Amplitudes Of Diffracted Waves Research Articles

Antipodal waveform observations of seismic waves diffracting and refracting at the base of Earth's outer core

We approach this analysis of the inner-outer core boundary (IOCB) with antipodal waveform data in the distance range 179.0°–180° to test the hypothesis whether propagation at the base of the outer core is commensurate with diffraction and/or refraction. The propagation paths observed cover about two-thirds of the IOCB surface. Seismic data from seven diameters are examined—Tonga to Algeria (station code TAM), Sulawesi to Amazon (PTGA), northern Chile to Hainan Island (QIZ), two between central Chile and the mainland China (XAN, ENH), and two diameters from New Zealand to both Portugal (PTO) and Spain (ECAL). The adequacy of global Earth models—both 1-D and 1-D core +3D mantle—in fitting the antipodal observations is found to be deficient. Lateral heterogeneity among the propagation paths are mapped and projected to the Earth's surface for context. We stack data to increase signal-to-noise, and model waveform data via the 3D spectral element method on the EarthSimulator4. With the exception of one path to ENH China, none of the global Earth models match amplitudes of diffracted waves or the stacked amplitudes relative to PKIKP, which travels along the antipodal diameter. Energetic arrivals observed in the waveform data set are modeled as a combination of refraction within a low velocity zone (LVZ) at the base of the outer core, and diffraction around this structure. The observations at the base of the outer core may be further subdivided between paths with a thin low velocity zone (QIZ, XAN, PTGA), and those with a thicker, slower LVZ (TAM). The former paths are characterized by a thin LVZ (20–50 km thick, layer–gradient, respectively) with a velocity of approximately 10.0 km/s. The second group of paths shows large amplitude, complicated waveforms for TAM, corroborated by data from PTO and ECAL. TAM waveforms are modeled with a negative gradient from 100 to 50–75 km thickness above the IOCB, to a basal velocity of 8.8 km/s at IOCB. We also modeled the prospect of non-zero rigidity at the bottom of the outer core, but found no evidence of sensitivity in the core data. The structures fitting the data correlate across the inner–outer core boundary, with a velocity anomaly in PKPbc data juxtaposed immediately above the TAM path beneath SE Asia. More generally, the TAM IOCB path when projected up to the core-mantle boundary (CMB) apparently aligns with regions between the large low shear velocity provinces (LLSVP) at the CMB. Other paths with thinner or negligible LVZ at the IOCB partially underlie the LLSVP when projected to the CMB.

Acousto-optical modulation of airy light beams in crystals

The non-collinear diffraction of Airy light beams on ultrasound in the regime of weak and strong acousto-optical (AO) interaction is investigated. Expressions are obtained for the complex amplitudes of diffracted waves in the general case of an anisotropic medium. It is shown that with an increase in the dimensionless parameter of the Airy beam, the efficiency of Bragg diffraction decreases. At the same time, however, the shape of the modulation curve remains unchanged. It is established that the greatest efficiency of anisotropic Bragg diffraction is achieved for a case when the diffracted Airy beam propagates along the front of the ultrasonic wave. It is shown that the efficiency of AO diffraction of Airy beam on an amplitude-modulated wave is determined by the diffraction efficiencies of constant components in a diffracted beam.

Fourier optics modeling of interference microscopes.

We propose a practical theoretical model of an interference microscope that includes the imaging properties of optical systems with partially coherent illumination. We show that the effects on measured topography of a spatially extended, monochromatic light source at low numerical apertures can be approximated in a simplified model that assumes spatially coherent light and a linear, locally shift-invariant transfer function that accounts for optical aberrations and the attenuation of diffracted plane wave amplitudes with increasing spatial frequencies. Simulation of instrument response using this model agrees with methods using numerical pupil-plane integration and with an experimental measurement of surface topography.

X-Ray Diffraction Tomography Recovery of the 3D Displacement-Field Function of the Coulomb-Type Point Defect in a Crystal

A successive approach to the solution of the inverse problem of the X-ray diffraction tomography (XRDT) is proposed. It is based on the semi-kinematical solution of the dynamical Takagi–Taupin equations for the σ-polarized diffracted wave amplitude. Theoretically, the case of the Coulomb-type point defect in a single crystal Si(111) under the exact conditions of the symmetric Laue diffraction for a set of the tilted X-ray topography 2D-images (2D projections) is considered provided that the plane-parallel sample is rotated around the diffraction vector [bar{{bf{2}}}20]. The iterative simulated annealing (SA) and quasi-Newton gradient descent (qNGD) algorithm codes are used for a recovery of the 3D displacement-field function of the Coulomb-type point defect. The computer recovery data of the 3D displacement-field function related to the one XRDT 2D projection are presented. It is proved that the semi-kinematical approach to the solution of the dynamical Takagi–Taupin equations is effective for recovering the 3D displacement-field function even for the one XRDT 2D projection.

Photonic crystal-based compact high-power vacuum electronic devices

This paper considers how the finite dimensions of a photonic crystal placed inside a resonator or waveguide affect the law of electron beam instability. The dispersion equations describing e-beam instability in the finite photonic crystal placed inside the resonator or waveguide (a bounded photonic crystal) are obtained. Two cases are considered: the conventionally considered case, when diffraction is suppressed, and the case of direct and diffracted waves having almost equal amplitudes. The instability law is shown to be responsible for increase of increment of instability and decrease of length, at which instability develops, for the case when amplitude of diffracted wave is comparable with that of direct one, that happens in the vicinity of $\pi$-point of dispersion curve. Application of photonic crystals for development of THz sources at electron beam current densities available at modern accelerators is discussed.

X-ray two-wave dynamical diffraction analogue of Talbot effect

Imaging of periodic objects in free space brings us to the Talbot effect in optics, for x-rays and for matter waves. X-ray dynamical diffraction imaging of periodic objects inside a crystal has scientific and applied interest. Thus, we arrive at the concept of the dynamical diffraction Talbot effect in a crystal. This work is the first attempt to investigate the Talbot effect inside a medium. Using the Green function formalism, an exact formula for the diffracted wave amplitude is found. By means of ‘paraxial’ approximation, which is an analogue of the paraxial approximation in optics, it is shown that the dynamical diffraction Talbot effect takes place. Expressions for polarization sensitive Talbot and corrected Talbot distances are obtained. We analyse the influences of absorption, Bragg filtration of harmonics and polarization on the dynamical diffraction Talbot effect. For the first time, simulated Talbot carpets inside the crystal are obtained, which show that the predictions, obtained by ‘paraxial’ approximation, are true. We present the dynamical diffraction Talbot carpets observation method by means of a wedge-shaped crystal. The dynamical diffraction Talbot effect can be used for investigation of objects and crystal defects and deformations. Dynamical diffraction Talbot effect investigations in optics, for electrons and neutrons, are possible.

To the Solution of the Inverse Problem of X-Ray Topo-Tomography. Computer Algorithms and 3D Reconstruction on the Example of a Crystal with a Point Defect of Coulomb Type

A successive approach to the solution of the inverse problem of diffraction X-ray topo-tomography is proposed. It is based on the semikinematic solution to the Takagi–Taupin equations for the amplitude of diffracted σ-polarized wave. An example of a point defect of Coulomb type in a Si(111) single-crystal plate under conditions of symmetric Laue diffraction and a set of oblique two-dimensional topographic projections corresponding to the rotation of plane-parallel sample crystal around the diffraction vector [ $$\bar {2}20$$ ] is considered. Iterative algorithms for simulated annealing (SA) and quasi-Newton-type algorithms are used for computer reconstruction of the three-dimensional function of the displacement field around a point defect. The results of computer simulation of the displacement field function based on the data for one 2D projection, corresponding to the point defect image in an X-ray topogram in classical X-ray diffraction topography, are presented.

Wave diffraction by a circular crack in an ice sheet floating on water of finite depth

The problem of wave diffraction by a circular crack in an ice sheet floating on water of finite depth is considered. The fluid flow is described by the linear velocity potential theory, while the infinitely extended ice sheet is modeled as a thin elastic plate with uniform properties. At the crack, zero bending moment and shear force conditions are enforced. The solution starts from the Green function for ice sheet without the crack. This is then used to obtain an integral equation, in which the jumps of the displacement and slope across the crack are the unknowns. For a circular crack, the unknowns are expanded into the Fourier series in the circumferential direction. Through imposing the boundary conditions at the crack, a matrix equation is obtained for the unknowns, which is then truncated and solved. Convergence study is undertaken with respect to the truncation, and it has been found that the series converges fast. A far field identity is used to verify the solution procedure and is found to be satisfied very accurately. Extensive results are provided, and their physical implications are discussed. These include the jumps of the displacement and slope across the crack, resonant motion, far field diffracted wave amplitude, and the deflection of the ice sheet.

Introduction to special section: Diffraction imaging

Diffraction imaging is gaining momentum as a possible tool to map discontinuities such as fracture swarms, corridors, and small offset faults in unconventional reservoirs prior to drilling. The amplitudes of diffracted waves are inherently much weaker than those of the specular reflections and are

X-ray Laue diffraction with allowance for two-dimensional curvature of the wavefront: The concept a locally plane wave

Symmetrical Laue diffraction in a perfect crystal with a plane entance surface is considered. The two dimensional curvature of the wave front of the incident beam is taken into account. Using the corresponding Green function, a general expression for the amplitude of diffracted wave in the crystal is presented. Based on this expression, the concept of a locally plane wave is analyzed taking into account two-dimensional curvature of the wave front. This concept is used for obtaining the rocking curves depending on the angles of deviation from the chosen exact Bragg direction in both the diffraction plane and in perpendicular direction. The obtained result is compared with the result of the standard dynamical diffraction theory.

Theory of magneto-optical effects in nanostructured ferromagnetics

We present a theory of light interaction with a magnetic nanostructured material consisting of a periodic array of ferromagnetic nanocylinders magnetized along the symmetry axis. Using a multiple scattering approach, we calculated the amplitudes of diffracted waves in terms of multipole expansion coefficients and used the developed formalism to describe the experimentally observed enhancement of the magneto-optical effects in magnetic nanostructures. The analysis provides a physical interpretation of the enhancement in terms of interference between the resonant and nonresonant scattering processes. The resonant scattering is caused by excitation of dipole surface-plasmon eigenmodes; the nonresonant one is associated with the monopole modes.

Wave diffraction on arc-shaped floating perforated breakwaters

An analytical method is developed to study the sheltering effects on arc-shaped floating perforated breakwaters. In the process of analysis, the floating breakwater is assumed to be rigid, thin, vertical, and immovable and located in water with constant depth. The fluid domain is divided into two regions by imaginary interface. The velocity potential in each region is expanded by eigenfunction in the context of linear theory. By satisfying continuity of pressure and normal velocity across the imaginary fluid interface, a set of linear algebraic equations can be obtained to determine the unknown coefficients for eigenfunction expansions. The accuracy of the present model was verified by a comparison with existing results for the case of arc-shaped floating breakwater. Numerical results, in the form of contour maps of the non-dimensional wave amplitude around the breakwater and diffracted wave amplitude at typical sections, are presented for a range of wave and breakwater parameters. Results show that the sheltering effects on the arc-shaped floating perforated breakwater are closely related to the incident wavelength, the draft and the porosity of the breakwater.

Diffraction of flexural waves by finite straight cracks in an elastic sheet over water

The diffraction of long-crested incident waves propagating within a thin flexible elastic sheet floating on water by narrow cracks is considered. The cracks are straight and each of finite length and must be parallel to one another. This arrangement lends itself to the use of Fourier transform methods, which allows the solution to a simpler problem to be used. For N cracks, 2N coupled integral equations results for 2N unknown functions related to the jump in displacement and slope across each crack as a function of distance along the cracks. These integral equations are hypersingular but, in approximating their solution using Galerkin's method, a judicious choice of trial function provides maximum simplification in the algebraic equations which result. Numerical results focus on the diffracted wave amplitudes, the maximum displacement of the elastic sheet and the stress intensity factor at the ends of the cracks. For two side-by-side cracks, large resonant motion can occur in the strip between the cracks.

X-ray multiple diffraction from crystalline multilayers: Application to a 90° Bragg reflection

A generalized solution to the problem of multiple-beam dynamical x-ray diffraction from single and layered crystals with vertical strain is presented. The new formalism embraces all possible diffraction geometries, including extreme cases of grazing and normal incidence and emergence. The expressions for the diffracted wave amplitudes were obtained as recursion formulas. Numerical simulations were implemented and shown to be computationally stable. The formalism was successfully tested using experimental data obtained from a SiC/Si single-layered structure in a 90 deg. diffraction geometry.

Fast calculation method for optical diffraction on tilted planes by use of the angular spectrum of plane waves.

A novel method for simulating field propagation is presented. The method, based on the angular spectrum of plane waves and coordinate rotation in the Fourier domain, removes geometric limitations posed by conventional propagation calculation and enables us to calculate complex amplitudes of diffracted waves on a plane not parallel to the aperture. This method can be implemented by using the fast Fourier transformation twice and a spectrum interpolation. It features computation time that is comparable with that of standard calculation methods for diffraction or propagation between parallel planes. To demonstrate the method, numerical results as well as a general formulation are reported for a single-axis rotation.

Higher‐order Laue zone line contrast in large‐angle convergent‐beam electron diffraction around a dislocation

The physical picture of higher-order Laue zone (HOLZ) line contrast in a large-angle convergent-beam electron diffraction pattern around a dislocation, which is used for determining the Burgers vector, was examined. To evaluate the analytical expression of diffracted wave amplitude, we introduced an approximate form of the atomic displacement field of a dislocation. We showed that the four features of the HOLZ line contrast, that is, splitting, fading, bending and periodical contrast can be explained by analysis of the atomic displacement field. The localized lattice plane bending around a dislocation core made a HOLZ line split, fade and bend. However, we found that the periodical contrast of a HOLZ line was produced by the change of phase difference of the atomic displacement field between the crystals above and below the slip plane across the dislocation line.

Boundary effects in the diffraction of light by volume gratings: II. Phase transmission gratings

The two-wave approximation of the method of characteristics modes is used to analyze theoretically the diffraction of light by volume phase transmission gratings for arbitrary proportions between the average refractive index of the grating and the refractive indices of the ambient media. Simple expressions are derived for calculating the amplitudes of the diffracted waves in the case where the angle of incidence of the primary wave coincides with the Bragg angle. The roles of the front and back boundaries of the grating are discussed separately, and a new method for separating the contribution of the boundary diffraction, which is due to the periodic inhomogeneity of the grating boundaries, is proposed. The experimental possibility of selecting partial Bloch waves in the measurements of the diffraction efficiency of gratings that have high reflection coefficients of both boundaries is demonstrated.

The Diffraction of Elastic Waves on the Tip of Slots

The diffraction of elastic waves on the tip of cracks has been widely used in detecting defects. It is always difficult to fabricate a real crack in the laboratory. In this investigation, a slot of width 0.3 mm is used as an alternative to simulate the fatigue crack. The slot of 0.3 mm width is embedded in a cylindrical block. The diffractions of elastic waves on the tip of the artificial slot are measured for three test blocks made of steel, aluminum, and brass, respectively. In the experiment, the amplitudes of diffracted waves at various receiving angles are recorded and compared with the theoretical predictions of diffraction on a fatigue crack tip. The influences of incident angle and frequency and material of the test block are presented. By examining the discrepancies of diffractions on the slots and cracks, one is able to evaluate the feasibility of replacing a real crack by an artificial slot, once the real crack is not available.

Diffraction characteristics of the ring zone plate as a representative of curvilinear zone plates

It is shown that curvilinear zone plates can be constructed as phase representatives of plane curvilinear light sources. The ring zone plate is one representative of this type of diffracting optical element. At the same time, it is an optical device which can be used as a lens substitute for ring lenses, especially in the UV region. The diffracted wave amplitude and the intensity distribution for both binary, positive and negative, computer-generated ring zone plates, are determined by the use of the Fresnel-Kirchoff approximation. It is found that the ring zone plate exhibits manifold real and virtual focal rings.

Raman-nath light diffraction by ultrasound in gyrotropic cubic crystals in an external electric field

UDC 535.42:534.8 Investigation of light diffraction by ultrasonic (US) waves in gyrotropic cubic crystals in an external electric field is of scientific and practical interest [1 I, as a number of crystals with a structure of sillenite (Bi12GeO20, Bil2SiO20, Bil2TiO20, etc.) possess the high specific rotation of a light polarization plane and at the same time a substantial electrooptical effect. While in uniaxial and biaxial crystals the electric field-induced anisotropy is manifested only for light propagations close to the optical axes, in a cubic crystal the anisotropy must be taken into account for any geometry of acoustooptical (AO) interaction [2 ]. It is interesting to investigate the simultaneous influence of gyrotropy and electroinduced anisotropy on AO diffraction in cubic crystals. The AO Raman-Nath diffraction in gyrotropic crystals has been considered in an approximation of a given polarization [3, 4 I. For calculation of the complex vector amplitudes of diffracted waves, they were decomposed into circularly polarized components. In [5 ], a system of equations of coupled waves is given that describes specific features of the Raman-Nath AO diffraction in gyrotropic cubic crystals with the strong light-ultrasound interaction without allowance for the influence of electroinduced anisotropy of the crystal in an external electric field. In [6, 7 ], consideration is given of the influence of an external electric field on the Bragg light diffraction by ultrasound in gyrotropic crystals. It is shown that the combined influence of gyrotropy and acousto- and electroinduced anisotropy causes a substantial change in the polarization and energy characteristics of diffracted waves. Some peculiar features of the Bragg AO diffraction by confined light beams in gyrotropic cubic crystals are discussed in [8 ]. A promising use of biaxial gyrotropic crystals for creation of AO filters is shown in [9 I. In the present work we investigate the Raman-Nath AO diffraction in gyrotropic cubic crystals in an external electric field by solving numerically a system of reduced equations obtained by the method of slowly changing amplitudes. From the Maxwell and material equations 12, 10 ] of a gyrotropic crystal in an external electric field a wave