Rayleigh-Sommerfeld Diffraction Research Articles

The Fresnel Approximation and Diffraction of Focused Waves

In this paper, diffraction of scalar waves by a screen with a circular aperture is explored, considering the incidence of either a collimated beam or a focused wave, a historical review of the development of the theory is presented, and the introduction of the Fresnel approximation is described. For diffraction by a focused wave, the general case is considered for both high numerical aperture and for finite values of the Fresnel number. One aim is to develop a theory based on the use of dimensionless optical coordinates that can help to determined the general behaviour and trends of different system parameters. An important phenomenon, the focal shift effect, is discussed as well. Explicit expressions are provided for focal shift and the peak intensity for different numerical apertures and Fresnel numbers. This is one application where the Rayleigh–Sommerfeld diffraction integrals provide inaccurate results.

Cohesive framework for non-line-of-sight imaging based on Dirac notation.

The non-line-of-sight (NLOS) imaging field encompasses both experimental and computational frameworks that focus on imaging elements that are out of the direct line-of-sight, for example, imaging elements that are around a corner. Current NLOS imaging methods offer a compromise between accuracy and reconstruction time as experimental setups have become more reliable, faster, and more accurate. However, all these imaging methods implement different assumptions and light transport models that are only valid under particular circumstances. This paper lays down the foundation for a cohesive theoretical framework which provides insights about the limitations and virtues of existing approaches in a rigorous mathematical manner. In particular, we adopt Dirac notation and concepts borrowed from quantum mechanics to define a set of simple equations that enable: i) the derivation of other NLOS imaging methods from such single equation (we provide examples of the three most used frameworks in NLOS imaging: back-propagation, phasor fields, and f-k migration); ii) the demonstration that the Rayleigh-Sommerfeld diffraction operator is the propagation operator for wave-based imaging methods; and iii) the demonstration that back-propagation and wave-based imaging formulations are equivalent since, as we show, propagation operators are unitary. We expect that our proposed framework will deepen our understanding of the NLOS field and expand its utility in practical cases by providing a cohesive intuition on how to image complex NLOS scenes independently of the underlying reconstruction method.

Technique for enhancing the accuracy of the Rayleigh-Sommerfeld convolutional diffraction through the utilization of independent spatial sampling.

The Rayleigh-Sommerfeld diffraction integral (RSD) is a rigorous solution that precisely satisfies both Maxwell's equations and Helmholtz's equations. It seamlessly integrates Huygens' principle, providing an accurate description of the coherent light propagation within the entire diffraction field. Therefore, the rapid and precise computation of the RSD is crucial for light transport simulation and optical technology applications based on it. However, the current FFT-based Rayleigh-Sommerfeld integral convolution algorithm (CRSD) exhibits poor performance in the near field, thereby limiting its applicability and impeding further development across various fields. The present study proposes, to our knowledge, a novel approach to enhance the accuracy of the Rayleigh-Sommerfeld convolution algorithm by employing independent sampling techniques in both spatial and frequency domains. The crux of this methodology involves segregating the spatial and frequency domains, followed by autonomous sampling within each domain. The proposed method significantly enhances the accuracy of RSD during the short distance while ensuring computational efficiency.

Fast scaled cylindrical holography based on scaled convolution

Recently, the 360°display of cylindrical holography has garnered significant attention. However, existing studies have been limited to concentric cylindrical surfaces with equal heights, which restricts the height of the objects. In this paper, a fast scaled cylindrical holography is proposed based on scaled convolution to break this constraint. Firstly, the scale cylindrical diffraction is derived using the Rayleigh–Sommerfeld diffraction formula for scaled cylindrical holography, in which the object’s height can be adjusted. Then, a method of fast scaled cylindrical diffraction is presented by scaled convolution based on scaled Fourier Transform, and it significantly reduces computational time, enhancing its practical feasibility. Furthermore, the proposed method is extended to 3D objects using a layer-oriented approach. Moreover, the effectiveness and feasibility of the proposed method are verified through numerical simulations and optical experiments. Therefore, the issue of the constraint on the object’s height in cylindrical holography is thoroughly discussed and effectively addressed for the first time. This breakthrough paves the way for the development of a zoom-able cylindrical holographic display with immense potential for applications in mixed reality and beyond.

Digital holographic interferometry method for tracking detector modules displacement

In high energy particle physics scattering experiments, the precision of the reconstructed particle tracks can be fundamental. For this reason, a method for detecting the displacement of tracking detector modules is developed. The modules are silicon planes mounted on a frame and used in the MUonE project, which aims at a precision measurement of the scattering angle of elastic muon-electron scattering. From the scattering angle, the hadronic contribution to the anomalous magnetic moment of the muon is extracted. To achieve the desired accuracy, the position of the tracking detector planes must be monitored. The allowable relative displacements must be less than 10 μ m. To meet the specifications and to monitor as large an area of the detector as possible, a digital holographic interferometer was developed. It is based on a novel lens-less design in off-axis holographic geometry. Light from a fiber-coupled laser source is split by a fiber beam splitter, with one output used to illuminate the detector plane and the other for the reference beam. The two beams produce an interference pattern on a CMOS image sensor. To obtain relative displacement information, successive images are superimposed on an initial reference image and reconstructed by solving the Rayleigh-Sommerfeld diffraction integral taking into account the spherical wavefronts of the beams. The interference fringes that appear in the reconstructed holographic image provide a measure of the relative displacement of the detector plane compared to the initial position. The performance of the reconstruction method used was verified with the proposed setup at a real tracking station.

Review of diffractive deep neural networks

In 2018, a UCLA research group published an important paper on optical neural network (ONN) research in the journal Science. It developed the world’s first all-optical diffraction deep neural network (DNN) system, which can perform MNIST dataset classification tasks at near-light-speed. To be specific, the UCLA research group adopted a terahertz light source as the input, established the all-optical diffractive DNN (D2NN) model using the Rayleigh-Sommerfeld diffraction theory, optimized the model parameters using the stochastic gradient descent algorithm, and then used 3D printing technology to make the diffraction grating and built the D2NN system. This research opened a new ONN research direction. Here, we first review and analyze the development history and basic theory of artificial neural networks (ANNs) and ONNs. Second, we elaborate D2NN as holographic optical elements (HOEs) interconnected by free space light and describe the theory of D2NN. Then we cover the nonlinear research and application scenarios for D2NN. Finally, the future directions and challenges of D2NN are briefly discussed. Hopefully, our work can provide support and help to researchers who study the theory and application of D2NN in the future.

Cascaded metasurfaces for high-purity vortex generation

We introduce a new paradigm for generating high-purity vortex beams with metasurfaces. By applying optical neural networks to a system of cascaded phase-only metasurfaces, we demonstrate the efficient generation of high-quality Laguerre-Gaussian (LG) vortex modes. Our approach is based on two metasurfaces where one metasurface redistributes the intensity profile of light in accord with Rayleigh-Sommerfeld diffraction rules, and then the second metasurface matches the required phases for the vortex beams. Consequently, we generate high-purity LGp,l optical modes with record-high Laguerre polynomial orders p = 10 and l = 200, and with the purity in p, l and relative conversion efficiency as 96.71%, 85.47%, and 70.48%, respectively. Our engineered cascaded metasurfaces suppress greatly the backward reflection with a ratio exceeding −17 dB. Such higher-order optical vortices with multiple orthogonal states can revolutionize next-generation optical information processing.

Intensity response model and measurement error compensation method for chromatic confocal probe considering the incident angle

Chromatic confocal measurement technology is extensively applied in precision surface profiling within the industrial domain. However, due to limited measurement space and the complex shape of surfaces, it becomes challenging to ensure that the optical axis is aligned with the normal direction of the surface, particularly in regions with steep slopes. Although the reflected light can be captured when the probe is tilted within the numerical aperture angle, it introduces measurement uncertainty. In this paper, we investigate the reflective characteristics of incident light and develop a chromatic confocal light intensity response model that accounts for the incident angle. This model is based on the Rayleigh-Sommerfeld diffraction and dispersion theory. Simulation results demonstrate that the tilt angle results in a shift in the peak wavelength of the reflected spectrum. To compensate for the measurement error caused by this wavelength shift, we establish a calibration curve representing the angle and wavelength variation for the chromatic confocal measurement system. Experimental findings reveal that after compensation, the measurement error attributable to the incident angle is reduced by approximately 72 %.

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.

Sub-micron weak phase particle characterization using the reconstructed volume intensities from in-line digital holography microscopy

Digital holography microscopy is a powerful technique for retrieving particle properties that are often challenging to obtain with other methods. However, analyzing holographic data using non-linear fitting algorithms and machine learning is difficult without prior information. In this paper, we present a novel technique for sizing and determining the refractive index of sub-micron, weak phase, spherical particles using the Rayleigh-Sommerfeld diffraction theory. Our method involves reconstructing the volume of synthetic holograms using the HoloPy software for particles of different sizes and material properties. We then fit a Gaussian function at the point of numerical re-focus, as given by the first local minima after the Gouy-phase shift, for size and sub-pixel position estimates. By retrieving the magnitude and location of the maximum scattered intensity, we build a scattered interpolant that relates all the parameters of interest. We demonstrate that our technique had a mean error in particle refractive index of 1.24 ± 1.74% with synthetic data and accurately estimates the size and refractive index of similar-sized nanoparticles in experiments. Compared to least-squares fitting, our method performs similarly with synthetic data but outperformed it with experimental data, showcasing its superior accuracy and reliability. Furthermore, we use our technique combined with function-fitting approaches. This hybrid method uses our proposed technique to approximate the initial conditions of non-linear fitting algorithms, leading to improved accuracy. Finally, we explore the potential of using the intensity cubes as well-defined thresholds to differentiate particles from interference caustics in a reconstructed volume. Overall, our results demonstrate the efficacy and accuracy of our technique for retrieving particle properties from holographic data, even when prior information was not available. This technique could have broad applications in the field of nanoparticle characterization.

Zone Plate Virtual Lenses for Memory-Constrained NLOS Imaging

Phasor Fields is a wave-based formulation that allows computing time-efficient NLOS reconstructions using Rayleigh-Sommerfeld diffraction kernels. Unfortunately, these kernels are heavy in memory, hampering the integration in memory-constrained devices. We propose alternative kernels based on zone plates that require 16 times less memory to store, offering an attractive trade-off.

Analytic plenoptic camera diffraction model and radial distortion analysis due to vignetting.

Using a mathematical approach, this paper presents a generalization of semi-analytical expressions for the point spread function (PSF) of plenoptic cameras. The model is applicable in the standard regime of the scalar diffraction theory while the extension to arbitrary main lens transmission functions generalizes a priori formalism. The accuracy and applicability of the model is well verified against the exact Rayleigh-Sommerfeld diffraction integral and a rigorous proof of convergence for the PSF series expression is made. Since vignetting can never be fully eliminated, it is critical to inspect the image degradation it poses through distortions. For what we believe is the first time, diffractive distortions in the diffraction-limited plenoptic camera are closely examined and demonstrated to exceed those that would otherwise be estimated by a geometrical optics formalism, further justifying the necessity of an approach based on wave optics. Microlenses subject to the edge diffraction effects of the main lens vignetting are shown to translate into radial distortions of increasing severity and instability with defocus. The distortions due to vignetting are found to be typically bound by the radius of the geometrical defocus in the image plane, while objects confined to the depth of field give rise to merely subpixel distortions.

A Full-Color Holographic System Based on Taylor Rayleigh–Sommerfeld Diffraction Point Cloud Grid Algorithm

Real objects-based full-color holographic display systems usually collect data with a depth camera and then modulate the input light source to reconstruct the color three-dimensional scene of the real object. However, at present, the main problems of the real-time high quality full-color 3D display are slow speed, low reconstruction quality, and high consumption of hardware resources caused by excessive computing. Based on the hybrid Taylor Rayleigh–Sommerfeld diffraction algorithm and previous studies on full-color holographic systems, our paper proposes Taylor Rayleigh–Sommerfeld diffraction point cloud grid algorithm (TR-PCG), which is to perform Taylor expansion on the radial value of Rayleigh–Sommerfeld diffraction in the hologram generation stage and modify the data type to effectively accelerate the calculation speed and ensure the reconstruction quality. Compared with the wave-front recording plane, traditional point cloud gridding (PCG), C-PCG, and Rayleigh–Sommerfeld PCG without Taylor expansion, the computational complexity is significantly reduced. We demonstrate the feasibility of the proposed method through experiments.

Physics-model-based neural networks for inverse design of binary phase planar diffractive lenses.

The inverse design approach has enabled the customized design of photonic devices with engineered functionalities through adopting various optimization algorithms. However, conventional optimization algorithms for inverse design encounter difficulties in multi-constrained problems due to the substantial time consumed in the random searching process. Here, we report an efficient inverse design method, based on physics-model-based neural networks (PMNNs) and Rayleigh-Sommerfeld diffraction theory, for engineering the focusing behavior of binary phase planar diffractive lenses (BPPDLs). We adopt the proposed PMNN to design BPPDLs with designable functionalities, including realizing a single focal spot, multiple foci, and an optical needle with size approaching the diffraction limit. We show that the time for designing single device is dramatically reduced to several minutes. This study provides an efficient inverse method for designing photonic devices with customized functionalities, overcoming the challenges based on traditional data-driven deep learning.

Generating superposed terahertz perfect vortices via a spin-multiplexed all-dielectric metasurface

Perfect optical vortices (POVs), characterized as a ring radius independent of topological charge (TC), possess extensive application in particle manipulation and optical communication. At present, the complex and bulky optical device for generating POVs has been miniaturized by leveraging the metasurface, and either spin-dependent or spin-independent POV conversions have been further accomplished. Nevertheless, it is still challenging to generate superposed POVs for incidences with orthogonal circular polarization. Here, a spin-multiplexed all-dielectric metasurface method for generating superposed POVs in the terahertz frequency range is proposed and demonstrated. By using the multiple meta-atom comprised structure as the basic unit, the complex amplitude of two superposed POVs is modulated, decoupled, and subsequently encoded to left- and right-handed circular polarization incidences. Furthermore, two kinds of metasurfaces are fabricated and characterized to validate this controlling method. It is demonstrated that the measured intensity and phase distributions match well with the calculation of the Rayleigh–Sommerfeld diffraction integral, and the radius of superposed POVs is independent of TCs. This work provides promising opportunities for developing ultracompact terahertz functional devices applied to complex structured light generation and terahertz communication, and exploring sophisticated spin angular momentum and orbital angular momentum interactions like the photonic spin-Hall effect.

Ring and Radius Sampling Based Phasor Field Diffraction Algorithm for Non-Line-of-Sight Reconstruction.

Non-Line-of-Sight (NLOS) imaging reconstructs occluded scenes based on indirect diffuse reflections. The computational complexity and memory consumption of existing NLOS reconstruction algorithms make them challenging to be implemented in real-time. This paper presents a fast and memory-efficient phasor field-diffraction-based NLOS reconstruction algorithm. In the proposed algorithm, the radial property of the Rayleigh Sommerfeld diffraction (RSD) kernels along with the linear property of Fourier transform are utilized to reconstruct the Fourier domain representations of RSD kernels using a set of kernel bases. Moreover, memory consumption is further reduced by sampling the kernel bases in a radius direction and constructing them during the run-time. According to the analysis, the memory efficiency can be improved by as much as 220×. Experimental results show that compared with the original RSD algorithm, the reconstruction time of the proposed algorithm is significantly reduced with little impact on the final imaging quality.

HTRSD: Hybrid Taylor Rayleigh-Sommerfeld diffraction.

Computing wave propagation is of the utmost importance in computational optics, especially three-dimensional optical imaging and computer-generated hologram. The angular spectrum method, based on fast Fourier transforms, is one of the efficient approaches; however, it induces sampling issues. We report a Hybrid Taylor Rayleigh-Sommerfeld diffraction (HTRSD) that achieves more accurate and faster wave propagation than the widely used angular spectrum method.

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.

Fiber spectrum analyzer based on planar waveguide array aligned to a camera without lens

We propose and experimentally demonstrate a fiber spectrum analyzer based on a planar waveguide chip butt-coupled with an input fiber and aligned to a standard camera without any free-space optical elements. The chip consists of a single-mode waveguide to connect with the fiber, a beam broadening area, and a waveguide array in which the lengths of the waveguides are designed for both wavelength separation and beam focusing. The facet of the chip is diced open so that the outputs of the array form a near-field emitter. The far field are calculated by the Rayleigh-Sommerfeld diffraction integral. We show that the chip can provide a focal depth on the millimeter scale, allowing relaxed alignment to the camera without any fine-positioning stage. Two devices with 120 and 220 waveguides are fabricated on the polymer waveguide platform. The measured spectral width are 0.63 nm and 0.42 nm, respectively. This simple and practical approach may lead to the development of a spectrum analyzer for fiber that is easily mountable to any commercial camera, thereby avoiding the complication for customized detectors as well as electronic circuits afterwards.

Rayleigh-Sommerfeld scalar diffraction by rotating apertures

We have analytically explored the Rayleigh-Sommerfeld scalar diffraction for various rotating apertures such as rotating single-slit, rotating double-slit, rotating linear diffraction grating, and rotating regular polygonal aperture. Here the apertures are considered to be uniformly rotating along the axes perpendicular to the plane of the apertures and the diffracted fields are analysed in the far-field zone. We have compared the time-averaged intensity distributions for the rotating apertures with the intensity distributions for static circular apertures of the same area. We also have obtained angular speed of rotation dependent Fraunhofer diffraction formulae for the rotating apertures, in particular, the rotating single-slit, the rotating double-slit, and the rotating linear diffraction grating.