Superposition Of Gaussian Beams Research Articles

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.

Axion-like-particle search with high-intensity lasers

We study ALP-photon-conversion within strong inhomogeneous electromagnetic fields as provided by contemporary high-intensity laser systems. We observe that probe photons traversing the focal spot of a superposition of Gaussian beams of a single high-intensity laser at fundamental and frequency-doubled mode can experience a frequency shift due to their intermittent propagation as axion-like-particles. This process is strongly peaked for resonant masses on the order of the involved laser frequencies. Purely laser-based experiments in optical setups are sensitive to ALPs in the $\mathrm{eV}$ mass range and can thus complement ALP searches at dipole magnets.

A new form of Gaussian beam function as the base function for generic wave field analysis.

Wave propagations are best described by three major approaches: the spherical wave approach represented by Rayleigh integral, the planar wave approach represented by spatial Fourier analysis, and the Gaussian beam approach. Spherical and planar wave approaches, both in form of double integrals, are analytically inconvenient and computationally expensive. In contrast, the Gaussian beam approach, with huge advantages in analytical operability and computational efficiency, has been quietly but firmly winning popularity. Dr. Breazeale’s article: A diffraction beam field expressed as the superposition of Gaussian beams, co-authored with Wen, is not only a most representative work for Gaussian beam approach but also a most cited academic work in sound field analysis and computation. As a memory of Dr. Breazeale’s pioneer role in the raise of Gaussian beam method, this article presents a new enhancement to the existing Gaussian beam method. A modified form of Gaussian beam function introduced as the base function set for wave field analysis greatly reduces the error rooted from the parabolic approximation, the number of Gaussian beams needed, and the run-time computation burden. The process for calculating the beam coefficients and beam parameters is also greatly simplified.

Recovery of High Frequency Wave Fields for the Acoustic Wave Equation

Computation of high frequency solutions to wave equations is important in many applications and notoriously difficult in resolving wave oscillations. Gaussian beams are asymptotically valid high frequency solutions concentrated on a single curve through the physical domain, and superposition of Gaussian beams provides a powerful tool to generate more general high frequency solutions to PDEs. An alternative way to compute Gaussian beam components such as phase, amplitude, and Hessian of the phase is to capture them in phase space by solving Liouville-type equations on uniform grids. Following [H. Liu and J. Ralston, Multiscale Model. Simul., to appear] we present a systematic construction of asymptotic high frequency wave fields from computations in phase space for acoustic wave equations; the superposition of phase space based Gaussian beams over two moving domains is shown to be necessary. Moreover, we prove that the kth order Gaussian beam superposition converges to the original wave field in the energy norm at the rate of $\epsilon^{\frac{k}{2}+\frac{1-n}{4}}$ in dimension n.

Recovery of High Frequency Wave Fields from Phase Space–Based Measurements

Computation of high frequency solutions to wave equations is important in many applications, and notoriously difficult in resolving wave oscillations. Gaussian beams are asymptotically valid high frequency solutions concentrated on a single curve through the physical domain, and superposition of Gaussian beams provides a powerful tool for generating more general high frequency solutions to PDEs. An alternative way to compute Gaussian beam components such as phase, amplitude, and Hessian of the phase is to capture them in phase space by solving Liouville-type equations on uniform grids. In this work we review and extend recent constructions of asymptotic high frequency wave fields from computations in phase space. We give a new level set method of computing the Hessian and higher derivatives of the phase. Moreover, we prove that the kth order phase space–based Gaussian beam superposition converges to the original wave field in $L^2$ at the rate of $\epsilon^{\frac{k}{2}-\frac{n}{4}}$ in dimension n.

基于倾斜多高斯光束叠加的均匀化照明整形方法

基于倾斜多高斯光束叠加的均匀化照明整形方法

Problems with power content beam parameter product

The M 2 beam propagation factor is widely used to characterize the quality of laser radiation and its propagation. When M 2 is defined by the second-moments, M 2 ⩾ 1 holds in the paraxial approach. For many applications it is more convenient to use the power content values (normally η = 86.5%), also proposed by ISO. For the corresponding power content M pc 2 factor, it is often assumed that M pc 2 ⩾ 1 also holds. We have demonstrated previously that for a superposition of two coherent Gauss–Laguerre modes with radial symmetry, the 86.5% value of M pc 2 < 1 [6]. In recent years, M pc 2 < 1 has also been presented experimentally for a superposition of axially shifted Gaussian beams [7]. The problems with power content M pc 2 for axial superposition of Gaussian beams are discussed. In this paper it is shown that the 86.5% power content M pc 2 value can not be smaller than one for a coherent superposition of axially shifted Gaussian beams with radial symmetry presented in Ref. [7]. A superposition of two Gaussian beams with different waists and without shift is also discussed, and the corresponding M pc 2 of such beam can be smaller than one, depending strongly on the power content value η. For low power content values η and a large (or very small) ratio of the two different waists M pc 2 approaches zero. These investigations demonstrate that M pc 2 is not a suitable parameter to characterize laser radiation.

Gaussian beam decomposition of high frequency wave fields

In this paper, we present a method of decomposing a highly oscillatory wave field into a sparse superposition of Gaussian beams. The goal is to extract the necessary parameters for a Gaussian beam superposition from this wave field, so that further evolution of the high frequency waves can be computed by the method of Gaussian beams. The methodology is described for R d with numerical examples for d = 2 . In the first example, a field generated by an interface reflection of Gaussian beams is decomposed into a superposition of Gaussian beams. The beam parameters are reconstructed to a very high accuracy. The data in the second example is not a superposition of a finite number of Gaussian beams. The wave field to be approximated is generated by a finite difference method for a geometry with two slits. The accuracy in the decomposition increases monotonically with the number of beams.

Efficiency of excitation of a quasioptical open resonator by a waveguide

We study theoretically and experimentally the efficiency of excitation of an open resonator by a waveguide. A two-dimensional model of a quasioptical two-mirror open resonator, which is excited through a coupling element in the form of an open end of a plane-parallel waveguide, is proposed. The field of the open resonator is represented as the superposition of Gaussian wave beams and its scattering on the coupling element is described by solving the problem of diffraction of waveguide waves and Gaussian wave beams on an open end of a flanged waveguide. Properties of the mirrors and the resonator loss are specified from physical considerations. The calculation results are compared with the experimental data, and a physical interpretation of the obtained results is offered. The factors which influence the efficiency of excitation of an open resonator are identified.

ELECTROMAGNETIC SCATTERING OF A FIELD KNOWN ON A CURVED INTERFACE USING CONFORMAL GAUSSIAN BEAMS

Asymptotic techniques have been successfully applied to compute electromagnetic wave radiation in various high-frequency engineering domains. Recent approaches based on Gaussian beams for tracking fields may overcome some problems inherent to the ray methods such as caustics. The efficiency of these methods is based on the ability to expand surface fields into a superposition of Gaussian beams. However, some difficulties may arise when the surface is curved. In this paper, we propose a new efficient way to expand fields on a curved surface into Gaussian beams. For this purpose, a new beam formulation called Conformal Gaussian Beam (CGB) is used. The CGBs have been developed to overcome the limitation of the expansion into paraxial Gaussian Beams. The analytical Plane-Wave Spectrum and far-field of a CGB are derived and compared with numerical calculations. A brief parameter analysis of the CGBs is realised.

Theoretical and experimental study of the nonlinear ultrasonic field radiated from a conical focused PVDF transducer

Conical transducer offers both good lateral resolution and large depth of detecting region for ultrasonic imaging. In this paper, the superposition of Gaussian beams is utilized to simulate the source distribution of the conical focused transducer. On the basis of the KZK equation, we obtain the analytical solution of the nonlinear sound field from the conical focused source. A conical focused PVDF transducer has been fabricated in our Lab and the sound field distributions of the fundamental and second harmonic components were experimentally measured. The measured results are in reasonably good agreement with the theoretical predictions.

Mountain Waves and Gaussian Beams

Gaussian beams are approximate solutions to hyperbolic partial differential equations that are concentrated on a curve in space-time. In this paper, we present a method for computing the stationary in time wave field that results from steady air flow over topography as a superposition of Gaussian beams. We derive the system of equations that governs these mountain waves as a linearization of the basic equations of fluid dynamics and show that this system is well-posed. Furthermore, we show that the approximate Gaussian beam stationary solution is close to a true time-dependent solution of the linearized system.

Computation of paraxial wave fields using transparent boundary conditions

The parabolic wave equation is solved numerically by applying transparent conditions used to confine the computational domain. A numerical implementation of the boundary conditions is proposed based on representing the incident wave as a superposition of Gaussian beams. A modification of the transparent conditions in the case of dielectric objects extending beyond the computational domain is described. Numerical examples are presented.

Paradigms for bit-oriented holographic information storage

One of the difficulties encountered during the many years of research on holographic information storage was the lack of an easy theoretical way to assess proposed paradigms. I exploit the fact that for bit-oriented holographic storage, Gaussian beams are usually involved. For this case I show that the reconstructed wave can be represented as a superposition of simple Gaussian beams, regardless of the exact recording condition, and a virtual source for this wave can be determined. This theoretical result is used to explore several holographic storage architectures, in particular thick volume holograms and layered volume holograms. Simulation results demonstrate the power of the method, show good correspondence with earlier experimental studies, and provide clues for further developments.

Scintillation index of flat-topped Gaussian beams

The scintillation index is formulated for a flat-topped Gaussian beam source in atmospheric turbulence. The variations of the on-axis scintillations at the receiver plane are evaluated versus the link length, the size of the flat-topped Gaussian source, and the wavelength at selected flatness scales. The existing source model that represents the flat-topped Gaussian source as the superposition of Gaussian beams is employed. In the limiting case our solution correctly matches with the known Gaussian beam scintillation index. Our results show that for flat-topped Gaussian beams scintillation is larger than that of the single Gaussian beam scintillation when the source sizes are much smaller than the Fresnel zone. However, this trend is reversed and scintillations become smaller than the Gaussian beam scintillations for flat-topped sources with sizes much larger than the Fresnel zone.

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.

Multi-Gaussian Ultrasonic Beam Modeling for Multiple Curved Interfaces—an ABCD Matrix Approach

ABSTRACT A multi-Gaussian beam model uses a superposition of Gaussian beams to simulate the waves radiated from an ultrasonic transducer. We show that propagation and reflection/transmission laws for Gaussian beams in fluids and elastic solids can be written in the form of A , B , C , D matrices that are analogous to the A, B, C, D scalars used in Gaussian optics. This representation leads to simple expressions for a Gaussian beam even after that beam has been transmitted or reflected at multiple curved interfaces and produces a highly modular multi-Gaussian beam model that is also computationally very efficient. Some examples of the use of this model for both planar and curved interfaces are given.

Influence of the beam shape on aperture measurements with the laser beam scanning technique

The realization of a number of radiometric and photometric units requires one to measure accurately the area of precision apertures. The BIPM employs the laser-beam scanning technique which is based on the generation of a uniform irradiance by a two-dimensional superposition of many regularly spaced laser beams.The existing literature describes the formation of the uniform irradiance as a superposition of Gaussian laser beams and derives a condition for the largest allowable lattice spacing to obtain the necessary uniformity. In this paper we analyse the effect of the use of real laser beams which are almost, but not completely, Gaussian by applying Fourier analysis. Experimental results of aperture measurements with different spacings of the laser beams are also shown. The results of the calculations and the measurements show a characteristic dependence of the measured area on the grid spacing, even for spacings sufficiently small according to the condition based on Gaussian theory. This can be explained by very small deviations of the actual beam shape from a perfect Gaussian profile.

Laser scanning for semiconductor mask pattern generation

The use of laser scanning to generate semiconductor masks is reviewed. Following a brief historical introduction that describes early pattern generator implementations, current and future industry mask requirements are described with the consequences for pattern generator design: the need for small features, tight CD control, and high pixel delivery rates. The system architecture of a current deep UV scanning system is described in detail along with important print strategies, such as grayscale printing and multipass error averaging. Several subsystem technologies are then explored with emphasis on the application to short wavelengths and multiple beams. Today, frequency-doubled lasers generate the 257-nm radiation used by DUV pattern generators; tomorrow, sum frequency generation will be required to reach the wavelengths at or below 200 nm. Acousto-optic modulation (AOM) technology is shown to scale favorably with shorter wavelengths and to have the bandwidth capability for future system. Acousto-optic beam deflection, polygonal mirror beam deflection, and the reduction of scan bow error through the use of an f /spl middot/ sin(/spl theta/) lens are examined. A section on scan optics and image formation presents the differences between partially coherent imaging as used by a wafer stepper and the incoherent superposition of Gaussian beams as used by a laser scanner. Partially coherent imaging is shown to have a sharper image slope but worse feature size linearity. This section also discusses the effect of finite AOM turn-on time on the aerial image in the scan direction.

Expansion of a Plane Wave into Gaussian Beams

An integral expansion which expresses a plane monochromatic wave as a superposition of Gaussian beams is found. The expansion can be used to solve many wave propagation problems in complicated structures, including laterally inhomogeneous media with curved interfaces.