Grazing-incidence Monk-Gillieson monochromator based on surface normal rotation of a varied-line-spacing grating

A geometric theory of a grazing-incidence varied-line-spacing plane-grating monochromator system whose scanning is made by a simple grating rotation about the grating normal has been developed for designing Monk-Gillieson monochromators capable of covering an energy range of 0.6-2.5 keV. Analytic expressions are given for the grating equations, focal conditions, dispersion, spectral image shape, and optimization of groove parameters. On the basis of the theory, two monochromator systems have been designed: system I for moderate resolution and system II for relatively high resolution. The validity of the analytic formulas and the expected performance of the designed systems have been evaluated by means of ray tracing. The results show that the analytic formulas are sufficiently accurate for practical applications and that systems I and II would provide resolving power of approximately 1450-600 and 7500-2000, respectively, in the wavelength region of 0.5-2.0 nm.     

Aspheric wave-front recording optics for holographic gratings

The geometric theory of aspheric wave-front recording optics is extended to include the fourth-order groove parameters that correspond to the fourth-order holographic terms in the light-path function. We derived explicit expressions of the groove parameters by analytically following an exact ray-tracing procedure for a double-element optical system that consists of a point source, an ellipsoidal mirror, and an ellipsoidal grating blank. Design examples of holographic gratings for an in-plane Eagle-type vacuum-UV monochromator are given to demonstrate the capability of the present theory in the design of aspheric wave-front recording optics.     

Merit function for the design of grating instruments

A merit function that is closely correlated with the rms spread of an infinite number of ray-traced spots by means of analytic spot-diagram formulas is critically evaluated in comparison with exact ray tracing. The analytic spot-diagram formulas are found to generate spot diagrams that are almost indistinguishable from ray-traced ones. The rms spread of ray-traced spots about the mean shows varying degrees of statistical dispersion depending on the number of rays traced, and it approaches the value given by the merit function as the number of rays is increased. All the results clearly show that the merit function behaves as defined and provides enough accuracy and a sufficiently short computing time for the design of highly sophisticated grating instruments.     

Analytic steady-state solutions to dynamic grating envelope

Abstract

The photorefractive holographic recording is a dynamic grating formulation process, where the grating is induced by the interference modulation of two beams, and the energies of the beams are exchanged with each other owing to their interaction in the induced grating. The grating modulation envelope is a comprehensive analysis for holographic dynamic recording and non-destructive readout, but analytic solutions can only be obtained by simultaneously solving the materials equations and the coupled wave equations. In this paper, based on Kukhtarev's model and Moharam's field function exact steady-state analytic expressions for dynamic grating envelopes are deduced for any constant values of light modulation depth and light excitation efficiency. Thus their approximations for straight formulation instead of parametric formulation are then obtained. These steady-state analytic solutions to envelopes describe the dynamic holographic recording and readout more clearly and exactly.     

橡胶筒承受径向载荷时的位移解析解

基于经典弹性理论，利用Euler方程组推导了橡胶筒承受径向载荷时的位移解析解，推导过程中未采用体积不可压缩假设,有限元计算表明，该解析解是正确的。     

Interpretation of Fringes Produced by Time‐Averaged Projection Moiré

Abstract: New exact formulas describing the observed shift of projected moiré grating lines on a surface of an object are derived for the paraxial model. These formulas enable to construct more accurate explicit relationship among the amplitude of oscillation, the pitch of the projected grating and the order of the fringe. Analytical derivations and numerical illustrations are used throughout the text to explain the process of formation of double‐exposure and time‐averaged projection moiré fringes.     

Comparison of holographic photopolymer materials by use of analytic nonlocal diffusion models

The one-dimensional diffusion equation governing holographic grating formation in photopolymers, which includes both nonlocal material response and generalized dependence of the rate of polymerization on the illuminating intensity, has been previously solved under the two-harmonic expansion assumption. The resulting analytic expressions for the monomer and polymer concentrations have been derived and their ranges of validity tested in comparison with the more accurate numerical four-harmonic case. We used these analytic expressions to carry out a study of experimental results presented in the literature over a 30-year period. Automatic fitting of the data with these formulas allows material parameters, including the nonlocal chain-length variance sigma, to be estimated. In this way, (i) a quantitative comparison of different materials can be made, and (ii) a standard form of experimental result presentation is proposed to facilitate such a procedure.     

Magnetostatic interaction fields for a three-dimensional array of ferromagnetic cubes

An exact analytic formula for the magnetostatic interaction energy of a three-dimensional array of ferromagnetic cubes is presented. The magnetization within each cube is assumed constant but varies in direction from cube to cube. From the expressions for the interaction energy, exact formulas for the effective magnetostatic interaction field are derived. This provides a useful tool for three-dimensional micromagnetic calculations.     

A Comparison of Diffraction Theories for Off-bragg Replay

Two first-order analytic solutions and a second-order analytic solution for the case of lossless-dielectric unslanted volume transmission gratings are compared for the case of weak grating index modulation and significantly off-Bragg replay. It is shown that the analytic solution predicted using the Beta-value method, which has previously been found to agree more closely with experimental results for the unslanted case than the first-order K -vector closure method (of Kogelnik), also agrees more closely with the analytic expression produced by the second-order coupled-mode equations of Kong for this case. A numerical comparison of the first order theories and the rigorous coupled wave theory gives a similar result. Thus the Beta-value method offers definite advantages over the Kogelnik K -vector closure method for the unslanted transmission geometry.     

Gaussian profile estimation in one dimension

We present several new results on the classic problem of estimating Gaussian profile parameters from a set of noisy data, showing that an exact solution of the maximum likelihood equations exists for additive Gaussian-distributed noise. Using the exact solution makes it possible to obtain analytic formulas for the variances of the estimated parameters. Finally, we show that the classic formulation of the problem is actually biased, but that the bias can be eliminated by a straightforward algorithm.     

Generalized Bruggeman Formula for the Effective Thermal Conductivity of Particulate Composites with an Interface Layer

Combining the well-known Bruggeman theory and Nan et al. results, formulas for predicting the effective thermal conductivity of anisotropic particulate composites with an interface layer are derived. These formulas are valid for a composite material containing arbitrarily oriented ellipsoidal particles with any aspect ratio, and they can be expected to be suitable mainly for large volume fractions, when the thermal interaction between neighboring particles needs to be considered. Results of the present approach are reduced to simpler formulas for some limiting cases in the particle shape. Theoretical analysis of the effective thermal conductivity as a function of volume fraction and shape of the particles is performed. Comparison of the obtained formulas with previously reported experimental data for the effective thermal conductivity is also presented.     

Transmittance analysis of diffraction phase grating

In order to accurately analyze and design the transmittance characteristic of a diffraction phase grating, the validity of both the scalar diffraction theory and the effective medium theory is quantitatively evaluated by the comparison of diffraction efficiencies predicted from both simplified theories to exact results calculated by the rigorous vector electromagnetic theory. The effect of surface profile parameters, including the normalized period, the normalized depth, and the fill factor for the precision of the simplified methods is determined at normal incidence. It is found that, in general, when the normalized period is more than four wavelengths of the incident light, the scalar diffraction theory is useful to estimate the transmittance of the phase grating. When the fill factor approaches 0.5, the error of the scalar method is minimized, and the scalar theory is accurate even at the grating period of two wavelengths. The transmittance characteristic as a function of the normalized period is strongly influenced by the grating duty cycle, but the diffraction performance on the normalized depth is independent of the fill factor of the grating. Additionally, the effective medium theory is accurate for evaluating the diffraction efficiency within an error of less than around 1% when no higher-order diffraction waves appear and only the zero-order waves exist. The precision of the effective medium theory for calculating transmittance properties as a function of the normalized period, the normalized groove depth, and the polarization state of incident light is insensitive to the fill factor of the phase grating.     

Exact and approximate solution of one mixed problem of the theory of heat conduction with the help of special functions

A special psi function has been introduced and with its help an exact and an approximate analytic representations of the solution of the mixed problem of the heat conduction theory for a rod have been obtained. The advantage of the approximate formula is its relative simplicity and the absence of quadratures. An efficient estimate of the approximate solution error has been found.     

A problem of elasticity theory for a space containing an inclusion of any stiffness in shear

We deduce the formulas for the evaluation of stresses in thin inclusions and their concentration in the matrix near its contour convenient for engineering applications. The influence of stiffness of the inclusion and its geometric parameters on the levels of stresses in the matrix and in the inclusion is investigated. Partial cases of the problem for an ellipsoidal cavity and an absolutely rigid ellipsoidal inclusion are analyzed. The formulas for the evaluation of the corresponding stress intensity factors are obtained.     

粗线纹光栅接长技术研究

光栅线位移传感技术在机床数控、数显中占有重要的地位，但其接长非常困难，以至于目前我国除粗线纹光栅外，能接长到３ｍ以上的位移传感器还没有实用化的。本文针对实用化接长技术需要，借助于粗线纹光栅特殊的测量原理，提出了一种新的接长方法，在理论上作了初步探讨，并作了实验验证。该接长方法乙用于工厂现场，接长６．４ｍ坐标。     

A New Closed-Form Solution to Light Scattering by Spherical Nanoshells

Light or electromagnetic wave scattered by a single sphere or a coated sphere has been considered as a classic Mie theory. There have been some further extensions that were made further based on the Mie theory. Recently, a closed-form analytical model of the scattering cross section of a single nanoshell has been considered. The present paper is documented further, based on the work in 2006 by Alam and Massoud, to derive another different closed-form solution to the problem of light scattered by the nanoshells using polynomials of up to order 6. Validation is made by comparing the present closed-form solution to the exact Mie scattering solution and also to the other closed-form solution by Alam and Massoud. This study is found to be, however, more generalized and also more accurate for the coated spheres of either tiny/small or medium sizes than that of Alam and Massoud. Therefore, the derived formulas can be used for accurately characterizing both surface plasmon resonances of nanoparticles (of small sizes) or nanoantenna near-field properties (of medium sizes comparable with half wavelength).     

On formulas for the Rayleigh wave velocity in pre-strained elastic materials subject to an isotropic internal constraint

In the present paper, formulas for the velocity of Rayleigh waves propagating along principal directions of prestrain of an elastic half-space subject to a pure homogeneous prestrain, and an isotropic internal constraint have been derived using the theory of cubic equation. They have simple algebraic form, and hold for any strain-energy function and any isotropic constraint. In undeformed state, these formulas recover the exact value of the Rayleigh wave speed in incompressible isotropic elastic materials. Some specific cases of strain-energy function and isotropic constraint are considered, and the corresponding formulas become totally explicit in terms of the parameters characterizing the material and the prestrains. The necessary and sufficient conditions for existence of Rayleigh wave are examined in detail. The use of obtained formulas for nondestructive evaluation of prestrains and prestresses is discussed.     

Compact cross-dispersion device based on a prism and a plane transmission grating

This paper presents a cross-dispersion prism-grating device using a plane transmission grating attached directly to a prism, which is different from traditional cross-dispersion grating-prism systems that are based on the reflection grating. Unlike conventional direct-vision grism or constant-dispersion grism in which both the prism and grating have the same dispersion direction, for this device the dispersion directions of the prism and grating are different. The analytical expressions for the cross-dispersion of this device are derived in detail and the formulas of the footprint of the dispersed spectra are given. The numerical results and ray-tracing simulations by ZEMAX are shown. The device provides a compact, small-sized and broadband cross-dispersion device used for the medium resolution spectrometer.     

Theory of fracture in canonical formulation

Starting from the hodograph equation as a limiting condition in functional form, where the velocity vector is a continuous function of the position vector with at least continuous first derivative, the theory of one to another dependent functions was applied and completed. No hypotheses have been made or empirical formulas used. Because these assumptions were sufficient to establish the equation of motion, we call it the canonical formulation of the theory of fracture. The results have been explained in Cartesian coordinates for the plane flow. Additional results were derived in special forms of the equation of motion expressed in tensors and in Cartesian coordinates. Then we have investigated the equation of motion, if the limiting condition is given in terms of stresses and finally the exact stress-strain-relations have been derived for the plane flow with an example for the ideal plastic body.     

Wide angle conoscopic interference patterns in uniaxial crystals

The fringe pattern obtained when a divergent (or convergent) beam goes through a sample of birefringent crystal between two crossed polarizers contains information that is inherent to the crystalline sample under study. The formation of fringe patterns is analyzed from distinct approaches and with different degrees of approximation considering cones of light of large numerical aperture. We obtain analytic explicit formulas of the phase shift on the screen and compare them with the exact numerical solution. The results obtained are valid for arbitrary orientation of the optical axis and are not restricted either to low birefringence or to small angles of incidence. Moreover, they enable the extraction of the main features related to the characterization of uniaxial crystal slabs, such as the optical axis tilt angle and the principal refractive indices.