Pearcey Beams Research Articles

Nonparaxial propagation of abruptly autofocusing circular Pearcey Gaussian beams

We introduce a new class of abruptly autofocusing circular Pearcey Gaussian beams (AAFCPGBs) which tend to be abruptly autofocusing circular Pearcey beams with a small distribution factor, or Gaussian beams with a larger distribution factor. The nonparaxial propagation of the AAFCPGBs is investigated by numerical calculation. The radiation force of the AAFCPGBs exerted on a Rayleigh particle is analyzed in detail.

Propagation of a radially polarized Pearcey beam in uniaxial crystals

We investigate the propagation properties and the radiation forces of a radially polarized Pearcey beam (RPPB) in uniaxial crystals analytically and numerically in this paper. Obtaining the propagation expression of an RPPB, we perform the intensity (in the x-direction, in the y-direction and in the synthesizing direction), the polarization direction, the energy flow and the angular momentum of an RPPB with the method of numerical calculation. The propagating properties of an RPPB in the free space are shown for comparison. The auto-focusing and the inversion of an RPPB propagating in uniaxial crystals are demonstrated. The influence of the extraordinary refractive index and the ordinary refractive index on an RPPB in the uniaxial crystals is discussed.

Autofocusing and spreading of partially coherent Butterfly beams modulated by flat-topped factor in atmospheric turbulence

By transferring higher hierarchy Butterfly catastrophe to optical field, the autofocusing behavior and beam spreading of partially coherent Butterfly beams modulated by flat-topped factor in atmospheric turbulence are investigated, where the input power is fixed. In the process of autofocusing behavior the main lobe remains its location near origin, which is different from the movement of main lobe in a lower hierarchy Pearcey beam. The turbulence does not change the autofocusing behavior, but accelerates the decaying in side lobes of butterfly structure. A higher coherence length can lead to a more concentrated intensity at autofocusing plane. The analysis of beam spreading also indicates that there exist critical values of propagation distance of zc. A smaller flat-topped factor tends to maintain a smaller spreading for z<zc, while for z>zc the opposite case can be found. Angular spread decreases with an increasing of coherence length, and a smaller beam order shows a larger far field divergence angle. The study of partially coherent Butterfly beams in turbulence provides the possibility of the atmospheric laser communication by using higher-order catastrophe optical field.

Focusing properties of circle Pearcey beams.

We introduce a new class of (2+1) dimensional circle Pearcey beams (CPBs) with the abruptly autofocusing (AAF) characteristics. Compared with circular Airy beams, CPBs can increase the peak intensity contrast, shorten the focus distance and, especially, eliminate the oscillation after the focal point. Furthermore, we discuss the influence of the optical vortices (including on-axis, off-axis, and vortex pairs) on the light intensity distribution of the CPBs during propagating.

The Poynting vector and angular momentum density of the autofocusing Butterfly-Gauss beams

Three different Butterfly-Gauss beams are proposed by introducing higher order Butterfly catastrophe to the field of optics, where an autofocusing Butterfly-Gauss beam with a special profile is emphatically studied and it can be regarded as the sum of two Half-Butterfly-Gauss beams. Based on the Collins integral formula, the autofocusing behavior, Poynting vector and angular momentum density of the corresponding Butterfly-Gauss beams during propagation in free space, focus system and chiral medium are investigated, respectively. The results show that the Butterfly-Gauss beam not only exhibits autofocusing behavior similar to the Pearcey beams or circular Airy beams, but also presents rotation in analogy with the Swallowtail beam during propagation in free space. In the focus system, the tail directions of special patterns in the focal plane can be controlled by scaling lengths. In the chiral medium, the greater chirality parameters correspond to the increasing of phase velocity of beam, which results in the fact that the distance of autofocusing plane becomes smaller, and it is easier to autofocus the beam. The proposed Butterfly-Gauss beams have the application possibility in the field of manipulating microparticles along intensity channels due to their special spatial structures in focal plane, where the Poynting vectors flow from center spot to the controlled tail.

Pearcey光束的形成及其聚焦特性

Pearcey光束的形成及其聚焦特性

Pearcey solitons in curved nonlinear photonic caustic lattices

Controlling artificial Pearcey and swallowtail beams allows the realization of caustic lattices in nonlinear photosensitive media at very low light intensities. We examine their functionality as 2D and 3D waveguiding structures and show the potential of exploiting these lattices as linear beam splitters, which we name a ‘Pearcey-Y-splitter’. For symmetrized Pearcey beams as auto-focusing beams, the formation of solitons in focusing nonlinearity is observed. Our original approach represents the first realization of caustic photonic lattices and can directly be applied in signal processing, microscopy and material lithography.

Stable and unstable Airy-related caustics and beams

Optical beams with an underlying caustic structure are stable under perturbation if the caustics belong to the catastrophe-theory classification; otherwise they are unstable. The original Airy beam in two spatial dimensions, with its curved caustic, is stable in this sense. But the separable Airy-product beam in three-dimensions is unstable: under separability-breaking perturbation, it unfolds into the hyperbolic umbilic diffraction catastrophe, which is stable. By including initial phase factors, a variety of new exact solutions of the paraxial wave equation can be generated, corresponding to Pearcey and higher-catastrophe beams with stable caustics, and with the associated diffraction catastrophes appearing in their canonical forms or as deformations of these.

Propagation of a Pearcey–Gaussian–vortex beam in free space and Kerr media

The propagation of a Pearcey–Gaussian–vortex beam (PGVB) has been investigated numerically in free space and Kerr media. In addition, we have done a numerical experiment for the beam in free space. A PGVB maintains the characteristics of auto-focusing, self-healing and form-invariance which are possessed by a Pearcey beam and a Pearcey–Gaussian beam. Due to the influence of the optical vortex, a bright speck occurs in front of the main lobe. Compared with a Pearcey beam and a Pearcey–Gaussian beam, a PGVB has the most remarkable intensity singularity and the phase singularity. It is worth noting that the impact of the vortex at the coordinate origins means that a PGVB in the vicinity carries no angular momentum or transverse energy flow. We have investigated and numerically simulated the transverse intensity of a PGVB in Kerr media. We find that the auto-focusing of a PGVB in a Kerr medium becomes stronger with increasing power.

Symmetric form-invariant dual Pearcey beams.

We introduce another type of Pearcey beam, namely, dual Pearcey (DP) beams, based on the Pearcey function of catastrophe theory. DP beams are experimentally generated by applying Fresnel diffraction of bright elliptic rings. Form-invariant Bessel distribution beams can be regarded as a special case of DP beams. Subsequently, the basic propagation characteristics of DP beams are identified. DP beams are the result of the interference of two half DP beams instead of two classical Pearcey beams. Moreover, we also verified that half DP beams (including special-case parabolic-like beams) generated by half elliptical rings (circular rings) are a new member of the family of form-invariant beams.

Effect of turbulent atmosphere on the on-axis average intensity of Pearcey–Gaussian beam

The propagation characteristics of the Pearcey–Gaussian (PG) beam in turbulent atmosphere are investigated in this paper. The Pearcey beam is a new kind of paraxial beam, based on the Pearcey function of catastrophe theory, which describes diffraction about a cusp caustic. By using the extended Huygens–Fresnel integral formula in the paraxial approximation and the Rytov theory, an analytical expression of axial intensity for the considered beam family is derived. Some numerical results for PG beam propagating in atmospheric turbulence are given by studying the influences of some factors, including incident beam parameters and turbulence strengths.

Construction of Bi-Pearcey beams and their mathematical mechanism

We present a theoretical expression in the form of the Pearcey function by deducing the Fresnel diffraction distribution of an elliptic line. Then, we numerically simulate and experimentally generate this kind of new Pearcey beams by using the Fresnel diffraction of optical ellipse line. This kind of beams can be referred to as Bi-Pearcey beams because their appearance of the topological structure is very similar to the combination of two face-to-face classical Pearcey beams. It is no doubt that so-called Bi-Pearcey beams are the new member of a family of form-invariant Pearcey beams. Subsequently, we also provide the theoretical mechanism of generating Bi-Pearcey beams based on the Zeeman catastrophe machine of catastrophic theory. By solving the critical equation of potential function of Bi-Pearcey beams generated by an ellipse line, we find that the optical morphogenesis of Bi-Pearcey beams is determined by the number of roots of the critical equation. The critical equation of potential function of Bi-Pearcey beams is a classical Cartan equation, which has at most three real roots. For the Fresnel diffraction of ellipse line, three real roots of the critical equation are corresponding to three stable points and represent three diffraction lines, hence they can be used to examine the optical topological structure of Bi-Pearcey beams. By choosing the appropriate control variable of Bi-Pearcey beams, two diffraction lines of an ellipse line overlap, and the strong caustic line of Bi-Pearcey beams is correspondingly generated when the two of the three real roots of the critical equation are equal. If the three real roots of the critical equation are all equal, the strongest cusps of Bi-Pearcey beams are generated, accordingly. Moreover, the equation of the caustic line and their positions of four cusps of Bi-Pearcey beams are given by solving the control variable equation of Bi-Pearcey beams. In conclusion, we elucidate the mathematical mechanism of topical morphogenesis of Bi-Pearcey beams based on catastrophic theory.

Pearcey光束的光学形态形成及其数学机理研究

Pearcey光束的光学形态形成及其数学机理研究

Generation of a family of Pearcey beams based on Fresnel diffraction catastrophes

Based on the theory of differential geometry and Fresnel diffraction catastrophes, we theoretically prove that the cusped caustic of Pearcey beams are the evolute of a parabola and thus identify the key factor determining the optical structure of Pearcey beams. We numerically simulate and experimentally generate a family of Pearcey beams with various optical topological structures using different parabolas. We then investigate their optical structures and propagation properties.

Half Pearcey laser beams

We obtain a new solution of the paraxial Helmholtz equation that describes a family of three-dimensional and two-dimensional form-invariant half-Pearcey beams (HP-beams). HP-beams generalize Pearcey beams obtained in Ring et al (2012) Opt. Express 20 18955, since these Pearcey beams can be considered as the sum of two first-order HP-beams. Three-dimensional HP-beams have angular spectra of plane waves, which are non-zero at a half parabola. For functions of HP-beam complex amplitudes, the orthogonality properties have been revealed. Using a spatial phase modulator, we generated superposition of HP-beams. For two-dimensional HP-beam acceleration and deceleration of trajectory has been shown for areas before and beyond the focal plane respectively.

Pearcey beams carrying orbital angular momentum

We have obtained a new solution of the paraxial Helmholtz equation, which describes a two-parameter family of structurally stable three-dimensional vortex circular Pearcey beams with their complex amplitude expressed through a degenerate hypergeometric function. The vortex Pearcey beams have an orbital angular momentum and the auto-focusing property, and propagate along an accelerating trajectory toward their focus, where the intensity ring of the beam is "sharply" reduced in diameter. An explicit expression has been obtained for the complex amplitude of vortex circular Pearcey-Gaussian beams, which also have the auto-focusing property.

Generation of a family of Pearcey beams and their optical structure

Based on differential geometry and optical diffraction theory, the determinants of optical structure of Pearcey beams are examined. We theoretically propose and experimentally observe a bundle of Pearcey beams with different optical topology structures. Besides, we have studied their properties and the results show that the structure of Pearcey beams can be flexibly controlled, hence they will be expected to play a new role in some research fields.

Virtual source of a Pearcey beam.

A virtual source that yields a family of a Pearcey wave is demonstrated. A closed-form expression is derived for the Pearcey wave that simplifies to the paraxial Pearcey beam (PB) in the appropriate limit. From the perturbative series representation of a complex-source-point spherical wave, an infinite series nonparaxial correction expression for a PB is obtained. The infinite series expression of a PB can give accuracy up to any order of the diffraction angle. By applying the integral representation of the Pearcey wave, the first three terms in the nonparaxial correction series to the paraxial PB are provided.

Structurally stable three-dimensional and two-dimensional laser half pearcey beams

We obtain a new solution of the paraxial Helmholtz equation that describes a family of three-dimensional and two-dimensional structurally stable half-Pearcey beams (HP-beams). HP-beams generalize Pearcey beams obtained in Opt. Express, 20, 18955 (2012), since these Pearcey beams can be considered as the sum of two first-order HP-beams. Three-dimensional HP-beams have angular spectrum of plane waves, which is non-zero at a half of parabola. For functions of HP-beams complex amplitudes, the orthogonality properties have been revealed. For two-dimensional HP-beam acceleration and deceleration of trajectory has been shown for areas before and beyond the focal plane respectively.

Auto-focusing and self-healing of Pearcey beams

We present a new solution of the paraxial equation based on the Pearcey function, which is related to the Airy function and describes diffraction about a cusp caustic. The Pearcey beam displays properties similar not only to Airy beams but also Gaussian and Bessel beams. These properties include an inherent auto-focusing effect, as well as form-invariance on propagation and self-healing. We describe the theory of propagating Pearcey beams and present experimental verification of their auto-focusing and self-healing behaviour.