Rayleigh-Sommerfeld Diffraction Integral 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.

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.

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.

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.

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.

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.

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.

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.

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.

Investigation and Enhancement of the Detectability of Flaws with a Coarse Measuring Grid and Air Coupled Ultrasound for NDT of Panel Materials Using the Re-Radiation Method

Non-destructive ultrasonic testing is utilized widely by industries for quality assurance. For sensitive materials or surfaces, non-contact, non-destructive testing methods are in demand. The air-coupled ultrasound (ACU) is one possible solution. This can be used to investigate large, panel-like objects for delaminations and other flaws. For a high detectability, fine measurement grids are required (typically < λ is used), which results in extremely long data acquisition times that are only practicable for laboratory applications. This paper aimed at reducing the required measurement grid points for obtaining high detectability evaluations. The novel method presented in this paper allows a measurement grid that is much coarser than the resulting grid. The method combines a software refinement of the measured data with the Rayleigh–Sommerfeld diffraction integral for the calculation of the pressure distribution on the object’s surface. This result allows the precise prediction of delaminations and flaws in the tested object. The presented method shows a decrease in the total investigation time by up to 98%.

Non-approximated Rayleigh-Sommerfeld diffraction integral: advantages and disadvantages in the propagation of complex wave fields.

Advantages and disadvantages of the non-approximated numerical implementation of the Rayleigh-Sommerfeld diffraction integral (RSD) are revisited. In this work, it is shown that as trade-off for its large computation load, the non-approximated RSD removes any limitation on the propagation range and does not introduce any artifact in the computed wave field. A non-approximated GPU implementation of the RSD is contrasted with the angular spectrum, the Fresnel transform, and a fast Fourier transform implementation of the RSD. The forecasted phase shift introduced in the propagated wave fields as light is diffracted on complementary apertures and utilized as a metric to quantify the performance of the tested methods. An application to numerical reconstructions with arbitrary shape and size of digital recorded holograms from digital lensless holographic microscopy is presented.

Non-line-of-sight imaging using phasor-field virtual wave optics.

Non-line-of-sight imaging allows objects to be observed when partially or fully occluded from direct view, by analysing indirect diffuse reflections off a secondary relay surface. Despite many potential applications1-9, existing methods lack practical usability because of limitations including the assumption of single scattering only, ideal diffuse reflectance and lack of occlusions within the hidden scene. By contrast,line-of-sight imaging systems do not impose any assumptions about the imaged scene, despite relying on the mathematically simple processes of linear diffractive wave propagation. Here we show that the problem of non-line-of-sight imaging can also be formulated as one of diffractive wave propagation, by introducing a virtual wave field that we term the phasor field. Non-line-of-sight scenes can be imaged from raw time-of-flight data by applying the mathematical operators that model wave propagation in a conventional line-of-sight imaging system. Our method yields a new class of imaging algorithms that mimic the capabilities of line-of-sight cameras. To demonstrate our technique, we derive three imaging algorithms, modelled after three different line-of-sight systems. These algorithms rely on solving a wave diffraction integral, namely the Rayleigh-Sommerfeld diffraction integral. Fast solutions to Rayleigh-Sommerfeld diffraction and its approximations are readily available, benefiting our method. We demonstrate non-line-of-sight imaging of complex scenes with strong multiple scattering and ambient light, arbitrary materials, large depth range and occlusions. Our method handles these challenging cases without explicitly inverting a light-transport model. We believe that our approach will help to unlock the potential of non-line-of-sight imaging and promote the development of relevant applications not restricted to laboratory conditions.

A unifying approach for the vectorial Rayleigh–Sommerfeld diffraction integrals

A unique approximation to the vectorial Rayleigh–Sommerfeld diffraction integrals (RSDI) is developed to study the diffraction properties of apertures illuminated with collimated, converging or diverging polarized waves. The apertures can be bigger or smaller than the wavelength of the illumination light. For circular apertures, it is shown that the approach can be expressed as a simple set of equations containing one-dimensional integrals.

Nonparaxial Propagation Properties of Specially Correlated Radially Polarized Beams in Free Space

A specially correlated radially polarized (SCRP) beam with unusual physical properties on propagation in the paraxial regime was introduced and generated recently. In this paper, we extend the paraxial propagation of an SCRP beam to the nonparaxial regime. The closed-form 3 × 3 cross-spectral density matrix of a nonparaxial SCRP beam propagating in free space is derived with the aid of the generalized Rayleigh–Sommerfeld diffraction integral. The statistical properties, such as average intensity, degree of polarization, and spectral degree of coherence, are studied comparatively for the nonparaxial SCRP beam and the partially coherent radially polarized (PCRP) beam with a conventional Gaussian–Schell-model correlation function. It is found that the nonparaxial properties of an SCRP beam are strikingly different from those of a PCRP beam. These nonparaxial properties are closely related to the correlation functions and the beam waist width. Our results may find potential applications in beam shaping and optical trapping in nonparaxial systems.

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.

One-dimensional integrals to calculate the two-dimensional Rayleigh–Sommerfeld diffraction integrals for non-rotationally symmetric functions and general polarizing illuminating fields

The recently reported (Alcalá Ochoa, 2018) approximation to the Rayleigh–Sommerfeld diffraction integrals (RSDI) is used to obtain a set of expressions to calculate the diffraction properties of non-rotationally symmetric optical systems illuminated with polarization fields. It is proved that the three two-dimensional integrals of RSDI can be expressed accurately from two to eight one-dimensional integrals depending of the type of polarization studied. It is proved also that previous developments based on Taylor’s approximations to the RSDI can be described by these new expressions.

Diffraction-limited superresolution ptychography in the Rayleigh-Sommerfeld regime.

Superresolution in lensless near-field ptychography is demonstrated via the application of a strongly curved illumination function. The reconstruction is performed using the Rayleigh-Sommerfeld diffraction integral, which is implemented via a pixel-size adjustable angular spectrum method. In this manner, the reconstructed object details, which are not only smaller than the pixel size of the sensor but even smaller than the smallest resolvable object detail defined by the effective NA of the 2D sensor, are enabled. The expected resolution, as predicted by the angles of scatter present in the optical configuration, is experimentally validated using a US air force resolution test target. The approach discussed here is not limited to ptychography; it can be extended to other coherent diffractive imaging modalities such as object scanning holography or optical diffraction tomography, so as to enable high-resolution near-field quantitative phase imaging.

Nonparaxial propagation of vector vortex beams diffracted by a circular aperture.

In terms of the vectorial Rayleigh-Sommerfeld diffraction integral, the analytical expressions of the arbitrary vector vortex Laguerre-Gaussian beams on the higher-order Poincaré sphere diffracted by a circular aperture propagating in the nonparaxial and paraxial regimes are presented. The cylindrical vector beam, circularly polarized vortex beam, and elliptically polarized vortex beam are viewed as the special cases of our general result. The analyses show that the nonparaxial evolution properties of the apertured vector vortex beams are determined by the waist width, the truncation parameter, the topological charge, and the ellipticity angle.

Flexible wide-field high-resolution scanning camera for continuous-wave acoustic holography.

We present a system for measuring the amplitude and phase profiles of the pressure field of a harmonic acoustic wave with the goal of reconstructing the volumetric sound field. Unlike optical holograms that cannot be reconstructed exactly because of the inverse problem, acoustic holograms are completely specified in the recording plane. We demonstrate volumetric reconstructions of simple arrangements of objects using the Rayleigh-Sommerfeld diffraction integral and introduce a technique to analyze the dynamic properties of insonated objects.