球形粒子在聚焦拉盖尔-高斯光束中的散射特性研究

研究了球形粒子在聚焦拉盖尔-高斯光束中的散射特性. 根据广义Mie理论, 推导出球形粒子在聚焦拉盖尔-高斯光束中散射系数的解析公式. 针对光束的电场分布及粒子散射强度进行了数值仿真, 讨论了散射强度随散射角、散射球粒子半径和拓扑荷的变化特性, 并通过散射系数解释了散射强度分布的振荡现象. 结果表明, 在聚焦拉盖尔-高斯光束照射下, 球形粒子的后向散射强度随着粒子半径的增大而逐渐增大; 后向散射强度开始增大时对应的粒子半径与拓扑荷有关. 通过与高斯光束的对比, 可以看出球形粒子在聚焦拉盖尔-高斯光束中散射特性的差异, 使其在粒径测量、光通信和大气后向散射探测等方面具有潜在应用价值.     

Induced focusing and conversion of a Gaussian beam into an elliptic Gaussian beam

We have presented an investigation of the induced focusing in Kerr media of two laser beams, the pump beam and the probe beam, which could be either Gaussian or elliptic Gaussian or a combination of the two. We have used variational formalism to derive relevant beam-width equations. Among several important findings, the finding that a very week probe beam can be guided and focused when power of both beams are well below their individual threshold for self-focusing, is a noteworthy one. It has been found that induced focusing is not possible for laser beams of any wavelength and beam radius. In case both beams are elliptic Gaussian, we have shown that when power of both beams is above a certain threshold value then the effective radius of both beams collapses and collapse distance depends on power. Moreover, it has been found that induced focusing can be employed to convert a circular Gaussian beam into an elliptic Gaussian beam.     

Analysis and computations of measurement system effects in ultrasonic scattering experiments.

A model that characterizes the effects of beams and waveforms on the measurement of ultrasonic scattering is analyzed in detail. The analysis obtains a wideband expression for the system function in terms of an integration over spatial- and temporal-frequency variables. The temporal-frequency integration is reduced to a convolution in the direction of the scattering vector when the temporal frequencies are concentrated in a narrow band around a central frequency. The spatial-frequency integration is simplified to a straight line path when the spatial frequencies in the angular spectra of the emitter are concentrated around a point on the axis of the emitter and the spatial frequencies of the detector sensitivity pattern are similarly concentrated around a point on the axis of the detector. Expressions that result from the temporal and spatial approximations are evaluated analytically for circularly symmetric Gaussian spatial apertures and Gaussian temporal waveforms. In addition, numerical results are obtained to compare the effects of circularly symmetric Gaussian, exponential, and uniform spatial aperture functions on the weight that beam patterns have on measurements of scattering. The results may be used to design experiments from which intrinsic parameters of scattering media can be obtained by an appropriate normalization to remove measurement system effects from the data.     

Quantitative comparison of self-healing ability between Bessel–Gaussian beam and Airy beam

The self-healing ability during propagation process is one of the most important properties of non-diffracting beams. This ability has crucial advantages to light sheet-based microscopy to reduce scattering artefacts, increase the quality of the image and enhance the resolution of microscopy. Based on similarity between two infinite-dimensional complex vectors in Hilbert space, the ability to a Bessel–Gaussian beam and an Airy beam have been studied and compared. Comparing the evolution of the similarity of Bessel–Gaussian beam with Airy beam under the same conditions, we find that Bessel–Gaussian beam has stronger self-healing ability and is more stable than that of Airy beam. To confirm this result, the intensity profiles of Bessel–Gaussian beam and Airy beam with different similarities are numerically calculated and compared.     

Optimum and standard beam widths for numerical modeling of interface scattering problems

Gaussian beams provide a useful insonifying field for surface or interface scattering problems such as encountered in electromagnetics, acoustics and seismology. Gaussian beams have these advantages: (i) They give a finite size for the scattering region on the interface. (ii) The incident energy is restricted to a small range of grazing angles. (iii) They do not have side lobes. (iv) They have a convenient mathematical expression. The major disadvantages are: (i) Insonification of an interface is nonuniform. The scattered field will depend on the location of the scatterers within the beam. (ii) The beams spread, so that propagation becomes an integral component of the scattering problem. A standard beam parameterization is proposed which keeps propagation effects uniform among various models so that the effects of scattering only can be compared. In continuous wave problems, for a given angle of incidence and incident amplitude threshold, there will be an optimum Gaussian beam which keeps the insonified area as small as possible. For numerical solutions of pulse beams, these standard parameters provide an estimate of the smallest truncated domain necessary for a physically meaningful result.     

Identification of embedded horizontal cracks in beams using measured mode shapes

Mode shapes (MSs) have been extensively used to detect structural damage. This paper presents two new non-model-based methods that use measured MSs to identify embedded horizontal cracks in beams. The proposed methods do not require any a priori information of associated undamaged beams, if the beams are geometrically smooth and made of materials that have no stiffness discontinuities. Curvatures and continuous wavelet transforms (CWTs) of differences between a measured MS of a damaged beam and that from a polynomial that fits the MS of the damaged beam are processed to yield a curvature damage index (CDI) and a CWT damage index (CWTDI), respectively, at each measurement point. It is shown that the MS from the polynomial fit can well approximate the measured MS and associated curvature MS of the undamaged beam, provided that the measured MS of the damaged beam is extended beyond boundaries of the beam and the order of the polynomial is properly chosen. The proposed CDIs of a measured MS are presented with multiple resolutions to alleviate adverse effects caused by measurement noise, and cracks can be identified by locating their tips near regions with high values of the CDIs. It is shown that the CWT of a measured MS with the n-th-order Gaussian wavelet function has a shape resembling that of the n-th-order derivative of the MS. The crack tips can also be located using the CWTs of the aforementioned MS differences with second- and third-order Gaussian wavelet functions near peaks and valleys of the resulting CWTDIs, respectively, which are presented with multiple scales. A uniform acrylonitrile butadiene styrene (ABS) cantilever beam with an embedded horizontal crack was constructed to experimentally validate the proposed methods. The elastic modulus of the ABS was determined using experimental modal analysis and model updating. Non-contact operational modal analysis using acoustic excitations and measurements by two laser vibrometers was performed to measure the natural frequencies and MSs of the ABS cantilever beam, and the results compare well with those from the finite element method. Numerical and experimental crack identification can successfully identify the crack by locating its tips.     

Planar dropsizing by elastic and fluorescence scattering in sprays too dense for phase Doppler measurement

An attempt was made to measure, non-intrusively, average droplet sizes in a dense cooling spray of water. The small droplet size and high number density presented severe problems to conventional nonintrusive measurement methodology with phase Doppler anemometry (PDA). A recently developed optical technique, with more promise for measurements in dense sprays, laser sheet dropsizing (LSD), was tried with more success. Sources of error were considered and the uncertainty of the drop sizes measured by LSD was estimated at ±7%, neglecting multiple scattering, dropsize distribution effects and the contributions of droplets at the edge of the laser beam. The greatest of the known contributions to uncertainty is the calibration of the technique against PDA. The greatest of the unknown contributions is likely to be multiple scattering in such dense sprays. Received: 1 March 2000 / Revised version: 25 May 2000 / Published online: 20 September 2000     

An ultrasonic measurement model using a multi-Gaussian beam model for a rectangular transducer

To date, ultrasonic measurement models have primarily treated systems where circular transducers are used. Recently, however, a highly efficient ultrasonic beam model for a rectangular transducer has also become available where the transducer is represented as a superposition of a relatively few Gaussian beams. Thus, using the multi-Gaussian beams, we developed ultrasonic measurement models for systems where a rectangular transducer is employed. In this paper, we describe the developed models including the beam model, the efficiency factor for a rectangular transducer and far-field scattering models for some standard scatterers. Furthermore, the accuracy of the proposed model is verified by the comparison of the model-based predictions to the experimental measurements.     

High-Resolution Phase Doppler Particle Sizing based on Hilbert transform signal processing

A new phase Doppler anemometry (PDA) signal processing method based on a Hilbert transform algorithm is introduced and analysed. By generating a 90° phase-shifted burst signal in the time domain, the envelope of the Doppler burst can be determined. In addition, this envelope is approximated by a Gaussian exponential function. The difference of the maxima of these Gaussian approximations for two related PDA bursts gives an estimate of the time difference between these time shifted signals. With the introduction of this estimation method, the restriction to the [0,360°] interval resulting from conventional signal analysis may be avoided in many cases. To investigate the dependence on SNR, burst position, burst frequency and sampling rate, results of computer simulations are presented. The feasibility of the method is demonstrated briefly by experimental results. Phase differences of more than 2000° arising from the measurement of monodisperse droplets by a conventional PDA setup could be determined.     

均匀椭球粒子对拉盖尔-高斯光束的散射特性研究

基于广义Mie理论, 研究了椭球粒子对在轴入射的拉盖尔-高斯光束的散射特性. 通过局域近似法求解椭球坐标系中的波束因子, 计算得到了波束因子之间满足的普遍关系. 对散射强度随椭球粒子不同尺寸参数和扁圆程度的变化特性进行了数值计算, 并针对不同拓扑荷时的散射强度进行了对比分析. 结果表明: 当椭球粒子尺寸在与入射光波长可比拟的范围内变化时, 散射强度随尺寸参数的增大而增大, 随椭球长短轴之比和拓扑荷的增大而减小. 本文的理论研究能够为拉盖尔-高斯光束在粒径测量、大气激光通信、 大气遥感等领域的应用提供更准确的粒子模型和参考价值. 关键词： 椭球粒子 拉盖尔-高斯光束 波束因子 散射强度     

Arbitrary direction incident Gaussian beam scattering by multispheres

Based on spherical vector wave functions and their coordinate rotation theory,the field of a Gaussian beam in terms of the spherical vector wave functions in an arbitrary unparallel Cartesian coordinate system is expanded.The beam shape coefficient and its convergence property are discussed in detail.Scattering of an arbitrary direction Gaussian beam by multiple homogeneous isotropic spheres is investigated.The effects of beam waist width,sphere separation distance,sphere number,beam centre positioning,and incident angle for a Gaussian beam with two polarization modes incident on various shaped sphere clusters are numerically studied.Moreover,the scattering characteristics of two kinds of shaped red blood cells illuminated by an arbitrary direction incident Gaussian beam with two polarization modes are investigated.Our results are expected to provide useful insights into particle sizing and the measurement of the scattering characteristics of blood corpuscle particles with laser diagnostic techniques.     

Fast Gaussian wavepacket transforms and Gaussian beams for the Schrödinger equation

This paper introduces a wavepacket-transform-based Gaussian beam method for solving the Schrödinger equation. We focus on addressing two computational issues of the Gaussian beam method: how to generate a Gaussian beam representation for general initial conditions and how to perform long time propagation for any finite period of time. To address the first question, we introduce fast Gaussian wavepacket transforms and develop on top of them an efficient initialization algorithm for general initial conditions. Based on this new initialization algorithm, we address the second question by reinitializing the beam representation when the beams become too wide. Numerical examples in one, two, and three dimensions demonstrate the efficiency and accuracy of the proposed algorithms. The methodology can be readily generalized to deal with other semi-classical quantum mechanical problems.     

Investigation of ultrashort pulse parameters under SRS temporal compression

Based on numerical simulations, we analyze the compression of femtosecond pulses by forward stimulated Raman scattering (SRS). We have revealed the main patterns and determined the optimal conditions of this process to achieve the highest conversion efficiency. The influence of diffraction effects on the generation and compression of Gaussian beams is investigated.     

多艾里光束合成自聚焦光束的实验实现

多光束合成和单光束聚焦一直是提高激光束功率和功率密度的两个重要方法. 结合艾里光束在自由空间中沿弯曲路径传播的特性, 从数值模拟和实验两个方面, 研究了利用多个一维艾里光束合成自由空间自聚焦光束的方法, 并对所得到的模拟和实验结果进行了对比. 采用分步束传播法, 分别模拟了由四个和八个一维艾里光束合成的自聚焦光束在自由空间中的传播过程, 给出了自聚焦光束在传播过程中横向和纵向的光强分布和变化趋势. 采用计算全息和空间光调制器技术实验, 实现了多个一维艾里光束合成的自聚焦光束. 实验中分别测量了四个和八个一维艾里光束合成的自聚焦光束的横向光强分布. 实验结果和理论结果符合得较好. 另外, 为了进一步增大自聚焦光束的功率, 可以增加参与合成的一维艾里光束的数量. 同时, 自聚焦光束的焦距可以通过调整各个一维艾里光束的相对位置进行调节.     

Behaviour of scattering from a rough surface at small grazing angles

The particular problem of wave scattering at low grazing angles is of great interest because of its importance for the long-distance propagation of radio waves along the Earth's surface, radar observation of near surface objects, as well as solving many other fundamental and applied problems of remote sensing. One of the main questions is: how do the scattering amplitude and specific cross section behave for extremely small grazing angles? We consider the process of wave scattering by a statistically rough surface with the Neumann boundary condition. This model corresponds to sound scattering from a perfectly 'hard' surface (for example, the interface between air and the sea surface) or 'vertically' polarized electromagnetic waves scattered by a perfectly conducting one-dimensional (i.e. cylindrical) surface when the magnetic field vector is directed along the generating line of this cylindrical surface. We assume that the surface roughness is sufficiently small (in the sense of the Rayleigh parameter) and the surface is rigorously statistically homogeneous and therefore, infinite. We confine ourselves only to the first-order approximation of small perturbation theory and therefore consider every act of wave scattering in the Born approximation when the Bragg scattering process takes place. Only one resonant Fourier component of surface roughness is responsible for the scattering in a given direction. However, we take into account the attenuation of incident and scattered waves due to the multiple scattering processes on the path 'before' and 'after' a scattering event in a given direction. Also we consider every one of these multiple scattering events only in the Born approximation. The main result we have obtained is that for small grazing angles the scattering cross section of the diffuse component decreases as the second power of the grazing angles with respect to the incident and scattered directions, and as the fourth power of the grazing angle for the backscattering (radar) situation. Generalizing our results from plane-wave scattering to finite beams allows us to obtain the criterion on the beamwidth. For sufficiently narrow beams the multiple scattering processes do not play any role because of a short 'interaction path', and only single Bragg scattering determines the scattering amplitude (which does not tend to zero for small grazing angles). However, for sufficiently wide beams the result obtained for infinite plane waves becomes valid: due to the above-mentioned multiple scattering processes, the scattering amplitude tends to zero for small grazing angles. Consequently, the behaviour of the scattering cross section for small grazing angles depends on the radiation pattern width of the transmitting and receiving antennae: for sufficiently wide beams the scattering cross section decreases to zero at small grazing angles, but for narrow beams it tends to the finite non-zero value.     

Propagation analysis of flattened circular Gaussian beams with a misaligned circular aperture in turbulent atmosphere

The propagation properties of flattened Gaussian beam with a misaligned circular aperture in turbulent atmosphere have been studied by using the extended Huygens-Fresnel formula. From the study and numerical calculation, the effects of aperture parameters on the propagation properties of flattened Gaussian beams in turbulent atmosphere have been illuminated. The results show that angle misalignments and lateral displacements of aperture create unsymmetrical average intensity distribution at any cross section. The intensity distributions are much more sensitive to the lateral displacement than to the angle misalignments. And the propagation properties of different flattened Gaussian beams in turbulent atmosphere are also compared.     

Transducer profile effect on the second harmonic level

The measurement of nonlinear parameter of the propagating medium using finite amplitude techniques is based on the detection of the second harmonic generated nonlinearly in the investigated medium. This method requires an analytical expression for the second harmonic. Analytical expressions have been derived for the Gaussian source. For other shapes than Gaussian, a set of Gaussian beams can be used to approximate the pressure distribution at the source. Gaussian coefficients, in the literature, are provided for a uniform source. However, the sources used in many applications radiate non-uniformly because of the manner the piezoelectric element is fixed and because of Lamb waves generated in transducer’s active element. This is of a great importance to derive an analytical expression for the second harmonic for different profile “excitation” of the transducer. Our model is based on the quasilinear theory and a set of Gaussian beams. We used the K-Prony method in order to compute the Gaussian coefficients for each of the uniform, exponential, elliptic and Bessel sources. Using the obtained Gaussian coefficients we showed that the second harmonic magnitude is varying respectively to the used source’s profile. For the measurement of the nonlinear parameter one needs to compute the appropriate values of the Gaussian parameters according to the profile of the used source. One can also use the Gaussian parameters for the uniform source with a correction.     

A Bessel beam laser Doppler velocimeter

We present a laser Doppler velocimeter that is based on the use of two intersecting Bessel beams instead of the conventionally used Gaussian beams. Due to the quasi non-divergent nature of the Bessel beams, a short effective measurement volume of about 40 μm length could be realized with a low fringe spacing variation of 0.3%. Both of these advantageous values were experimentally verified in flow measurements.     

Accuracy of Phase-Doppler Anemometry in the Presence of Flames

The objectives of the present study were to clarify the influence of flames on phase-Doppler anemometry (PDA) and to consider quantitatively the accuracy of measured sizes and velocities using a known-size polystyrene particle and a Bunsen burner placed in the optical path in PDA simulating the variation of refractive index in a combusting field. It was found that fluctuations of a flame had some influence on the measurement, especially on size, while the velocities obtained in these experiments showed little influence of the flame. Furthermore, a compensation procedure for the error was discussed by comparing the scattered data in size measurement with the displacement of incident laser beams. As a result, it was found that the displacement of the laser beams can be related to the degree of errors caused by the influence of a flame on the optical path in PDA.     

Assessment of Pulsed Gasoline Fuel Sprays by Means of Qualitative and Quantitative Laser‐based Diagnostic Methods

The combination of qualitative measuring techniques such as imaging, with quantitative drop sizing techniques like Laser Diffraction and Phase Doppler Analyzer (PDA), has been applied for assessing the sprays formed by injectors for gasoline direct injection (DI) engines. Both, the sizing instruments as well as the imaging, are offering temporal resolution in order to investigate the important features of pulsed DI sprays. Using a combination of the spatially integrating Laser Diffraction instrument with strobe illuminated dual view 2D‐imaging, the overall spray properties have been assessed. Having the 2D information of the global spray shape in two perpendicular directions allows one to immediately correlate the concentration and drop size measurement results of the Laser Diffraction instrument with the global spray appearance. Thus, the changes of the spray pattern can be related with the sizing information as the spray propagates away from the injector. For injector design improvements, however, it is required to achieve a higher spatial resolution and especially to measure closer to the injector exit orifice than the Laser Diffraction allows. By using a Phase Doppler Analyzer, the different phases of the injection event, i.e. opening of the injector, main spray and closing phase of the injector, can be distinguished from each other. However, in sprays, where the spray geometry is changing with time, the PDA can suffer due to its high spatial resolution, yielding results that are difficult to interpret.Assisting the PDA with a simultaneous imaging technique of similar spatial resolution creates a very robust experimental approach. By visualizing the plane perpendicular to the PDA probe volume, i.e. the crossing of the PDA laser beams on the spray image itself, a very precise adjustment of the PDA probe volume with respect to the spray rather than the nozzle can be achieved. This becomes critical when getting to the near orifice area at distances closer than 10mm. The synchronized images also bring additional information to the point measurement provided by the PDA. It becomes easier to choose which particular phase of the spray formation the user wants to characterize. Finally, more confidence in the interpretation of PDA data from locations close to the injector tip is reached.