Propagation dynamics of finite-energy Airy beams in nonlocal nonlinear media

We investigate periodic inversion and phase transition of normal and displaced finite-energy Airy beams propagating in nonlocal nonlinear media with the split-step Fourier method. Numerical simulation results show that parameters such as the degree of nonlocality and amplitude have profound effects on the intensity distribution of the period of an Airy beam. Nonlocal nonlinear media will reduce into a harmonic potential if the nonlocality is strong enough, which results in the beam fluctuating in an approximately cosine mode. The beam profile changes from an Airy profile to a Gaussian one at a critical point, and during propagation the process repeats to form an unusual oscillation. We also briefly discus the two-dimensional case, being equivalent to a product of two one-dimensional cases.     

Relation between Airy beams and triple-cusp beams

Airy beams and triple-cusp beams are two kinds of accelerating beams. The propagation characteristics and internal topological structures of accelerating Airy beams are well understood because of the developed mathematical theory about Airy function. However, limited information is available about the optical characteristics of accelerating triple-cusp beams. In this work, the relationship between Airy beams and triple-cusp beams is examined theoretically and experimentally. Results reveal some important optical characteristics of triple-cusp beams based on the optical characteristics of Airy beams. These findings are expected to provide a foundation for future applications of triple-cusp beams.     

Specular and vortical Airy beams

A new kind of truncated Airy beams is investigated and discussed. These beams are a superposition of shifted and truncated Airy functions and its specular counterparts, where zeroes or extremal points of the Airy function are chosen as a truncation point. The specular Airy beams are smooth at the truncation point and produce a diffraction pattern similar to Hermite-Gaussian modes. Under propagation in Fresnel zone, specular Airy beams demonstrate a symmetrical acceleration in opposite sides and the beam divergence is proportional to the traveled distance squared. The astigmatic mode converter transforms a two-dimensional specular Airy beam into a quasi-annular field with a nonzero orbital angular momentum. Vortical Airy beams based on truncated Airy functions are also discussed. These beams are similar to Laguerre-Gaussian modes, while their annular structure is changed during propagation.     

Reduced side-lobe Airy beams

The primary interest in the finite-energy Airy beam derives from the special properties possessed by the maximum intensity at its central lobe. However, the defining spatial dependence, the Airy function, is an oscillatory function that consists of decaying side lobes. For some applications these side lobes may create deleterious effects. Fortunately, a nonsymmetric apodization of the beam in Fourier space is shown to enhance the central lobe as the side lobes are reduced and clipped. The properties of the central lobe are unaffected over a large proportion of the propagation path.     

Ballistic dynamics of Airy beams

We demonstrate both theoretically and experimentally that optical Airy beams propagating in free space can perform ballistic dynamics akin to those of projectiles moving under the action of gravity. The parabolic trajectories of these beams as well as the motion of their center of gravity were observed in good agreement with theory. The possibility of circumventing an obstacle placed in the path of the Airy beam is discussed.     

Product of three Airy beams

A two-dimensional field that is a product of three Airy beams is proposed and investigated. It is shown that the Fourier image of this field has a cubic phase and a radially symmetric intensity with a super-Gaussian decrease. Propagation of the product of three Airy beams in a Fresnel zone is investigated numerically.     

Airy beams beyond the paraxial approximation

The behavior of the vectorial Airy beams beyond the paraxial approximation is investigated. Indeed, closed-form (even though non exact) expressions for the electric components of the fields generated by the same boundary conditions, which should pertain to the scalar Airy beams, are obtained on the basis of the vectorial Rayleigh-Sommerfeld diffraction integrals under suitable approximations. Such expressions may accompany more complete approaches, like that in Opt. Expr. 17, 22432 (2009), where a fully numerical analysis of the propagation of exponentially smoothed Airy beams has been presented, faithfully reproducing the conditions of their experimental demonstration as reported in Phys. Rev. Lett. 99, 213907 (2007). Comments on other well known approaches to the investigation of the nonparaxial propagation of definite paraxial beams are also given.     

Observation of accelerating Airy beams

We report the first observation of Airy optical beams. This intriguing class of wave packets, initially predicted by Berry and Balazs in 1979, has been realized in both one- and two-dimensional configurations. As demonstrated in our experiments, these Airy beams can exhibit unusual features such as the ability to remain diffraction-free over long distances while they tend to freely accelerate during propagation.     

Accelerating finite energy Airy beams

We investigate the acceleration dynamics of quasi-diffraction-free Airy beams in both one- and two-dimensional configurations. We show that this class of finite energy waves can retain their intensity features over several diffraction lengths. The possibility of other physical realizations involving spatiotemporal Airy wave packets is also considered.     

Fractional Fourier transform of Airy beams

An analytical expression of an Airy beam passing through a fractional Fourier transform (FRFT) system is presented. The effective beam size of the Airy beam in the FRFT plane is also derived. The influences of the order of FRFT, the modulation parameter, and the transverse scale on the normalized intensity distribution and the effective beam size of an Airy beam in the FRFT plane are examined. The order of FRFT controls the effective beam size and the orientation of the beam spot. The modulation parameter affects the effective beam size and the number of lateral side lobes. The normalized intensity distribution and the effective beam size are both proportional to the transverse scale.     

The Gouy phase of Airy beams

The phase behavior of Airy beams is studied, and their Gouy phase is defined. Analytic expressions for the idealized, infinite-energy type beam are derived. They are shown to be excellent approximations for finite-energy beams generated under typical experimental conditions.     

Airy beams from a microchip laser

It is theoretically shown that an end-pumped microchip laser formed by a thin laser crystal with plane-plane but slightly tilted facets can emit, under appropriate pumping conditions and near a crystal edge, a truncated self-accelerating Airy output beam.     

Propagation properties and radiation force of circular Airy Gaussian vortex beams in strongly nonlocal nonlinear medium

We study the abruptly autofocusing and autodefocusing properties of the circular Airy Gaussian vortex(CAi GV)beams in strongly nonlocal nonlinear medium for the first time through numerical simulations.The magnitude of topological charges and the position of the vortex could change not only the light spot pattern but also the intensity contrast.Meanwhile,we can change the position of the autofocusing and autodefocusing planes by changing the parameter of the incident beam.Furthermore,we can control the peak intensity contrast through choosing properly the truncation factor.As for the radiation force,we study the gradient and the scattering forces of CAi GV beams on Rayleigh dielectric sphere.Our analyses demonstrate that the radiation force can be enhanced by choosing proper parameters of CAi GV beams.     

Airy trajectory engineering in dynamic linear index potentials

We study the propagation of Airy beams in transversely linear index potentials with a gradient that is dynamically changing along the propagation direction. We find exact solutions in the case of Airy and apodized (Gaussian and exponentially) Airy beams in 1+1 and 2+1 dimensions. More important, we find that the Airy beam can follow any predefined path, in which case the potential gradient is determined as a function of this path.     

Plasmonic Airy beams with dynamically controlled trajectories

We report the experimental generation and dynamic trajectory control of plasmonic Airy beams (PABs). The PABs are created by directly coupling free-space Airy beams to surface plasmon polaritons through a grating coupler on a metal surface. We show that the ballistic motion of the PABs can be reconfigured in real time by either a computer addressed spatial light modulator or mechanical means.     

The deflected angle and reflected displacement of Airy beams

The deflected angle of airy beams is defined to describe their bending degree. Based on this parameter, we analyze the lateral displacement of airy beams and the changes of deflected angles during reflection process. In multi-reflections of airy beams, we find that deflected angles increase linearly with the times of reflection, which is useful to study the propagation properties of airy beam in parallel plate waveguides and other waveguides. Also we can use this parameter to control the incident and reflected routes to realize airy beam bypassing obstacles and reflecting back at predictable location.     

Airy光束在Kerr介质中的传输特性

通过对非线性薛定谔方程的研究,得出Airy光束在Kerr介质中的崩塌功率及有效束宽演化的解析表达式。经过数值计算发现,Airy光束在聚焦的Kerr介质中,其主瓣在开始传播时始终是会聚的;当输入功率小于临界崩塌功率时,Airy光束主瓣的中心部分出现局部崩塌。在不同的Kerr介质中, Airy光束的形状和传输轨道均能保持不变,如同在自由空间中传播,但光场大小的分布,随着不同的Kerr介质会发生改变:在Kerr的聚焦介质中,光场向中心聚焦;而在散焦的Kerr介质中,光场会发散。     

Evolution of Airy beams in a chiral medium

We study the evolution of Airy beams in chiral media. We numerically demonstrate that the circularly polarized Airy beams of opposite handedness can be separated when traveling through the chiral media. The constructive interference of the Airy beams in the near- and far-zone is explicitly analyzed by varying the chirality parameter of the medium. It is interesting to find that the self-acceleration of the beam is sensitive to the interference term.     

Airy光束在Kerr介质中的传输特性

通过对非线性薛定谔方程的研究,得出Airy光束在Kerr介质中的崩塌功率及有效束宽演化的解析表达式。经过数值计算发现,Airy光束在聚焦的Kerr介质中,其主瓣在开始传播时始终是会聚的;当输入功率小于临界崩塌功率时,Airy光束主瓣的中心部分出现局部崩塌。在不同的Kerr介质中, Airy光束的形状和传输轨道均能保持不变,如同在自由空间中传播,但光场大小的分布,随着不同的Kerr介质会发生改变:在Kerr的聚焦介质中,光场向中心聚焦;而在散焦的Kerr介质中,光场会发散。     

Propagation of Airy beams through a hard-edged aperture

Based on the fact that a hard-edged aperture function can be expanded into a finite sum of complex Gaussian functions, we obtain the analytical formula for the propagation of the Airy beams in free space and through the fractional Fourier transformation (FrFT) system. According to the derived formulas, the propagation properties of the Airy beams in free space and through the FrFT plane with a hard-edged aperture are illustrated numerically.