Vector Diffraction Research Articles

Creation of pure longitudinal super-oscillatory spot.

We present a method that creates a super-oscillatory focal spot of a tightly focused radially polarized beam using the concept of a phase mask. Using vector diffraction theory, we report a super-oscillatory focal spot that is much smaller than the diffraction limit and the super-oscillation criterion. The proposed mask works as a special polarization filter that enhances the longitudinal component and filters out the transverse component of radial polarization at focus, permitting the creation of a pure longitudinal super-oscillatory focal spot.

Diffractive optical element design based on vector diffraction theory and improved PSO-SA algorithm

The existence of the beam vector effect requires rigorous electromagnetic computational methods for the optimization of subwavelength diffractive optical elements (DOEs). To design DOEs with high performance, we proposed an optimization strategy combining a global optimization algorithm with the finite difference time domain (FDTD) method. First, scalar initial solutions are obtained by the Gerchberg–Saxton algorithm. Then, regarding the diffraction efficiency, the uniformity error and figure of merit are calculated by FDTD as evaluation functions, and the optimization of the initial solutions is implemented by improved particle swarm optimization and continues with the simulated annealing algorithm. After several numerical simulations, the analysis results confirm the robustness of the algorithm and the reduction of the computational cost. The experimental data demonstrate that the correlation of diffraction energy orders between the simulated design and fabricated sample are above 85%. This method efficiently improves the DOEs design.

Optical lattices engineered by vector polarization and multisector amplitude modulation

Optical lattices have several applications including super-resolution imaging, lithography, and light-tweezers. Compared with the conventional multibeam interference method, optical lattices in a tightly focused light fields are presented by cylindrical vector polarization illumination and multisector amplitude modulation. The polarization and phase difference of the illumination beam, including the size and relative position of the sectors, were analyzed using the vector diffraction theory. The obtained results indicate that the primitive cell shape of optical lattices can be controlled by the polarization direction of the illumination beam when the relative positions of sectors in the amplitude modulation mask are set. In addition, the period and primitive cell shape of optical lattices with hyper or complex constructions can be controlled by the relative positions of sectors, while the optical lattice zone primarily depends on the sector size. By combining polarization and amplitude modulation in a high numerical aperture optical system, these engineered optical lattices are potentially beneficial in expanding their applications.

Dynamical Modulation of Transverse Orbital Angular Momentum in Highly Confined Spatiotemporal Optical Vortex

Spatiotemporal optical vortices (STOVs) have attracted numerous attention from researchers in recent years due to their intriguing characteristics with transverse orbital angular momentum (OAM) in the spatiotemporal domain. In this work, we numerically analyze the tightly focusing characteristics of higher-order STOVs and present a method to dynamically modulate the transverse OAM in highly confined STOVs. Richards–Wolf vectorial diffraction theory was employed to simulate the three-dimensional spatiotemporal distribution of the focused STOV corresponding to the incident wave packet of topological charge of −2. The simulation results show that the higher-order spatiotemporal vortices in the transversely polarized components of the focused wave packets split into two first-order vortices with topological charge of −1 when the waist radius of the incident wave packet was larger than 40% of the pupil radius of the focusing lens, and the spacing of the two split vortices could be tailored by adjusting the waist radius of the incident wave packet. Meanwhile, the incident spatial waist radius also affected the tilt angle of the phase singularity trace in the z-polarized component of the focused field. The presented method provides a flexible way to dynamically engineer the spatiotemporal vortices in the tightly focused wave packet and may find potential applications in nanophotonics, light–matter interaction, quantum information processing, etc.

Design of single-layer color echelle grating optical waveguide for augmented-reality display.

We proposed a single-layer color echelle grating combined optical waveguide structure for an augmented-reality display. In this structure, we used echelle gratings with super-wavelength periodic scale as in-coupling, relay, and out-coupling elements. The combined propagation of three light beams in the waveguide was realized by overlapping different high diffraction orders of the RGB three primary colors, and deflection of the beam direction between gratings was achieved by conical diffraction generated by the inclined grating. Using the vector diffraction theory, the structural parameters and tolerance ranges of the three types of gratings were optimized, rendering average diffraction efficiencies of the three primary colors of the in-coupling, relay, and out-coupling gratings greater than 74%, 21%, and 35%, respectively. As a result, we obtained dual-channel one-dimensional pupil dilation of the original image and a field-of-view angle of h18.9° × v36.87°.

Separating and transporting of particles using tightly focused radially polarized Bessel–Gaussian beam with conical phase

The intensity distributions near the focus for radially polarized Bessel–Gaussian beam with conical phase by a high numerical aperture objective are computed based on the vector diffraction theory. Results show that the circular focal spot can be changed from one to two on the focal plane by adjusting the truncation parameter β. By increasing the phase modulation parameter L, the focal pattern appeared focal shift. According to the intensity distribution, the forces acting on a particle are calculated. The stability of particle trapping is analyzed. It is shown that a tightly focused radially polarized Bessel–Gaussian beam with conical phase is applicable to trapping, separating and transporting of particles.

Vortex Circular Dichroism: An experimental technique to assess the scalar/vectorial regime of diffraction

Background: In classical electrodynamics, light-matter interactions are modelled using Maxwell equations. The solution of Maxwell equations, which is typically given by means of the electric and magnetic field, is vectorial in nature. Yet it is well known that light-matter interactions can be approximately described in a scalar (polarization independent) way for many optical applications. While the accuracy of the scalar approximation can be theoretically computed, to the best of our knowledge, it has never been determined experimentally. Here, we show that the vectoriality of diffraction can be probed with a new technique: Vortex Circular Dichroism(VCD). Methods: We measure the differential transmission of left and right circularly polarized vortex beams through a set of single circular nano-apertures with diameters ranging from 150 to 1950 nm. We observe that VCD > 0 for smaller diameters, VCD ≲ 0 for intermediate values and VCD ≈ 0 for larger values of the diameter. We also carry out Mie Theory simulations for spheres with the same diameters as the nanoholes and observe that the theoretical and experimental VCD values follow the same trend line. Results: We relate VCD ≠ 0 to a vectorial diffraction, and VCD ≈ 0 to a scalar one. This is corroborated by the simulations, which show that a diffraction process characterized by a VCD ≈ 0 (VCD ≠ 0) is polarization-independent (polarization-dependent). Conclusions: Overall, our results give a wealth of evidence that VCD allows for the experimental assessment of the scalar/vectorial regime of diffraction.

Towards optical toroidal wavepackets through tight focusing of the cylindrical vector two dimensional spatiotemporal optical vortex.

Spatiotemporal optical vortices (STOVs) carrying transverse orbital angular momentum (OAM) are of rapidly growing interest for the field of optics due to the new degree of freedom that can be exploited. In this paper, we propose cylindrical vector two dimensional STOVs (2D-STOVs) containing two orthogonal transverse OAMs in both x-t and y-t planes for the first time, and investigate the tightly focusing of such fields using the Richards-Wolf vectorial diffraction theory. Highly confined spatiotemporal wavepackets with polarization structure akin to toroidal topology is generated, whose spatiotemporal intensity distributions resemble the shape of Yo-Yo balls. Tightly focused radially polarized 2D-STOVs will produce wavepackets towards transverse magnetic toroidal topology, while the focused azimuthally polarized 2D-STOVs will give rise to wavepackets towards transverse electric toroidal topology. The presented method may pave a way to experimentally generate the optical toroidal wavepackets in a controllable way, with potential applications in electron acceleration, nanophotonics, energy, transient light-matter interaction, spectroscopy, quantum information processing, etc.

Estimation of Zernike polynomials for a highly focused electromagnetic field using polarimetric mapping images and neural networks

In this communication, we present a method to estimate the aberrated wavefront at the focal plane of a vectorial diffraction system. In contrast to the phase, the polarization state of optical fields is simply measurable. In this regard, we introduce an alternative approach for determining the aberration of the wavefront using polarimetric information. The method is based on training a convolutional neural network using a large set of polarimetric mapping images obtained by simulating the propagation of aberrated wavefronts through a high-NA microscope objective; then, the coefficients of the Zernike polynomials could be recovered after interrogating the trained network. On the one hand, our approach aims to eliminate the necessity of phase retrieval for wavefront sensing applications, provided the beam used is known. On the other hand, the approach might be applied for calibrating the complex optical system suffering from aberrations. As proof of concept, we use a radially polarized Gaussian-like beam multiplied by a phase term that describes the wavefront aberration. The training dataset is produced by using Zernike polynomials with random coefficients. Two thousand random combinations of polynomial coefficients are simulated. For each one, the Stokes parameters are calculated to introduce a polarimetric mapping image as the input of a neural network model designed and trained for predicting the polynomial coefficients. The accuracy of the neural network model is tested by predicting an unseen dataset (test dataset) with a high success rate.

Digital Mask를 이용한 STED Lithography 기술의 분석적 검증

In this study, a digital mask-based STED lithography technique was proposed, and the effective resolution of the proposed system was calculated analytically. The proposed STED lithography system uses two spatial light modulators to modulate the phase of an excitation beam and a depletion beam, respectively. The excitation spatial light modulator acts as a digital mask to form the patterning image, while the depletion spatial light modulator creates an image surrounding the projected excitation image. Thus, photopolymerization is suppressed by stimulated emission occurring in the depletion focus region surrounding the excitation focus, thereby improving lithographic resolution. Electromagnetic field and intensity distribution were calculated, and the resolution of the proposed STED lithography was simulated based on vectorial diffraction theory. An effective resolution of 72 nm was calculated under the condition of ISUBs/SUB 0.1, which is similar to the resolution of conventional STED lithography at 70 nm. These results analytically confirmed that the proposed STED lithography system could pattern a two-dimensional region simultaneously without the loss of resolution compared to conventional STED lithography.

Effect of optical spatial coherence on localized spin angular momentum density in tightly focused light [Invited

Optical coherence is one of the most fundamental characteristics of light and has been viewed as a powerful tool for governing the spatial, spectral, and temporal statistical properties of optical fields during light-matter interactions. In this work, we use the optical coherence theory developed by Emil Wolf as well as the Richards-Wolf's vectorial diffraction method to numerically study the effect of optical coherence on the localized spin density of a tightly focused partially coherent vector beam. We find that both the transverse spin and longitudinal spin, with the former induced by the out-of-phase longitudinal field generated during strong light focusing and the latter induced by the vortex phase in the incident beam, are closely related to the optical coherence of the incident beam, i.e., with the decrease of the transverse spatial coherence width of the incident beam, the magnitude of the spin density components decreases as well. The numerical findings are interpreted well with the two-dimensional degrees of polarization between any two of the three orthogonal field components of the tightly focused field. We also explore the roles of the topological charge of the vortex phase on enhancing the spin density for the partially coherent tightly focused field. The effect of the incident beam's initial polarization state is also discussed.

Inverse Design of Focused Vector Beams for Mode Excitation in Optical Nanoantennas

We propose a free-space, inverse design of nanostructure's effective mode-matching fields via a backward propagation of tightly focused vector beams to the pupil plane of an aplanatic system of high numerical aperture. First, we study the nanostructure's eigenmodes without considering any excitation fields and then extract the modal near fields in the focal plane. Each modal field is then taken as the desired focal field, the band-limited waves of which are backward propagated to the pupil plane via a reversal of the Richards--Wolf vector diffraction formula. The pupil fields can be designed to be genuinely paraxial by associating the longitudinal electric/magnetic field component with the radial one on the reference sphere. The inversely designed pupil field in turn is propagated forwardly into the focal region to generate the designed focal field, whose distribution over the nanostructure's surface is used to evaluate the overlap between the designed focal field and the modal fields, i.e., the modal expansion coefficients. Studies for a silicon nanodisk monomer, dimer, and tetramer demonstrate the ability of our inverse approach to design the necessary tightly focused vector field that can effectively and exclusively match a certain eigenmode of interest. Compared with the forward beam-shaping method, the inverse design approach tends to yield quantitatively more precise mode-matching field profiles. This work can have a significant impact on optical applications that rely on controllable and tunable mode excitation and light scattering.

Generation of a sub-wavelength sized optical needle with arbitrary longitudinal rotation.

We show that under tight focusing conditions, arbitrarily rotating the longitudinally polarized optical needle in space is possible. Applying the Richards and Wolf vector diffraction methods, the explicit expressions underlying the simultaneous control depth of focus (DoF), intensity suppression of the sidelobes, as well as the orientation of the optical needle can be obtained, and then the strength vectors of the three-dimensional electromagnetic fields can be calculated. Calculations reveal that the sidelobe suppression ratio reaches 5.35% of the principal lobe, the optimal DoF is 5.1λ, and the maximum length is about 18.2λ. In addition, we specify the necessary conditions for rotating the sub-wavelength sized optical needle in the longitudinal field. Such an optical needle with controllable length, purity, and orientation can provide a flexible approach and additional degree of freedom for 3D precise fabrication and some other potential application areas.

Generation and Modulation of Controllable Multi-Focus Array Based on Phase Segmentation.

A Circular-Sectorial Phase Segmentation (CSPS) noniterative method for effectively generating and manipulating muti-focus array (MFA) was proposed in this work. The theoretical model of the CSPS was built up based on vectorial diffraction integral and the phase modulation factor was deduced with inverse fast Fourier transform. By segmenting the entrance pupil into specified regions, which were sequentially assigned with the values carried out by phase modulation factor, the methodology could generate flexible MFAs with desired position and morphology. Subsequently, the CSPS was investigated in parallelized fabrication with a laser direct writing system. The positioning accuracy was greater than 96% and the morphologic consistency of the parallelly fabricated results was greater than 92%.

Dielectric Metalens for Superoscillatory Focusing Based on High-Order Angular Bessel Function.

The phenomenon of optical superoscillation provides an unprecedented way to solve the problem of optical far-field label-free super-resolution imaging. Numerous optical devices that enable superoscillatory focusing were developed based on scalar and vector diffraction theories in the past several years. However, these reported devices are designed according to the half-wave zone method in spatial coordinates. In this paper, we propose a dielectric metalens for superoscillatory focusing based on the diffraction of angular Bessel functional phase modulated vector field, under the inspiration of the tightly autofocusing property of a radially polarized high-order Bessel beam. Based on this kind of metalens with a numerical aperture (NA) of 0.9, the linearly polarized light is converted into a radially polarized one and then focus into a superoscillating focal spot with the size of 0.32λ/NA. This angular spectrum modulation theory involved in this paper provides a different way of designing superoscillatory devices.

Toward High‐Efficiency Ultrahigh Numerical Aperture Freeform Metalens: From Vector Diffraction Theory to Topology Optimization (Laser Photonics Rev. 16(10)/2022)

Toward High‐Efficiency Ultrahigh Numerical Aperture Freeform Metalens: From Vector Diffraction Theory to Topology Optimization (Laser Photonics Rev. 16(10)/2022)

Optimization of Multi-Layer Data Recording and Reading in an Optical Disc

Storage capacity of a conventional optical disc can be enhanced significantly by recording data within the three-dimensional volume. However, spherical aberrations and light-scattering limit the number of layers that can be efficiently recorded and retrieved. In this paper, we show that by optimizing the multi-layer data recording and reading parameters, the number of fabricated layers could be tripled to 60 in commercial grade poly-methyl-methacrylate polycarbonate and reduce the read laser power by a factor of 7 by the 20th layer. Influence of spherical aberration on write laser spot size was studied by using vector diffraction theory and compared with experimental values obtained by monitoring the threshold energy of the write laser with depth. Scattering losses of the read laser were studied by simulating the propagation of light through multi-layered micro-structured material. The main advantage of our technique is its practical implementation, as it refrains from the use of index-matching liquids and specialized optics during the read–write process.

Creating axial bifocal spots by solving the binary phase zone plate in inverse in 4Pi focusing system

We propose an approach to obtain bifocal spots with variable axial distance in 4Pi focusing system. The approach is to inversely solve the radii of a binary phase zone plate (BPZP) according to the conceived 4Pi focal field distribution. The ring radius of the BPZP solved in reverse has an analytic expression by the Richards-Wolf vector diffraction theory, which is directly related to the axial offset distance of the bifocal spots. With the BPZP designed by the analytic expression to modulate the incident radially polarized beam, bifocal spots with variable axial distance can be achieved in the 4Pi focusing region, and two focal spots have the same size. Moreover, some special focal fields, such as optical chain and optical cage, can also be created by selecting suitable axial offset distance to design the BPZP for modulating radially polarized beam. In addition, axial multi-spherical spots can also be generated by superposition of a BPZP and an annular obstruction.

Concise and explicit expressions for typical spatial-structured light beams beyond the paraxial approximation.

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.

Optical vortex fields with an arbitrary orbital angular momentum orientation.

Optical vortex fields with a tilted phase singularity line are associated with a tilted orbital angular momentum (OAM). In this Letter, we propose a method to generate optical vortex fields with arbitrary OAM orientation based on the time-reversal method, vectorial diffraction theory, and a 4Pi optical configuration. The ability to control the 3D OAM orientation may find applications in optical tweezing, light-matter interaction, and spin-orbital coupling.