Sommerfeld Diffraction Research Articles

A computer algebra derivation of the Rayleigh–Sommerfeld diffraction formula by a plane screen

The Rayleigh–Sommerfeld diffraction formula is derived without resorting to generalized or Green functions starting from the plane wave or angular spectrum representation of the scalar field after a plane screen. All the steps are clarified by means of computer algebra software.

Nonparaxial propagation of flat-topped beam diffracted in the present of a rectangular aperture

On the basis of the vectorial Rayleigh–Sommerfeld diffraction integral, an analytical nonparaxial propagation equations of vectorial flat-topped beams through a rectangular aperture are derived. The far-field and paraxial propagation equations of flat-topped beams through a rectangular aperture are treated as the special cases. It is found that the beam parameters w0x, w0y and two truncation parameter a, b affect the propagation properties of flat-topped beams.

Nonparaxial propagation of a multi-Gaussian Schell-model beam

Based on the vectorial Rayleigh–Sommerfeld diffraction integral formulas, analytical expressions for the elements of 3 × 3 cross-spectral density matrix of polarized multi-Gaussian Schell-model (MGSM) beam’s nonparaxial propagation in free space are derived, and used to investigate its propagation properties. As a special case of nonparaxial propagation, the paraxial case has also been considered by the reduced paraxial approximation expressions. The distributions of spectral density, spectral degree of coherence, and spectral degree of polarization of nonparaxial MGSM beam are illustrated by numerical examples. Results show that the statistical properties of MGSM beam’s nonparaxial propagation in free space are closely related to the initial beam’s correlation width related parameter and transverse width related parameter These characteristics of MGSM beam might find applications in laser scanning, optical trapping, and free space optical communication.

Nonparaxial propagation of a rectangular multi-Gaussian Schell-model beam

Abstract According to the vectorial Rayleigh–Sommerfeld diffraction integral formulas, analytical expressions for the elements of 3×3 cross-spectral density matrix of polarized Rectangular Multi-Gaussian Schell-Model (RMGSM) beam’s nonparaxial propagation in free space are derived, and the paraxial analytical expression has also been presented as a special case of nonparaxial propagation. Its statistical properties, including the distributions of spectral density, spectral degree of coherence, and spectral degree of polarization of nonparaxial RMGSM beam are numerically demonstrated. Results reveal that the statistical properties of RMGSM beam’s nonparaxial propagation in free space are closely related to the initial beam’s transverse width σ and correlation widths δ x and δ y . These characteristics of RMGSM beam might be useful in optical material surface processing, imaging, and free space optical communication.

Propagation of a Laguerre–Gaussian correlated Schell-model beam beyond the paraxial approximation

Recently, Laguerre–Gaussian correlated Schell-model (LGCSM) beam was introduced theoretically [Mei and Korotkova, Opt. Lett. 38(2), (2013), 91–93] and generated experimentally [Wang et al., Opt. Lett. 38(11), (2013), 1814–1816]. In this paper, we treat the propagation of a LGCSM beam beyond the paraxial approximation. Based on the generalized Rayleigh–Sommerfeld diffraction integral, analytical expressions for the intensity and spectral degree of coherence of a nonparaxial LGCSM beam propagating in free space are obtained, and the corresponding results of a paraxial LGCSM beam are also derived as a special case. Our numerical results show that the nonparaxial propagation properties of a LGCSM beam are closely related to the initial beam parameters, such as the beam waist width, the coherence width and the mode order. Furthermore, in the far field, flat-topped, hollow intensity profiles can be formed through varying the initial beam parameters, which is different from that of a nonparaxial Gaussian Schell-model beam.

A novel method for the design of diffractive optical elements based on the Rayleigh–Sommerfeld integral

The original design method for diffractive optical elements (DOEs) is limited to cases of small-angle diffraction due to the Fresnel or Fraunhofer diffraction integral. In this paper, we propose a new method based on the Rayleigh–Sommerfeld diffraction integral, which does not have this limit. In this method, the target intensity distribution is first modified via coordinate transformation and intensity adjustment. Then, the modified Gerchberg–Saxton algorithm is used to achieve the phase distribution of the DOE. To verify this method, simulations and experiments are performed. The results show that the original method is effective only when the full diffraction angle of the DOE is below 25°. Conversely, this method can achieve the target with both small and large diffraction angles.

A class of interface problems for the Helmholtz equation in

Motivated by the Sommerfeld diffraction problem, we consider interface problems for the n‐dimensional Helmholtz equation in (due to xn>0 or xn<0, respectively, with main interest in n = 3) where the interface Γ = ∂Ω is identified with and divided into two parts, Σ and Σ′, with different transmission conditions of first and second kind. These two parts are half‐spaces of (half‐planes for n = 3) and more general sets in the first part of the paper. The aim of this work is to construct resolvent operators acting from the interface data into the energy space H1(Ω) and representing the solution explicitly. The approach is based upon a factorization conception for Wiener–Hopf operators (according to the interface equations), the so‐called Wiener–Hopf factorization through an intermediate space, that includes Simonenko's well‐known ‘generalized factorization of matrix functions in Lp spaces’ and avoids an interpretation of the factors as unbounded operators. Copyright © 2015 John Wiley & Sons, Ltd.

Vector Fourier optics of anisotropic materials

The Fourier optics technique is founded on the transfer function derived from the scalar wave equation and thus has traditionally been applied to monochromatic scalar fields propagating paraxially in isotropic lossless materials. We review scalar and vector diffraction theory to show that the scalar Fourier optics technique can be seamlessly extended to the case of time-dependent nonparaxial and evanescent vector fields propagating in anisotropic and/or absorbing materials. The missing piece is shown to be the Fourier-domain vector boundary conditions that have recently been derived from the real-domain vector Rayleigh–Sommerfeld diffraction integral. These boundary conditions, complemented by the anisotropic transfer functions provided by the roots of the Booker quartic, combine the rigor of Green’s function integrals with the intuitive simplicity and general applicability of Fourier optics. We illustrate the power of this technique via analytically simple solutions to traditionally complex problems such as Fresnel transmission between arbitrary media, uniaxial conoscopy, and biaxial conical refraction. The accuracy of vector near-field diffraction is validated via comparison with the finite-difference time-domain method.

A new kind of superimposing morphology for enhancing the light scattering in thin film silicon solar cells: Combining random and periodic structure

In this article, a new type of superimposing morphology comprised of a periodic nanostructure and a random structure is proposed for the first time to enhance the light scattering in silicon-based thin film solar cells. According to the framework of the Reyleigh—Sommerfeld diffraction algorithm and the experimental results of random morphologies, we analyze the light-scattering properties of four superimposing morphologies and compare them with the individual morphologies in detail. The results indicate that the superimposing morphology can offer a better light trapping capacity, owing to the coexistence of the random scattering mechanism and the periodic scattering mechanism. Its scattering property will be dominated by the individual nanostructures whose geometrical features play the leading role.

Fast nonparaxial scalar focal field calculations.

An efficient algorithm for calculating nonparaxial scalar field distributions in the focal region of a lens is discussed. The algorithm is based on fast Fourier transform implementations of the first Rayleigh-Sommerfeld diffraction integral and assumes that the input field at the pupil plane has a larger extent than the field in the focal region. A sampling grid is defined over a finite region in the output plane and referred to as a tile. The input field is divided into multiple separate spatial regions of the size of the output tile. Finally, the input tiles are added coherently to form a summed tile, which is propagated to the output plane. Since only a single tile is propagated, there are significant reductions of computational load and memory requirements. This method is combined either with a subpixel sampling technique or with a chirp z-transform to realize smaller sampling intervals in the output plane than in the input plane. For a given example the resulting methods enable a speedup of approximately 800× in comparison to the normal angular spectrum method, while the memory requirements are reduced by more than 99%.

Generation of vector beams in planar photonic crystal cavities with multiple missing-hole defects.

We propose a novel method to generate vector beams in planar photonic crystal cavities with multiple missing-hole defects. Simulating the resonant modes in the cavities, we observe that the optical fields in each defect have different phase and polarization state distributions, which promise the compositions of vector beams by the scattered light from the defects. The far-field radiation patterns of the cavity modes calculated via the Sommerfeld diffraction theory present vector beams possessing hollow intensity profiles and polarization singularities. In addition, the extraction efficiencies of the vector beams from the cavities could be improved by modifying the air-holes surrounding the defects. This planar photonic crystal cavity-based vector beam generator may provide useful insights for the on-chip controlling of vector beams in their propagations and interactions with matter.

Nonparaxial circular grating diffraction properties of radially polarized beams

The Rayleigh–Sommerfeld diffraction integral is employed to study the circular grating nonparaxial diffraction properties of the radially polarized beams, and the analytical expression for the diffraction electric field is derived. The analyses indicate that the properties of the nonparaxial diffraction field strongly depend on the grating parameters and beam waist. When the radially polarized beam is diffracted by the sub-wavelength grating, only the 0th diffraction ring is obtained, or the distinct higher order diffraction rings will emerge. Compared with the paraxial diffraction, the nonparaxial diffraction intensity field is weaker. But when the beam waist is larger than 3λ the paraxial approximation is valid; thus their distinction will be negligible. At last, it shows that in the radially polarized beams nonparaxial diffraction field, the longitudinal component not only leads to stronger diffraction intensity than the azimuthally polarized beams, but also makes the polarization degenerate.

A Riemannn surface approach for diffraction from rational wedges

This paper aims at the explicit analytical representation of acoustic, electromagnetic or elastic, time-harmonic waves diffracted from wedges in R3 in a correct setting of Sobolev spaces. Various problems are modelled by Dirichlet or Neumann boundary value problems for the 2D Helmholtz equation with complex wave number. They have been analyzed before by several methods such as the Malinzhinets method using Sommerfeld integrals, the method of boundary integral equations from potential theory or Mellin transformation techniques. These approaches lead to results which are particularly useful for asymptotic and numerical treatment. Here we develop new representation formulas of the solutions which are based upon the solutions to Sommerfeld diffraction problems. We make use of symmetry properties, which require a generalization of these formulas to Riemann surfaces in order to cover arbitrary rational angles of the wedge. The approach allows us to prove well-posedness in suitable Sobolev spaces and to obtain explicit solutions in a new, perhaps surprising, form provided the angle is rational, i.e., α = πm/n where m,n ∈ N . Mathematics subject classification (2010): 35J05, 78A45, 45E10, 47B35.

The effect of frequency on secondary wave mode generation for ultrasonic health monitoring of fatigue damaged plate structures

ABSTRACTA computational frequency analysis of secondary wave mode generation is considered in this paper. Secondary wave mode conversion at the tip of a through‐thickness slit in a circular isotropic plate is analysed similar to a classical Sommerfeld's diffraction problem. The analysis is useful for quantitative non‐destructive evaluation of fatigue cracked plate structures that involve propagation of a single wave mode (a vertically polarised symmetric Lamb mode) and measurement of the amplitude of a new mode (a horizontally polarised shear mode). A two‐dimensional finite element model is first introduced as a baseline reference for the non‐dispersive scenario. This model is used to study the effect that P‐wave angle of incidence has on the mode converted S‐wave from the tip of the slit. A dispersive three‐dimensional finite element model of the circular plate is then introduced at a number of frequency‐thickness products within the range 0.5–3 MHz.mm−1. Within this frequency–thickness range, significant changes to the through‐thickness displacement profiles of the incident S0 Lamb mode occur. The secondary SH0 plate wave mode generated by mode conversion from the 45o incident S0 Lamb mode at the tip of a slit is analysed to establish the efficiency of the mode conversion. Visualisations of the wave fields demonstrate that at frequency–thickness products higher than the first symmetric Lamé mode (where there is no in‐plane surface displacement), the in‐plane through‐thickness displacement profile of the incident S0 Lamb mode becomes incompatible with that of the SH0 plate mode, and the mode conversion phenomenon is thus severely compromised.

Rayleigh–Sommerfeld diffraction formula in k space

An angular spectrum representation in three dimensions is used to develop three-dimensional Fourier forms of the first and second Rayleigh-Sommerfeld diffraction formulae and the Kirchhoff diffraction formula. For forward-propagating waves, these reduce to three-dimensional Fourier representations for diffraction in the forward half-space.

TOWARDS WAVE DISTURBANCE IN PORTS COMPUTED BY A DETERMINISTIC CONVOLUTION-TYPE MODEL

Among the wide range of potential applications of the convolution-type approach to deterministic wave modeling, this paper looks into the challenge of complex shaped domains. The canonical case of diffraction around a semiinfinite vertical barrier, the 'Sommerfeld diffraction' case, is first studied. Focusing on locally constant water depth, the convolution method is related to a boundary integral representation by which the impulse response function representing the convolution kernel is related to a Green's function for the Laplace equation. This provides a framework for determining the impulse response function by solving a local, three-dimensional Laplace problem prior to the time-stepping of the wave transformation problem. For the Sommerfeld case, numerical results for the impulse response function near the barrier are computed numerically and compared with an analytical solution. For complex-shaped domains, numerical determination of the impulse response functions is the only solution. A very preliminary example of application to wave disturbance in a real port is given.

On numerical integration with high-order quadratures: with application to the Rayleigh–Sommerfeld integral

The paper focusses on the advantages of using high-order Gauss–Legendre quadratures for the precise evaluation of integrals with both smooth and rapidly changing integrands. Aspects of their precision are analysed in the light of Gauss’ error formula. Some “test examples” are considered and evaluated in multiple precision to $$\approx $$ 200 significant decimal digits with David Bailey’s multiprecision package to eliminate truncation/rounding errors. The increase of precision on doubling the number of subintervals is analysed, the relevant quadrature attribute being the precision increment. In order to exemplify the advantages that high-order quadrature afford, the technique is then used to evaluate several plots of the Rayleigh–Sommerfeld diffraction integral for axi-symmetric source fields defined on a planar aperture. A comparison of the high-order quadrature method against various FFT-based methods is finally given.

The diffraction by a small aperture

We study the diffractions by one small circular aperture, and numerically calculate the diffraction intensity distribution with the help of the finite-difference time-domain (FDTD) technique. The diffraction patterns are presented, the variation of the near-field diffraction with the radius and the propagation distance can be observed clearly. The one-dimensional distributions on the axis, along the boundary shadow and the transverse direction are discussed too. These results are also compared with to ones obtained according to Rayleigh–Sommerfeld diffraction and the Fresnel approximation. We think that this work does not only enrich the diffraction theory of the circular aperture, but it is also helpful for the applications of the small holes in near field.

Nonparaxial propagation of spatially and spectrally partially coherent electromagnetic Cosh-Gaussian pulsed beams

Based on the generalized Rayleigh–Sommerfeld diffraction integral, the analytical expression for 3×3 cross-spectral density matrix of nonparaxial spatially and spectrally partially coherent electromagnetic Cosh-Gaussian (ChG) pulsed beams propagating in free space is derived, and used to formulate the spectral density and spectral degree of polarization of electromagnetic pulsed beams at the z-plane. It is found that the parameters f and fαα are the key parameters in determining the nonparaxiality of spatially and spectrally partially coherent electromagnetic ChG pulsed beams. And the decentered parameter, pulse duration and temporal coherence length can change the nonparaxial behavior of the electromagnetic ChG pulsed beams. The effect of decentered parameter, pulse duration and temporal coherence length on the spectral density and spectral degree of polarization of electromagnetic ChG pulsed beams is illustrated through numerical calculations. Propagation of nonparaxial spatially and spectrally partially coherent electromagnetic Gaussian Schell-model pulsed beams can be treated as a special case when the decentered parameter of electromagnetic ChG pulsed beams approaches to zero.

Evaluation of the power-law impulse response for liver in the time- and frequency-domains

Ultrasound attenuation in biological media often follows a frequency-dependent power-law relationship. Power-law attenuation is accompanied by dispersion in which phase velocity also changes as a function of frequency. The power-law wave equation has exact time- and frequency-domain Green's function solutions that are used in numerical evaluations of the Rayleigh–Sommerfeld diffraction integral. These Green's functions contain stable probability density functions that are evaluated in the time-domain using the STABLE toolbox and in the frequency-domain by evaluating the characteristic function of a stable distribution. The impulse response for a circular piston is evaluated on-axis in the time- and frequency-domain using values for human liver. The results show the accuracy of the time-domain impulse response calculation is dependent upon adequate spatial sampling of the piston face. The frequency-domain impulse response exhibits aliasing caused by wrap around of the heavy tail of the stable distribution. Frequency-domain accuracy is dependent upon the spatial sampling of the piston face and the density of the frequency samples. Numerical evaluations show that the frequency- and time-domain calculations converge to the same result. [This work was supported in part by NIH Grant R01 EB012079.]