Angular Spectrum Representation Research Articles

Goos-Hänchen shift of an Airy beam reflected in an epsilon-near-zero metamaterial

We numerically investigated lateral Goos-Hänchen shifts a two-dimensional Airy beam experiences when it is reflected at the interface between air and an epsilon-near-zero metamaterial. Angular spectrum theory is employed to decompose Airy beam into plane waves, followed by beam vectorization to overcome the paraxial propagation assumption. The reflected light field can then be obtained by integrating plane waves weighted by Fresnel reflection coefficients. Results show that Airy beam experiences nonzero GH shift at normal incidence due to asymmetric angular spectrum. Influences of beam parameters and permittivity are further investigated. Our study provides foundation for deep understandings of Airy beam in epsilon-near zero metamaterials.

Abruptly autofocusing property and optical manipulation of circular Airy beams

By employing the angular spectrum representation, we successfully derive the partial-wave expansion coefficients of the circular Airy beams (CABs) with different polarizations, based on which the scattering of a spherical particle in the CABs is solved exactly. Special attention is focused on exploring the potential applications of the CABs in the abruptly autofocusing (AAF) property and optical manipulation of microparticle by optical force. It is found that both the linear and circular polarizations are the best candidates to implement the AAF property robustly against strong disturbance by a large-sized particle. In addition, although the CABs can be intensively focused to a small region, resulting in an abrupt increase of light intensity by two orders of magnitude at the focal point and enabling three-dimensional stable trapping of a Rayleigh particle near the focal point, the usual CABs fail to generate an optical force outweighing the Brownian force to achieve a stable transverse trapping in the region before the focal point. On the other hand, a Mie particle can be confined transversely in the primary ring and accelerated along the curved trajectory of CABs to the focal point, then pushed further all the way through the focal point rather than being trapped therein, and eventually accelerated further along a straight trajectory in the direction of light propagation over 100 wavelengths, due to the weak diffraction characteristic of a morphed Bessel-like beam of CABs. The exotic curved-straight trajectory transport of particles by the CABs may find applications in the case where the transport path is partly blocked.

Generation of Bessel beam sources in FDTD.

In this paper, a straightforward approach is presented to generate Bessel beam sources in three-dimensional finite-difference time-domain (FDTD) method. Based on the angular spectrum representation (ASR), the incident Bessel beam is described as a superposition of plane waves whose wavevectors covering a conical surface. This decomposition of Bessel beam is then approximated by a finite collection of plane waves, which are injected into FDTD simulation domain using the total-field/scattered-field (TF/ST) method. The present method's correctness and accuracy are verified by comparing the reconstructed field in FDTD with the original field. Far-field scattered diagrams of a dielectric sphere and a spheroid particle illuminated by a zero-order or a higher-order Bessel beam are calculated using FDTD. The results are compared with those calculated using the generalized Lorenz-Mie theory (GLMT) and surface integral equation method (SIEM). Very good agreements have been achieved, which partially indicate the correctness of our method. Internal and near-surface field distributions for a two-layer hemisphere particle, which are illuminated by Bessel beams, are also displayed to show the potentials of this approach in solving scattering problems of complex particles. This approach can also be applied to generate other structured beam sources in FDTD, which provides an access to solve structured beam scattering by complex particles using FDTD.

Validation of vectorial theories for the focusing of high numerical aperture Fresnel zone plates

Abstract To study the applicability of vectorial angular spectrum (VAS) theory and vectorial Debye–Wolf diffraction integral for the focusing of high numerical aperture (NA) Fresnel Zone Plates (FZPs), these two vectorial methods have been rigorously examined by the three-dimensional finite-difference time-domain (FDTD) method under general and specific conditions. It is found that for a low total annulus number and when the wavelength dispersion or obstruction are existing, Debye–Wolf diffraction integral has failed to describe the light field distribution in the focal region. On the contrary, VAS theory can generally predict the focusing properties. However, when a FZP only contains one or two transparent annuli, both VAS and Debye–Wolf diffraction theories are invalid. Wavelength dispersion and purpose obstruction can produce the multi-focus or focal shift, which can be used for the axial wavelength scanning, the precise measurement of laser wavelength, or optical trapping.

Electromagnetic exploration of focusing properties of high-numerical-aperture micro-Fresnel zone plates

Focusing properties of high-numerical-aperture micro-Fresnel zone plates (FZP) have been rigorously investigated at the wavelength from deep ultraviolet to visible based on the combination of the vectorial angular spectrum theory and three-dimensional finite-difference time-domain method. New insights are found. The axial spot size evaluated by full-width-at-half-maximum is in direct proportion to the inverse of the total annulus number. Linearly negative focal shift phenomenon can be observed due to the wavelength dispersion over a finite range of tens of nanometers. A proper wavelength deviation can lead to the maximum focal intensity. A simple method has been demonstrated to probe either multi-focus or a light needle using a composite microstructure based on high-NA FZPs. The findings of this study expand the common understandings of the classic widely-used FZPs.

A new method to produce optical nano-needle

A method is described to design a microstructure comprised of multi-Fresnel zone plate (FZP) fragments for shaping an optical needle with arbitrary length. Thus, a microstructure comprised of three planar FZP fragments with different focal lengths f1, f2, and f3 is designed to form a long optical needle by delicate interference of coherent light beams diffracted from these three FZP fragments. For a 74.34-μm-diameter microstructure illuminated with a linearly x-polarized beam, a 7.87-λ-long optical needle is produced at a distance of 12.31 λ away from the mask surface. The sizes of transverse beam are 0.97 and 0.4 λ in x and y directions, respectively. For this work, the vectorial angular spectrum (VAS) theory is employed to describe the electric field of light behind the microstructure, as well as the three-dimensional finite-difference time-domain (3D FDTD) method is adopted to further verify the results obtained.

Intensity distribution of diffractive axicon with the optical angular spectrum theory

Diffractive axicon is lightweight and easy to be manufactured, it is also efficiently used to produce diffraction-free Bessel beam. But, deducing its expression of the field distribution of light is difficult in theory and the theoretical equation expression has approximation by using the Huygen-Fresnel diffraction integral. Here, the rigorous optical angular spectrum theory that follows the Helmholtz equation is used. The propagation theory of angular-spectrum with fewer approximation, will result in more accurate results. A mathematical model to analyze the field distribution of light of the diffractive axicon is set up. The diffraction intensity distribution of the diffractive axicon with the monochromatic Gaussian beam illumination and multicolor illumination is numerically calculated and discussed. The results of numerical calculation with a monochromatic laser source show that not all the intensity distribution in the focal depth of field has the characteristics of the diffraction-free Bessel beam, and the on-axis diffraction intensity shows direct correlation of the lateral intensity distribution of the incident waves and the period of the diffractive axicon. The extended depth of field of the diffractive axicon with multicolor illumination is increasing as the wavelength reducing. It is expected that our study can provide very useful information for analyzing the diffractive axicon which exists in the optical trapping systems and laser-processing applications and imaging systems.

Analysis of influence of thermally induced aberrations on laser beam quality factor (M2) in a high-power laser system

In a high-power laser system, thermally induced aberrations, which can influence the laser beam quality, are caused by thermal deformations of the optical elements when they are irradiated by the laser. We evaluate the mechanism responsible for the influence of thermally induced aberrations on the laser beam quality factor (M2) using the finite element method and angular spectrum theory. The thermally induced aberrations are decomposed into their constituent Zernike terms, showing that the influence of the thermally induced aberrations on M2 comprises a mixed contribution of different aberrations. The M2 of a fiber laser is simulated and measured when the laser transmits through organic glass. The simulation results have a good consistency with the experimental results, indicating that the method used in this study is effective when analyzing the influence of thermally induced aberrations on M2.

Vectorial structures of linear-polarized Butterfly-Gauss vortex beams in the far zone

By introducing the Butterfly catastrophe to optics, the far-zone vectorial structures of Butterfly-Gauss beam with vortex and non-vortex are studied using the angular spectrum representation and stationary phase method. The influence of topological charge, linear-polarized angle, off-axis distance and scaling length on the far-zone vectorial structures, especially in the Poynting vector and angular momentum density of the corresponding beam is emphasized. The results show that the embedded optical vortex at source plane lead to special dark zones in the far zone, where the number of dark zone equals the absolute value of topological charge of optical vortex. Furthermore, the symmetry and direction of the special dark zones can be controlled by off-axis distance and scaling length, respectively. The linear-polarized angle adjusts only the Poynting vectors of TE and TM terms, but it does not affect those of whole beam. Finally, the vectorial expressions also indicate that the total angular momentum density is certainly zero owing to the far-zone stable structures rather than rotation behaviors.

Accurate experimental detection method for characterizing superoscillatory lenses made from multiannular metasurfaces

To verify the superoscillatory optical diffraction focusing characteristics of multiannular metasurfaces (MAMs), we propose an experimental detection method. It consists of two parts, the reflection positioning optical system and transmission detection imaging system. The best focal plane of the superoscillatory optical diffraction focus for the MAM can be precisely positioned through the reflection positioning optical system, which is the core part of the experimental detection method and is based on the structured illumination optical sectioning principle. A typical MAM is designed using vectorial angular spectrum (VAS) theory and a genetic algorithm. It was fabricated by focused ion-beam milling. The three-dimensional finite-difference time-domain method is used to verify the intensity distribution of the focus predicted by the VAS theory. For the designed 14-μm-diameter MAM at a wavelength of 640 nm, the simulation result broadly agrees with the experimental result obtained from the transmission microscopic imaging system. The proposed detection method can be used in fields such as optical diffraction focusing and subwavelength resolution imaging.

Enhanced photonic spin Hall effect due to controllable permittivity of alloy film

Photonic spin Hall effect (SHE) is an interesting transport phenomenon, and has attracted growing attention. The spin-dependent splitting of photonic SHE as a weak effect is just tens of nanometers so that it can usually be detected indirectly with the weak measurement techniques. To detect it directly and use it properly, many efforts have been devoted to enhancing the photonic SHE. Recently, the surface plasmon resonance (SPR) excited by a pure nanometal structure is used to enhance the photonic SHE. However, the pure metal permittivities are limited, therefore the regulation of the photonic SHE is also restricted. It is worth mentioning that the alloy made from the pure metal with different composition proportions can achieve the artificial control of permittivity. More importantly, the alloy can also be used to manipulate the SPR. In this paper, we systematically investigate the photonic SHE in a nanoalloy structure composed of BK7 glass, alloy film and air in order to realize the enhancement of photonic SHE. First of all, the resonant angle of SPR varying with the permittivity of alloy is studied by using the angular spectrum theory of beam. It is found that the resonant angle of the SPR is mainly influenced by the real part of the permittivity of alloy, while the imaginary part has little influence on it. The resonant angle of SPR will increase with the increase of the real part of the permittivity. Secondly, the spin-dependent splitting is studied by changing the alloy permittivity when the incident angle is set to be a resonant angle. We find that the distribution of the larger spin-dependent splitting at the resonant angle is zonal. The optimal permittivity of alloy film is ε2=-2.8 + 1.6i, and the alloy can be composed of Ag and Ni according to the Bruggerman theory. Under the condition of the optimal permittivity, the spin-dependent splitting reaches about 1.2×105 nm at a resonant angle of 51.5°, which is about 40 times larger than the previous result in a pure nanometal structure. Finally, when the incident angle is fixed at 44.1°, it is revealed that the spin-dependent splitting varying with the permittivity is axially symmetric and spherical radiation is centered at a maximum value. The farther away from the center, the smaller the corresponding beam shift is. The alloy permittivity in the spherical radiation center is ε2=-10.6 + 1.2i, which can be composed of Au and Ag. The value of spin-dependent splitting reaches about 8000 nm, which is greatly improved when compared with the previous maximum value 3000 nm in a pure nanometal structure. These findings can effectively enhance the photonic SHE and provide theoretical basis for the research and development of nanophotonic devices such as the SPR-based sensor.

Vectorial structure of arbitrary vector vortex beams diffracted by a circular aperture in the far field

In terms of the vector angular spectrum representation and the method of stationary phase, the analytical vectorial structure of the arbitrary vector vortex Laguerre–Gaussian beam on the higher-order Poincaré sphere diffracted by a circular aperture is obtained in the far field. The far-field energy flux distributions of the arbitrary apertured vector vortex beams including cylindrical vector beam, circularly polarized vortex beam, and elliptically polarized vortex beam are numerically demonstrated from the vectorial structure. The analyses show that the topological charge, the truncation parameter, and the ellipticity angle play an important role in the vectorial structure and the evolution properties of the apertured vector vortex beam in the far field. This research is useful for manipulating the vector vortex beams by adjusting states of polarization on the higher-order Poincaré sphere.

Rigorous full-wave calculation of optical forces on dielectric and metallic microparticles immersed in a vector Airy beam.

Based on the generalized Lorenz-Mie theory and the Maxwell stress tensor approach we present the first rigorous full-wave solution of the optical forces acting on spherical microparticles immersed in a two-dimensional vector Airy beam beyond the paraxial approximation. The critical aspect lies in evaluating efficiently and accurately the partial wave expansion coefficients of the incident Airy beam, which are achieved by using the vector angular spectrum representation for a variety of polarizations. The optical field distributions are then simulated to show the self-accelerating and self-healing effects of the Airy beam. The dielectric and gold microparticles are shown to be trapped within the main lobe or the nearby side-lobes mostly by the transverse gradient optical force while driven forward along the parabolic trajectory of the Airy beam by the longitudinal scattering force. It is thus demonstrated theoretically that the vector Airy beam has the capability of precisely transporting both dielectric and metallic microparticles along the prespecified curved paths.

Energy flux density and angular momentum density of radial Pearcey-Gauss vortex array beams in the far field

The longitudinal and transverse energy flux density (EFD) and angular momentum density (AMD) of the radial Pearcey-Gauss vortex array beams are studied by using the vector angular spectrum representation and stationary phase method, where the influence of topological charge, initial phase index, noncanonical strength and the number of beamlet on far-field vectorial structures of the corresponding beam is emphasized. The results show that the topological charge of center optical vortex is always equal to initial phase index for different noncanonical strength. The longitudinal EFDs exhibit different structures by varying initial phase index and the number of beamlet. For an odd-N, the topological charge of center optical vortex can be determined by distinguishing the longitudinal EFD, however for an even-N the topological charge cannot be identified due to the equivalence of longitudinal EFD in the case of positive and negative initial phase index m. The AMD tends to split more branches from the positive and negative direction with increasing the absolute value of topological charge l, The inner structures of AMD are also affected by different beamlets while the zero value of AMD is always retained.

Real-time complex amplitude reconstruction method for beam quality M2 factor measurement.

We present a real-time complex amplitude reconstruction method for determining the beam propagation ratio M2 of laser beams based on the transport of intensity equation (TIE). In this work, a synchronous acquisition system consisting of two identical CCDs is established. Once two beam intensity images at different cross-section positions along the optical axis are captured simultaneously by the system, the complex amplitude of the laser beam can be rapidly reconstructed using TIE algorithm. Then the beam intensity distribution at any section position along its propagation direction can be obtained by using angular spectrum (AS) theory. The beam quality M2 factor is therefore calculated utilizing the second-order moments and hyperbola fitting methods, which conform to the ISO standard. The suitability of this method is verified by the numerical analysis and experiments with the He-Ne and high-power fiber laser sources, respectively. The experimental technique is simple and fast, which allows to investigate laser beams under conditions inaccessible to other methods.

Unraveling beam self-healing.

We show that, contrary to popular belief, diffraction-free beams may not only reconstruct themselves after hitting an opaque obstacle but also, for example, Gaussian beams. We unravel the mathematics and the physics underlying the self-reconstruction mechanism and we provide for a novel definition for the minimum reconstruction distance beyond geometric optics, which is in principle applicable to any optical beam that admits an angular spectrum representation. Moreover, we propose to quantify the self-reconstruction ability of a beam via a newly established degree of self-healing. This is defined via a comparison between the amplitudes, as opposite to intensities, of the original beam and the obstructed one. Such comparison is experimentally accomplished by tailoring an innovative experimental technique based upon Shack-Hartmann wave front reconstruction. We believe that these results can open new avenues in this field.

Numerical visualization of aperiodic scalar optical wave fields

In numerical wave optic simulations using the conventional angular spectrum method, numerical inter-periodic interference occurs as a result of the inherent periodicity of the discrete Fourier transform. In this paper, we propose an aperiodic angular spectrum representation of a scalar optical wave field without numerical inter-periodic interference in an aperiodic area. The visualization of holographic three-dimensional image light field is presented for the validity of the proposed method. It is believed that the proposed method can be broadly applied to visualization of scalar optical wave field represented by the angular spectrum plane wave bases.

Influence of anisotropic turbulence on the orbital angular momentum modes of Hermite-Gaussian vortex beam in the ocean.

Applying the angular spectrum theory, we derive the expression of a new Hermite-Gaussian (HG) vortex beam. Based on the new Hermite-Gaussian (HG) vortex beam, we establish the model of the received probability density of orbital angular momentum (OAM) modes of this beam propagating through a turbulent ocean of anisotropy. By numerical simulation, we investigate the influence of oceanic turbulence and beam parameters on the received probability density of signal OAM modes and crosstalk OAM modes of the HG vortex beam. The results show that the influence of oceanic turbulence of anisotropy on the received probability of signal OAM modes is smaller than isotropic oceanic turbulence under the same condition, and the effect of salinity fluctuation on the received probability of the signal OAM modes is larger than the effect of temperature fluctuation. In the strong dissipation of kinetic energy per unit mass of fluid and the weak dissipation rate of temperature variance, we can decrease the effects of turbulence on the received probability of signal OAM modes by selecting a long wavelength and a larger transverse size of the HG vortex beam in the source's plane. In long distance propagation, the HG vortex beam is superior to the Laguerre-Gaussian beam for resisting the destruction of oceanic turbulence.

Controllable design of super-oscillatory lenses with multiple sub-diffraction-limit foci

The conventional multifocal optical elements cannot precisely control the focal number, spot size, as well as the energy distribution in between. Here, the binary amplitude-type super-oscillatory lens (SOL) is utilized, and a robust and universal optimization method based on the vectorial angular spectrum (VAS) theory and the genetic algorithm (GA) is proposed, aiming to achieve the required focusing performance with arbitrary number of foci in preset energy distribution. Several typical designs of multifocal SOLs are demonstrated. Verified by the finite-difference time-domain (FDTD) numerical simulation, the designed multifocal SOLs agree well with the specific requirements. Moreover, the full-width at half-maximum (FWHM) of the achieved focal spots is close to λ/3 for all the cases (λ being the operating wavelength), which successfully breaks the diffraction limit. In addition, the designed SOLs are partially insensitive to the incident polarization state, functioning very well for both the linear polarization and circular polarization. The optimization method presented provides a useful design strategy for realizing a multiple sub-diffraction-limit foci field of SOLs. This research can find its potentials in such fields as parallel particle trapping and high-resolution microscopy imaging.

Investigation of axial and transverse focal spot sizes of Fresnel zone plates.

An axial spot size formula is given for a standard binary Fresnel zone plate (FZP) illuminated with a linearly polarized beam. The axial spot size, characterized by the full width at half-maximum (FWHM), can be accurately expressed by the Jiang-Wilson formula, viz., dz=0.9λ0/[η-(η2-NA2)1/2], when the number of transparent annuli is no less than 3. λ0 is the illumination light wavelength, η is the refractive index, and NA is the equivalent numerical aperture of a FZP. The transverse spot sizes are near linear with the radial width of the outmost annulus when the corresponding numerical aperture is not very large. This study is carried out based on vectorial angular spectrum theory and has been rigorously confirmed by the three-dimensional finite-difference time-domain method. This finding provides the theoretical basis for applying FZPs in a variety of applications, such as nanolithography, nanoscopy, and high-density data storage.