A new polar nephelometer for measurement of atmospheric aerosols

The angular dependence of light scattered by aerosol particles in the atmosphere—the aerosol phase function—is one of the key properties of every radiative transfer and climate model. Side-scattering in the atmosphere is currently believed to be underestimated by most of the radiative transfer models in certain cases. The aerosol phase function can be measured with polar nephelometers.An innovative polar nephelometer able to measure the phase function of ambient aerosol directly is presented in this publication. The performance of the device was simulated completely using ray-tracing techniques. The results of these simulations are used to interpret the measurement data precisely. The measurements with the new polar nephelometer are fully automated. The quality of the measurement results was verified using different approaches. The values of the statistical and systematic error in measuring atmospheric aerosol are about 4% each. Scattering angles from 10° to 160° were measured and it is shown that with the same design this range can be extended to 3-177° in steps of 0.16° with an aperture of less than 1.8°.Measurements of the aerosol phase function are presented and compared with data taken with an integrating nephelometer and measurements of the particulate matter concentration. All data correlate very well. The described polar nephelometer can measure the phase function and the scattering coefficient of atmospheric aerosol with high accuracy and can be used for continuous monitoring measurements as well as in field campaigns.     

Toward an ideal integrating nephelometer

The Integrating Sphere Integrating Nephelometer is a novel and unique reciprocal nephelometer that uses an integrating sphere with attached truncation-reduction tubes to contain the sample volume and to integrate the scattered light. Its main advantage compared with current integrating nephelometers is a sevenfold reduction in truncation angle, which reduces errors in measured scattering from large particles. Additional features include improved sampling efficiency for large particles and a well-defined operating wavelength.     

Numerical experiments on the efficiency of local grid refinement based on truncation error estimates

Local grid refinement aims to optimise the relationship between accuracy of the results and number of grid nodes. In the context of the finite volume method no single local refinement criterion has been globally established as optimum for the selection of the control volumes to subdivide, since it is not easy to associate the discretisation error with an easily computable quantity in each control volume. Often the grid refinement criterion is based on an estimate of the truncation error in each control volume, because the truncation error is a natural measure of the discrepancy between the algebraic finite-volume equations and the original differential equations. However, it is not a straightforward task to associate the truncation error with the optimum grid density because of the complexity of the relationship between truncation and discretisation errors. In the present work several criteria based on a truncation error estimate are tested and compared on a regularised lid-driven cavity case at various Reynolds numbers. It is shown that criteria where the truncation error is weighted by the volume of the grid cells perform better than using just the truncation error as the criterion. Also it is observed that the efficiency of local refinement increases with the Reynolds number. The truncation error is estimated by restricting the solution to a coarser grid and applying the coarse grid discrete operator. The complication that high truncation error develops at grid level interfaces is also investigated and several treatments are tested.     

旋转波片偏振仪波片方位误差研究

旋转波片Stokes偏振仪是最常用的测量光束偏振态的仪器。波片快轴方位误差是影响旋转波片Stokes偏振仪的主要误差源之一。为了研究波片方位偏差对测量精度的影响,提出了一种描述波片快轴方位误差向最终的偏振测量误差传递的数学模型,并引入协方差矩阵法表征偏振测量误差。根据这一模型,获得最优的偏振仪配置参数。在推导过程中,假设波片方位误差服从同一高斯分布。基于此误差模型,得到如下结论：(1)由波片方位误差引入的测量误差与光强测量次数N成反比;(2)测量误差独立于入射光强度,但是依赖于入射光偏振态(s₁, s₂, s₃)和波片的位相延迟量δ;(3)波片位相延迟量在(103.22°, 116.13°)范围内时波片方位误差引入的测量误差最小。最后,经过仿真实验证明,所得解析结果与仿真模拟结果相一致。     

Error analysis for polychromatic speckle contrast measurements

Revival of interest in speckle technologies raises a curial question on the accuracy calculation of speckle measurements. In particular, the accuracy calculation of speckle contrast, an important metrics of a stochastic process, is quite different from the accuracy calculation of a typical intensity measurement. Speckle contrast depends more on stochastic characteristics of a process rather than on hardware characteristics. In this article, we consider errors introduced by the limited number of available speckle and by intensity saturation for monochromatic and polychromatic speckles. Equations for these types of errors were derived. Particularly, we show that the error due to limited number of speckles is inversely proportional to the square root of the speckle number, in the similar way as the average intensity error. For the error due to limited dynamic range of a recording device, the truncation effect increases the error as the mean intensity increased. Monochromatic speckles are more sensitive to such truncation than polychromatic speckles. We report the optimal mean intensity in terms of the pixel depth of a recording device and the total number of speckle recorded. In addition, a recommendation to minimize the saturation error for a typical camera is included.     

Refinement of the method for determining of the angular size of the halo from the plane wave passed through a statistically inhomogeneous screen

A two-position sensing scheme was used to determine the dependence of the angular size of the halo around a sensing beam on the hole size in an opaque screen. It was shown that the angular size of the halo is inversely proportional to the transverse hole size. This allowed us to refine the general scenario of microstructure measurements for the screen with random transmittance modulation. It was shown that the estimate of the error in the angular size of the plane wave halo is controlled by the sum of the errors of the angular sizes of the undistorted beam and the beam passed through the statistically inhomogeneous screen.     

拼接式钢带光栅编码器测角误差分析与修正

针对望远镜对高精度测角技术的要求,分析了机械安装偏心、加工椭圆度误差、轴系晃动等对新型拼接式钢带光栅编码器测角的影响,并制定了相位差为90°的4读数头软件细分测角方案。在机械安装偏心为10μm、加工椭圆度为2μm、轴系晃动为0.6″的实验平台上进行了逆时针和顺时针测角实验,两组实验测角误差RMS值分别为0.393″和0.488″。实验表明：该测角方案能消除机械安装偏心、加工椭圆度误差等对测角数据的影响,实现了亚角秒级测角。     

High order accurate, explicit, difference formulas for the classical wave equation

This article presents a simple and yet very novel approach to developing difference schemes for wave equations. The schemes that are developed are explicit in nature. The schemes are of such generality that one can transform from one difference scheme to another with only the slightest of computational effort. The schemes exhibit dispersive errors. The errors can be minimized, however, by increasing the order of truncation error. Numerical results are presented for two linear model equations with truncation error ranging up to O(h⁵). Numerical results are also presented for a system of shallow water equations. By choosing the appropriate a for a first order linear equation (a defines the geometry of an element) we may generate stable schemes for an arbitrary Courant number.     

Analysis of 2-D motion tracking in ultrasound with dual transducers

We study displacement and strain measurement error of dual transducers (two linear arrays, aligned orthogonally and coplanar). Displacements along the beam of each transducer are used to obtain measurements in two-dimensions. Simulations (5 MHz) and experiments (10 MHz) are compared to measurements with a single linear array, with and without angular compounding. Translation simulations demonstrate factors of 1.07 larger and 8.0 smaller biases in the axial and lateral directions respectively, for dual transducers compared to angular compounding. As the angle between dual transducers decreases from 90° to 40°, for 1% compression simulations, the lateral RMS error ranges from 2.1 to 3.9 μm compared to 9 μm with angular compounding. Simulation of dual transducer misalignment of 1 mm and 2° result in errors of less than 9 μm. Experiments demonstrate factors of 3.0 and 5.2 lower biases for dual transducers in the axial and lateral directions respectively compared to angular compounding.     

Correction factors for a total scatter/backscatter nephelometer

Total aerosol scattering and backscattering atmospheric values are typically obtained with an integrating nephelometer. Due to design limitations, measurements do not cover the full (0°–180°) angular range, and correction factors are necessary. The effect of angle cutoff is examined for a number of particle size distributions and refractive indices. Scattering data for sub-micron particles can be corrected by the use of a modified Anderson approximation, while data for larger particle distributions can be approximated by a function of the effective size parameter. Correction factors for the hemispheric backscatter ratio are found to be small if nonsphericity is assumed. Such approximations will help more accurate corrections for angle range, particularly at large size parameter values.     

Spectral accuracy in fast Ewald-based methods for particle simulations

A spectrally accurate fast method for electrostatic calculations under periodic boundary conditions is presented. We follow the established framework of FFT-based Ewald summation, but obtain a method with an important decoupling of errors: it is shown, for the proposed method, that the error due to frequency domain truncation can be separated from the approximation error added by the fast method. This has the significance that the truncation of the underlying Ewald sum prescribes the size of the grid used in the FFT-based fast method, which clearly is the minimal grid. Both errors are of exponential-squared order, and the latter can be controlled independently of the grid size. We compare numerically to the established SPME method by Essmann et al. and see that the memory required can be reduced by orders of magnitude. We also benchmark efficiency (i.e. error as a function of computing time) against the SPME method, which indicates that our method is competitive. Analytical error estimates are proven and used to select parameters with a great degree of reliability and ease.     

基于泽尼克多项式进行面形误差拟合的频域分析

获得泽尼克多项式的频谱信息是正确利用该多项式进行误差拟合的关键。推导出了泽尼克多项式的傅里叶变换公式,在频域中分析了不同阶数该多项式的径向频谱信息和幅角频谱信息,得到了有限项泽尼克多项式能够有效表达面形误差的最大径向空间频率和角频率。基于频域分析理论,利用泽尼克多项式对不同口径局部误差进行了拟合,并利用齐戈（Zygo）干涉仪对带有不同面形误差的光学元件进行了试验分析。结果表明,当误差的径向空间频率或角频率超出泽尼克多项式所能表达的频谱范围时,拟合误差迅速变大。     

Design of an optimal wave-vector filter for enhancing the resolution of reconstructed source field by near-field acoustical holography (NAH)

In near-field acoustical holography using the boundary element method, the reconstructed field often diverges due to the presence of small measurement errors. In order to handle this instability in the inverse problem, the reconstruction process should include some form of regularization for enhancing the resolution of source images. The usual method of regularization has been the truncation of wave vectors associated with small singular values, although the determination of an optimal truncation order is difficult. In this article, an iterative inverse solution technique is suggested in which the mean-square error prediction is used. A statistical estimation of the minimum mean-square error between measured pressures and the model solution is required for yielding the optimal number of iterations. The continuous curve of an optimal wave-vector filter is designed, for suppressing the high-order modes that can produce large reconstruction errors. Experimental results from a baffled radiator reveal that the reconstruction errors can be reduced by this form of regularization, by at least 48% compared to those without any regularization. In comparison to results using the optimal truncation method of regularization, the new scheme is shown to give further reductions of truncation error of between 7% and 39%, for the example in this article.     

拼接式编码器测角误差分析及试验

根据望远镜测角系统的精度需求,分析了影响拼接式编码器测角精度的误差源,以及多读数头测角消差原理,确定了采用相位相差90°的4读数头的测角方式。在某望远镜方位轴系转台进行了逆时针和顺时针方向的测角试验,通过对测角数据进行谐波分析并补偿后,得到两组实验测角误差RMS分别为0.34"、0.38"。实验表明,相位相差90°的均布4读数头的测角方式消除了由轴系误差、编码器安装位置误差和钢带安装盘部分圆度误差对测角精度的影响,实现了亚角秒级测角的目的。研究结果可用于大口径望远镜设计阶段的误差分析与分配、预估测角精度,为降低设计和加工误差提供参考。     

Doubling initialization revisited

In previous work, the author studied errors arising solely from initial-layer approximations in doubling. It now appears that misleading conclusions may be drawn from that work, because it failed to consider the interaction of angular and initial-layer error. That interaction is such that decreasing initial-layer error (a) often has no effect on total error, or (b) sometimes increases total error. ONe concludes from this that, contrary to accepted practice, it is advisable to use an intial layer of fairly large optical depth, in order to strike a balance between angular and initial-layer error. The diamond initialization still seems generally superior, but only dramatically so for high orders of angular approximation.     

Convergence characteristics of a domain decomposition scheme for approximation of quantum forces

We investigate the convergence characteristics of a domain decomposition scheme to approximately compute quantum forces in the context of semiclassical Bohmian mechanics. The study is empirical in nature. Errors in the approximate quantum forces are compiled while relevant parameters in the numerical scheme are systematically changed. The compiled errors are analyzed to extract underlying trends. Our analysis shows that the number of Bohmian particles used in discretization has relatively weak influence on the error, while the length scale of interaction among subdomains controls the error in most cases. More precisely, the overall numerical error decreases as the length scale of interaction among subdomains decreases, if the error due to the truncation of the tail of the probability density distribution is adequately controlled. Our results suggest that it may be necessary to develop an efficient method to effectively control the error due to the truncation of the tail. Further studies, especially rigorous mathematical ones, should follow to understand and improve the behavior of the scheme.     

Mutual alignment errors due to localized distortion in free-space laser communication links

Instead of Zernike polynomials, truncated Gaussian function is proposed to express localized wave-front deformation in studying mutual alignment errors in free-space laser communication links, which simplifies the calculation. Mutual alignment errors include pointing and tracking errors which depend on transmitting and receiving optical system, respectively. It is shown that both pointing and tracking errors depend on three parameters (the center value A, the radius a and the distance d) of Gaussian localized distortion and change regularly as they increase. The maximum pointing and tracking errors always appear around A = 0.4λ (rms = π). Moreover, pointing error is more sensitive to localized wave-front deformation than tracking error. Beam truncation ratio has great influence on pointing error, and obscuration ratio has less influence on pointing and tracking errors except that antenna secondary mirror shelters parts of the localized deformation. To reduce the influence of localized aberrations, the principles how to choose the optical devices with large aperture are given, and a method that aligns the pointing direction to compensate pointing and tracking errors is suggested. The work will contribute to the design of free-space laser communication systems.     

拼接光栅对压缩系统中的误差补偿

 根据失调拼接光栅对压缩系统模型,推导了拼接误差引起的角色散公式。针对光栅拼接中的转角误差和光栅常数误差,得到了误差间的相互补偿关系式,并对补偿方式进行了讨论。研究结果表明：绕光栅平面内垂直于刻线方向的转角在一定范围内可以补偿绕光栅厚度方向旋转造成的误差;绕光栅刻线方向的转角可以补偿光栅常数不同造成的误差;绕光栅平面内垂直于刻线方向或绕光栅厚度方向的转角在一定范围内也可以补偿由光栅常数不同造成的误差。前两种补偿方式的补偿效果较好。