Beam Approximation Research Articles

Gaussian Beam Shaping and Multivariate Analysis in Plasmonic Sensing.

This work demonstrates a novel strategy to improve the sensing performance of a prism-coupled surface plasmon resonance system by Gaussian beam shaping and multivariate data analysis. The propagation of the beam along the optical system has been studied using the Gaussian beam approximation to design the incident beam such that the beam waist is aligned precisely and that stability is assured at the metal-dielectric interface. This renders a collimated incident beam, hence least angular dispersion, yielding a stronger and sharper plasmonic resonance. Moreover, we use the multivariate analysis method partial least squares that combines multiple features of the surface plasmon resonance curve and allows for a more precise analysis of the plasmonic response. Compared to univariate analysis, partial least squares improves typical sensing performance parameters remarkably. The combination of both aspects, beam shaping and multivariate analysis, overcomes current limitations of plasmonic detection systems. Thereby, we improve analytical sensitivity by a factor of 16, decrease the prediction error of the concentration of an unknown analyte by a factor of 11, and enhance resolution to the order of 5 × 10-7 RIU in angular interrogation.

Numerical and analytical study for ultrasonic testing of internal delamination defects considering surface roughness

The improvement of detection accuracy and reliability of micro delamination defect is restricted by the rough surface. The flat bottom holes (FBHs) are frequently employed as reference targets to evaluate the sensitivity of ultrasonic testing for internal defects. A roughness-modified analytical model for ultrasonic testing of FBHs is established based on the principle of multi-Gaussian beam and phase-screen approximation. The signal of reference reflector is obtained from two-dimensional ultrasonic simulation model. The amplitude changes of echo signals and noises of FBHs with different diameters and depths under rough surfaces are presented. The analyses results indicate that the root-mean-square (rms) height plays a dominant role in the amplitude change of signals compared with the correlation length. The reflected wave amplitude of FBHs decreases nonlinearly with an increase of roughness whereas the amplitude of noise increases slightly. Subsequently, a method is proposed further combining the noise amplitude acquired from numerical simulation and the echo signal amplitude obtained by the analytical model, which is to predict and estimate the detection accuracy of internal defects under different surface roughness. The parallel experiments are performed on several samples with different roughness to validate the evaluation method.

Optimization of sum frequency output from intra-cavity sum frequency generation in singly resonant optical parametric oscillator

Theoretical investigation and optimization of output characteristics of intra-cavity sum frequency generation between pump and the signal waves in singly resonant optical parametric oscillators (IC-SFG-SROs) are presented. Under the collimated Gaussian beam approximation, simple models considering different spatial orderings of SRO and SFG processes of IC-SFG-SROs are set up. The calculations based on experimental parameters are in good agreement with experimental results. Output characteristics of IC-SFG-SROs operated at other wavelengths are optimized and predicted. For given experimental conditions (pump wavelength, pump power and SRO crystal length), the optimum length of SFG crystal and SF power are calculated, respectively. Furthermore, when a periodically poled crystal with a cascaded pattern is used as the SRO crystal and SFG crystal, the optimum SFG crystal length and the SF power are also calculated. These calculations and models provide guidelines for designing and optimizing continuous-wave tunable IC-SFG-SROs.

A Comparative Study of the Construction of Positron Emission Tomography/Computed Tomography Facilities in Three South African Hospitals

Introduction: Development of higher energy modalities such as positron emission tomography/computed tomography (PET/CT), has led to more complex shielding problems. This is due to several factors, such as the radiopharmaceutical relatively high-administered activity, high patient throughput, and high energies of 511 kilo-electron volt (keV) positron annihilation photons. Therefore, this study aimed to compare three different methods used to determine the required shielding thicknesses of PET/CT facilities. Material and Methods: The required shielding thicknesses for three facilities were determined by using three different shielding methods, i.e. narrow beam, broad beam and Monte Carlo approximation. The design goal was chosen as 6 mSv/year for radiation workers and 1 mSv/year for the public. In addition, occupancy factors (T) were established, and all calculations had a use factor (U) of 1. The workload (W) of facilities and thicknesses of all barriers were then calculated for the three facilities. Results: For narrow beam approximation the average required thicknesses obtained were 6.16 mm lead, 5.12 cm concrete and 2.95 cm iron. Broad beam approximation required an average of 7.55 mm lead, 8.01 cm concrete and 2.96 cm iron thicknesses. Monte Carlo approximation required 7.62 mm lead, 10.59 cm concrete and 2.94 cm iron thicknesses. Conclusion: The narrow beam approximation demonstrated the least shielding thickness required for the materials used in this study, which can lead to under-shielding. The broad beam and Monte Carlo approximations demonstrated higher required shielding thickness although there were discrepancies between these two approximations for lead, concrete, and iron.

Investigation of Delamination from an Elastic Base Using a Model with Two Coefficients of Subgrade Reaction

Currently, various structural components have coatings of different purposes that are multilayer or gradient structures. During operation these elements are often subjected to compressive loads, as a result of which sometimes the upper part of the coating is delaminated. To analyze this process, we use a simplified model of infinite structure, namely, a modified beam approximation that takes into account two coefficients of subgrade reaction similarly to the Pasternak model. The coating is subjected to compressive prestresses. Within the framework of the previously proposed model, delamination of the coating from the elastic base is investigated. We consider a simple case when the coefficients of subgrade reaction in the deamination zone are equal to zero. At the first stage, we conduct nondimensionalization of the model under study; obtain and study transcendental equation to find the critical load; prove the existence of a solution within some region of parameter variation; construct approximate formulas for the critical load; and indicate the correlation with the model parameters. The effect of the second coefficient of subgrade reaction on the critical load value is shown. At the second stage, based on the variational formulation, the problem of infinite coating is reduced to the problem on a finite beam with modified boundary conditions. The results obtained on the basis of a simplified model are compared with a numerical solution of the transcendental equation obtained at the first stage. At the final third stage, the problem on the supercritical behavior of the coating is investigated by minimizing the non-quadratic potential energy functional. Using the Ritz method and setting power-exponential functions corresponding to an infinite coating as coordinate, a solution of the problem on supercritical deformation is constructed. A study of the solution depending on the coefficients of subgrade reaction is carried out.

Three Dimensional Radiative Effects in Passive Millimeter/Sub-Millimeter All-sky Observations

This study was conducted to quantify the errors prompted by neglecting three-dimensional (3D) effects, i.e., beam-filling and horizontal photon transport effects, at millimeter/sub-millimeter wavelengths. This paper gives an overview of the 3D effects that impact ice cloud retrievals of both current and proposed (Ice Cloud Imager) satellite instruments operating at frequencies of ≈186.3 and ≈668 GHz. The 3D synthetic scenes were generated from two-dimensional (2D) CloudSat (Cloud Satellite) observations over the tropics and mid-latitudes using a stochastic approach. By means of the Atmospheric Radiative Transfer Simulator (ARTS), three radiative transfer simulations were carried out: one 3D, one independent beam approximation (IBA), and one-dimensional (1D). The comparison between the 3D and IBA simulations revealed a small horizontal photon transport effect, with IBA simulations introducing mostly random errors and a slight overestimation (below 1 K). However, performing 1D radiative transfer simulations results in a significant beam-filling effect that increases primarily with frequency, and secondly, with footprint size. For a sensor footprint size of 15 km, the errors induced by neglecting domain heterogeneities yield root mean square errors of up to ≈4 K and ≈13 K at 186.3 GHz and 668 GHz, respectively. However, an instrument operating at the same frequencies, but with a much smaller footprint size, i.e., 6 km, is subject to smaller uncertainties, with a root mean square error of ≈2 K at 186.3 GHz and ≈7.1 K at 668 GHz. When designing future satellite instruments, this effect of footprint size on modeling uncertainties should be considered in the overall error budget. The smallest possible footprint size should be a priority for future sub-millimeter observations in light of these results.

Beam Approximation for Dynamic Analysis of Launch Vehicles Modelled as Stiffened Cylindrical Shells

A beam approximation method for dynamic analysis of launch vehicles modelled as stiffened cylindrical shells is proposed. Firstly, an initial beam model of the stiffened cylindrical shell is established based on the cross-sectional area equivalence principle that represents the shell skin and its longitudinal ribs as a beam with annular cross-section, and the circumferential ribs as lumped masses at the nodes of the beam elements. Then, a fine finite element model (FE model) of the stiffened cylindrical shell is constructed and a modal analysis is carried out. Finally, the initial beam model is improved through model updating against the natural frequencies and mode shapes of the fine FE model of the shell. To facilitate the comparison between the mode shapes of the fine FE model of the stiffened shell and the equivalent beam model, a weighted nodal displacement coupling relationship is introduced. To prevent the design parameters used in model updating from converging to incorrect values, a pre-model updating procedure is added before the proper model updating. The results of two examples demonstrate that the beam approximation method presented in this paper can build equivalent beam models of stiffened cylindrical shells which can reflect the global longitudinal, lateral and torsional vibration characteristics very well in terms of the natural frequencies.

Asymptotic Properties of Linear Field Equations in Anti-de Sitter Space

We study the global dynamics of the wave equation, Maxwell’s equation and the linearized Bianchi equations on a fixed anti-de Sitter (AdS) background. Provided dissipative boundary conditions are imposed on the dynamical fields we prove uniform boundedness of the natural energy as well as both degenerate (near the AdS boundary) and non-degenerate integrated decay estimates. Remarkably, the non-degenerate estimates “lose a derivative”. We relate this loss to a trapping phenomenon near the AdS boundary, which itself originates from the properties of (approximately) gliding rays near the boundary. Using the Gaussian beam approximation we prove that non-degenerate energy decay without loss of derivatives does not hold. As a consequence of the non-degenerate integrated decay estimates, we also obtain pointwise-in-time decay estimates for the energy. Our paper provides the key estimates for a proof of the non-linear stability of the anti-de Sitter spacetime under dissipative boundary conditions. Finally, we contrast our results with the case of reflecting boundary conditions.

Gaussian beam sphere optics in condensed matter research

Transparent spheres in which the radius R of the sphere is larger than or equal to the wavelength λ of light have currently become of particular interest for quite a number of physical problems (2D colloidal crystal structures composed of transparent spheres, photonic crystals and meta-materials) and applications (laser lithography, fiber-optic systems, super resolution microscopy, biomedical applications, and others). The optical properties of a “full” sphere, when the light aperture r is of order R and the sphere is a natural spherical aberration, have not been investigated in ample detail yet. Strong spherical aberration makes focusing nontrivial. Usually, the exact solution of sphere optics is obtained using the Mie theory, the generalized Lorenz–Mie theory (GLMT), DFT, DDFT codes, which do not give much of a physical insight, as it requires summation of a large number of terms in a multipole expansion even for moderate sphere sizes. In this work we present an algorithm for describing the focusing properties of a transparent sphere using the Gaussian beam approximation that gives a good description of the field in the region of lens caustic. The algorithm developed for the spherical systems was used to create a code for calculating based on standard computation systems spheres with diameters ranging from ∼λ to thousands of λ. The results obtained were compared with the data of some authors obtained earlier by more sophisticated methods.

Experimental Study of Polarized Radar Scattering from the Tree Canopy at the Ka-Band

In this study, experimental measurements were carried out to estimate the polarized backscattering coefficient at the Ka-band through deciduous and coniferous trees. By utilizing a ground-based scatterometer, the returned signal was obtained in three polarized states at an incident angle ranging from 30 to 80 degrees. The scattering coefficients were estimated from the radar equation for the distributed target under narrow beam approximation. Radiometric calibration was conducted using a metal sphere, and a set of corner reflectors was used as the calibration target to ensure measurement accuracy. Both frequency and angular response of backscattering coefficients with three polarizations were analyzed for the cases of deciduous (banyan) and coniferous (pine) trees. The results indicated that the dynamic range for HV polarization is higher than that of HH and VV polarizations for both banyan and pine trees. Comparatively, the dynamic range for the pine tree is larger than that of the banyan tree.

An architectural understanding of natural sway frequencies in trees.

The relationship between form and function in trees is the subject of a longstanding debate in forest ecology and provides the basis for theories concerning forest ecosystem structure and metabolism. Trees interact with the wind in a dynamic manner and exhibit natural sway frequencies and damping processes that are important in understanding wind damage. Tree-wind dynamics are related to tree architecture, but this relationship is not well understood. We present a comprehensive view of natural sway frequencies in trees by compiling a dataset of field measurement spanning conifers and broadleaves, tropical and temperate forests. The field data show that a cantilever beam approximation adequately predicts the fundamental frequency of conifers, but not that of broadleaf trees. We also use structurally detailed tree dynamics simulations to test fundamental assumptions underpinning models of natural frequencies in trees. We model the dynamic properties of greater than 1000 trees using a finite-element approach based on accurate three-dimensional model trees derived from terrestrial laser scanning data. We show that (1) residual variation, the variation not explained by the cantilever beam approximation, in fundamental frequencies of broadleaf trees is driven by their architecture; (2) slender trees behave like a simple pendulum, with a single natural frequency dominating their motion, which makes them vulnerable to wind damage and (3) the presence of leaves decreases both the fundamental frequency and the damping ratio. These findings demonstrate the value of new three-dimensional measurements for understanding wind impacts on trees and suggest new directions for improving our understanding of tree dynamics from conifer plantations to natural forests.

Model crack with a scalable linear parameter

The cracked body strain problem is considered in a linearly elastic formulation on the basis of the interaction layer (IL) concept for loading with opening mode or shear. The stress state in the IL is determined on the basis of the average-thickness characteristics of the stressed and strained state (SSS). The proposed problem formulation includes the linear parameter (LP). The wedge force-to-IL thickness relation is derived proceeding from the problem’s analytical solution to a beam approximation. It is shown that it is possible to use the energy product of the linear size and the increment in the layer’s specific free energy as the generic criterion of destruction. The established association of the specimen size with the critical force ensures is found out to ensure the independence of the critical force from the IL thickness to a desired degree of precision. The resulting and the conventional solution for a notch in the form of mathematical cut are compared to determine the assumptions on which Griffith’s criterion concurs with the EP criterion.

Nonlinear acoustics at Moscow State University

The rapid development of nonlinear acoustics, which began in the second half of the 20th century, had several distinct centers in different countries, and one of them was M. V. Lomonosov Moscow State University (MSU) in Russia. From MSU came one of the basic equations in nonlinear acoustics—the Khokhlov-Zabolotskaya (KZ) equation, the 50th anniversary of which is celebrated this year [Sov. Phys. Acoust. 15(1), 35–40 (1969)]. By the 1960–1970s, the work conducted by Academician Rem Khokhlov and his colleagues made MSU a leading international center for nonlinear acoustics and, even more influentially, for nonlinear optics. Nonlinear optics at that time underwent rapid growth related to the invention of lasers, and many phenomena in optics had analogs in acoustics. One of them was a parabolic approximation for wave beams, which led to formulation of the KZ equation after translation to acoustics with consideration of the details of nonlinear effects in weakly dispersive media. Along with the development of theoretical foundations, initial experiments on nonlinear acoustic phenomena were carried out at MSU. The current paper highlights these studies, as well as contemporary work and achievements on nonlinear acoustics at Moscow State University.

Singular knot bundle in light.

As the size of an optical vortex knot, imprinted in a coherent light beam, is decreased, nonparaxial effects alter the structure of the knotted optical singularity. For knot structures approaching the scale of wavelength, longitudinal polarization effects become non-negligible, and the electric and magnetic fields differ, leading to intertwined knotted nodal structures in the transverse and longitudinal polarization components, which we call a knot bundle of polarization singularities. We analyze their structure using polynomial beam approximations and numerical diffraction theory. The analysis reveals features of spin-orbit effects and polarization topology in tightly focused geometry, and we propose an experiment to measure this phenomenon.

Airy-like beams with parabolic accelerating trajectories in non-paraxial case

Abstract We propose a new type of the Airy-like beams that can propagate along arbitrary parabolic accelerating trajectories without requiring the paraxial condition. Through the geometrical relation between predesigned parabolic caustic curve and local light rays emitting from initial plane, we derive the explicit expression of initial phase function of Airy-like beams and introduce an amplitude decay function to guarantee beams’ invariable intensity feature along accelerating trajectories. Furthermore, we compare the initial phase and complex amplitude distributions of the proposed beams with those of traditional Airy beams and numerically calculate the propagation of the two kinds of beams using Fourier angular spectrum formula. The results show that: (1) traditional paraxial Airy beams are the asymptotic approximation of Airy-like beams in paraxial condition, and that only propagate along parabolic accelerating trajectory in paraxial condition and ultimately lose their non-diffracting and accelerating features in the far field, (2) unlike the traditional Airy beams, our proposed Airy-like beams also possess these features even in non-paraxial condition and can maintain them up to the far field.

Quantum motion with trajectories: beyond the Gaussian beam approximation

A quantum model based on the Euler–Lagrange variational approach is proposed. Similarly to classical transport, our approach maintains the description of particle motion in terms of trajectories in a configuration space. Our method is designed to provide some corrections to the classical motion of nearly localized particles due to quantum phenomena. We describe the spatial localization of the particles by extending the usual ansatz which is at the basis of the Gaussian beam method. We discuss the completeness of our ansatz and the connection of our results with the Bohm trajectories approach. Our method finds application in the simulation of the motion of light nuclei in ab initio calculations.

On the validity of the use of a localized approximation for helical beams. II. Numerical aspects

The description of an electromagnetic beam for use in light scattering theories may be carried out by using an expansion over vector spherical wave functions with expansion coefficients expressed in terms of Beam Shape Coefficients (BSCs). A celebrated method to evaluate these BSCs has been the use of a localized approximation. In Part I of the present work devoted to formal aspects of the issue, we have demonstrated, using what is known as the N-procedure, that the use of a localized approximation is likely to be of limited validity in the case of helical beams exhibiting a nonzero topological charge. In the present Part II devoted to numerical aspects, we confirm the previous statement by relying on the comparison between exact and localized values of the BSCs in the case of Laguerre–Gauss beams. As a by-product we shall exhibit new examples of helical beams which are nonvortex beams.

Laser-beam scintillations for weak and moderate turbulence

The scintillation index is obtained for the practically important range of weak and moderate atmospheric turbulence. To study this challenging range, the Boltzman-Langevin kinetic equation, describing light propagation, is derived from first principles of quantum optics based on the technique of the photon distribution function (PDF) [G. P. Berman et al., Phys. Rev. A 74, 013805 (2006)]. The paraxial approximation for laser beams reduces the collision integral for the PDF to a two-dimensional operator in the momentum space. Analytical solutions for the average value of PDF as well as for its fluctuating constituent are obtained using an iterative procedure. The calculated scintillation index is considerably greater than that obtained within the Rytov approximation even at moderate turbulence strength. The relevant explanation is proposed.

DEFORMATIONS AND STRESSES IN THE CONSOLE SYSTEM «FILM – SUBSTRATE»

The purpose of this publication is analysis of the stress-strain state in the console system «film – substrate». Deformations of the cantilever system «film – substrate» under the impact of stresses occurring upon film deposition are analyzed in beam approximation. General formulas are derived to calculate the angular and linear deformation of the system at various values of the module of elasticity of the materials of the substrate and film. In contrast to the well-known Stoney formula, the expression obtained in this paper for the angular deformation of the cantilever system is valid for different values of the film and substrate thickness. The Stoney formula is the limiting case of the derived general formula valid when the thickness of the film is several orders of magnitude less than the substrate thickness. The influence of various elastic moduli on the angular and linear deformation of the console, as well as on the stress distribution in the «film – substrate» system, is estimated. It is shown that if the substrate thickness is 100 times larger than the film thickness, the angular deformation changes approximately by a factor of two at 2-fold change in the ratio the moduli of elasticity. The measuring capabilities of the cantilever converter of stresses in the film into the angular deformation of the end of the console are analyzed. The sensitivity of the converter is estimated as a function of the ratio of film and substrate thicknesses. It is shown that the obtained expressions for the displacements and stresses can be used in studying temperature deformations and stresses in the «film – substrate» system. To do this, the stress in the film should be replaced by the product of the modulus of elasticity and the temperature change and the difference in the coefficients of thermal expansion of the materials of the film and substrate.

Determination of the fibre orientation distribution of a mineral wool network and prediction of its transverse stiffness using X-ray tomography

A method to determine the orientation and diameter distributions of mineral wool fibre networks using X-ray tomography and image analysis is presented. The method is applied to two different types of mineral wool: glass wool and stone wool. The orientation information is obtained from the computation of the structure tensor, and the diameter is estimated by applying a greyscale granulometry. The results of the image analysis indicate the two types of fibres are distributed in a 2D planar arrangement with the glass wool fibres showing a higher degree of planarity than the stone wool fibres. The orientation information is included in an analytical model based on a Euler–Bernoulli beam approximation. The model enables prediction of the transverse stiffness. It is indicated that the glass wool transverse stiffness is lower than the stone wool transverse stiffness. Comparison with experimental results confirms the assumption that the underlying deformation mechanism of mineral wool is the bending of fibre segments between bonds.