Arbitrary accelerating micron-scale caustic beams in two and three dimensions

Sidelobe suppression for accelerating beams

Although the control of trajectory, amplitude and beam-width in accelerating beams have been extensively investigated, sidelobes manipulation of such beams, which is required in many applications, has been surprisingly under-researched. This paper presents an approach for the generating of accelerating beams with significantly reduced sidelobes. The proposed method encompasses a two-step angular spectrum design, including employing a general model to establish the phase distribution and applying a stochastic parallel gradient descent (SPGD) algorithm to optimize the binary amplitude modulation. Experimental results confirm that the sidelobe intensity of accelerating beams can be reduced by over 50% with our method, thereby enhancing their applicability in many fields, such as micro-machining, particle manipulation, and optical communication.

Extended-depth-of-focus wavefront design from pseudo-umbilical space curves

Designing extended-depth-of-focus wavefronts is required in multiple optical applications. Caustic location and structure analysis offer a powerful tool for designing such wavefronts. An intrinsic limitation of designing extended-depth-of-focus wavefronts is that any smooth surface, with a non-constant mean curvature, unavoidably introduces a separation between caustic sheets, which is proportional to the ratio of change of the mean curvature along a curve embedded in the wavefront. We present a method to obtain extended-depth-of-focus wavefronts where the mean curvature variation ratio is reduced thanks to using a long circle-involute space curve effectively filling the wavefront surface. Additionally, we present a variant of the method in which the wavefront is modified within a small tubular neighborhood of the circle involute in order to partially meet the umbilical condition along that tubular region. Finally, we provide some numerical results showing the potential of our method in an application example.

Propagating analysis of Airy beams via complex phase space and ray method

A complex phase space and ray method is proposed to present intuitive interpretation of unique intensity profiles, parabolic accelerating trajectories, and distinctive phase shifts of Airy beam and its exponentially decaying version (i.e., finite-energy Airy beam). In the complex phase space, finite-energy Airy beam manifests itself as a complex parabolic phase space curve (PSC) which represents a cluster of light rays with complex wave vectors. The complex ray cluster converges to a complex parabolic caustic curve in complex coordinate space. For infinite-energy Airy beam, phase space, PSC, light ray and caustic curve change to real values. In the paraxial condition, Airy beams can maintain parabolic form of PSC and keep constant ray density, which guarantees the non-diffraction property of Airy beam and approximate non-diffraction property of finite-energy Airy beam. From the evolution of vertexes of parabolic PSC along a parabolic trajectory in phase space, one can also give the parabolic caustic curve for Airy beam and the complex parabolic caustic curve for exponentially decaying Airy beam. Further, the special phase and decay factors in the propagating solution of complex amplitude of Airy beams can be directly derived from the phase shifts of light ray cluster along transverse and longitudinal directions. The proposed phase space and ray method can present intuitive comprehension to distinctive propagating characteristics of Airy beams including intensity and phase, without resort to solving wave equation or diffracting integral formula.

Airy-type X-ray states generated using 3/2 flat diffractive optics

X-rays have developed into an essential tool in variety of fields, such as biology, materials, chemistry, and physics etc. Numerous X-ray types, including the orbital angular momentum (OAM), the Laguerre-Gauss, and the Hermite-Gauss states, have been proposed. This greatly enhances the depth of application of X-ray. The X-ray states described above are mostly produced by binary amplitude diffraction elements. In light of this, this paper proposes a flat X-ray diffraction grating based on caustic theory to generate Airy-type X-ray. It is proved by the simulation of multislice method that the proposed grating can generate the Airy beam in the X-ray field. The results show that the generated beams have a secondary parabolic trajectory deflection with the propagation distance, which is consistent with the theory. Inspired by the success of Airy beam in light-sheet microscope, the Airy-type X-ray can be anticipated to enable novel image capability for bio or nanoscience.

Controllable self-rotating array beam with an arc-shaped accelerating trajectory

In this study, a modified interfering vortex phase mask (MIVPM) is proposed to generate a new type of self-rotating beam. The MIVPM is based on a conventional and stretched vortex phase for generating a self-rotating beam that rotates continuously with increasing propagation distances. A combined phase mask can produce multi-rotating array beams with controllable sub-region number. The combination method of this phase was analyzed in detail. This study proves that this self-rotating array beam has an effectively enhanced central lobe and reduced side lobe owing to adding a vortex phase mask compared with a conventional self-rotating beam. Furthermore, the propagation dynamics of this beam can be modulated by varying the topological charge and constant a. With an increase in the topological charge, the area crossed by the peak beam intensity along the propagation axis increases. Meanwhile, the novel self-rotating beam is used for optical manipulation under phase gradient force. The proposed self-rotating array beam has potential applications in optical manipulation and spatial localization.

Shaping the transmission trajectory of vortex beam by controlling its radial phase

Vortex beam carrying orbital angular momentum (OAM), which features a helical phase front, has shown its potential applications in diverse areas, especially in free-space optical (FSO) communications. However, when generating vortex beams, the radial phase distribution is usually disregarded in previous reports. In this paper, by controlling the radial phase distribution, we propose a method for the generation of vortex beams with arbitrary convex trajectories. By using this method, we successfully generate vortex beams with different predesigned trajectories with high accuracy. Moreover, we also demonstrate the transmission of the radial phase-controlled vortex beams in FSO links for different scenarios in simulation. Firstly, we generate vortex beams with different OAM states (l=+1, + 3, and +6), which have the same ring diameter at the receiver side. Secondly, we generate three vortex beams (l=+3) with the same ring diameter at different transmission distances (z = 100 m, 200 m, and 300 m). Finally, by carefully controlling the radial phase of the vortex beam, we generate vortex beams that can almost keep the same ring diameter for a long distance. The proposed method for shaping the transmission trajectory of vortex beams may pave the way for more applications in OAM-based FSO communications.

Caustics of the axially symmetric vortex beams: analysis and engineering

We demonstrate that our theoretical scheme developed in the previous study on the caustics of the abruptly autofocusing vortex beams [Xiao et al., Opt. Express29, 19975 (2021)10.1364/OE.430497] is universal for all the axially symmetric vortex beams. Further analyses based on this method show the complex compositions of the vortex caustics in real space. Fine features of the global caustics are well reproduced, including their deviations from the trajectories of the host beams. Besides, we also show the possibility of tailoring the vortex caustics in paraxial optics based on our theory. The excellent agreements of our theoretical results with both numerical and experimental results confirm the validity of this scheme.

Dual projectile beams

Accelerating beams, of which the Airy beam is an important representative, are characterized by intensity maxima that propagate along curved trajectories. In this work we present a simple approach to directly generate accelerating beams with controllable trajectories by means of binary phase structures that consist of only a π phase step modulation in comparison to previous studies where two-dimensional cubic phase modulations for example are required, and which have practical limitations due to their challenging fabrication with phase plates or diffractive optical elements (DOEs), or the spatially extended system needed for their generation at the Fourier plane. In our approach, two intensity maxima are formed that propagate along root parabolic trajectories in contrast to Airy and higher order caustic beams that propagate along a parabolic curve, hence we call these beams Dual Projectile Beams (DPBs). By tailoring a step or slit phase patterns with additional Fresnel lenses, we either generate hollow-core or abruptly focusing beams and control their curvatures. Moreover, using DPBs as a simpler complement to complex structured light fields, we demonstrate their versatility at the example of their interaction with nonlinear matter, namely the formation of a spatial soliton in a photorefractive material. We show that the formed solitary state propagates almost unchanged for a distance of several Rayleigh lengths. This light matter interaction can be regarded as a light beam deceleration. The simplicity of this approach makes these beams suitable for integrated optics and high-power laser applications using DOEs or meta-surfaces.

Terahertz Airy beam generated by Pancharatnam-Berry phases in guided wave-driven metasurfaces

Metasurface antennas scatter traveling guided waves into spatial waves, which act as extendable subsources to overcome the size limitation on emission sources. With the use of a Pancharatnam-Berry phase metasurface stimulated by a circularly polarized wave in a waveguide, the local phase distributions of scattered spatial waves can be made consistent with those of an Airy beam, thereby allowing the generation of high-quality Airy beams. In a slab waveguide, circularly polarized waves are synthesized through superposition of in-plane transverse electric modes. Simulations demonstrate that a 20 mm × 20 mm footprint all-dielectric guided wave-driven metasurface generates a 2D Airy beam at a frequency of 0.6 THz. Furthermore, we employ a metasurface deposited on a strip waveguide to generate a 1D Airy beam under direct stimulation by the fundamental transverse electric mode. Our work not only provides a large-scale emitter, but it also suggests promising potential applications in on-chip imaging and holography.

Generation and control of tornado waves by means of ring swallowtail vortex beams

Tornado waves (ToWs), which refer to a light that accelerates and twists over both the radial and the angular directions, have gained a great deal of interest since the concept was introduced by Brimis et al [Opt. Lett.45, 280 (2020)10.1364/OL.45.000280]. In this paper, we superimpose two pairs of ring swallowtail vortex beams (RSVBs) to generate ToWs and we call them tornado swallowtail waves (ToSWs). Each pair consists of RSVBs while carrying orbital angular momentum of opposite helicity and slightly different with the radius of the main ring of RSVBs. The waves spiral forward and reveal intensity maxima, exhibiting a tornado-like intensity profile during propagation. Meanwhile, the angular acceleration of the ToSWs is illustrated via tracing the angular position of the high-intensity main lobes. It is found that ToSWs present very high values of angular acceleration. Compared with typical tornado waves, ToSWs are more diverse and tunable, giving a new degree of freedom to tailor the propagation dynamics due to the flexibility of the swallowtail diffraction catastrophe. In addition, we confirm such waves experimentally and the results match well with the numerical ones. Also, we demonstrate the ability of optical manipulation of ToSWs for the first time in that they allow for particles not only to be trapped but also to be rotated. Finally, we analyze the poynting vectors and power exchange of ToSWs to demonstrate convincingly the physical mechanism.

Manipulation of curved beams using beam-domain optimization

An efficient scheme for the design of aperture fields (distributed sources) that radiate arbitrary trajectory curved (accelerating) beams, with enhanced controllability of various beam features, is presented. The scheme utilizes a frame-based phase-space representation of aperture fields to overcome the main hurdles in the design for large apertures: First, it uses the a-priory localization of caustic beams to significantly reduce the optimization problem's variable space, to that of few Gaussian window coefficients accurately capturing those beams. Then, the optimization problem is solved in the reduced (local) spectral domain. We adopt a linearization approach that enables the solution by sequential application of conventional convex optimization tools, which are naturally compatible with the proposed phase-space representation. The localized nature of the Gaussian windows' radiation is used also for fast field evaluation at a greatly reduced number of optimization constraint points. The significant enhancement in the controllability over the various beam parameters is demonstrated through a range of examples.

Design of autofocusing beams based on accelerating beams

Autofocusing Airy-like beams have been designed based on the self-accelerating beams along arbitrary polynomial paths. The evolutions of the autofocusing beam in free space and turbulent atmosphere have been studied. Results show that the energy concentration of the autofocusing beam near the focal plane has a close relationship with its accelerating trajectory. The spot size of the autofocusing beam in the focal plane varies with the acceleration trajectory. The influence of turbulence on different autofocusing beams is different and has a relation with the degree of the curvature of the accelerating beam. The autofocusing beam propagating along a small curvature path has better resistance against turbulence.

Experimental visualization of various cross sections through a butterfly caustic

Optical caustics and wavefronts of butterfly beams (BBs) derived by using a catastrophe theory determined by potential functions depending on the state and control variables are reported. Due to the high dimensionality for the control variables, BBs can be manipulated into various optical light structures. It is also demonstrated that these curious beams have relatively simple Fourier spectra that can be described as polynomials, and another way to generate BBs from the Fourier spectrum's perspective is provided. The dynamics for BBs are investigated by potential functions. Our experimental results agree well with the theoretical predictions. In addition to micro-manipulation and machining, these novel, to the best of our knowledge, caustic beams will pave the way for creating waveguide structures since they display high-intensity formations that evolve along curved trajectories.

Mirror and Circular Symmetry of Autofocusing Beams

This article demonstrates the crucial importance of the symmetrization method for the formation of autofocusing beams. It is possible to impart autofocusing properties to rather arbitrary distributions, for example, truncated and inverted classical modes (such as Hermite–Gaussian, Laguerre–Gaussian, and Bessel modes) or shift the fundamental Gaussian beam by inserting mirror or circular symmetry. The most convenient for controlling autofocusing characteristics is the truncated sinus function with a power-law argument dependence. In this case, superlinear chirp beams (with power q > 2) exhibit sudden and more abrupt autofocusing than sublinear chirp beams (with power 1 < q < 2). Comparison of the different beams’ propagation is performed using fractional Fourier transform, which allows obtaining the field distribution in any paraxial region (both in the Fresnel and Fraunhofer diffraction regions). The obtained results expand the capabilities of structured beams in various applications in optics and photonics.

Propagation-invariant space-time caustics of light

Caustics are responsible for a wide range of natural phenomena, from rainbows and mirages to sparkling seas. Here, we present caustics in space-time wavepackets, a class of pulsed beams featuring strong coupling between spatial and temporal frequencies. Space-time wavepackets have attracted much attention with their propagation-invariant intensity profiles that travel at tunable superluminal and subluminal group velocities. These intensity profiles, however, have been largely restricted to an X-shape or similar pattern. We show that space-time caustics combine the propagation invariance of space-time wavepackets with the flexible design of caustics, allowing for customizable intensity patterns in space-time wavepackets. Our method directly provides the phase distribution needed to realize user-designed caustic patterns in space-time wavepackets. We show that space-time caustics can feature in a broad range of intriguing optical phenomena, including backward traveling caustics formed from purely forward propagating waves, and nondiffracting beams that evolve with time. Our findings should open the doors to an even wider range of structured light with spatiotemporal coupling.

Compact devices for generating multi-focus autofocusing optical beams in free space

We introduce a new type of optical beam that can autofocus multiple times in free space, namely, a multi-focus autofocusing optical beam. Based on the superposed caustic method, we design and construct one-dimensional autofocusing optical beams by tailoring and overlapping two paraxial accelerating optical beams. The two accelerating optical beams are shaped in cosine trajectories symmetrically, thus enabling an on-axis multi-focus property. Furthermore, we fabricate compact devices utilizing diffractive optical elements on a quartz plate patterning the required phase for generations of three-focus and four-focus autofocusing optical beams in free space. We expect that this work may bring autofocusing optical beams further potential applications in particle transporting, material processing, and lithography technology.

Caustics of Non-Paraxial Perfect Optical Vortices Generated by Toroidal Vortex Lenses

In this paper, we consider the comparative formation of perfect optical vortices in the non-paraxial mode using various optical elements: non-paraxial and parabolic toroidal vortex lenses, as well as a vortex axicon in combination with a parabolic lens. The theoretical analysis of the action of these optical elements, as well as the calculation of caustic surfaces, is carried out using a hybrid geometrical-optical and wave approach. Numerical analysis performed on the basis of the expansion in conical waves qualitatively confirms the results obtained and makes it possible to reveal more details associated with diffraction effects. Equations of 3D-caustic surfaces are obtained and the conditions of the ring radius dependence on the order of the vortex phase singularity are analyzed. In the non-paraxial mode, when small light rings (several tens of wavelengths) are formed, a linear dependence of the ring radius on the vortex order is shown. The revealed features should be taken into account when using the considered optical elements forming the POV in various applications.

Caustic Interpretation of the Abruptly Autofocusing Vortex beams

We propose an effective scheme to interpret the abruptly autofocusing vortex beam. In our scheme, a set of analytical formulae are deduced to well predict not only the global caustic, before and after the focal plane, but also the focusing properties of the abruptly autofocusing vortex beam, including the axial position as well as the diameter of focal ring. Our analytical results are in excellent agreement with both numerical simulation and experimental results. Besides, we apply our analytical technique to the fine manipulation of the focusing properties with a scaling factor. This set of methods would be beneficial to a broad range of applications such as particle trapping and micromachinings.

Shaping convex edges in borosilicate glass by single pass perforation with an Airy beam

We demonstrate curved modifications with lengths of up to 2 mm within borosilicate glass produced by single 1030 nm picosecond laser shots with an Airy beam profile. Plasma ignition in the sidelobes of the beam as well as surface damage prove to be the crucial limitations for confined bulk energy deposition on a curved trajectory. A combined experimental and numerical analysis reveals optimum laser parameters for confined bulk energy deposition. This way, we achieved single pass perforation of a 525 µm thick glass sheet and separation by a subsequent etching step, resulting in a well-defined convex edge down to a radius of curvature of 774 µm.

Swallowtail-type diffraction catastrophe beams

We demonstrate a universal approach for generating high-order diffraction catastrophe beams, specifically for Swallowtail-type beams (abbreviated as Swallowtail beams), using diffraction catastrophe theory that was defined by potential functions depending on the control and state parameters. The three-dimensional curved caustic surfaces of these Swallowtail catastrophe beams are derived by the potential functions. Such beams are generated by mapping the cross sections of the high-order control parameter space to the corresponding transverse plane. Owing to the flexibility of the high-order diffraction catastrophe, these Swallowtail beams can be tuned to a diverse range of optical light structures. Owing to the similarity in their frequency spectra, we found that the Swallowtail beams change into low-order Pearcey beams under given conditions during propagation. Our experimental results are in close agreement with our simulated results. Such fantastic catastrophe beams that can propagate along curved trajectories are likely to give rise to new applications in micromachining and optical manipulation, furthermore, these diverse caustic beams will pave the way for the tailoring of arbitrarily accelerating caustic beams.

Multifunctional focusing and accelerating of light with a simple flat lens

The wavefronts emerging from phase gradient metasurfaces are typically sensitive to incident beam properties such as angle, wavelength, or polarization. While this sensitivity can result in undesired wavefront aberrations, it can also be exploited to construct multifunctional devices which dynamically vary their behavior in response to tuning a specified degree of freedom. Here, we show how incident beam tilt in a one dimensional metalens naturally offers a means for changing functionality between diffraction limited focusing and the generation of non-paraxial accelerating light beams. This attractively offers enhanced control over accelerating beam characteristics in a simple and compact form factor.

Designing the phase and amplitude of scalar optical fields in three dimensions

The ability to generate any arbitrarily chosen optical field in a three-dimensional (3D) space, in the absence of any sources, without modifying the index of refraction, remains an elusive but much-desired capability with applications in various fields such as optical micromanipulation, imaging, and data communications, to name a few. In this work, we show analytically that it is possible to generate any desired scalar optical field with predefined amplitude and phase in 3D space, where the generated field is an exact duplicate of the desired field in case it is a solution of Helmholtz wave equation, or if the existence of such field is strictly forbidden, the generated field is the closest possible rendition of the desired field in amplitude and phase. The developed analytical approach is further supported via experimental demonstration of optical beams with exotic trajectories and can have a significant impact on the aforementioned application areas.

Manipulation and control of 3-D caustic beams over an arbitrary trajectory

We present an algorithm for manipulating and controlling 3-D field patterns, with energy confined to the narrow vicinity of predefined 3-D trajectories in free-space, which are of arbitrary curvature and torsion. This is done by setting the aperture field's phase to form smooth caustic surfaces that include the desired trajectory. The aperture amplitude distribution is constructed to manipulate both the on-axis intensity profile and the off-axis beam-width, and is updated iteratively. Once the aperture distribution is calculated, the radiation from a finite sampled aperture is computed numerically using a Fast Fourier Transform-based scheme. This allows for both verification of the design and examination of its sensitivity to parameters of realistic discrete implementation. The algorithm is demonstrated for the cases of an Airy beam of a planar trajectory, as well as for helical and conical-helical trajectory beams.

Nonlinear Beam Shaping in Domain Engineered Ferroelectric Crystals

Domain engineered ferroelectric crystals are a type of microstructure functional material that is used widely in the area of nonlinear optics. Herein, research processes in the area of nonlinear beam shaping using domain engineered crystals in the past decade are reviewed. The newly developed design methods, such as the nonlinear Huygens-Fresnel principle, nonlinear volume holography, and caustic design, which have analogs in linear optics, are introduced. Using the proposed methods for nonlinear beam shaping, multiple function integration, generation of Airy beams, and arbitrary curved trajectories are realized. As an extra degree of freedom, orbital angular momentum of light beams is generated and manipulated through domain engineering. Discussions and future directions in this field are presented.

Recent Advances in Integrated Photonic Jet-Based Photonics

The study of accelerating Airy-family beams has made significant progress, not only in terms of numerical and experimental investigations, but also in conjunction with many potential applications. However, the curvature of such beams (and hence their acceleration) is usually greater than the wavelength. Relatively recently, a new type of localized wave beams with subwavelength curvature, called photonic hooks, was discovered. This paper briefly reviews the substantial literature concerning photonic jet and photonic hook phenomena, based on the photonic jet principle. Meanwhile, the photonic jet ensemble can be produced by optical wave diffraction at 2D phase diffraction gratings. The guidelines of jets’ efficient manipulation, through the variation of both the shape and spatial period of diffraction grating rulings, are considered. Amazingly, the mesoscale dielectric Janus particle, with broken shape or refractive index symmetry, is used to generate the curved photonic jet—a photonic hook—emerging from its shadow-side surface. Using the photonic hook, the resolution of optical scanning systems can be improved to develop optomechanical tweezers for moving nanoparticles, cells, bacteria and viruses along curved paths and around transparent obstacles. These unique properties of photonic jets and hooks combine to afford important applications for low-loss waveguiding, subdiffraction-resolution nanopatterning and nanolithography.

Paraxial optical fields whose intensity pattern skeletons are stable caustics

We construct exact solutions to the paraxial wave equation in free space characterized by stable caustics. First, we show that any solution of the paraxial wave equation can be written as the superposition of plane waves determined by both the Hamilton-Jacobi and Laplace equations in free space. Then using the five elementary stable catastrophes, we construct solutions of the Hamilton-Jacobi and Laplace equations, and the corresponding exact solutions of the paraxial wave equation. Therefore, the evolution of the intensity patterns is governed by the paraxial wave equation and that of the corresponding caustic by the Hamilton-Jacobi equation.

Free-space data-carrying bendable light communications

Bendable light beams have recently seen tremendous applications in optical manipulation, optical imaging, optical routing, micromachining, plasma generation and nonlinear optics. By exploiting curved light beams instead of traditional Gaussian beam for line-of-sight light communications, here we propose and demonstrate the viability of free-space data-carrying bendable light communications along arbitrary trajectories with multiple functionalities. By employing 39.06-Gbit/s 32-ary quadrature amplitude modulation (32-QAM) discrete multi-tone (DMT) signal, we demonstrate free-space bendable light intensity modulated direct detection (IM-DD) communication system under 3 different curved light paths. Moreover, we characterize multiple functionalities of free-space bendable light communications, including bypass obstructions transmission, self-healing transmission, self-broken trajectory transmission, and multi-receiver transmission. The observed results indicate that bendable light beams can make free-space optical communications more flexible, more robust and more multifunctional. The demonstrations may open a door to explore more special light beams enabling advanced free-space light communications with enhanced flexibility, robustness and functionality.

Generation of spirally accelerating optical beams

We developed a generalized spectral phase superposition approach for generating accelerating optical beams along arbitrary trajectories. Such beams can be customized by predefining an appropriate superimposed phase pattern that consists of multiple sub-phases. We generated a spirally accelerating beam in a three-dimensional space and developed an algorithm to improve the uniformity of the intensity along the trajectory by introducing phase-shift factors. We also experimentally verified our numerical simulations. The proposed approach breaks the conventional convex trajectory restrictions. These various accelerating beams would pave the way for optically moving particles along a desired trajectory. The generation of such arbitrary accelerating beams is likely to give rise to new applications in flexible optical manipulation, wave front control, and optical transportation and guidance of particles.

Broadband Generation of Photonic Spin-Controlled Arbitrary Accelerating Light Beams in the Visible

Bending light along arbitrary curvatures is a captivating and popular notion, triggering unprecedented endeavors in achieving diffraction-free propagation along a curved path in free-space. Much effort has been devoted to achieving this goal in homogeneous space, which solely relies on the transverse acceleration of beam centroid exerted by a beam generator. Here, based on an all-dielectric metasurface, we experimentally report a synthetic strategy of encoding and multiplexing acceleration features on a freely propagating light beam, synergized with photonic spin states of light. Independent switching between two arbitrary visible accelerating light beams with distinct acceleration directions and caustic trajectories is achieved. This proof-of-concept recipe demonstrates the strength of the designed metasurface chip: subwavelength pixel size, independent control over light beam curvature, broadband operation in the visible, and ultrathin scalable planar architecture. Our results open up the possibility of creating ultracompact, high-pixel density, and flat-profile nanophotonic platforms for efficient generation and dynamical control of structured light beams.

Construction, characteristics, and constraints of accelerating beams based on caustic design

Caustic methods have been proposed for wavefront design to enable light beams propagating along curved trajectories, namely accelerating beams. Here we elaborate the complete construction, remarkable characteristics, and hidden constraints of these methods. It is found that accelerating beams based on the caustic design have not only a well-known curved intensity distribution but also a linear phase distribution along the caustic proportional to the curved length, as if light field indeed moved along the caustic. Moreover, with this characteristic, further light-ray analyses are implemented to illustrate the constraints of caustic design in different cases. We expect our work will clarify some confusion on the effectiveness and applicability of caustic methods, and thus facilitate the design of accelerating beams for various applications.

Synchrotron radiation from an accelerating light pulse

Synchrotron radiation-namely, electromagnetic radiation produced by charges moving in a curved path-is regularly generated at large-scale facilities where giga-electron volt electrons move along kilometer-long circular paths. We use a metasurface to bend light and demonstrate synchrotron radiation produced by a subpicosecond pulse, which moves along a circular arc of radius 100 micrometers inside a nonlinear crystal. The emitted radiation, in the terahertz frequency range, results from the nonlinear polarization induced by the pulse. The generation of synchrotron radiation from a pulse revolving about a circular trajectory holds promise for the development of on-chip terahertz sources.

Nonsymmetric curved beams within a symmetric caustic skeleton

Nonsymmetric curved beams having a symmetric caustic skeleton are presented. They arise from a finite jump in the symmetric spectral phase that breaks the symmetry of the beam intensity without altering its associated caustic curve. These nonsymmetric beams can be represented as a superposition of two caustic beams whose wave fields have well-defined even and odd symmetries with weight coefficients dependent on the phase jump. In this approach, the phase jump acts as a measure of the beam asymmetry degree that can be easily controlled in experiments. This scheme is a promising step towards optical cryptography and quantum optics applications.

Dynamic generation and modulation of acoustic bottle-beams by metasurfaces

Acoustic bottle-beams have been realized by acoustic metasurfaces (AMs) composed of space-coiling subunits. By manipulating the transmitted acoustical phase, the special AM can generate two intersecting accelerating beams along the designed convex trajectories, forming the acoustic bottle-beam. The transmitted acoustic bottle-beams are investigated theoretically and demonstrated numerically. We find that the shape and area of the acoustic bottle-beam could be statically controlled by designing the AM as well as dynamically modulated by the incident angles. In addition, the highly efficient acoustic focusing could be obtained at the convergence point of the bottle-beams, which also could be adjusted dynamically by the incident angles. It is further found that this focusing is robust against the obstacle scattering. The realization and manipulation of acoustic bottle-beams may have potential applications in biomedical imaging/therapy and non-destructive evaluation.

The generation of three-dimensional curved beams based on holographic metasurface

A reflective metasurface is proposed to create the three-dimensional curved beams based on holographic technique. In the design of the structure of holographic metasurface, the amplitude and phase distribution is obtained combining the generalized law of reflection and the method for producing two-dimensional Bessel beams. Furthermore, the holographic metasurface is implemented by the varying metallic square patch on the grounded dielectric substrate. The final experimental measurements confirm the theoretical analysis of fully vectorial integral equation and the full-wave simulation.

Independent amplitude and trajectory/beam-width control of nonparaxial beams

We show that it is possible to generate non-paraxial optical beams with pre-engineered trajectories and designed maximum amplitude along these trajectories. The independent control of these two degrees of freedom is made possible by engineering both the amplitude and the phase of the optical wave on the input plane. Furthermore, we come to the elegant conclusion that the beam width depends solely on the local curvature of the trajectory. Thus, we can generate beams with pre-defined amplitude and beam-width by appropriately selecting the local curvature. Our theoretical results are in excellent agreement with numerical simulations. We discuss about methods that can be utilized to experimentally generate such beam. Our work might be useful in applications where precise beam control is important such as particle manipulation, filamentation, and micromachining.

Controlling Surface Plasmons Through Covariant Transformation of the Spin-Dependent Geometric Phase Between Curved Metamaterials

General relativity uses curved space-time to describe accelerating frames. The movement of particles in different curved space-times can be regarded as equivalent physical processes based on the covariant transformation between different frames. In this Letter, we use one-dimensional curved metamaterials to mimic accelerating particles in curved space-times. The different curved shapes of structures are used to mimic different accelerating frames. The different geometric phases along the structure are used to mimic different movements in the frame. Using the covariant principle of general relativity, we can obtain equivalent nanostructures based on space-time transformations, such as the Lorentz transformation and conformal transformation. In this way, many covariant structures can be found that produce the same surface plasmon fields when excited by spin photons. A new kind of accelerating beam, the Rindler beam, is obtained based on the Rindler metric in gravity. Very large effective indices can be obtained in such systems based on geometric-phase gradient. This general covariant design method can be extended to many other optical media.

Ring-Airy beams at the wavelength limit

We demonstrate that paraxial ring-Airy beams can approach the wavelength limit, while observing a counterintuitive, strong enhancement of their focal peak intensity. Using numerical simulations, we show that this behavior is a result of the coherent constructive action of paraxial and nonparaxial energy flow. A simple theoretical model enables us to predict the parameter range over which this is possible.

Hyperbolic accelerating beams and their relation with Hermite-Gaussian beams

We derive the initial distributions of phase and complex amplitude of accelerating beams with arbitrary predesigned hyperbolic trajectories using the caustic-design method and explore the relation between these beams and Hermite-Gaussian beams. The results show the hyperbolic accelerating beams are a larger class of beams than Hermite-Gaussian beams. When the bending parameter is an integer, the hyperbolic accelerating beams have a similar initial complex amplitude distribution and almost the same propagating characteristics as Hermite-Gaussian beams. Through the analysis of the ray-based method, we also derive an approximate expression for the initial complex amplitude of Hermite-Gaussian beams after introducing an amplitude distribution function. Although the proposed approximate expressions of complex amplitude are more complex than the usually used Hermite-Gaussian function, they explicitly indicate the information on local amplitude, wave vector, and internal ray structure (including caustics) of these beams and thus provide us clearer geometrical insights into these beams.

Controllable multiple plasmonic bending beams via polarization of incident waves

Plasmonic bending beams, which preserve their spatial shapes while propagating along curved trajectories in metal-dielectric interface, offer important applications in the fields of fiber sensor, optical trapping, and micro-nano manipulation. In this work, circular hole array, as a local point-like sources of surface plasmon polaritons, is designed on the metal film to generate multiple plasmonic bending beams. The electric field intensity of multiple plasmonic bending beams is controlled by polarization angle of input light. In addition, the electric filed intensity of multiple plasmonic bending beams relies on circle hole radius. These findings provide guidance in the design and optimization of plasmonic bending beam generators.

Scaling the abruptly autofocusing beams in the direct-space

We propose a simple technique to scale the abruptly autofocusing beams in the direct space by introducing a scaling factor in the phase. Analytical formulas are deduced based on optical caustics, explicitly revealing how the scaling factor controls location, peak intensity, and size of the focal spot. We demonstrate that the multiplication of a scaling factor on the phase is equivalent to the axial-scaling transformation under the paraxial approximation. Further numerical and experimental results confirm theoretical predictions. In addition, amplitude modulation using phase-only holograms is used to maintain the peak intensity level of the focal spots.

Cherenkov Radiation Control via Self-accelerating Wave-packets

Cherenkov radiation is a ubiquitous phenomenon in nature. It describes electromagnetic radiation from a charged particle moving in a medium with a uniform velocity larger than the phase velocity of light in the same medium. Such a picture is typically adopted in the investigation of traditional Cherenkov radiation as well as its counterparts in different branches of physics, including nonlinear optics, spintronics and plasmonics. In these cases, the radiation emitted spreads along a “cone”, making it impractical for most applications. Here, we employ a self-accelerating optical pump wave-packet to demonstrate controlled shaping of one type of generalized Cherenkov radiation - dispersive waves in optical fibers. We show that, by tuning the parameters of the wave-packet, the emitted waves can be judiciously compressed and focused at desired locations, paving the way to such control in any physical system.

Practical algorithm for custom-made caustic beams

We present a practical algorithm for designing an aperture field (source) that propagates along a predefined generic beam trajectory that consists of both convex and concave sections. We employ here the mechanism that forms the well-known Airy beam in which the beam trajectory follows a smooth convex caustic of the geometric optics rays and generalize it for a class of beams that are referred to as "caustic beams" (CBs). The implementation is based on "back-tracing" rays from the predefined beam trajectory to the source's aperture to form its phase distribution. The amplitude is set in order to form a uniform smooth amplitude of the CBs along their trajectories. Several numerical examples are included.

Autofocusing Airy beams generated by all-dielectric metasurface for visible light

Conventional method to generate autofocusing Airy (AFA) beam involves the optical Fourier transform (FT) system, which has a fairly long working distance due to the focal length of FT lens, presence of spatial light modulator (SLM) and auxiliary total reflection mirrors. Here, we propose an extremely compact design to generate high-efficiency AFA beam at visible frequency by using metasurface which is composed of a single layer array of amorphous titanium dioxide (TiO₂) elliptical nanofins sitting on the fused-silica substrate. Numerical simulations show that the designed structures are capable of precisely controlling the deflection of Airy beam and tuning the focal length of AFA beam. We further numerically demonstrate that the phase modulation of AFA beam could combine with the concept of vortex light field to produce vortical AFA beam. We anticipate that such device can be useful in the ultra-compact integrated optic system, biomedical nanosurgery and optical trapping applications.

Intensity-symmetric accelerating caustic beams

We construct and generate symmetric accelerating caustic beams (ACBs) by using 3/2-order phase-only masks with elliptical contour based on optical caustics and diffraction theory. The symmetric ACBs are a type of bimodal accelerating caustic beam with two quasi-constant intensity peaks, very similar to the combination of two face-to-face Airy-like beams judging by appearance. Their fundamental optical morphology and force properties of particles in ACBs are subsequently provided. The unique optical properties of ACBs can be exploited for practical uses, such as accelerating electrons and clearing micrometer-sized particles as a laser micrometer-sized "water pump" instead of a laser micrometer-sized "snowblower" of accelerating Airy beams.

Geometrical connection between catacaustics and kinematics of planar motion of a rigid solid

Unnoticed and hidden optomechanical analogies between kinematics of planar motion of a rigid solid and catacaustics generated by mirror reflection on smooth profiles in geometrical optics are discussed. A concise and self-consistent theory is developed, which intends to explain and clarify many partial aspects covered by the literature.

Optical Bloch oscillations and Zener tunneling of Airy beams in ionic-type photonic lattices

We report on the existence of optical Bloch oscillations (OBOs) and Zener tunneling (ZT) of Airy beams in ionic-type photonic lattices with a refractive index ramp. Different from their counterparts in uniform lattices, Airy beams undergoing OBOs show an alternatively switched concave and convex trajectory as well as a periodical revival of input beam profiles. Moreover, the ionic-type photonic lattice established in photorefractive crystal exhibits a reconfigurable lattice structure, which provides a flexible way to tune the amplitude and period of the OBOs. Remarkably, it is demonstrated that the band gap of the lattice can be readily controlled by rotating the lattice inducing beam, which forces the ZT rate to follow two significant different decay curves amidst decreasing index gradient. Our results open up new possibilities for all-optical switching, routing and manipulation of Airy beams.

Anomalous Refraction of Acoustic Guided Waves in Solids with Geometrically Tapered Metasurfaces

The concept of a metasurface opens new exciting directions to engineer the refraction properties in both optical and acoustic media. Metasurfaces are typically designed by assembling arrays of subwavelength anisotropic scatterers able to mold incoming wave fronts in rather unconventional ways. The concept of a metasurface was pioneered in photonics and later extended to acoustics while its application to the propagation of elastic waves in solids is still relatively unexplored. We investigate the design of acoustic metasurfaces to control elastic guided waves in thin-walled structural elements. These engineered discontinuities enable the anomalous refraction of guided wave modes according to the generalized Snell's law. The metasurfaces are made out of locally resonant toruslike tapers enabling an accurate phase shift of the incoming wave, which ultimately affects the refraction properties. We show that anomalous refraction can be achieved on transmitted antisymmetric modes (A_{0}) either when using a symmetric (S_{0}) or antisymmetric (A_{0}) incident wave, the former clearly involving mode conversion. The same metasurface design also allows achieving structure embedded planar focal lenses and phase masks for nonparaxial propagation.

Indefinite Plasmonic Beam Engineering by In-plane Holography

Recent advances in controlling the optical phase at the sub-wavelength scale by meta-structures offer unprecedented possibilities in the beam engineering, holograms, and even invisible cloaks. In despite of developments of plasmonic beam engineering for definite beams, here, we proposed a new holographic strategy by in-plane diffraction process to access indefinite plasmonic beams, where a counterintuitive oscillating beam was achieved at a free metal surface that is against the common recognition of light traveling. Beyond the conventional hologram, our approach emphasizes on the phase correlation on the target, and casts an in-depth insight into the beam formation as a kind of long depth-of-field object. Moreover, in contrast to previous plasmonic holography with space light as references, our approach is totally fulfilled in a planar dimension that offers a thoroughly compact manipulation of the plasmonic near-field and suggests new possibilities in nanophotonic designs.

Quasi-Airy beams along tunable propagation trajectories and directions

We present a theoretical and experimental exhibit that accelerates quasi-Airy beams propagating along arbitrarily appointed parabolic trajectories and directions in free space. We also demonstrate that such quasi-Airy beams can be generated by a tunable phase pattern, where two disturbance factors are introduced. The topological structures of quasi-Airy beams are readily manipulated with tunable phase patterns. Quasi-Airy beams still possess the characteristics of non-diffraction, self-healing to some extent, although they are not the solutions for paraxial wave equation. The experiments show the results are consistent with theoretical predictions. It is believed that the property of propagation along arbitrarily desired parabolic trajectories will provide a broad application in trapping atom and living cell manipulation.

Nonparaxial abruptly autofocusing beams

We study nonparaxial autofocusing beams with pre-engineered trajectories. We consider the case of linearly polarized electric optical beams and examine their focusing properties, such as contrast, beam width, and numerical aperture. Such beams are associated with larger intensity contrasts, can focus at smaller distances, and have smaller spot sizes as compared to the paraxial regime.