# Guided Self-Accelerating Airy Beams--A Mini-Review
## 1 Introduction  
Diffraction is a fundamental phenomenon in physical optics, due to which beams bend and spread, and the peak intensity of the beam decreases upon propagation. Sometimes it is beneficial, as in the diffraction gratings, sometimes a nuisance, as in the diffraction limit. In situations where diffraction is not desired and needs to be overcome, the nondiffracting beams come to the fore. These beams are a class of nondispersive solutions of the Helmholtz wave equation that display exotic characteristics: they are nondiffracting, self-accelerating, and self-healing, among other properties. The representative nondiffracting beams include the radially symmetric Bessel beams and the asymmetric Airy beams. In comparison with the Bessel beam, the most remarkable feature of an Airy beam is the self-acceleration in free space [1; 2; 3]. The Airy beam concept originated from quantum mechanics. In 1979, [PERSON] and [PERSON] demonstrated that the Airy function is an eigenmode of the linear Schrodinger equation [1] and that it is the only nontrivial solution that does not expand with time and it accelerates in space. The paraxial wave equation--the wave equation in the paraxial approximation--has the same form as the linear Schrodinger equation. Based on this mathematical similarity between optics and quantum mechanics, one can get not only Airy beams and Airy pulses, but also other types of waves which satisfy the paraxial wave equation, such as surface plasmon polaritons [4; 5], acoustic waves [6]and water waves [7; 8], among others. Notably, the nondiffracting feature of an Airy beam comes from its infinite transverse extension and power, since the ideal Airy function is not square integrable. This feature is similar to the simple plane wave. Hence, to possess finite energy and become a physical quantity, the Airy beam must be truncated. In optics, this is simply achieved by an exponential aperture, as first put forward in 2007, by [PERSON] et al. [2; 9]. The truncated Airy beam can still propagate for a long distance, preserving major characteristics of an ideal Airy beam, but eventually it will diffract and lose its unique structure and properties. Nonetheless, this method of generation makes the Airy beam experimentally available and of wide interest for applications in optical beam manipulation. However, this method also limits the light energy utilization and the stability of beams in the nonlinear domain. Still, by exploiting these unique characteristics of Airy beams, various application possibilities have been explored or implemented, for example, for Airy plasma guiding [10], routing surface plasmon polaritons [11], image signal transmission [12; 13], laser filamentation [14], optical micromanipulation [15; 16], optical trapping [17; 18; 19; 20], light bullet generation [21; 22; 23; 24], electron acceleration [25], and other applications [25; 26; 27; 28; 29]. In addition to research on Airy beams in free space and linear media, work has also been extended to nonlinear (NL) media and regimes. In the most common optical NL media, i.e., the Kerr, saturable and quadratic media, the nonlinearity is spatially modulated, so that the beam diffraction can be effectively balanced by the nonlinearity through a soliton-like beam generation process. In 2009, [PERSON] et al. produced an Airy beam in an asymmetrically modulated quadratic optical NL medium by the three-wave mixing process [30]. This novel nonlinear generation method not only produced an Airy beam at a new wavelength and a higher energy, but also provided new possibilities for manipulating the dynamics of Airy beams in NL media [31; 32; 33; 34; 35]. According to the nonlinear Schrodinger equation, the nonlinearity plays a nontrivial role in controlling the persistence as well as the breakdown of Airy beams [36; 37; 38]. It has been demonstrated that the main lobe of an Airy beam experiences self-phase modulation in NL media that results in the self-focusing or trapping of the beam [33].