Matrix method to find a new set of Zernike coefficients from an original set when the aperture radius is changed

A matrix method is developed that allows a new set of Zernike coefficients that describe a surface or wave front appropriate for a new aperture size to be found from an original set of Zernike coefficients that describe the same surface or wave front but use a different aperture size. The new set of coefficients, arranged as elements of a vector, is formed by multiplying the original set of coefficients, also arranged as elements of a vector, by a conversion matrix formed from powers of the ratio of the new to the original aperture and elements of a matrix that forms the weighting coefficients of the radial Zernike polynomial functions. In developing the method, a new matrix method for expressing Zernike polynomial functions is introduced and used. An algorithm is given for creating the conversion matrix along with computer code to implement the algorithm.     

Wave-front sensing by use of a Green's function solution to the intensity transport equation

A method for reconstructing an unknown wave front from measurements of its intensity distribution on two planes along the direction of propagation is described. The method solves the intensity transport equation by use of Neumann boundary conditions, leading to a solution that requires only matrix multiplication. The method provides real-time wave-front reconstruction with high accuracy and is easily reposed to permit reconstruction of the wave front in any orthonormal basis set.     

Polarization imaging by use of digital holography

We present what we believe to be a new digital holographic imaging method that is able to determine simultaneously the distributions of intensity, phase, and polarization state at the surface of a specimen on the basis of a single image acquisition. Two reference waves with orthogonal polarization states interfere with the object wave to create a hologram that is recorded on a CCD camera. Two wave fronts, one for each perpendicular polarization state, are numerically reconstructed in intensity and phase. Combining the intensity and the phase distributions of these two wave fronts permits the determination of all the components of the Jones vector of the object-wave front. We show that this method can be used to image and measure the distribution of the polarization state at the surface of a specimen, and the obtained results indicate that precise quantitative measurements of the polarization state can be achieved. An application of the method to image the birefringence of a stressed polymethyl methacrylate sample is presented.     

Solution to the shearing problem

Lateral shearing interferometry is a promising reference-free measurement technique for optical wave-front reconstruction. The wave front under study is coherently superposed by a laterally sheared copy of itself, and from the interferogram difference measurements of the wave front are obtained. From these difference measurements the wave front is then reconstructed. Recently, several new and efficient algorithms for evaluating lateral shearing interferograms have been suggested. So far, however, all evaluation methods are somewhat restricted, e.g., assume a priori knowledge of the wave front under study, or assume small shears, and so on. Here a new, to our knowledge, approach for the evaluation of lateral shearing interferograms is presented, which is based on an extension of the difference measurements. This so-called natural extension allows for reconstruction of that part of the underlying wave front whose information is contained in the given difference measurements. The method is not restricted to small shears and allows for high lateral resolution to be achieved. Since the method uses discrete Fourier analysis, the reconstructions can be efficiently calculated. Furthermore, it is shown that, by application of the method to the analysis of two shearing interferograms with suitably chosen shears, exact reconstruction of the underlying wave front at all evaluation points is obtained up to an arbitrary constant. The influence of noise on the results obtained by this reconstruction procedure is investigated in detail, and its stability is shown. Finally, applications to simulated measurements are presented. The results demonstrate high-quality reconstructions for single shearing interferograms and exact reconstructions for two shearing interferograms.     

Potential and vortex features of optical speckle fields and visualization of wave-front singularities

The feasibility of noninterferometric methods to measure phase distribution in a laser beam cross section for visualization of the vortex dislocations of an optical speckle-field wave front is analyzed. Peculiarities of the phase retrieved from the measured intensity distribution (the phase problem in optics) and from the wave-front slopes measured by a Hartmann sensor are discussed. A concept of the vortex and the potential parts of the phase is introduced. An analytic formula to retrieve the potential phase from the measured intensity has been obtained. We show that the considered means of measurements allow the positions of the dislocation centers to be sensed and the spatial configuration of the intensity zero lines to be reconstructed.     

Wave-front sensing from subdivision of the focal plane with a lenslet array

A wave-front sensing scheme based on placing a lenslet array at the focal plane of the telescope with each lenslet reimaging the aperture is analyzed. This wave-front sensing arrangement is the dual of the Shack-Hartmann sensor, with the wave front partitioned in the focal plane rather than in the aperture plane. This arrangement can be viewed as the generalization of the pyramid sensor and allows direct comparisons of this sensor with the Shack-Hartmann sensor. We show that, as with the Shack-Hartmann sensor, when subdividing in the focal plane, the quality of the wave-front estimate is a trade-off between the quality of the slope measurements over each region in the aperture and the resolution to which the slope measurements are obtained. Open-loop simulation results demonstrate that the performance of the lenslet array at the focal plane is equivalent to that of the Shack-Hartmann sensor when no modulation is applied to the lenslet array. However, when the array is modulated in a manner akin to that of the pyramid sensor, subdivision at the focal plane provides advantages when compared with the Shack-Hartmann sensor.     

Sparse matrix approximation method for an active optical control system

We develop a sparse matrix approximation method to decompose a wave front into a basis set of actuator influence functions for an active optical system consisting of a deformable mirror and a segmented primary mirror. The wave front used is constructed by Zernike polynomials to simulate the output of a phase-retrieval algorithm. Results of a Monte Carlo simulation of the optical control loop are compared with the standard, nonsparse approach in terms of accuracy and precision, as well as computational speed and memory. The sparse matrix approximation method can yield more than a 50-fold increase in the speed and a 20-fold reduction in matrix size and a commensurate decrease in required memory, with less than 10% degradation in solution accuracy. Our method is also shown to be better than when elements are selected for the sparse matrix on a magnitude basis alone. We show that the method developed is a viable alternative to use of the full control matrix in a phase-retrieval-based active optical control system.     

Wave-front measurement errors from restricted concentric subdomains

In interferometry and optical testing, system wave-front measurements that are analyzed on a restricted subdomain of the full pupil can include predictable systematic errors. In nearly all cases, the measured rms wave-front error and the magnitudes of the individual aberration polynomial coefficients underestimate the wave-front error magnitudes present in the full-pupil domain. We present an analytic method to determine the relationships between the coefficients of aberration polynomials defined on the full-pupil domain and those defined on a restricted concentric subdomain. In this way, systematic wave-front measurement errors introduced by subregion selection are investigated. Using vector and matrix representations for the wave-front aberration coefficients, we generalize the method to the study of arbitrary input wave fronts and subdomain sizes. While wave-front measurements on a restricted subdomain are insufficient for predicting the wave front of the full-pupil domain, studying the relationship between known full-pupil wave fronts and subdomain wave fronts allows us to set subdomain size limits for arbitrary measurement fidelity.     

Electromagnetic scattering by an aggregate of spheres

We present a comprehensive solution to the classical problem of electromagnetic scattering by aggregates of an arbitrary number of arbitrarily configured spheres that are isotropic and homogeneous but may be of different size and composition. The profile of incident electromagnetic waves is arbitrary. The analysis is based on the framework of the Mie theory for a single sphere and the existing addition theorems for spherical vector wave functions. The classic Mie theory is generalized. Applying the extended Mie theory to all the spherical constituents in an aggregate simultaneously leads to a set of coupled linear equations in the unknown interactive coefficients. We propose an asymptotic iteration technique to solve for these coefficients. The total scattered field of the entire ensemble is constructed with the interactive scattering coefficients by the use of the translational addition theorem a second time. Rigorous analytical expressions are derived for the cross sections in a general case and for all the elements of the amplitude-scattering matrix in a special case of a plane-incident wave propagating along the z axis. As an illustration, we present some of our preliminary numerical results and compare them with previously published laboratory scattering measurements.     

一种传感器结构对水中爆炸冲击波影响的数值模拟研究?

为了了解一种流线型传感器结构对水中爆炸冲击波传播的影响规律,利用有限元方法对这种流线型传感器结构附近的水中爆炸冲击波传播过程进行数值模拟,研究分析了这种流线型传感器结构对水中爆炸冲击波传播的影响规律。研究结果表明,这种流线型传感器的前端锥体结构会使传感器结构表面冲击波增强,冲击波压力增大。传感器结构外径尺寸越大,对冲击波的增强作用越明显,前端锥体结构锥角高度不同的传感器结构对冲击波增强作用没有显著差异。     

Polarization ellipse and Stokes parameters in geometric algebra

In this paper, we use geometric algebra to describe the polarization ellipse and Stokes parameters. We show that a solution to Maxwell's equation is a product of a complex basis vector in Jackson and a linear combination of plane wave functions. We convert both the amplitudes and the wave function arguments from complex scalars to complex vectors. This conversion allows us to separate the electric field vector and the imaginary magnetic field vector, because exponentials of imaginary scalars convert vectors to imaginary vectors and vice versa, while exponentials of imaginary vectors only rotate the vector or imaginary vector they are multiplied to. We convert this expression for polarized light into two other representations: the Cartesian representation and the rotated ellipse representation. We compute the conversion relations among the representation parameters and their corresponding Stokes parameters. And finally, we propose a set of geometric relations between the electric and magnetic fields that satisfy an equation similar to the Poincaré sphere equation.     

Effects of Translational Nonequilibrium in the Shock Wave Front in Dense Gases and Liquids

On the basis of the generalized Enskog kinetic equation, the problem of the structure of a shock wave in dense reacting gases and liquids is solved. It is shown that the physicochemical processes in the zone of translational nonequilibrium (ZTN) in the shock wave front do not obey the Arrhenius kinetics. The rates of high-threshold processes in the ZTN in the shock wave front may exceed these rates in equilibrium behind the front by a factor of 10⁶ and more. An analytical expression is obtained for the rate constant of nonequilibrium processes in the front. Comparison of the results of calculation of the yield of products with broken double bond in the benzene molecule with the experimental data reveals that the formation of these products proceeds by the mechanism of nonadiabatic collisions realized under highly nonequilibrium conditions.     

Class of basis functions for use in optical systems analysis

A method is described for modeling the effects of spatial apertures on optical sensor systems. The method consists of defining a set of basis functions that is obtained by partitioning the aperture image plane into a series of rectangular regions and replacing the field in each rectangular subregion with an orthogonal function series approximation. Each orthogonal function has a finite extent that is matched to the aperture image. The individual functions are propagated by application of the Fresnel approximation of the Rayleigh-Sommerfeld diffraction formula to other ranges, and the resultant functions are shown to be valid basis functions for defining a field at any other range. The technique is applied to a scattering problem using complex Fourier series.     

A comparative study of the scalability of alternative metamodelling techniques

Metamodels, also known as surrogate models, can be used in place of computationally expensive simulation models to increase computational efficiency for the purposes of design optimization or design space exploration. The accuracy of these metamodels varies with the scale and complexity of the underlying model. In this article, three metamodelling methods are evaluated with respect to their capabilities for modelling high-dimensional, nonlinear, multimodal functions. Methods analyzed include kriging, radial basis functions, and support vector regression. Each metamodelling technique is used to model a set of single output functions with dimensionality ranging from fifteen to fifty independent variables and modality ranging from one to ten local maxima. The number of points used to train the models is increased until a predetermined error threshold is met. Results show that kriging metamodels perform most consistently across a variety of functions, although radial basis functions and support vector regression are very competitive for highly multimodal functions and functions with large local gradients, respectively. Support vector regression metamodels consistently offer the shortest build and prediction times when applied to large scale multimodal problems.     

Use of artificial neural networks for Hartmann-sensor lenslet centroid estimation

For adaptive optical systems to compensate for atmospheric-turbulence effects, the wave-front perturbation must be measured with a wave-front sensor (WFS). A Hartmann WFS typically divides the optical aperture into subapertures and then measures the slope of the wave front within each subaperture. Hartmann WFS slope measurements are based on estimating the location of the centroid of the image that is formed from a guide star within each subaperture. Conventional techniques for centroid estimation involve the use of a linear estimator and conversion tables. Neural networks provide nonlinear solutions to this problem. We address the use of neural networks for estimating the location of the centroid from the subaperture image. We find that neural networks provide more accurate estimates over a larger dynamic range and with less variance than do the conventional linear centroid estimator.     

Scattering of a plane wave by an anisotropic ferrite-coated conducting sphere

On the basis of the three-dimensional spherical vector wave functions in ferrite anisotropic media, and the fact that the first and second spherical vector wave functions in ferrite anisotropic media satisfy the same differential equations, the electromagnetic fields in homogeneous ferrite anisotropic media can be expressed as the addition of the first and second spherical vector wave functions in ferrite anisotropic media. Applying the continue boundary condition of the tangential component of electromagnetic fields in the interface between the ferrite anisotropic medium and free space, and the tangential electric field vanishing in the interface of the conducting sphere, the expansion coefficients of electromagnetic fields in terms of spherical vector wave function in ferrite medium and the scattering fields in free space can be derived. The theoretical analysis and numerical result show that when the radius of a conducting sphere approaches zero, the present method can be reduced to that of the homogeneous ferrite anisotropic sphere. The present method can be applied to the analyses of related microwave devices, antennas and the character of radar targets.     

A normal mode expansion for piezoelectric plates and certain of its applications

The equations of motion of an arbitrary piezoelectric plate are represented by a set of second-order differential equations involving only the transverse spatial coordinates. This is achieved by expanding the thickness dependence in a set of basis functions derived from the solutions to the plate problem at cutoff. Techniques are presented for constructing approximate plate equations using only chosen mode amplitudes; such equations predict the true cutoff frequencies and give dispersion curves which are rigorously correct up to terms of order k(2). The coefficients in these equations can be computed analytically, and techniques for doing this are presented. Comparisons with dispersion curves calculated by partial wave analysis are given both for quartz and for lithium niobate. The theory provides a quite general basis for modeling devices such as trapped energy resonators.     

Numerical calculation of a converging vector electromagnetic wave diffracted by an aperture by using Borgnis potentials. I. General theory

A method is proposed, on the basis of the vector electromagnetic theory, for the numerical calculation of the diffraction of a converging electromagnetic wave by a circular aperture by using Borgnis potentials as auxiliary functions. The diffraction problem of vector electromagnetic fields is simplified greatly by solving the scalar Borgnis potentials. The diffractive field is calculated on the basis of the boundary integral equation, taking into consideration the contribution of the field variables on the diffraction screen surface, which is ignored in the Kirchhoff assumption. An example is given to show the effectiveness and suitability of this method and the distinctiveness of the diffractive fields caused by the vector characteristics of the electromagnetic fields.     

Huygens's principle and rays in uniaxial anisotropic media. I. Crystal axis normal to refracting surface

Huygens's principle is used to derive equations for tracing the extraordinary ray in a uniaxial crystal when the crystal axis is normal to the refracting surface. Snell's law is used to trace ordinary rays that lead to an array of ordinary spherical wavelets centered on the refracting surface. Each spherical wavelet is then replaced by an extraordinary wavelet in the shape of a rotationally symmetric ellipsoid whose major axis is in the direction of the crystal axis. The envelope of these is the extraordinary wave front; the extraordinary ray is a vector from the center of the wavelet to its point of contact with the extraordinary wave front. The inverse problem is also solved, yielding expressions for the ordinary ray in terms of the extraordinary ray, making it possible to trace the extraordinary ray out of the system by using a fictive ordinary ray. We found the geometrical invariant for the uniaxial crystals.     

Optimal selection of commercial sensors for linear model representation of daylight spectra

The average spectral power distribution of a set of measured daylight spectra, E(av)(lambda), is used for preliminary screening to select optimal sensor sets for daylight recovery. Spectra quite different from E(av)(lambda) are applied to the screened sets to obtain minimum total spectral error, which is closely related to recovery metrics but not to the coefficient of error. All basis functions should be utilized to make these two errors equal, to predict precisely the best sensor set, and to extend a set of few sensors to a set of many sensors. These are not acquirable by an exhaustive full search method.