Rayleigh-Sommerfeld Diffraction Theory Research Articles

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.

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 %.

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.

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.

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.

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.

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.

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.

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.

Hybrid cryptosystem based on diffraction transfer function and phase-truncated Fourier transform encryption

A hybrid cryptosystem based on phase-truncated Fourier transform (PTFT) encryption and diffraction transfer function is designed. The designed cryptosystem cannot be decrypted by phase-retrieval-based attacks to which conventional phase-truncated Fourier transform cryptosystems are proven vulnerable. Compared with conventional PTFT encryption, the encryption process of this proposed cryptosystem needs no extra devices. With the help of Rayleigh–Sommerfeld diffraction theory, the inverse diffraction optically is realized and the decryption process can be performed using a liquid crystal spatial light modulator. Unlike conventional PTFT cryptosystem, no useful information can be retrieved in case of lacking any of ciphertext, private keys and diffraction parameters. The optical setup and numerical simulation results are provided to verify the effectiveness, security and robustness of proposed hybrid cryptosystem.

Generalized Rayleigh-Sommerfeld Diffraction Theory for Metasurface-Modulating Paraxial and Non-Paraxial Near-Field Pattern Estimation

In this paper, different diffraction theories for estimating the diffraction field patterns modulated by metasurfaces are firstly revisited. Further reformulation of these theories is performed to better reveal their inherent mechanisms and differences. To compute the metasurface-modulating paraxial and/or non-paraxial diffraction field patterns within the near-field region, including the evanescent area, a universal pattern-propagation Eigenfactor is introduced to generalize Rayleigh-Sommerfeld diffraction theory. To investigate its applicability and accuracy, a representative monofocal metasurface with an ultrahigh numerical aperture of 0.96, together with two coplanar and non-coplanar multifocal holographic metasurfaces, are constructed as illustrative examples. Their near-field patterns are calculated by the generalized Rayleigh-Sommerfeld (GRS) diffraction integral and compared with those extracted by the finite-different time-domain full wave analysis, generalized Huygens-Fresnel principle, and Huygens's Principle. It is demonstrated that within the near-field region including the non-paraxial and evanescent area, the GRS diffraction integral provides the best and satisfactory agreement with the full wave simulation, and thus offers a more accurate and efficient tool for quantitative analysis and iterative optimization.

An accurate design of graphene oxide ultrathin flat lens based on Rayleigh-Sommerfeld theory

Graphene oxide (GO) ultrathin flat lenses have provided a new and viable solution to achieve high resolution, high efficiency, ultra-light weight, integratable and flexible optical systems. Current GO lenses are designed based on the Fresnel diffraction model, which uses a paraxial approximation for low numerical aperture (NA) focusing process. Herein we develop a lens design method based on the Rayleigh-Sommerfeld (RS) diffraction theory that is able to unambiguously determine the radii of each ring without the optimization process for the first time. More importantly, the RS design method is able to accurately design GO lenses with arbitrary NA and focal length. Our design is experimentally confirmed by fabricating high NA GO lenses with both short and long focal lengths. Compared with the conventional Fresnel design methods, the differences in ring positions and the resulted focal length are up to 13.9% and 9.1%, respectively. Our method can be further applied to design high performance flat lenses of arbitrary materials given the NA and focal length requirements, including metasurfaces or other two-dimensional materials.

Is the Rayleigh-Sommerfeld diffraction always an exact reference for high speed diffraction algorithms?

In several areas of optics and photonics the behavior of the electromagnetic waves has to be calculated with the scalar theory of diffraction by computational methods. Many of these high-speed diffraction algorithms based on a fast-Fourier-transformation are approximations of the Rayleigh-Sommerfeld-diffraction (RSD) theory. In this article a novel sampling condition for the well-sampling of the Riemann integral of the RSD is demonstrated, the fundamental restrictions due to this condition are discussed, it will be demonstrated that the restrictions are completely removed by a sampling below the Abbe resolution limit and a very general unified approach for applying the RSD outside its sampling domain is given.

Optimization of Iterative Algorithms for Focusing Light Through Scattering Media

A light beam transmitted through scattering media usually turns into disordered speckles. Fortunately, there are iterative algorithms to focus light transmitted through scattering media using a spatial phase modulator (SPM). In this work, we optimized the conventional iterative algorithms to achieve greater efficiency via theory and simulations. Two independent processes, characterizing the transmission matrix (TM) of the scattering media and calculating the phase compensation of the SPM, are included in our optimized algorithm. While the TM of the scattering media is determined through the interference effect and the light reciprocity principle, the phase compensation of the SPM is calculated using the Rayleigh–Sommerfeld diffraction theory. By addressing these two processes separately, accelerating the process of characterizing the TM of the scattering media, and theoretically calculating the phase compensation of the SPM, our optimized algorithm is helpful for fast imaging through scattering media.

Dynamic Transmit-Receive Beamforming by Spatial Matched Filtering for Ultrasound Imaging with Plane Wave Transmission.

During conventional ultrasound imaging, the need for multiple transmissions for one image and the time of flight for a desired imaging depth limit the frame rate of the system. Using a single plane wave pulse during each transmission followed by parallel receive processing allows for high frame rate imaging. However, image quality is degraded because of the lack of transmit focusing. Beamforming by spatial matched filtering (SMF) is a promising method which focuses ultrasonic energy using spatial filters constructed from the transmit-receive impulse response of the system. Studies by other researchers have shown that SMF beamforming can provide dynamic transmit-receive focusing throughout the field of view. In this paper, we apply SMF beamforming to plane wave transmissions (PWTs) to achieve both dynamic transmit-receive focusing at all imaging depths and high imaging frame rate (>5000 frames per second). We demonstrated the capability of the combined method (PWT + SMF) of achieving two-way focusing mathematically through analysis based on the narrowband Rayleigh-Sommerfeld diffraction theory. Moreover, the broadband performance of PWT + SMF was quantified in terms of lateral resolution and contrast from both computer simulations and experimental data. Results were compared between SMF beamforming and conventional delay-and-sum (DAS) beamforming in both simulations and experiments. At an imaging depth of 40 mm, simulation results showed a 29% lateral resolution improvement and a 160% contrast improvement with PWT + SMF. These improvements were 17% and 48% for experimental data with noise.

基于瑞利-索末菲衍射理论的近场大衍射角衍射光学元件的设计

基于瑞利-索末菲衍射理论的近场大衍射角衍射光学元件的设计

Numerical study of a Fresnel zone plate based lens for a 2 μm × 2 μm CMOS image sensor pixel

Fresnel zone plate (FZP) topology is proposed to be used as the lens media for the light concentration in a 2μm×2μm CMOS (complementary metal oxide semiconductor) image sensor pixel. According to the pixel size scaling, the width of each ring of the FZP becomes smaller than the incident light’s wavelength. Rayleigh-Sommerfeld diffraction theory is used to analyze dielectric scattering effects on the interfaces. Both the analytical and finite difference time domain (FDTD) numerical simulations agree on the focal distance of 2.7μm. Moreover, angular response and crosstalk of the final CMOS image sensor (CIS) are calculated and analyzed. It is shown that the FZP based pixel has lower crosstalk at angles above 19°, compared with similar microlens (μlens) based pixels. Furthermore, wider response is observed in comparison with the μlens based pixels.