Angular Spectrum Representation Research Articles

Overmodulation and phase correction of virtual images propagated by holographic curved waveguide for near-eye display

Generally, current type of waveguide used for near-eye displays is planar waveguide, which belongs to flat panel display. Compared to display system in three-dimensional space, flat panel display limits people's view of the 3-D world and is not conducive to the integration of virtual and reality. This paper proposes a holographic curved waveguide system for near-eye display. In order to analyze the propagation process of light beam in waveguide, an iterative algorithm is established based on angular spectrum theory to get the intensity distribution from each total internal reflection point. According to the simulation results and experimental phenomenon, we have developed phase compensation measures for overmodulation during beam propagation in the waveguide. The research shows that using curved waveguide can achieve beam transmission with less aberration after introducing an appropriate phase compensation in a distance, and it can also achieve beam transmission with lossless energy within a large radius range. Experiments have proven that the final coupling effect is basically consistent with the simulation results, this means that our proposal about curved waveguide has potential application value in near-eye display systems.

Propagation of elliptical Gaussian vortex beam based on angular spectrum representation

Vortex beams with hollow elliptical ring structure have attracted broad interest in applications such as micro-particle trapping and manipulation. The paper investigates the dependence of the hollow elliptic ring structure of vortex beams on the beam order and the topological charge, while propagating in the near- and far-field zones. The electric field of the beam is formulated by using the angular spectrum method, which satisfies rigorously Maxwell's equation. The electric field, linear momentum and orbital angular momentum are calculated. Numerical results show that the hollow elliptical ring structure approximately remains unchanged over a finite propagation distance, accompanied by a rotation whose angle depends on the topological charge. In the far-field zone, the ring structure is broken, leading to two bright spots and several dark islands. It is found that the beam profile in the near-field zone is mainly related to the beam order but in the far-field zone is primarily affected by the topological charge. The results presented in the paper may be useful in predicting and explaining the interaction between the elliptic hollow Gaussian vortex beam and the micro-particles, such as the movement of micro-particles along the orbit and the rolling rotation.

Polarization-insensitive high-numerical-aperture metalens for wide-field super-resolution imaging.

The development of super-oscillatory lens (SOL) offers opportunities to realize far-field label-free super-resolution microscopy. Most microscopes based on a high numerical aperture (NA) SOL operate in the point-by-point scanning mode, resulting in a slow imaging speed. Here, we propose a high-NA metalens operating in the single-shot wide-field mode to achieve real-time super-resolution imaging. An optimization model based on the exhaustion algorithm and angular spectrum (AS) theory is developed for metalens design. We numerically demonstrate that the optimized metalens with an NA of 0.8 realizes the imaging resolution (imaging pixel size) about 0.85 times the Rayleigh criterion. The metalens can achieve super-resolution imaging of an object with over 200 pixels, which is one order of magnitude higher than the unoptimized metalens. Our method provides an avenue toward single-shot far-field label-free super-resolution imaging for applications such as real-time imaging of living cells and temporally moving particles.

Analytical calculation of beam profile and orbital angular momentum spectrum of Laguerre Gaussian beams reflected from a graphene plasmonic structure.

In this paper, Laguerre Gaussian (LG) beams with different topological charges are used for excitation of surface plasmon polaritons (SPPs) through a graphene layer inserted in the Otto-configuration. By utilizing the angular spectrum representation (ASR) and Lorenz-gauge vector potential, an explicit analytical expression is derived for the electromagnetic fields of the reflected beam. At the optimal excitation state of graphene SPPs, the reflected beam exhibits a distinctive field profile characterized by two identical crescent-shaped lobes separated by a vertical strip with null intensity. Furthermore, in the absence of external magnetic field, the orbital angular momentum (OAM) spectrum of the reflected beam at the optimal excitation of SPPs reveals the annihilation of central OAM mode and the generation of two equal OAM sidebands, regardless of the incident OAM topological charge. Furthermore, the phase distributions of electric field of the reflected beam confirm the existence of OAM sidebands in the vicinity of optimal SPPs excitation. As the system is taken away from the optimal excitation of SPPs by introduction of an external magnetic field or increasing the chemical potential or increasing the incident angle, both central and sideband modes appear in the OAM spectrum of the reflected beam. In this case, when the topological charge of the incident wave increases, the weight of central OAM mode decreases while the weight of sidebands increases. In contrast, in the presence of external magnetic field, at the optimal excitation of SPPs, both central OAM and sidebands modes exist in the reflected beam such that the weight of central modes increases with the external magnetic field. This effect is also confirmed by plotting the phase distributions of the reflected beam at different external magnetic fields and for different incident topological charges. Therefore, the manipulation of graphene plasmons characteristics leads to the control of OAM sideband generation.

Needle of longitudinally polarized light using the circular Airy beam.

An optical needle is created using a radially polarized circular Airy beam with a conical angle, stemmed from the auto-focusing property of light beams. The utilization of the angular spectrum representation serves to illustrate the field distributions of the optical needle, and an explicit formula is provided to describe the angular spectrum of the light beam. The findings suggest that the optical needle exhibits a long depth of focus and well uniformity, and the full width at half maximum of the transverse field distribution is approximately 0.38 λ beyond the diffraction limit. The uniformity of the optical needle can be tailored by adjusting the width of the primary ring, the decay parameter, and the conical angle. Additionally, the depth of focus of the optical needle significantly improves as the radius of the primary ring increases while still maintaining well uniformity. It may find applications in high-resolution optical imaging and optical manipulation.

Generation of a sub-wavelength optical needle by wavelength shifting and center masking of a Fresnel zone plate

In this paper, a method to generate a sub-wavelength optical needle is proposed and demonstrated by wavelength shifting and center masking in the Fresnel zone plate (FZP). In theory, the vectorial angular spectrum (VAS) theory combined with genetic algorithm (GA) is used for the design of the center-masking FZP for generating optical needle, and finite-difference time-domain (FDTD) method is used for theoretical validation. In experiment, an amplitude-type center-masking FZP with a processing error of 5 nm is fabricated by focused ion beam etching (FIB), and the focusing intensity distribution of the optical needle is measured based on a self-made device. Finally, a sub-wavelength optical needle in far field is obtained, featuring a depth of focus of 7.16 µm, a central focal length of 26.87 µm, and a minimum full width at half maximum of 500 nm and 467 nm in x and y directions, respectively. This study provides both theoretical and experimental foundations for the practical application of FZPs.

On evanescent waves and blowing-ups of the finite series technique in spherical wave expansion of shaped beams

In generalized Lorenz-Mie theory, the beam shape coefficients of Gauss-like beams obtained with finite series method blow up when the wave order is high. The blowing-ups caused by the loss of significant digits may be suppressed by increasing numerical precisions. Once the elimination of such blowing-ups is carried out, blowing-up phenomena still occur whose origin is still not clear to us. The work presented here attempts to provide an explanation on this issue. Based on the angular spectrum representation of the beam field, it is found that the blowing-up originates from the contribution of evanescent waves.

General diffraction integral for converging freeform vectorial fields

Herein it is presented a new vector diffraction integral capable to compute the diffraction patterns for aspheric and freeform wavefronts, with different pupil shapes, and different polarization states. The aforementioned vector diffraction integral is based on the energy conservation theorem, the far field approximation, and the angular spectrum representation of optical fields. The integral is validated through illustrative examples whose results are as expected and coincides with the related works.

Focusing performances of high numerical aperture Fresnel zone plates under different immersion media

The focusing performances of high-numerical-aperture (NA) Fresnel zone plates (FZP) under different immersion media are investigated by combining the vectorial angular spectrum (VAS) theory and the finite-difference time-domain method. It has been found that as the refractive index deviation increases, the focus approximately linearly shifts along the positive z-axis. Simultaneously, the transverse size of the focusing spot gradually decreases, and super-resolution focusing is realized. However, the peak intensity declines with an increase in the refractive index. In addition, when the deviation of the refractive index of the actual immersion medium from the design value is less than about 30%, VAS theory can correctly estimate the focusing light field distribution, except when the FZP has only one or two transparent annuli. On the contrary, VAS theory cannot achieve an accurate prediction when the deviation is larger. These findings will benefit the practical application of high-NA FZPs.

Efficient post-processing of electromagnetic plane wave simulations to model arbitrary structured beams incident on axisymmetric structures

The study of an optical beam interacting with material structures is a fundamental of nanophotonics. Computational electromagnetic solvers facilitate the rapid calculation of the scattering from material structures with arbitrary geometry and complexity, but have limited efficiency when employing structured excitation fields. We have developed a post-processing method and package that can efficiently calculate the full three-dimensional electric and magnetic fields for any optical beam incident on a particle or structure with at least one axis of continuous rotational symmetry, called an axisymmetric body (such as a sphere, cylinder, cone, torus or surface). Provided an initial batch of plane wave simulations is computed, this open-source package combines data from computational electromagnetic solvers in a post-processing fashion using the angular spectrum representation to create arbitrarily structured beams, including vector vortex beams. Any and all possible incident beams can be generated from the initial batch of PWSs, without the need for further simulations. This allows for efficiently performing parameter sweeps such as changing the angle of illumination or translating the particle position relative to the beam, all in post-processing, with no need for additional time-consuming simulations. We demonstrate some applications by numerically calculating optical force and torque maps for a spherical plasmonic nanoparticle in a tightly focused Gaussian beam, a plasmonic nanocone in an azimuthally polarised beam and compute the fields of a non-paraxial Laguerre–Gaussian vortex beam reflecting on a multilayered surface. We believe this package, called BEAMS, is a valuable tool for rapidly quantifying electromagnetic systems that are beyond traditional analytical methods.

A Systematic Summary and Comparison of Scalar Diffraction Theories for Structured Light Beams

Structured light beams have recently attracted enormous research interest for their unique properties and potential applications in optical communications, imaging, sensing, etc. Since most of these applications involve the propagation of structured light beams, which is accompanied by the phenomenon of diffraction, it is very necessary to employ diffraction theories to analyze the obstacle effects on structured light beams during propagation. The aim of this work is to provide a systematic summary and comparison of the scalar diffraction theories for structured light beams. We first present the scalar fields of typical structured light beams in the source plane, including the fundamental Gaussian beams, higher-order Hermite–Gaussian beams, Laguerre–Gaussian vortex beams, non-diffracting Bessel beams, and self-accelerating Airy beams. Then, we summarize and compare the main scalar diffraction theories of structured light beams, including the Fresnel diffraction integral, Collins formula, angular spectrum representation, and Rayleigh–Sommerfeld diffraction integral. Finally, based on these theories, we derive in detail the analytical propagation expressions of typical structured light beams under different conditions. In addition, the propagation of typical structured light beams is simulated. We hope this work can be helpful for the efficient study of the propagation of structured light beams.

Reflection and refraction of higher-order Hermite-Gaussian beams: a vector wave analysis.

Hermite-Gaussian beams, as a typical kind of higher-order mode laser beams, have attracted intensive attention because of their interesting properties and potential applications. In this paper, a full vector wave analysis of the higher-order Hermite-Gaussian beams upon reflection and refraction is reported. The explicit analytical expressions for the electric and magnetic field components of the reflected and refracted Hermite-Gaussian beams are derived with the aid of angular spectrum representation and vector potential in the Lorenz gauge. Based on the derived analytical expressions, local field distributions of higher-order Hermite-Gaussian beams reflection and refraction at a plane interface between air and BK7 glass are displayed and analyzed.

Applying the angular spectrum representation to calculate the optical force density generated in dielectrics by tightly focused laser beams

The Angular Spectrum Representation (ASR) is applied to describe semi-analytically the optical force densities acting on linear dielectric media when a quasi-monochromatic tightly focused Gaussian beam is applied. This method is seen to be inherently faster than conventional finite-difference schemes. Numerical simulations of the optical force densities were also performed and found to be in agreement with the literature, providing a complementary tool for the study of opto-mechanical effects in matter.

Rigorous full-wave calculation of optical forces on microparticles immersed in vector Pearcey beams.

We present the electromagnetic fields of vector Pearcey beams by employing the vector angular spectrum representation. The beams maintain the inherent properties of autofocusing performance and inversion effect. Based on the generalized Lorenz-Mie theory and Maxwell stress tensor approach, we derive the partial-wave expansion coefficients of arbitrary beams with different polarization and the rigorous solution to evaluate the optical forces. Furthermore, we investigate the optical forces experienced by a microsphere placed in vector Pearcey beams. We study the effects on the longitudinal optical force arising from the particle size, permittivity and permeability. This exotic curved trajectory transport of particles by vector Pearcey beams may find applications in the case where the transport path is partly blocked.

Modeling of a multireflector terahertz imaging system using a geometrical optics model based on angular spectrum theory.

We propose a simulation method for a multireflector terahertz imaging system. The description and verification of the method are based on an existing active bifocal terahertz imaging system at 0.22 THz. Using the phase conversion factor and angular spectrum propagation, the computation of the incident and received fields requires only a simple matrix operation. The phase angle is used to calculate the ray tracking direction, and the total optical path is used to calculate the scattering field of defective foams. Compared with the measurements and simulations of aluminum disks and defective foams, the validity of the simulation method is confirmed in the field of view of 50 cm × 90 cm at 8 m. This work aims to develop better imaging systems by predicting their imaging behavior for different targets before manufacturing.

Research on spatial frequency shift super-resolution imaging based on evanescent wave illumination

<sec>In spite of the success of fluorescence microscopes (such as stimulated emission depletion microscopy, stochastic optical reconstruction microscopy and photoactivated localization microscopy) in biomedical field, which have realized nanometer scale imaging resolution and promoted the great development of bio-medicine, the super-resolution imaging method for non-fluorescent sample is still scarce, and the resolution still has a big gap to nanometer scale. Among existing methods, structured illumination microscopy, PSF engineering, super-oscillatory lens and microsphere assisted nanoscopy are more mature and widely used. However, limited by the theory itself or engineering practice, the resolutions of these methods are hard to exceed 50 nm, which limits their applications in many fields. Enlightened by synthetic aperture technique, researchers have proposed spatial frequency shift super-resolution microscopy through shifting and combining the spatial frequency spectrum of imaging target, which is a promising super-resolution imaging scheme, for its resolution limit can be broken through continually. Currently, owing to the limitation of the refractive index of optical material, the wavelength of illumination evanescent wave is hard to shorten when this wave is generated at prism surface via total internal reflection, which determines the highest resolution of this spatial frequency shift super-resolution imaging system. Another deficiency of this scheme is the difference in imaging resolution among different directions, for the image has the highest resolution only in the direction along the wave vector of illumination evanescent wave; while, the image has the lowest resolution in the direction perpendicular to the wave vector, which is the same as that obtained by far-field illumination.</sec><sec>In order to solve the above thorny questions, a new model of generating the evanescent wave is proposed, which can generates an omnidirectional evanescent wave with arbitrary wavelength based on the phase modulation of nano-structure, and solve the both problem in imaging system at the same time. To verify the our scheme, we set up a complete simulation model for spatial frequency shift imaging scheme, which includes three parts: the generation of evanescent wave and the interaction of the evanescent wave with the nano-structures at imaging target, which can be simulated with FDTD algorithm; the propagation of light field from near-field to far-field region, from the sample surface to the focal plane of objective lens, which can be calculated with angular spectrum theory; the propagation of light field from the focal place to the image plane, which can be worked out with Chirp-Z transform.</sec><sec>Firstly, with this complete simulation model, we compare the resolution of microscopy illuminated by evanescent wave with that by propagating wave. The experimental results verify the super-resolution imaging ability of evanescent wave illumination and the influence of prism refractive index. The higher the refractive index, the shorter the wavelength of evanescent wave is and the higher the resolution of spatial frequency shift imaging system. Secondly, we demonstrate the resolution difference in a series of directions with a three-bar imaging target rotated to different directions. The result shows that the highest imaging resolution occurs in the direction of illumination evanescent wave vector, and the lowest resolution appears in the direction perpendicular to the wave vector. Finally, we simulate the evanescent wave generated by nano-strcuture and demonstrate its properties of wavelength and vector direction. When applied to near-field illumination super-resolution imaging, the omnidirectional evanescent wave solves the both problems in the model of total internal reflection from the prism surface.</sec><sec>Therefore, the advantages of our scheme are higher imaging resolution and faster imaging speed, no need for multi-direction and multiple imaging, and also image post-processing. In this study, a new spatial frequency shift super-resolution imaging method is proposed, which lays a theoretical foundation for its applications.</sec>

Reflection of the individual multipoles fields from a flat substrate in the angular spectrum approach

In this report, we give the derivation of the electromagnetic field of multipoles over a substrate, using the angular spectrum representation. The multipole series up to the electric quadrupole is considered. If necessary, this approach can easily be extended to multipoles of any order. As a result, analytical expressions for the scattered and reflected multipole fields as an infinite sum of plane waves of different polarizations are obtained.

Scattering by a low-velocity charge with applied magnetic fields in a Laguerre–Gaussian beam

A method to analyze the scattering characteristics of a Laguerre–Gaussian (LG) vortex beam by a low-velocity charge in a steady applied magnetic field is investigated in this paper. Based on the plane wave angular spectrum representation and the electromagnetic radiation theory of a moving charge, the expressions of the scattered field and scattered power per unit solid angle are presented. Considering a low-velocity electron, the distribution of the scattered power per unit solid angle is numerically simulated. The effects of electron motion parameters, beam parameters, and applied magnetic field are discussed in detail. The results show that the longitudinal motion of the electron, which is parallel to the direction of the applied magnetic field, makes the distribution of the scattered power no longer maintain radial consistency but has a concave–convex distribution. Due to the periodicity of the circular motion of the electron in the applied magnetic field, the scattering power distribution pattern consists of many annular lobes. The spectral characteristics of the backscattered field are analyzed by fast Fourier transform. It is confirmed that the scattered field satisfies the rotational and the linear Doppler effects, and the accuracy of the formula provided in this paper is verified. This work lays a foundation for further study of the scattering characteristics of plasma on vortex electromagnetic waves and helps to promote the in-depth development of vortex beams in the field of plasma diagnosis.

Achromatic and wide field of view metalens based on the harmonic diffraction and a quadratic phase.

Although metalens has made breakthroughs in various imaging applications due to its ultrathin, lightweight, and multi-functionality, simultaneously achieving wide field of view (WFOV) and achromatic imaging remains a challenge. Here, we demonstrate a harmonic metalens with a quadratic phase profile that enables WFOV imaging and achromatic imaging at certain discrete wavelengths. First, we quantitatively explain why the quadratic phase enables WFOV imaging using its Fourier Transform (FT). Second, we derive the complex-amplitude transmittance formula of a harmonic metalens. The derived formula integrating with the Angular Spectrum theory can calculate the transmitted field distributions of a harmonic metalens. Third, we propose an achromatic WFOV metalens based on the harmonic diffraction and quadratic phase at a large numerical aperture (NA=0.76), which enables achromatic imaging at wavelengths λ = 600 nm, 1200 nm with a wide field of view (FOV) of 100°. The proposed scheme will undoubtedly expand the applications of metalens in the imaging area.

Theoretical study of the lateral displacement of random fields at interfaces.

The lateral displacement of a totally reflected light beam from the position expected by geometrical optics has been known for some time. The effect, known as the Goos-Hänchen shift, may be understood in terms of the phase changes acquired upon reflection at the interface by the plane wave components of the angular spectrum representation of the incident field. In this work, we study the shifts that occur in the reflection of random fields from flat dielectric interfaces. We find that, similar to the well-known effect for beams, random fields can also present lateral shifts in reflection experiments, and that the shifts acquired by speckle patterns are the same as those of an equivalent beam. For tapered speckle beams, we show that the shifts of the mean intensity do not depend on the average beam size, but mainly on speckle size.