy beam is also limited. The propagation of the attenuated Airy beam is depicted by the solution  
\[\psi(x,z)=\text{Ai}\left(x-\frac{z^{2}}{4}+iaz\right)\exp\left[\frac{i}{12}\left( 6a^{2}z-12 iax+6 iaz^{2}+6 xz-z^{3}\right)\right], \tag{7}\]  
as shown in Figure 1b. Compared with the ideal Airy beam, the tail of the truncated beam quickly decays during propagation, which makes the nondiffracting and self-accelerating properties preserved only over a finite distance. In Figure 1c, the healing property of the Airy beam is displayed. The main lobe of the Airy beam is screened out initially, but it recovers quickly during propagation, due to the transfer of energy from the tail to the head of the beam [19; 51; 52].  
Similarly, the AAF exponentially apodized radially symmetric Airy beam is written as:  
\[\varphi_{0}(r)=Ai[\pm(r_{0}-r)]\exp[\pm a(r_{0}-r)], \tag{8}\]  
where \(r_{0}\) is the initial radius of the main lobe, and \(\pm\) corresponds to the inward or outward going beams, respectively.