Rayleigh-Sommerfeld Diffraction Research Articles

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.

Non-paraxial transformation of finite Airy Gaussian beam array in isotropic space

Based on vector Rayleigh-Sommerfeld diffraction integration, we investigate the non-paraxial transmission behavior of the finite Airy Gaussian beam array (FAGBA) in isotropic space. The intensity distribution of the x- component and longitudinal component of the FAGBA at different cross section of the free space are described numerically, respectively. The influences of truncation factor, distribution parameter, wavelength, beam waist radius and refractive index on the FAGBA evolution at several observation planes as well as its side transmission view in isotropic space are also discussed. Through the obtained analytical formulae and conclusions, the beam array transmission locus can be controlled more precisely in non-paraxial situation by altering these parameters to meet the practical usage, especially in area of optical trapping and optical tweezers.

Diffraction integral computation using sinc approximation

We propose a method based on sinc series approximations for computing the Rayleigh-Sommerfeld and Fresnel diffraction integrals of optics. The diffraction integrals are given in terms of a convolution, and our proposed numerical approach is not only super-algebraically convergent, but it also satisfies an important property of the convolution—namely, the preservation of bandwidth. Furthermore, the accuracy of the proposed method depends only on how well the source field is approximated; it is independent of wavelength, propagation distance, and observation plane discretization. In contrast, methods based on the fast Fourier transform (FFT), such as the angular spectrum method (ASM) and its variants, approximate the optical fields in the source and observation planes using Fourier series. We will show that the ASM introduces artificial periodic boundary conditions and violates the preservation of bandwidth property, resulting in limited accuracy which decreases for longer propagation distances. The sinc-based approach avoids both of these problems. Numerical results are presented for Gaussian beam propagation and circular aperture diffraction to demonstrate the high-order accuracy of the sinc method for both short-range and long-range propagation. For comparison, we also present numerical results obtained with the angular spectrum method.

FPGA Accelerator for Real-Time Non-Line-of-Sight Imaging

Non-line-of-sight (NLOS) imaging systems reconstruct hidden scenes using computational methods based on indirect light that diffusely reflected from relay walls. Due to the computation and memory requirements of reconstruction algorithms, real-time NLOS imaging for room-size scenes based on non-confocal data has long been challenging. This paper proposes a field programmable gate array (FPGA) accelerator for the recently proposed Rayleigh-Sommerfeld Diffraction (RSD)-based NLOS reconstruction method. In the proposed accelerator design, ring sampling and radius sampling techniques are proposed to reduce the memory requirements by reconstructing the RSD kernels with a set of kernel bases and ring sampling coefficients during the runtime. Based on that, a customized hardware architecture and the corresponding FPGA design for real-time RSD-based NLOS reconstruction is further proposed. Implementation results show that the proposed FPGA accelerator is capable of reconstructing NLOS scenes at 25 frames per second (FPS), running at a relatively slow clock frequency of 50 MHz. To the best knowledge of the authors, this is the first real-time enabled FPGA accelerator for room-size NLOS imaging with a resolution of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$128\times 128$ </tex-math></inline-formula> .

MEMS–VCSEL design method using a diffraction loss map

The micro–optical resonator of the microelectromechanical system(MEMS) tunable vertical cavity surface emitting laser(VCSEL), which is a gain–guided laser, consists of two mirrors fabricated on semiconductor chips of different materials. To understand the relationship between the curvature radius of the concave mirror and the diameter of the tunnel junction forming the active region, the diffraction loss map was obtained. The Fox–Li method was used with the integral kernel of the Rayleigh–Sommerfeld diffraction to simulate an optical resonator with a large Fresnel number. We derived the guideline for the actual bonding process of the different material chips by simulating with each parameter.

Hyperuniform scalar random fields for lensless, multispectral imaging systems.

We propose a novel framework for the systematic design of lensless imaging systems based on the hyperuniform random field solutions of nonlinear reaction-diffusion equations from pattern formation theory. Specifically, we introduce a new class of imaging point-spread functions (PSFs) with enhanced isotropic behavior and controllable sparsity. We investigate PSFs and modulated transfer functions for a number of nonlinear models and demonstrate that two-phase isotropic random fields with hyperuniform disorder are ideally suited to construct imaging PSFs with improved performances compared to PSFs based on Perlin noise. Additionally, we introduce a phase retrieval algorithm based on non-paraxial Rayleigh-Sommerfeld diffraction theory and introduce diffractive phase plates with PSFs designed from hyperuniform random fields, called hyperuniform phase plates (HPPs). Finally, using high-fidelity object reconstruction, we demonstrate improved image quality using engineered HPPs across the visible range. The proposed framework is suitable for high-performance lensless imaging systems for on-chip microscopy and spectroscopy applications.

Conical holographic display to expand the vertical field of view.

Recently, cylindrical holographic display technology as a 360-degree display technology has attracted much attention. However, all the studies are based on the field of view (FOV) in the azimuth direction, and the issue of FOV in the vertical direction has never been discussed. In this paper, a new holographic display is proposed to expand the vertical FOV by a conical holographic diffraction model, in which the object plane is the outer cylinder and the observation plane is a part of the cone. In this proposed method, the proposed diffraction model is firstly established by the Rayleigh-Sommerfeld diffraction formula, and then the convolution and FFT are used for a fast diffraction calculation. The correctness and effectiveness of our proposed method are verified by the simulation of Young's interference and the numerical reconstructions from the complex amplitude and encoded holograms, respectively. In addition, an accurate relationship between the conical inclination angle and the vertical FOV expansion is analyzed, and the simulation results show that our proposed method can significantly expand the vertical FOV by 0.4 of the original object. Therefore, the issue of the vertical FOV in cylindrical holography is deeply discussed and successfully addressed for the first time.

Long-GRIN-Lens Microendoscopy Enabled by Wavefront Shaping for a Biomedical Microdevice: An Analytical Investigation.

We analytically investigate the feasibility of long graded-index (GRIN)-lens-based microendoscopes through wavefront shaping. Following the very well-defined ray trajectories in a GRIN lens, mode-dependent phase delay is first determined. Then, the phase compensation needed for obtaining diffraction limited resolution is derived. Finally, the diffraction pattern of the lens output is computed using the Rayleigh–Sommerfeld diffraction theory. We show that diffraction-limited resolution is obtained for a 0.5 mm diameter lens with a length over 1 m. It is also demonstrated that different imaging working distances (WDs) can be realized by modifying the phase compensation. When a short design WD is used, a large imaging numerical aperture (NA) higher than 0.4 is achievable even when a low NA lens (NA = 0.1) is used. The long- and thin-GRIN-lens-based microendoscope investigated here, which is attractive for biomedical applications, is being prioritized for use in a clinical stage microdevice that measures three-dimensional drug responses inside the body. The advance described in this work may enable superior imaging capabilities in clinical applications in which long and flexible imaging probes are favored.

Environmentally robust immersion supercritical lens with an invariable sub-diffraction-limited focal spot.

Planar metalenses provide an effective way to break the diffraction barrier in the far field. Their physical mechanism and applications have been intensively studied in the past decade. These investigations on sub-diffraction-limited light modulations have only been applied to specified single immersion environments; however, changing immersion environments can severely degrade their focusing performance, limiting their application potential. In this work, we propose and experimentally demonstrate an environmentally robust immersion supercritical lens (SCL) that can work in various immersion environments. The design of such a lens is based on the vectorial Rayleigh-Sommerfeld diffraction theory combined with a multi-objective optimization algorithm. The sub-diffraction-limited focusing effect has been experimentally demonstrated in commonly used media, including air, water, and oil, with refractive indices of 1.0, 1.33, and 1.51, respectively. Moreover, such a lens can maintain its effective numerical aperture at a fixed value, bringing a unique advantage in that the lateral size of the focal spots exhibits a similar value of ${{317}}\;{{\pm}}\;{{7}}\;{\rm{nm}}$ in all three media. Our demonstration provides the feasibility of SCLs in various application scenarios with multi-immersion environments, such as bioimaging, light trapping, and optical storage.

GPU-accelerated wave-optical model for external-cavity diode lasers.

The spatial distribution of the optical feedback field in an external-cavity diode laser (ECDL) is an important parameter influencing its lifetime and filamentation. From a modeling point of view, the evaluation of the full diffraction integral is required to calculate the propagation of light through thick lenses such as fast axis collimation lenses or beam transformation systems used for high-power diode lasers. In this Letter, we illustrate an algorithm for the accelerated computation of the Rayleigh-Sommerfeld diffraction integral on a graphics processing unit. We further show a validation of the results by measurements on an ECDL.

Rayleigh-Sommerfeld scalar diffraction by apertures moving at relativistic speeds

We have analytically obtained the theoretical results for Rayleigh-Sommerfeld (R-S) scalar diffraction by apertures such as single-slit, double-slit, grating, and circular apertures, moving at relativistic speeds with velocities perpendicular to the direction of incidence. We also have studied diffraction by the single slit of an oscillatory shutter. Significantly, our study probes the relativistic transverse Doppler effect on the intensity pattern of the diffracted field.

A Geometrical Analysis of Diffraction Based on Fractal Inhomogeneous Space

Although the diffraction of light is a simple experiment in optics, no complete interpretation that satisfactorily describes diffraction in any instance has been developed. Classical physics is unable to describe diffraction phenomena by considering photons solely as particles. In addition, modern mathematical solutions based on the wave-particle duality, including Rayleigh–Sommerfeld diffraction theory, are merely approximations, and fail to provide a model that can be applied to the diffraction caused by both transparent and opaque barriers. This study proposes a diffraction model that can account for both single photons and larger particles, such as electrons, in an inhomogeneous space near the surface of the objects, including the edges of the apertures. Furthermore, a three-dimensional model for calculating the light intensity at any arbitrary observation point is presented. This model provides accurate diffraction simulations and is independent of the near and the far-field zones as well as the aperture material.

Refractive index change measurement by quantitative microscopy phase imaging for femtosecond laser written structures

Quantitative phase imaging has been widely exploited and developed during the last decade. This technique is efficient, allowing for quantitative and qualitative spatially resolved measurements of the refractive index profiles. However, there are some measurement limitations for thick samples when exceeding the depth of field-of-view of the imaging system. In this paper, we present an approach based on the complete Rayleigh–Sommerfeld diffraction theory, which allows quantitative retrieve of the refractive index change from the measured optical path difference, especially in the case of thick samples. This approach is validated by a comparison with experimental measurements of refractive index changes of distinct series of cylindrically shaped volume modifications inscribed by femtosecond laser irradiation in silica glass. The results presented hereby, enable quantitative measurement of refractive index change for thick embedded 3D structures.

A large field-of-view metasurface for complex-amplitude hologram breaking numerical aperture limitation

Abstract Owing to the potential to manipulate simultaneously amplitude and phase of electromagnetic wave, complex-amplitude holographic metasurfaces (CAHMs) can achieve improved image-reconstruction quality compared with amplitude-only and phase-only ones. However, prevailing design methods based on Huygens–Fresnel theory for CAHMs, e.g., Rayleigh–Sommerfeld diffraction theory (RSDT), restrict acquisition of high-precision reconstruction in a large field of view (FOV), especially in the small numerical aperture (NA) scenario. To this end, a CAHM consisting of Sine-shaped meta-atoms is proposed in a microwave region, enabled by a novel complex amplitude retrieval method, to realize large FOV holograms while breaking the large NA limitation. Calculations and full-wave simulations demonstrate that the proposed method can achieve superior-quality holograms, even for nonparaxial holograms in a relatively small NA scenario, thus improving FOV and aperture utilization efficiency of CAHMs. The reconstruction comparison of a complex multi-intensity field distribution between CAHM prototypes designed by our method and by RSDT further confirms this point. We also compare both theoretically and experimentally the CAHM by our method with the phase-only metasurface by weighted Gerchberg–Saxton algorithm. Superior-quality holograms with suppressed background noise and relieved deformation, promised by the extra amplitude manipulation freedom, is witnessed. Finally, due to its wavelength irrelevance, the proposed method is applicable to the entire spectrum, spanning from microwave to optics.

Nonparaxial propagation and the radiation forces of the chirped annular Bessel Gaussian beams

We propose a new class of abruptly autofocusing beams, the chirped annular Bessel Gaussian beams (CABGBs), which are based on the radial Bessel profile. By using the Rayleigh–Sommerfeld diffraction formula, the nonparaxial propagation of the CABGBs is investigated. We find that the initial chirp with negative value tends to intensify the abruptly autofocusing property by shortening the focal length, shrinking the focal spot size, and increasing the peak intensity contrast. The optical trapping effect of the CABGBs acting on a Rayleigh particle is analyzed, and we find that the particle can be stably trapped in three dimensions if the parameters of the CABGBs are appropriate. If the absolute value of the negative chirp increases, the trapping position is shifted toward the input plane, and both the longitudinal and transverse trapping forces rise. A higher magnitude of the radiation forces can be reached if the incident power increases.

Diffraction-Limited Imaging with a Graphene Metalens*Supported by the Scholarship of China Scholarship Council (Grant No. 201706030189), the Industrial Transformation Training Centres Scheme (Grant No. IC180100005), and the National Natural Science Foundation of China (Grant No. 61935001).

Planar graphene metalens has demonstrated advantages of ultrathin thickness (200 nm), high focusing resolution (343 nm) and efficiency (>32%) and robust mechanical strength and flexibility. However, diffraction-limited imaging with such a graphene metalens has not been realized, which holds the key to designing practical integrated imaging systems. In this work, the imaging rule for graphene metalenses is first derived and theoretically verified by using the Rayleigh-Sommerfeld diffraction theory to simulate the imaging performance of the 200 nm ultrathin graphene metalens. The imaging rule is applicable to graphene metalenses in different immersion media, including water or oil. Based on the theoretical prediction, high-resolution imaging using the graphene metalens with diffraction-limited resolution (500 nm) is demonstrated for the first time. This work opens the possibility for graphene metalenses to be applied in particle tracking, microfluidic chips and biomedical devices.

Nonparaxial phasor-field propagation.

Growing interest in non-line-of-sight (NLoS) imaging, colloquially referred to as "seeing around corners", has led to the development of phasor-field (P-field) imaging, wherein the field envelope of amplitude-modulated spatially-incoherent light is manipulated like an optical wave to directly probe a space that is otherwise shielded from view by diffuse scattering. Recently, we have established a paraxial theory for P-field imaging in a transmissive geometry that is a proxy for three-bounce NLoS imaging [J. Dove and J. H. Shapiro, Opt. Express27(13) 18016 (2019)]. Our theory, which relies on the Fresnel diffraction integral, introduces the two-frequency spatial Wigner distribution (TFSWD) to efficiently account for specularities and occlusions that may be present in the hidden space and cannot be characterized with P-field formalism alone. However, because the paraxial assumption is likely violated in many, if not most, NLoS scenarios, in the present paper we overcome that limitation by deriving a nonparaxial propagation formula for the P field using the Rayleigh-Sommerfeld diffraction integral. We also propose a Rayleigh-Sommerfeld propagation formula for the TFSWD and provide a derivation that is valid under specific partial-coherence conditions. Finally, we report a pair of differential equations that characterize free-space TFSWD propagation without restriction.

Multi-directional beam steering using diffractive neural networks

The modern-day resurgence of machine learning has encouraged researchers to revisit older problem spaces from a new perspective. One promising avenue has been implementing deep neural networks to aid in the simulation of physical systems. In the field of optics, densely connected neural networks able to mimic wave propagation have recently been constructed. These diffractive deep neural networks (D2NN) not only offer new insights into wave propagation, but provide a novel tool for investigating and discovering multi-functional diffractive elements. In this paper, we derive an efficient GPU-friendly D2NN methodology based on Rayleigh-Sommerfeld diffraction. We then use the implementation to virtually forge cascades of optical phase masks subject to different beam steering conditions. The input and output conditions we use to train each D2NN instance is based on commercial electro-optic modulated waveguide systems to encourage experimental follow-on. In total, we analyze the beam steering efficacy of 27 individual D2NN instances which explore different permutations of input sources, mask cascades, and output steering targets.

Design of infrared microspectrometer based on phase-modulated axilenses

We design and characterize a novel axilens-based diffractive optics platform that flexibly combines efficient point focusing and grating selectivity and is compatible with scalable top-down fabrication based on a four-level phase mask configuration. This is achieved using phase-modulated compact axilens devices that simultaneously focus incident radiation of selected wavelengths at predefined locations with larger focal depths compared with traditional Fresnel lenses. In addition, the proposed devices are polarization-insensitive and maintain a large focusing efficiency over a broad spectral band. Specifically, here we discuss and characterize modulated axilens configurations designed for long-wavelength infrared (LWIR) in the 6 µm–12 µm wavelength range and in the 4 µm–6 µm midwavelength infrared (MWIR) range. These devices are ideally suited for monolithic integration atop the substrate layers of infrared focal plane arrays and for use as compact microspectrometers. We systematically study their focusing efficiency, spectral response, and cross-talk ratio; further, we demonstrate linear control of multiwavelength focusing on a single plane. Our design method leverages Rayleigh–Sommerfeld diffraction theory and is validated numerically using the finite element method. Finally, we demonstrate the application of spatially modulated axilenses to the realization of a compact, single-lens spectrometer. By optimizing our devices, we achieve a minimum distinguishable wavelength interval of Δ λ = 240 n m at λ c = 8 µ m and Δ λ = 165 n m at λ c = 5 µ m . The proposed devices add fundamental spectroscopic capabilities to compact imaging devices for a number of applications ranging from spectral sorting to LWIR and MWIR phase contrast imaging and detection.