Vector wave analysis of Airy beams upon reflection and refraction.

We report on the propagation dynamics of Airy light beams under nonparaxial conditions. The partial waves forming the Airy beam can be divided into two parts, the first of which contains only propagating waves, while the second part consists of evanescent waves. In this Letter we propose the concept of the evanescent Airy beam. We analyze the structure of the ideal evanescent Airy beam, the initial profile of which has the Airy form, while its spectral decomposition consists of only evanescent partial waves. Also, we discuss the refraction of the Airy beam through an interface and investigate the field of the transmitted Airy beam.

Airy beams in microwave and optical region have gained wide attention of researchers due to their remarkable non-diffractive, self-bending and self-healing properties. However, the analysis of Airy beams in terahertz wave band is less. In this letter, we proposed an Airy beam multifunctional terahertz metasurface which can achieve multiple functions under different polarized waves. Under the linearly polarized waves incidence, the designed metasurface generate Airy beams and autofocusing Airy beam. Meanwhile, the same metasurface can also realize diversified functions including focusing, multi-vortex beam, and focusing vortex beam under circularly polarized wave incidence. Our presented structure has potential applications in important scenarios such as biomedicine, imaging, holography, and terahertz multifunctional devices.

Self-accelerating beam is a kind of light beam capable of self-bending in free space without any external potential, of which a typical one is the well-known Airy beam. Such a beam has gained great attention for its extraordinary properties, including nondiffracting, self-accelerating and self-healing, which may have versatile applications in the delivery and guiding of energy, information and objects using light, such as particle manipulation, micro-machining, optical routing, super-resolution imaging, etc. However, since Airy beam can only propagate along parabolic trajectory, which reduces the flexibility in practical applications, thus how to design accelerating beams propagating along arbitrary trajectory is still a crucial problem in this area. One scheme is to keep on finding other analytical solutions of the wave equation besides Airy beam, such as semi-Bessel accelerating beams, Mathius beams, and Weber beams, moving along circular, elliptical, or parabolic trajectories, but it becomes increasingly difficult to find out any more solutions. A more effective solution to this problem is based on the caustic method, which associates the predesigned trajectory with an optical caustics and then obtains the necessary initial field distribution by performing a light-ray analysis of the caustics. This method has been implemented in real space and Fourier space based on Fresnel diffraction integral and angular-spectrum integral, respectively. It has been found recently that they can be unified by constructing Wigner distribution function in phase space. Based on the caustic method, accelerating beams were constructed to propagate along arbitrary convex trajectories in two-dimensional space at first. With continuous development of this method, the types of accelerating beams available have been extending from convex trajectories to nonconvex trajectories, from two-dimensional trajectories to three-dimensional trajectories, and from one main lobe to multiple main lobes, which opens up more possibilities for emerging applications based on accelerating beams. In future, previous researches and applications based on Airy beams will certainly be generalized to all these new types of accelerating beams, and owing to the great flexibility in designing accelerating beams, more application scenarios may emerge in this process with huge development potential. Thus in this paper, we review the principle and progress of the caustic method in designing accelerating beams.

Airy beam holds great promise for wide applications because of its unique features of being diffraction‐free, self‐accelerating, and self‐healing. Much efforts are devoted to develop concise and efficient methods to generate and manipulate accelerating Airy beams. In this paper, a novel method is proposed to generate single or paired Airy beams by encoding the geometric phase of dielectric metasurface. Cubic phase, together with a linearly gradient phase, is imposed on the metasurface to control the acceleration and initial launching velocity of accelerating Airy beams. Spin‐dependent geometric phase facilitates the generation of spin‐dependent Airy beams by simply controlling the polarization of incident light. High quality of the generated Airy beam is testified by characterization of its diffraction‐free, self‐accelerating, and self‐healing propagation. Despite the loss for silicon in visible region, generation efficiency as high as 63% is achieved for the metasurface Airy beam generator. The results demonstrate the possibility to manipulate complex structured optical field with all‐dielectric metasurface and broaden the potential application of Airy beams into compact and integrated systems.

Finite Element Methods for the Solution of 3D Eddy Current Problems

Explicit analytical expressions for the electromagnetic field components of typical structured light beams

The electromagnetic impedance boundary value problem is very important in the electromagnetic literature [1, 2]. We adapt the decoupled potential integral equation DPIE [3] for the problem of electromagnetic scattering from impenetrable objects with impedance boundary condition. Our representation of the electric and magnetic fields is based on the standard vector and scalar potentials A, φ in the Lorenz gauge, yet we establish new boundary conditions on the potentials themselves rather than on the fields. By doing this we develop well conditioned second-kind Fredholm integral equations that have no low frequency breakdown, no high density mesh breakdown and no spurious resonances. Like in the original DPIE for perfect electric conductors, the vector and scalar equations are decoupled. The scattered scalar potential φ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">scat</sup> depends on the incoming scalar potential φ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">in</sup> , and the scattered vector potential A <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">scat</sup> depends on the incoming vector potential A <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">in</sup> . Like in the original DPIE for perfect electric conductors, this formulation is valid for multiply connected geometries and also valid not only in the static limit but also for arbitrary ω ≥ 0.

The concepts of scalar and vector potentials are introduced and the electric and magnetic fields are shown to be derived from specific forms of these potentials. The choice of these forms is restricted by gauge considerations, and the Lorenz gauge is introduced as the one most applicable for radiation. Using this, the wave equations prescribing the potentials in terms of the source conditions are presented. The modifications of vector and scalar potentials to account for speed of light and causality lead to the concept of “retarded time.” The potentials can be expressed in terms of moments of the source along with concepts of “near,” “intermediate,” and “far” zones to facilitate derivation of approximate expressions for the potentials evaluated at appropriate distances from the source. Finally, expressions for the vector potential in terms of the electric and magnetic dipole, and electric quadrupole moments of the source in the approximation zones are presented.

Elliptical airy vortex beams (EAVBs) can spontaneously form easily identifiable topological charge focal spots. They are used for topological charge detection of vortex beams because they have the abruptly autofocusing properties of circular airy vortex beams and exhibit unique propagation characteristics. We study the use of the dynamic phase and Pancharatnam–Berry phase principles for generation and modulation of EAVBs by designing complex-amplitude metasurface and phase-only metasurface, at an operating wavelength of 1500 nm. It is found that the focusing pattern of EAVBs in the autofocusing plane splits into |m|+1 tilted bright spots from the original ring, and the tilted direction is related to the sign of the topological charge number m. Due to the advantages of ultra-thin, ultra-light, and small size of the metasurface, our designed metasurface device has potential applications in improving the channel capacity based on orbital angular momentum communication, information coding, and particle capture compared to spatial light modulation systems that generate EAVBs.

Based on the vector angular spectrum representation and stationary phase method, analytical far-field vectorial expressions of a noncanonical vortex dipole Airy beam, namely, a pair of noncanonical vortices with opposite charges embedded in an Airy beam are derived and used to investigate the phase singularities and energy flux distributions of the corresponding beam in the far-field regime, where the noncanonical characteristic of vortex is stressed. It is shown that the noncanonical strength, off-axis distance, and aperture coefficient affect the position and number of phase singularities, and the motion, creation, and annihilation of phase singularities are found by adjusting these parameters. For a low aperture coefficient, the energy flux distributions exhibit different numbers and orientations of lobes by varying the noncanonical strength. With increasing aperture coefficient, the symmetries of lobes are broken, and the energy flux distributions gradually become ellipses and the directions of their major axes vary with different noncanonical strengths. Finally, the energy flux distributions of an Airy beam carrying a single noncanonical vortex are discussed and compared.

Structured light beams with distinct spatial inhomogeneity of amplitude, phase, and polarization have garnered tremendous attention in recent years. A better understanding of the vectorial structure of such beams is helpful to reveal their important and interesting features for further applications. In this paper, explicit analytical expressions for the electric field components of typical spatial-structured light beams, including fundamental Gaussian beams, Hermite-Gaussian beams, Laguerre-Gaussian beams, Bessel/Bessel-Gaussian beams, and Airy beams, beyond the paraxial approximation are derived on the basis of the vectorial Rayleigh-Sommerfeld diffraction integrals. Compared with the existing expressions in the literature, the expressions given in this paper are very concise. Using the derived analytical expressions, distributions of the electric field components of these typical structured light beams are displayed and analyzed.

We demonstrate the nonlinear control of self-accelerating Airy beams. We show that under both self-focusing/defocusing nonlinearities, the ballistic trajectory of an Airy beam persists while its spatial spectrum reshapes, leading to distance-dependent negative/positive spectral defects.

We demonstrated a self-bending point spread function (SB-PSF) based on self-accelerating Airy beams for three-dimensional (3D) super-resolution fluorescence microscopy. The approach offers several-fold improvement over previously reported 3D imaging methods.

Spin-splitting and spin Hall effect of reflected Airy beams on the surface of topological insulator

We numerically and experimentally demonstrate that a three-Airy autofocusing beam can be generated by superposing three deformed two-dimensional (2D) Airy beams with a triangle symmetry. When the initial angle between two wings of the deformed 2D Airy beams increases, such a three-Airy autofocusing beam exhibits that the focusing length decreases and the intensity contrast at the focal point changes. Moreover, after introducing an optical vortex phase, this three-Airy autofocusing beam displays a transverse rotation in propagation. The rotation angle is determined by the topological charge of the vortex and the initial wing angle. Our results may have some potential applications in optical manipulation.