Wavelength Plane Research Articles

Use of electron backscatter diffraction patterns to determine the crystal lattice. Part 1. Where is the Bragg angle?

The derivation of a crystal structure and its phase-specific parameters from a single wide-angle backscattered Kikuchi diffraction pattern requires reliable extraction of the Bragg angles. By means of the first derivative of the lattice profile, an attempt is made to determine fully automatically and reproducibly the band widths in simulated Kikuchi patterns. Even under such ideal conditions (projection centre, wavelength and lattice plane traces are perfectly known), this leads to a lattice parameter distribution whose mean shows a linear offset that correlates with the mean atomic number Z of the pattern-forming phase. The consideration of as many Kikuchi bands as possible reduces the errors that typically occur if only a single band is analysed. On the other hand, the width of the resulting distribution is such that higher image resolution of diffraction patterns, employing longer wavelengths to produce wider bands or the use of higher interference orders is less advantageous than commonly assumed.

Numerical simulation of optical propagation using sinc approximation.

This paper further develops a recently proposed method for computing the diffraction integrals of optics based on sinc series approximation by presenting a numerical implementation, parameter selection criteria based on rigorous error analysis, and example optical propagation simulations demonstrating those criteria. Unlike fast Fourier transform (FFT)-based methods that are based on Fourier series, such as the well-known angular spectrum method (ASM), the sinc method uses a basis that is naturally suited to problems on an infinite domain. As such, it has been shown that the sinc method avoids the problems of artificial periodicity inherent in the ASM. After a brief review of the method, the detailed error analysis we provide confirms its super-algebraic convergence and verifies the claim that the accuracy of the method is independent of wavelength, propagation distance, and observation plane discretization; it depends only on the accuracy of the source field approximation. Based on this analysis, we derive parameter selection criteria for achieving a prescribed error tolerance, which will be valuable to potential users. Numerical simulations of Gaussian beam and optical phased array propagation verify the high-order accuracy and computational efficiency of the proposed algorithms. To facilitate the reproduction of numerical results, we provide a Matlab code that implements our numerical approach for the Fresnel diffraction integral. For comparison, we also present numerical results obtained with the ASM as well as the band-limited angular spectrum method.

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.

Maximum w‐cyclic holey group divisible packings with block size three and applications to optical orthogonal codes

AbstractIn this paper we investigate combinatorial constructions for ‐cyclic holey group divisible packings with block size three (3‐HGDPs). For any positive integers with , the exact number of base blocks of a maximum ‐cyclic 3‐HGDP of type is determined. This result is used to determine the exact number of codewords in a maximum three‐dimensional optical orthogonal code with at most one optical pulse per spatial plane and per wavelength plane.

Production of photonic nanojets by using pupil-masked 3D dielectric cuboid

Photonic nanojets can be created via plane wave irradiation of multi-shaped mesoscale dielectric particles, and a waist of full-width at half-maximum (FWHM) smaller than the diffraction limit can be achieved in this process. In this paper, photonic nanojet produced by a pupil-masked 3D dielectric cuboid lens is numerically investigated under the irradiation of 532 nm wavelength plane wave. It is found that a pupil-masked cuboid lens is not only able to produce photonic nanojets with shorter FWHMs, but also increase its maximal intensity at certain masking ratios on the receiving surface. This phenomenon is different from the result of the spherical-lens reported in previous publications, and is attributed to the convergence of power flow and near-field numerical aperture (NA) increase after analysis of simulated power flow diagrams.

Formula omitted]-cyclic holey group divisible designs and their application to three-dimensional optical orthogonal codes

Motivated by the application to three-dimensional optical orthogonal codes, we consider the constructions for a w-cyclic holey group divisible design of type (u,wv) with block size k, which is denoted by w-cyclic k-HGDD of type (u,wv). The necessary conditions of such a design, namely u,v≥k, (u−1)(v−1)w≡0(modk−1) and u(u−1)v(v−1)w≡0(modk(k−1)), are shown to be sufficient for k=3. As an application, we give a complete solution of the existence problem of perfect three-dimensional optical orthogonal codes of weight three with both at most one-pulse per spatial plane and at most one-pulse per wavelength plane properties.

Multidimensional Instability of Dust‐Acoustic Solitary Waves in a Magnetized Hot Dusty Plasma with Superthermal Electrons and Ions

AbstractMultidimensional instability of dust‐acoustic solitary wave (DASW) in magnetized dusty plasma with superthermal electrons and ions and micron size hot dust particles is investigated. The Zakharov‐Kuznetsov (ZK) equation, describing the small but finite amplitude DASW, was derived using the reductive perturbation method and its solitary answers was introduced. Effects of electrons and ions superthermality as well as the external magnetic field on the nature of DASW are discussed in detail. Dispersion relation, threshold condition, and growth rate of multidimensional instability of DASW were derived using small‐k (long wavelength plane wave) perturbation expansion method. We found that the direction and strength of external magnetic field extremely affect the growth rate and instability criterion. Results show that growth rate of instability decreases with increasing the number of superthermal electrons and increases with increasing the number of superthermal ions. (© 2015 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Multi-dimensional instability of obliquely propagating ion acoustic solitary waves in electron-positron-ion superthermal magnetoplasmas

The solitary structures of multi–dimensional ion-acoustic solitary waves (IASWs) have been considered in magnetoplasmas consisting of electron-positron-ion with high-energy (superthermal) electrons and positrons are investigated. Using a reductive perturbation method, a nonlinear Zakharov-Kuznetsov equation is derived. The multi-dimensional instability of obliquely propagating (with respect to the external magnetic field) IASWs has been studied by the small-k (long wavelength plane wave) expansion perturbation method. The instability condition and the growth rate of the instability have been derived. It is shown that the instability criterion and their growth rate depend on the parameter measuring the superthermality, the ion gyrofrequency, the unperturbed positrons-to-ions density ratio, the direction cosine, and the ion-to-electron temperature ratio. Clearly, the study of our model under consideration is helpful for explaining the propagation and the instability of IASWs in space observations of magnetoplasmas with superthermal electrons and positrons.

Stability of ion acoustic solitary waves in a magnetized plasma consisting of warm adiabatic ions and non-thermal electrons having vortex-like velocity distribution

AbstractSchamel's modified Korteweg-de Vries–Zakharov–Kuznetsov (S-ZK) equation, governing the behavior of long wavelength, weak nonlinear ion acoustic waves propagating obliquely to an external uniform static magnetic field in a plasma consisting of warm adiabatic ions and non-thermal electrons (due to the presence of fast energetic electrons) having vortex-like velocity distribution function (due to the presence of trapped electrons), immersed in a uniform (space-independent) and static (time-independent) magnetic field, admits solitary wave solutions having a sech4 profile. The higher order stability of this solitary wave solution of the S-ZK equation has been analyzed with the help of multiple-scale perturbation expansion method of Allen and Rowlands (Allen, M. A. and Rowlands, G. 1993 J. Plasma Phys. 50, 413; 1995 J. Plasma Phys. 53, 63). The growth rate of instability is obtained correct to the order k2, where k is the wave number of a long wavelength plane wave perturbation. It is found that the lowest order (at the order k) instability condition is strongly sensitive to the angle of propagation (δ) of the solitary wave with the external uniform static magnetic field, whereas at the next order (at the order k2) the solitary wave solutions of the S-ZK equation are unstable irrespective of δ. It is also found that the growth rate of instability up to the order k2 for the electrons having Boltzmann distribution is higher than that of the non-thermal electrons having vortex-like distribution for any fixed δ.

Multi-dimensional instability of multi-ion acoustic solitary waves in a degenerate magnetized plasma

The multi-dimensional instability of obliquely propagating multi-ion acoustic (MIA) solitary structures was studied theoretically by the small-k (long wavelength plane wave) perturbation expansion technique in an ultra-relativistic degenerate magnetized plasma, which consists of inertia less electrons, inertial ions and stationary arbitrarily charged heavy ions. The Zakharov–Kuznetsov equation is derived by the reductive perturbation method and its solitary wave solution is analyzed. The basic properties of small but finite-amplitude MIA solitary waves have been modified significantly by the combined effects of the degenerate electron number density, heavy ion number density, external magnetic field and obliqueness. The underlying physics of the MIA solitary waves, which are relevant to space plasma situations, and the basic features, such as amplitude, width and growth rate, are briefly discussed.

Ion‐Acoustic Waves Instability in a Three Components Magneto‐Plasma with Nonthermal Electrons

AbstractA theoretical investigation has been made on obliquely propagating ion‐acoustic (IA) solitary structures in a three components magneto‐plasma containing cold inertial ions, Boltzmann distributed positrons, and hot non‐thermal electrons. The Zakharov‐Kuznetsov equation has been derived by the reductive perturbation method, and its solitary wave solution has been analyzed. Multi‐dimensional instability has also studied by the small‐k (long wave‐length plane wave) perturbation expansion technique, which is found to exist in such a plasma. The effects of the external magnetic field, nonthermal electrons, obliqueness and temperature ratio have significantly modified the basic properties of small but finite‐amplitude IA solitary waves, such as amplitude, width, instability criterion and the growth rate. The present investigation contributes to the physics of the nonlinear IA waves in space and laboratory electron‐positron‐ion magneto‐plasmas in which wave damping produces an electron tail. (© 2013 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Ion-acoustic solitary waves and their multi-dimensional instability in a magnetized degenerate plasma

A rigorous theoretical investigation has been made on Zakharov-Kuznetsov (ZK) equation of ion-acoustic (IA) solitary waves (SWs) and their multi-dimensional instability in a magnetized degenerate plasma which consists of inertialess electrons, inertial ions, negatively, and positively charged stationary heavy ions. The ZK equation is derived by the reductive perturbation method, and multi-dimensional instability of these solitary structures is also studied by the small-k (long wave-length plane wave) perturbation expansion technique. The effects of the external magnetic field are found to significantly modify the basic properties of small but finite-amplitude IA SWs. The external magnetic field and the propagation directions of both the nonlinear waves and their perturbation modes are found to play a very important role in changing the instability criterion and the growth rate of the unstable IA SWs. The basic features (viz., amplitude, width, instability, etc.) and the underlying physics of the IA SWs, which are relevant to space and laboratory plasma situations, are briefly discussed.

Dust-ion-acoustic solitary waves and their multi-dimensional instability in a magnetized nonthermal dusty electronegative plasma

A rigorous theoretical investigation has been made on multi-dimensional instability of obliquely propagating electrostatic dust-ion-acoustic (DIA) solitary structures in a magnetized dusty electronegative plasma which consists of Boltzmann electrons, nonthermal negative ions, cold mobile positive ions, and arbitrarily charged stationary dust. The Zakharov-Kuznetsov (ZK) equation is derived by the reductive perturbation method, and its solitary wave solution is analyzed for the study of the DIA solitary structures, which are found to exist in such a dusty plasma. The multi-dimensional instability of these solitary structures is also studied by the small-k (long wave-length plane wave) perturbation expansion technique. The combined effects of the external magnetic field, obliqueness, and nonthermal distribution of negative ions, which are found to significantly modify the basic properties of small but finite-amplitude DIA solitary waves, are examined. The external magnetic field and the propagation directions of both the nonlinear waves and their perturbation modes are found to play a very important role in changing the instability criterion and the growth rate of the unstable DIA solitary waves. The basic features (viz. speed, amplitude, width, instability, etc.) and the underlying physics of the DIA solitary waves, which are relevant to many astrophysical situations (especially, auroral plasma, Saturn’s E-ring and F-ring, Halley’s comet, etc.) and laboratory dusty plasma situations, are briefly discussed.

Dust-ion-acoustic solitary waves and their multi-dimensional instability in a magnetized dusty electronegative plasma with trapped negative ions

A theoretical investigation has been made on obliquely propagating dust-ion-acoustic solitary waves (DIASWs) in magnetized dusty electronegative plasma containing Boltzmann electrons, trapped negative ions, cold mobile positive ions, and arbitrarily charged stationary dust. The reductive perturbation method has been employed to derive the modified Zakharov-Kuznetsov (MZK) equation which admits solitary wave solution under certain conditions. The multi-dimensional instability of these solitary waves is also studied by the small-k (long wavelength plane wave) perturbation-expansion technique. The basic properties (speed, amplitude, width, instability, etc.) of small but finite amplitude DIASWs are significantly modified by the effects of external magnetic field, obliqueness, polarity of dust, and trapped negative ions. The implications of our results in space and laboratory plasmas are briefly discussed.

Reactive-to-Radiating Near-Field Propagation Behavior Measured on a Ka-Band Horn Antenna

The spatial and temporal evolution of the propagating fields in the transition region between the reactive and radiating near-field distances of a standard gain horn antenna operating within the Ka-band are presented. Electric-field distributions with amplitude and phase information are measured with minimal invasiveness by a vector-stabilized, electrooptic field-imaging system. Acquiring a complete vector-imaging data set, a comprehensive near-field propagation chart-which shows the detailed field development at each wavelength plane with full phase progression-is presented for an antenna frequency of 30 GHz.

Three-dimensional instability of dust ion-acoustic solitary waves in a magnetized dusty plasma with two different types of nonisothermal electrons

The nonlinear propagation of dust ion-acoustic solitary waves (DIASWs) in a magnetized dusty plasma which consists of two different types of nonisothermal electrons, hot adiabatic inertial ions fluid and immobile negatively charged dust particles is studied. The modified Zakharov–Kuznetsov (MZK) equation, describing the small but finite amplitude DIASWs, is derived using a reductive perturbation method. The combined effects of the external magnetic field, obliqueness (i.e., the propagation angle), and the two-temperature nonisothermal electrons, which are found to significantly modify the basic properties of DIASWs, are explicitly examined. The three-dimensional instability of DIASWs is also analyzed using the small-k (long wavelength plane wave) perturbation expansion technique. The results show that the external magnetic field, the propagation angle, and the two-temperature nonisothermal electrons have strong effects on the instability criterion as well as the growth rate.

Steady, unsteady and transient vortex-induced vibration predicted using controlled motion data

In this study, we represent transient and unsteady dynamics of a cylinder undergoing vortex-induced vibration, by employing measurements of the fluid forces for a body controlled to vibrate sinusoidally, transverse to a free stream. We generate very high-resolution contour plots of fluid force in the plane of normalized amplitude and wavelength of controlled oscillation. These contours have been used with an equation of motion to predict the steady-state response of an elastically mounted body. The principal motivation with the present study is to extend this approach to the case where a freely vibrating cylinder exhibits transient or unsteady vibration, through the use of a simple quasi-steady model. In the model, we use equations which define how the amplitude and frequency will change in time, although the instantaneous forces are taken to be those measured under steady-state conditions (the quasi-steady approximation), employing our high-resolution contour plots.The resolution of our force contours has enabled us to define mode regime boundaries with precision, in the amplitude–wavelength plane. Across these mode boundaries, there are discontinuous changes in the fluid force measurements. Predictions of free vibration on either side of the boundaries yield distinct response branches. Using the quasi-steady model, we are able to characterize the nature of the transition which occurs between the upper and lower amplitude response branches. This regime of vibration is of practical significance as it represents conditions under which peak resonant response is found in these systems. For higher mass ratios (m* > 10), our approach predicts that there will be an intermittent switching between branches, as the vortex-formation mode switches between the classical 2P mode and a ‘2POVERLAP’ mode. Interestingly, for low mass ratios (m* ~ 1), there exists a whole regime of normalized flow velocities, where steady-state vibration cannot occur. However, if one employs the quasi-steady model, we discover that the cylinder can indeed oscillate, but only with non-periodic fluctuations in amplitude and frequency. The character of the amplitude response from the model is close to what is found in free vibration experiments. For very low mass ratios (m* < 0.36 in this study), this regime of unsteady vibration response will extend all the way to infinite normalized velocity.

On the stability of obliquely propagating dust ion-acoustic solitary waves in hot adiabatic magnetized dusty plasmas

Obliquely propagating dust ion acoustic solitary waves (DIASWs) are investigated in hot adiabatic magnetized dusty plasmas consisting of hot adiabatic inertial ions, hot adiabatic inertialess electrons, and negatively/positively charged static dust grains. Using a reductive perturbation method, a nonlinear Zakharov–Kuznetsov equation is derived. The effects of the concentration of negatively/positively charged dust particles and ion-neutral collision on the basic characteristics of DIASWs are studied. The three-dimensional stability of these waves is examined by the use of small-k (long wavelength plane wave) perturbation expansion technique. It is shown that the instability criterion and their growth rate depend on external magnetic field, obliqueness, the concentration of charged dust grains, ion-neutral, and ion-dust collisions.

Prediction of vortex-induced vibration response by employing controlled motion

In order to predict response and wake modes for elastically mounted circular cylinders in a fluid flow, we employ controlled-vibration experiments, comprised of prescribed transverse vibration of a cylinder in the flow, over a wide regime of amplitude and frequency. A key to this study is the compilation of high-resolution contour plots of fluid force, in the plane of normalized amplitude and wavelength. With such resolution, we are able to discover discontinuities in the force and phase contours, which enable us to clearly identify boundaries separating different fluid-forcing regimes. These appear remarkably similar to boundaries separating different vortex-formation modes in the map of regimes by Williamson & Roshko (J. Fluids Struct., vol. 2, 1988, pp. 355–381). Vorticity measurements exhibit the 2S, 2P and P + S vortex modes, as well as a regime in which the vortex formation is not synchronized with the body vibration. By employing such fine-resolution data, we discover a high-amplitude regime in which two vortex-formation modes overlap. Associated with this overlap regime, we identify a new distinct mode of vortex formation comprised of two pairs of vortices formed per cycle, where the secondary vortex in each pair is much weaker than the primary vortex. This vortex mode, which we define as the 2POVERLAP mode (2PO), is significant because it is responsible for generating the peak resonant response of the body. We find that the wake can switch intermittently between the 2P and 2PO modes, even as the cylinder is vibrating with constant amplitude and frequency. By examining the energy transfer from fluid to body motion, we predict a free-vibration response which agrees closely with measurements for an elastically mounted cylinder. In this work, we introduce the concept of an ‘energy portrait’, which is a plot of the energy transfer into the body motion and the energy dissipated by damping, as a function of normalized amplitude. Such a plot allows us to identify stable and unstable amplitude-response solutions, dependent on the rate of change of net energy transfer with amplitude (the sign of dE*/dA*). Our energy portraits show how the vibration system may exhibit a hysteretic mode transition or intermittent mode switching, both of which correspond with such phenomena measured from free vibration. Finally, we define the complete regime in the amplitude–wavelength plane in which free vibration may exist, which requires not only a periodic component of positive excitation but also stability of the equilibrium solutions.

Wave propagation in a sandwich structure

The propagation of elastic waves in a sandwich structure with two thin stiff face-plates and a thick compliant core is considered in this paper. A complete description of the dispersion relation with no restrictions on frequency and wavelength is provided. This is accomplished by transforming the wave equation to a Hamiltonian system and then using a transfer matrix approach for solving the Hamiltonian system. To provide insight, particular regimes of the frequency–wavelength plane are then considered. First, an explicit formula is derived for all natural frequencies at the long wavelength limit. It is shown that all waves with finite limiting frequency have zero group velocity, while those with vanishing limiting frequency correspond to longitudinal, shear and flexural waves. The displacement of the flexural waves are reminiscent of Mindlin plates, and an asymptotic procedure to find the shear correction factor is presented. Second, the lowest branch of the dispersion relation is studied in detail and mode shapes are used to motivate explicit but accurate description of this lowest branch. This approximate model is anticipated to be useful in simulations of large structures with sandwich structures.