Diffraction Of Plane Wave Research Articles

Plane wave diffraction by a semi‐infinite parallel‐plate waveguide with partial material loading: The case of H polarisation

AbstractThe analysis of diffraction by a semi‐infinite parallel‐plate waveguide with partial material loading is rigorously carried out using the Wiener–Hopf technique for the H‐polarised plane wave incidence. In solving the Wiener–Hopf equations, the authors apply the Modified Residue Calculus Technique (MRCT) to achieve highly accurate solutions. The scattered field in real space is explicitly derived by performing the inverse Fourier transform of the solution in the transform domain. Within the waveguide, the scattered field is represented in terms of the waveguide TM modes, while the external field is asymptotically evaluated by applying the saddle‐point method to yield a far field expression. The authors present representative numerical examples of the radar cross section for various physical parameters and discuss the far‐field scattering characteristics in detail. Comparisons with the E‐polarisation are also given.

Towards to Theory of the X-ray Diffraction Tomography of Crystals with Nano-Sized Defects

X-ray diffraction tomography is an innovative method that is widely used to obtain 2D-phase-contrast diffraction images and their subsequent 3D-reconstruction of structural defects in crystals. The most frequent objects of research are linear and helical dislocations in a crystal, for which plane wave diffraction images are the most informative, since they do not contain additional interference artifacts unrelated to the images of the defects themselves. In this work the results of modeling and analysis of 2D plane wave diffraction images of a nano-dimensional Coulomb-type defect in a Si(111) thin crystal are presented based on the construction of numerical solutions of the dynamic Takagi-Taupin equations. An adapted physical expression for the elastic displacement field of the point defect, which excludes singularity at the defect location in the crystal, is used. A criterion for evaluating the accuracy of numerical solutions of the Takagi-Taupin equations is proposed and used in calculations. It is shown that in the case of the Coulomb-type defect elastic displacement field, out of the two difference algorithms for solving the Takagi-Taupin equations used in their numerical solution, only the algorithm for solving the Takagi-Taupin equations where the displacement field function enters in exponential form is acceptable in terms of the required accuracy-duration of the calculations.

Plane wave diffraction by an interface of anomalously transmitting and impedance half planes

Diffraction of E-polarized plane waves by a junction of anomalously transmitting and impedance half planes is examined by employing the method of transition boundary (MTB). Firstly, initial fields that exist in space independent of the geometry are determined. In that situation, the scatterers causing diffraction are excluded. Secondly, total geometrical optics (GO) fields are written. Then, subtraction of initial fields from the total geometrical field will yield the scattered GO fields. Lastly, diffracted fields are found by using the scattered GO fields expression at the transition boundaries. The diffracted fields expression contains the diffracted fields of both half planes. The diffracted fields are examined for some parameters of the problem.

Single-frame double-grating interferometry with two inherent-phase-shifted interferograms.

In this work, we introduce a novel self-referencing, common-path, double-grating interferometry method for studying slowly varying phase samples. Two plane wave diffraction orders of the gratings, namely (0,+1) and (+1,0), with a certain phase difference, interfere with each other in a single frame. When a phase sample is applied in the middle of the impinging beam, two simultaneous inherent phase-shifted interferograms are generated on either side of the interference pattern. In one interferogram, the sample phase is added to (0,+1), while in the second one, the sample phase is added to (+1,0). Consequently, the phase of the first interferogram increases by the amount of the sample phase, while in the second interferogram, it decreases by the same amount. Without a phase sample, both interferograms have uniform intensity and value since the two interfering beams have the same phase difference in both patterns. We observe that the intensity changes in the two interferograms due to the phase sample, depending on the initial phase difference between the two interfering beams, can be equal or unequal, and in certain circumstances, they can be even complementary. We introduce a specific phase difference between the interfering orders by precisely controlling the separation between two diffraction gratings. This allows us to extract the sample's phase information from the resulting pair of interferograms. This setup enables us to obtain the sample phase without applying an additional phase shift(s) between the interfering beams. The method was applied to a thermal lens induced in a nonlinear liquid sample containing absorbent nanoparticles. The proposed method is characterized by its simplicity, accuracy, and insensitivity to vibrations, making it well-suited for analyzing dynamic samples with millimeter scales, such as nearly transparent organisms. The theory, simulations, and experimental results presented in this paper are found to be consistent. Unlike conventional methods, our approach does not necessarily require a reference interferogram. Additionally, when the absolute value of phase changes over the sample area and time is less than π, the raw phase pattern precisely matches the reconstructed phase pattern of the sample, eliminating the need for a phase-reconstructing algorithm. The phase pattern can be accurately calculated from the interferograms using an arcsine function without needing a fast Fourier transform. Compared to Fourier-based methods, the phase extraction process for each frame in our approach is 60 times faster. As a result, this technique operates in real-time for |φ(x, y;t) | < π.

Diffraction by a set of collinear cracks on a square lattice: An iterative Wiener–Hopf method

The diffraction of a time-harmonic plane wave on collinear finite defects in a square lattice is studied. This problem is reduced to a matrix Wiener–Hopf equation. This work adapts the recently developed iterative Wiener–Hopf method to this situation. The method was motivated by wave scattering in continuous media but it is shown here that it can also be employed in a discrete lattice setting. The numerical results are validated against a different method using discrete Green’s functions. Unlike the latter approach, the complexity of the present algorithm is shown to be virtually independent of the length of the cracks.

NATURAL ELECTROMAGNETIC MODES OF A COMPOSITE OPEN STRUCTURE INVOLVING A PERFECTLY CONDUCTING STRIP GRATING, AN INHOMOGENEOUS FERRITE LAYER, AND A MONOLAYER OF GRAPHENE

Subject and Purpose. Сonsidered are the natural modes and their correspondent eigenfrequencies of a composite structure which is nonuniform along one of the coordinates and consists of a lossy ferromagnetic layer placed in a static magnetic field. The layer involves a perfectly conducting strip grating at one of its boundaries and a graphene monolayer at the other. Methods and Methodology. The above stated problem can be approached within the analytical regularization procedure developed for dual series equations. The latter concern a broad class of diffraction problems which include, in particular, the diffraction of monochromatic plane waves on strip gratings placed at the boundary of a gyromagnetic medium. The amplitudes of the electromagnetic eigenmodes can be obtained from the infinite set of homogeneous linear algebraic equations solvable within a truncation technique. The roots of the system’s determinant represent complex-valued eigenfrequencies of the system under investigation. The material parameters adopted in our computations for the ferromagnetic layer correspond to such of yttrium iron garnet. Results. A number of numerical programs have been developed which permit analyzing the dependences of wave field eigenfunctions and complex eigenfrequencies upon geometrical parameters of the structure (such as grating slot width and period, and thickness of the lossy layer), as well as on electrodynamic parameters of the ferromagnet and graphene characteristics, specifically the chemical potential and relaxation energy of electrons. A number of behavioral regularities have been established, as well as the effect of non-uniformity of ferrite layer parameters upon the structure’s eigenfrequencies and wave field eigenfunctions. Conclusions. The structure under study has been shown to be is an open oscillatory system with a set of complex-valued natural frequencies demonstrating finite points of accumulation. The real parts of these eigenfrequencies lie in a certain interval determined by characteristic frequencies of the ferrite layer, while the imaginary parts are negative, such that the correspondent natural modes show an exponential decay with time. The grating edges represent the mirrors which the natural surface oscillations are reflected from, being supported at that by the ferromagnetic medium. The results obtained in this paper can be useful for creating the elemental base for microwave devices and the devices themselves.

Construction of Integral Representations of Fields in Problems of Diffraction by Penetrable Bodies of Revolution

Based on integral representations with densities distributed along a segment of the symmetry axis, a representation of the solution of the boundary value problem of plane wave diffraction by a local penetrable body of revolution with smooth surface is constructed and justified. The resulting integral representation allows one to avoid resonances of the interior domain when analyzing the scattering frequency characteristics.

Analytical solution for diffraction of magnetoelastic plane waves by a rigid strip in self-reinforced medium: a contour integration method

This study analytically addresses the diffraction of magnetoelastic plane waves by a rigid strip in a self-reinforced infinite elastic medium. Formulation of the studied model includes mixed type boundary value problem and has been solved by the method of integral equation. The expressions of vertical diffracted displacement and normal stress are established in the closed form by employing the contour integration technique. For the sake of validation, the deduced expressions of vertical diffracted displacement and normal stress are reduced to the special cases, such as without magnetoelastic coupling as well as anisotropy and matched with the pre-established results. The diffraction patterns for the vertical displacement component in the considered material medium are calculated numerically, and its dependence on various affecting parameters viz. magnetoelastic coupling parameter, phase velocity, wave number of the propagating magnetoelastic plane wave, and distance has been depicted graphically for self-reinforced and isotropic materials. Moreover, the influence of anisotropy of the infinite medium has been portrayed remarkably by the comparative study.

On the Problem of Observing the Maximum of a Superposition Wave Field A.

The well-known Laue condition determining the intensity maximums of diffracted on an ideal crystal structure plane wave is discussed. It is proved that the Laue conditions, which are restrictions imposed on the values of the scattering wave vector, is more correctly to be considered as a pair of two conditions. The first conditions relate to the wave vector of an incident wave and the second conditions relate to the wave vector of an observation direction. In other words, to observe the maximums, it is not enough that only the difference (the scattering wave vector) of these two vectors satisfies the Laue condition. To observe the maximums, it is necessary that each of these vectors separately, i.e. the wave vector of the incident wave and the wave vector of the observation direction, satisfy the Laue condition. It is shown that such a doubling of the maximums conditions leads to a decrease in the number of observed maximums compared to the condition imposed only on the scattering wave vector.

Nonstationary plane wave incidence on the perfectly conducting structures. Part 2. Diffraction of a nonstationary plane wave on the perfectly conducting sphere

The paper considers the incidence of a nonstationary plane wave on a perfectly conducting sphere. To find the current on the surface of the sphere, the well-known solution of the problem in the frequency domain has been used and the transition from the real frequency to the complex variable has been performed. Expressions for calculating the current on the sphere have been obtained using the residue formula. Graphs and vector images of the current have been plotted, which make it possible to observe its changes during the propagation of the incident wave. You can specify the following time intervals, differing in current distribution. At 0 &lt; t &lt; a/c, the current on the illuminated part of the sphere is close to the current determined in the geometrical optics approximation. In this case, there is naturally no current in the shadow region. For a/c &lt; t &lt; a/c + (π/2)a/c, a diffraction current appears in the shadow region, which turns out to be small compared to the current in the illuminated region. In the a/c + (π/2)a/c &lt; t &lt; a/c + πa/c, the current distribution on the sphere surface rapidly approaches the steady-state distribution. In the steady state, the streamlines represent circles obtained by sectioning the sphere with planes orthogonal to the magnetic field of the incident wave. At t &gt; a/c + πa/c, the current practically does not change, and the steady-state distribution is preserved.

Analytical Regularization Approach to Plane Wave Diffraction From Circular Hole in Infinite Resistive Plane

In this paper, the diffraction of a plane wave from a circular hole in an infinite resistive plane is addressed. In contrast to the holed perfectly electrically conducting screens, neither the equivalence principle combined with the image theory nor the Babinet principle can be applied here, and the problem has to be solved directly. The method adopted in this paper belongs to the family of the methods of analytical preconditioning. The revolution symmetry allows us to reduce the problem to an infinite set of dual integral equations in the vector Hankel transform domain for suitable unknowns vanishing outside the hole. Such equations are transformed to matrix equations by means of the Helmholtz-Galerkin discretization technique. The orthogonal eigenfunctions of the most singular part of the integral operator, reconstructing the behavior of the fields at the edge and around the center of the hole, are selected as expansion functions, thus leading to fast converging Fredholm second-kind matrix equations, whose elements can be expressed as quickly evaluable proper integrals. Numerical results show the near-field and far-field characteristics for various cases; the in-house software code is validated by means of comparisons with general-purpose commercial software.

Nonstationary plane wave incidence on the perfectly conducting structures. Part 1. Diffraction of a nonstationary plane wave on the perfectly conducting cylinder

The paper considers the incidence of a nonstationary plane wave on a perfectly conducting cylinder. To find the current on the surface of the cylinder, the well-known solution of the problem in the frequency domain has been used and the transition from the real frequency to a complex variable has been performed. Expressions for calculating the current on the sphere have been obtained using the residue formula. Graphs of the current have been plotted, which make it possible to observe its changes during the propagation of the incident wave. You can specify the following time intervals, differing in current distribution. At 0 &lt; t &lt; a/c, the current on the illuminated part of the surface is close to the current determined in the geometrical optics approximation. In this case, there is naturally no current in the shadow region. For a/c &lt; t &lt; a/c + π/2a/c, a diffraction current appears in the shadow region, which turns out to be small compared to the current in the illuminated region. In the a/c + π/2a/c &lt; t &lt; a/c + πa/c, the current distribution on the cylinder surface rapidly approaches the steady-state distribution. In the steady state, the magnitude of the current is equal to the intensity of the magnetic field of the incident wave. At t &gt; a/c + πa/c, the current practically does not change, and the steady-state distribution is preserved.

Plane-Wave Diffraction from Resistive-Filled Circular Hole in Infinite Resistive Plane: An Analytically Regularizing Approach

The study of the electromagnetic diffraction from penetrable screens with apertures and/or inhomogeneities is of great relevance today due to the huge number of modern applications in which they are involved. In this paper, the analysis of the plane wave scattering from a resistive-filled circular hole in a resistive plane is addressed. The uniquely solvable boundary value problem for the Maxwell equations, obtained via imposing generalized boundary conditions, power boundedness condition, and Silver–Muller radiation condition, is equivalently formulated in terms of an infinite set of singular dual integral equations in the vector Hankel transform domain. The Helmholtz–Galerkin technique allows for the discretization and, simultaneously, analytical regularization of the obtained integral equations. Fast convergence is guaranteed by a suitable choice of the basis functions reconstructing the physical behavior of the fields at the discontinuity between the two involved media. Moreover, the full-wave nature of the proposed approach allows the direct assessment of near-field and far-field parameters.

Theoretical and Experimental Investigation of Plane Wave Diffraction from an Amplitude Modified Fresnel Zone Plate

Theoretical and Experimental Investigation of Plane Wave Diffraction from an Amplitude Modified Fresnel Zone Plate

Diffraction by a Semi-Infinite Parallel-Plate Waveguide with Five-Layer Material Loading: The Case of H-Polarization

In this paper, the plane wave diffraction from a semi-infinite parallel-plate waveguide with five-layer material loading is rigorously analyzed for H-polarization using the Wiener–Hopf technique. The Fourier transform of the scattered field is introduced and boundary conditions are applied in the transform domain to formulate the problem as simultaneous Wiener–Hopf equations, which are solved by the factorization and decomposition procedure leading to exact and approximate solutions. The scattered field in real space is explicitly derived by taking the Fourier inverse of the solution in the transform domain. For the region inside the waveguide, the scattered field is represented by the waveguide TM modes, and the field outside the waveguide is evaluated asymptotically by applying the saddle-point method to obtain a far-field expression. Numerical examples of the radar cross section (RCS) for various physical parameters are presented, and far-field scattering characteristics of the waveguide are discussed in detail.

Plane-Wave Diffraction by a Slit Formed by Two Semi-Infinite Parallel-Plate Waveguides—Part I: The Case of H Polarization

Plane-Wave Diffraction by a Slit Formed by Two Semi-Infinite Parallel-Plate Waveguides—Part I: The Case of H Polarization

Plane-Wave Diffraction by a Slit Formed by Two Semi-Infinite Parallel-Plate Waveguides—Part II: The Case of E Polarization

Plane-Wave Diffraction by a Slit Formed by Two Semi-Infinite Parallel-Plate Waveguides—Part II: The Case of E Polarization

Junction of Two Coplanar Resistive Half-Planes: A Uniform Asymptotic Solution for the Plane Wave Diffraction and Validation Tests

Junction of Two Coplanar Resistive Half-Planes: A Uniform Asymptotic Solution for the Plane Wave Diffraction and Validation Tests

H‐polarized plane wave diffraction by a slotted cylinder with different surface impedances: Solution by the analytical—Numerical approach

The study investigates the H-polarized plane wave diffraction by a slotted cylinder with different surface impedances employing an analytical-numerical approach. The solution is obtained by expressing the current distribution on the obstacle with Gegenbauer polynomials considering the edge conditions. To obtain the field and current distribution on the surface, the Leontovich boundary condition on each side of the slotted cylinder is employed. The results are compared with Dirichlet and Neumann boundary condition cases obtained previously. It is observed that the resonant field amplitudes and the location at the resonances strongly depend on the impedance values.

Approximating transmission and reflection spectra near isolated nondegenerate resonances

A linear scattering problem for which incoming and outgoing waves are restricted to a finite number of radiation channels can be precisely described by a frequency-dependent scattering matrix. The entries of the scattering matrix, as functions of the frequency, give rise to the transmission and reflection spectra. To find the scattering matrix rigorously, it is necessary to solve numerically the partial differential equations governing the relevant waves. In this paper, we consider resonant structures with an isolated nondegenerate resonant mode of complex frequency $\omega_\star$, and show that for real frequencies near $\omega_0 = \mbox{Re}(\omega_\star)$, the transmission and reflection spectra can be approximated using only the scattering matrix at $\omega_0$ and information about the resonant mode. We also present a revised temporal coupled-mode theory that produces the same approximate formulas for the transmission and reflection spectra. Numerical examples for diffraction of plane waves by periodic structures are presented to validate our theory.