高斯叠代法研究相控阵非线性声场

相控阵在聚焦超声治疗应用中不可避免地受到非线性影响,提出了采用高斯叠代法计算相控阵的非线性声场。在该方法中,利用预设焦点参数并应用伪逆矩阵算法得到阵元的激励参数;然后将阵元近似拟合成一组高斯声束的叠加,通过高斯声束叠代计算非线性声场。数值计算中以64阵元一维相控阵为研究对象;线性条件下,高斯叠代法结果与菲涅耳积分结果的误差低于0.5%,验证了该方法的可行性;单焦点及双焦点模式的相控阵非线性声场结果表明非线性效应能提高焦点聚焦性能,并且非线性效应与激励声压及激励频率成正比。      

任意半径圆形电流在径向成层介质中产生的电磁场

采用递推矩阵方法计算径向成层介质中的二维Green函数.根据层界面处电场和磁场的连续性条件得到确定待定系数的矩阵方程组并通过递推方法快速求解.只需改变方程组中源项元素的位置,就可以方便地得到当源点和场点在任意层时的二维Green函数,并进而得到具有任意半径的圆形电流在介质中产生的电磁场.本文给出的Green函数具有表达方式简洁的优点.     

Propagation properties of Gaussian beams through the anamorphic fractional Fourier transform system with an eccentric circular aperture

As known, it is important for the propagation of Gaussian beams in optics. In this paper, based on the expanding a hard-edge circular aperture function as a finite sum of complex Gaussian functions and the scalar Rayleigh-Sommerfeld diffraction formula, an approximation analytical solution for Gaussian beams propagating through the anamorphic fractional Fourier transform system with an eccentric circular aperture is performed. Then, the detailed numerical calculation for the two-dimensional Gaussian beams in the above-mentioned optical system is presented. The simulation also shows that different location and size of aperture result in the change of diffracted field, including location, intensity and width. All these characteristics help us understand Gaussian beams propagation better.     

Two-dimensional linearly anisotropic scattering in a semi-infinite cylindrical medium exposed to a laser beam

A modification of Ambarzumian's method is used to develop the integro-differential equations for the source function, flux, and intensity at the boundary of a two-dimensional, semi-infinite cylindrical medium which scatters linearly. The incident radiation is collimated, normal to the top surface of the medium, and is dependent only on the radial coordinate. The radial variation is assumed to be a Bessel function or a Gaussian distribution. The Gaussian boundary condition is used to simulate a laser beam. Numerical results are presented in graphical and tabular forms for both boundary conditions. Results for forward and backward scattering phase functions are compared with those for isotropic scattering. A method is presented for extending these results to the problem of a strongly anisotropic phase function which is made up of a spike in the forward direction superimposed on a linear phase function.     

Gaussian beam decomposition of high frequency wave fields using expectation–maximization

A new numerical method for approximating highly oscillatory wave fields as a superposition of Gaussian beams is presented. The method estimates the number of beams and their parameters automatically. This is achieved by an expectation–maximization algorithm that fits real, positive Gaussians to the energy of the highly oscillatory wave fields and its Fourier transform. Beam parameters are further refined by an optimization procedure that minimizes the difference between the Gaussian beam superposition and the highly oscillatory wave field in the energy norm.     

Generation of the basis sets for multi-Gaussian ultrasonic beam models--an overview

By using a small number of Gaussian basis functions, one can synthesize the wave fields radiated from planar and focused piston transducers in the form of a superposition of Gaussian beams. Since Gaussian beams can be transmitted through complex geometries and media, such multi-Gaussian beam models have become powerful simulation tools. In previous studies the basis function expansion coefficients of multi-Gaussian beam models have been obtained by both spatial domain and k-space domain methods. Here, we will give an overview of these two methods and relate their expansion coefficients. We will demonstrate that the expansion coefficients that have been optimized for circular piston transducers can also be used to generate improved field simulations for rectangular probes. It will also be shown that because Gaussian beams are only approximate (paraxial) solutions to the wave equation, a multi-Gaussian beam model is ultimately limited in the accuracy it can obtain in the very near field.     

圆锥面PVDF聚焦换能器的非线性声场理论及实验研究

圆锥面聚焦换能器可在超声成像中获得较好径向分辨率的同时提高探测深度.利用高斯声源函数叠加法来近似表示圆锥面聚焦声源的分布函数，结合近轴近似的KZK方程，得到了圆锥面聚焦换能器在损耗媒质中产生的基波、二次谐波声场的解析解.在实验上制作了PVDF圆锥面聚焦换能器，测量了圆锥面聚焦换能器的基波及二次谐波声场，实验结果和理论计算相符.     

Quasi-Gaussian Bessel-beam superposition: application to the scattering of focused waves by spheres

A superposition of zero-order Bessel beams is examined that closely resembles an idealized paraxial Gaussian beam, provided the superposition is not tightly focused. Plots compare wavefield properties in the focal region and in the far field for different values of kw(0), the product of the wavenumber k, and the focal-spot-radius w(0). The superposition (which is an exact solution of the Helmholtz equation) has the important property that the scattering by an isotropic sphere can be calculated without any approximations for the commonly considered case of linear waves propagating in an inviscid fluid. The nth partial wave amplitude is similar to the case of plane-wave illumination except for a weighting factor that depends on incomplete gamma functions. An approximation for the weighting factor is also discussed based on a generalization of the Van de Hulst localization principle for a sphere of radius a at the focus of a Gaussian beam. Examples display differences between the directionality of the scattering with the plane wave case even though for the cases displayed, ka does not exceed 2 and w(0)∕a is not less than 2. Properties of tightly focused wavefields and the partial wave weighting factors are discussed.     

Topological order and conformal quantum critical points

We discuss a certain class of two-dimensional quantum systems which exhibit conventional order and topological order, as well as quantum critical points separating these phases. All of the ground-state equal-time correlators of these theories are equal to correlation functions of a local two-dimensional classical model. The critical points therefore exhibit a time-independent form of conformal invariance. These theories characterize the universality classes of two-dimensional quantum dimer models and of quantum generalizations of the eight-vertex model, as well as and non-abelian gauge theories. The conformal quantum critical points are relatives of the Lifshitz points of three-dimensional anisotropic classical systems such as smectic liquid crystals. In particular, the ground-state wave functional of these quantum Lifshitz points is just the statistical (Gibbs) weight of the ordinary two-dimensional free boson, the two-dimensional Gaussian model. The full phase diagram for the quantum eight-vertex model exhibits quantum critical lines with continuously varying critical exponents separating phases with long-range order from a deconfined topologically ordered liquid phase. We show how similar ideas also apply to a well-known field theory with non-Abelian symmetry, the strong-coupling limit of 2+1-dimensional Yang–Mills gauge theory with a Chern–Simons term. The ground state of this theory is relevant for recent theories of topological quantum computation.     

Stokes beams formed by superposition of a completely unpolarized and a completely polarized Gaussian Schell-model beam

Analytic expressions and computed examples are given to elucidate the coherence and polarization properties of Stokes beams, i.e. beams formed by superposition of a completely unpolarized and a completely polarized electromagnetic Gaussian Schell-model beam. We found that superposition of such two beams cannot form a Stokes beam with a constant state of polarization on propagation. An additional constraint on the source plane parameters of the two Gaussian Schell-model beams is proposed. The resultant Stokes beam with a constant state of polarization on propagation is found to be a Gaussian Schell-model beam with the same variances as the two constituent Gaussian Schell-model beams. However, the modulus of the Gaussian intensity distributions across the source planes of these beams may be different.     

A generalization of image theory to predict the interaction of multipole fields with plane surfaces

A derivation and computational scheme, based on exact image theory, for the field produced by the interaction of an outgoing vector wave harmonic with an infinite-extent plane surface is presented. The method represents the angular-dependent Fresnel reflection coefficients of the surface as Laplace transforms of a spatially dependent function, which results in the reflected field appearing as a superposition of image sources located at complex points along the normal axis within the surface medium. Exact, analytical formulas are given for the transformed reflection coefficients for arbitrary surface refractive index, and an efficient computation scheme for evaluation of the scattered field coupling between a particle and the surface is presented.     

Two-dimensional isotropic scattering in a finite thick cylindrical medium exposed to a laser beam

Graphical and tabular results are presented for the back-scattered intensity from a finite two-dimensional cylindrical medium exposed to a Gaussian beam of radiation. Also, results for the source function and flux at the boundaries are presented. The influence of optical thickness and albedo are most pronounced at large optical radii. The semi-infinite results can be used to approximate the finite case for small optical radii. Ranges for single, double, and multiple scattering are discussed. For locations far from the incident beam, the results can be expressed in terms of universal functions independent of beam size. A method is presented for extending the isotropic results to the anisotropic case where the phase function is made up of a spike superimposed on an otherwise isotropic phase function.     

Convergence of general beams into Gaussian intensity profiles after propagation in turbulent atmosphere

It is shown that a general shaped laser beam will eventually approach a Gaussian average intensity profile after propagation in turbulent atmosphere. In our formulation, source field at the exit plane of the laser is taken as the product of arbitrary functions of source transverse coordinates with Gaussian exponential modulations. Following the expansion of the arbitrary functions in terms of Hermite polynomials, the average receiver intensity expression is derived using the extended Huygens-Fresnel principle and the conditions for the intensity profile to assume a Gaussian shape are stated. The results are illustrated by simulating various source field distributions.     

柱形分布的电荷产生的电势

将无限长线电荷看成二维平面上的一个“源”,利用电势叠加原理计算了无限长柱形均匀分布的电荷产生的电势.这些柱体的截面电荷分布包括可解析表示和不可求解表示两种,后者可以通过数值计算给出结果.因此这种方法实际上可以计算任意截面柱形电荷分布的电势.通过计算,展示了各种电荷分布所产生电场的特性和共同特征.     

Intensity distribution and coherence property for the broadband Gaussian Schell-model array beams in free space

The intensity distribution and the equal-time complex degree of coherence for the broadband Gaussian Schell-model array beams with different bandwidth propagating in free space are analyzed by the numerical calculation. Results show that the intensity distribution for the broadband Gaussian Schell-model array beams by uncorrelated superposition is rarely affected by the bandwidth; the intensity distribution in the far field for the broadband Gaussian Schell-model array beams by correlated superposition is affected by the bandwidth more obviously than that in the near field. With the increasing of the propagation distance, differences in the equal-time complex degree of coherence for the broadband Gaussian Schell-model array beams with different bandwidth become prominent.     

Computation of AC losses in high-temperature superconductors

A formulation for the computation of AC losses in technical HTS conductors by using commercial FEM packages developed for two-dimensional computation of electromagnetic problems is presented. The formulation takes into account the real current density–electric field characteristic of a conductor and the spatial dependence of the current density. Having presented the formulation, example runs comparing transport current loss behaviour between HTS and LTS conductors are given.     

利用部分相干光和同心角偏差透镜列阵实现二维靶面均匀辐照

提出利用部分相干光通过透镜列阵系统实现靶而的均匀辐照, 透镜列阵能获得边缘陡峭且顶部较平坦的准近场焦斑, 透镜列阵的适当同心度偏差又使各子束的斑纹相互稍微错开, 使干涉斑纹变密, 而入射的激光是空间相干性减少了的部分相干光, 可进一步抑制了透镜列阵系统焦斑的小尺度不均匀性. 使用广义衍射积分理论, 对高斯-谢尔光束通过透镜列阵光学系统的焦斑光强分布进行了详细的二维模拟研究, 比较了完全相干光与部分相干光经过透镜列阵的匀滑效果. 数值模拟表明, 应用同心度偏差和适当离焦时, 可实现焦斑均匀性和能量利用率同时达到最佳的效果.     

Shadowing function with single reflection from anisotropic Gaussian rough surface. Application to Gaussian, Lorentzian and sea correlations

In this paper, the monostatic (transmitter and receiver are located at the same place) and bistatic (transmitter and receiver are distinct) statistical shadowing functions from an anisotropic two-dimensional randomly rough surface are presented. This parameter is especially important in the case of grazing angles for computing the bistatic scattering coefficient in optical and microwave frequencies. The objective of this paper is to extend the previous work (Bourlier C, Berginc G and Saillard J 2002 Waves Random Media 12 145-74), valid for a one-dimensional surface, to a two-dimensional anistropic surface by considering a joint Gaussian process of surface slopes and heights separating two points of the surface. The monostatic average (statistical shadowing function average over the statistical variables) shadowing function is then performed in polar coordinates with respect to the incidence angle, the azimuthal direction and the surface height two-dimensional autocorrelation function. In addition, for a bistatic configuration, it depends on the incidence angle and azimuthal direction of the receiver. For Gaussian and Lorentzian correlation profiles and practically important power-type spectra such as the Pierson-Moskowitz sea roughness spectrum, the numerical solution, obtained from generating the surface Gaussian elevations (Monte Carlo method), is compared with the uncorrelated and correlated models. The results show that the correlation underestimates the shadow slightly, whereas the uncorrelated results weakly overpredict the shadow and are close to the numerical solution.     

A high resolution spatially adaptive vortex method for separating flows. Part I: Two-dimensional domains

A grid-free high-resolution spatially-adaptive vortex method for two-dimensional incompressible flow in bounded domains is presented. The computational algorithm is based on operator splitting in which convection and diffusion are handled separately every time step. In the convection step, computational elements are convected with velocities obtained by fast approximations of the Biot–Savart superposition with second-order Runge–Kutta time integration scheme. Diffusion is performed using the smooth redistribution method that employs a Gaussian basis function for vorticity in the interior. Near solid walls, the core functions are modified to conserve circulation. The no-slip boundary condition is enforced by creating of a vortex sheet that is redistributed to neighboring elements using the redistribution method. The proposed method enables accurate and smooth recovery of the vorticity and does not require explicit use of vortex images or occasional re-meshing. Algorithms for reduction in computational cost by accurately removing elements in overcrowded regions and for spatial adaptivity that allows for variable core sizes and variable element spacing are presented. Computations of flow around an impulsively started cylinder for Reynolds number values of 1000, 3000, and 9500 are preformed to investigate various aspects of the proposed method.