Beam implementation in a nonorthogonal coordinate system: application to the scattering from random rough surfaces

The C method is known to be one of the most efficient and versatile tools established for modeling diffraction gratings. Its main advantage is the use of a coordinate system in which the boundary conditions apply naturally and are, ipso facto, greatly simplified. In the context of scattering from random rough surfaces, we propose an extension of this method in order to treat the problem of diffraction of an arbitrary incident beam from a perfectly conducting (PEC) rough surface. For that, we were led to revisit some numerical aspects that simplify the implementation and improve the resulting codes.     

Boundary integral equations from Hamilton's principle for surface acoustic waves under periodic metal gratings

The procedure describes the derivation of boundary integral equations for surface acoustic waves propagating under periodic metal strip gratings with piezoelectric films. It takes into account the electrical and mechanical perturbations, including the effects of mass loading caused by the gratings with an arbitrary shape. First, an integral equation is derived with line integrals on the boundaries within one period. This derivation is based on Hamilton's principle and uses Lagrange's method of multipliers to alleviate the continuous conditions of the displacement and the electric potential on the boundaries. Second, boundary integral equations corresponding to each substrate, piezoelectric film, metal strip, and free space region are obtained from the integral equation using the Rayleigh-Ritz method for admissible functions. With this procedure, it is not necessary to make any assumptions for separation of the boundary conditions between two neighboring regions. Consequently, we clarify the theoretical basis for the analytical procedure using boundary integral equations for longitudinal LSAW modes.     

Bending Analysis of Classical Symmetric Laminated Composite Plates by the Strip Element Method

A new formulation of strip element method based on classical laminated plate theory is derived for the bending analysis of laminated composite plates. In this method, an infinite-length plate is first considered and is discretized into a set of strip elements in the width direction. The principle of minimum potential energy is applied to obtain the ordinary differential equations, which are functions of only the coordinate in the length direction. These differential equations can then be solved analytically. The boundary conditions on the length coordinate direction are finally used to determine the deflection distribution in the plate. The strip element solutions are presented for a rectangular laminated composite plate with various boundary conditions and load cases. The solutions are compared with those of the Rayleigh-Ritz method, and very good agreement is obtained.     

Propagation of longitudinal leaky surface waves under periodic metal grating structure on lithium tetraborate

The dispersion properties of longitudinal leaky surface waves propagating under the periodic Al strip grating on lithium tetraborate (Li(2)B(4)O(7); LBO) are described theoretically and experimentally for applications of the mode to high frequency SAW devices. A theoretical method developed here is based on Floquet's theorem using space harmonics as an orthogonal function set and real boundary integral equations derived from the method of weighted residuals for a period of each region, i.e., substrate, metal, and free space. The boundary integral equations are solved by using the Galerkin procedure. The periodic strip gratings with both single-electrodes and double-electrodes are investigated, considering the convergency of the numerical computation for the number of the space harmonics. As a result, the propagation loss for shorted gratings was found to be relatively low in the thickness range of the Al strip below about 1% for the single-electrodes and 2% for the double-electrodes, although it greatly increases for a thickness over 2% for the single-electrodes and 3% for the double-electrodes.     

Off-Bragg analysis of the diffraction efficiency of reflection photorefractive holograms

Formulas are given for the calculation of diffraction efficiency of reflection-type gratings recorded in a photorefractive medium. The analysis incorporates the coupled-wave theory that was developed for photorefractive hologram gratings. This analysis takes into account grating slant with respect to the medium surface, light absorption during reconstruction, any incident angle of the reference beam, and any photorefractive phase shift. General solutions for signal and reference wave functions are given in a closed-form expression by use of a hypergeometric function. The optimum media parameters and recording conditions for high diffraction efficiency are obtained by the derived formulas. The diffraction properties for off-Bragg conditions are also discussed.     

Analysis of Far-field Diffraction Characteristics of Wavelength-sized Surface Relief Gratings

The rigorous boundary matching technique was used to analyse the far-field diffraction from wavelength-sized surface relief gratings for TE polarization. Diffraction characteristics of such surface relief gratings were studied as functions of complex refractive index, groove depth and the angle of incidence. The surface field at the bottom of the grooves is more severely influenced by the edges and walls for metallic material than dielectric ones. The reflectivities of gold gratings show significant fluctuation for different incident wavelengths. This can be clearly seen from the fact that the zeroth-order diffraction efficiency DE ₀ of gold is lower than that of aluminium at u = 0·4 wm but higher at u = 1·0 wm. It is also shown that the depth of the grooves has significant influences on the diffraction efficiencies, an optimum depth can be chosen to minimize the specular component. However, this optimum depth value is found to be different for the gratings of different materials, e.g. aluminium and gold, although the gratings have exactly the same size and shape. The effect of the incident angle, particularly at critical angles when certain diffracted waves are just about to be cut off was discussed. The zeroth-order diffraction efficiencies are found to change noticeably at the critical angles.     

Highly efficient transmission gratings in fused silica for chirped-pulse amplification systems

We report on highly efficient transmission gratings in fused silica with a grating period of 800 nm generated by electron-beam lithography. At a wavelength of 1060 nm, 95% diffraction efficiency is achieved under Littrow conditions. The damage threshold, extremely enhanced compared with conventional gold-coated diffraction gratings, makes these gratings the key elements in high average power (>100 W) femtosecond fiber chirped-pulse amplification systems.     

Dichromated pullulan diffraction gratings: influence of environmental conditions and storage time on their properties

Investigations of the environmental stability of diffraction gratings, recorded in dichromated pullulan (DCP), are reported. Profile changes of DCP surface relief gratings, under high humidity conditions, were analyzed using an atomic force microscope. It was found that the profile was not altered, while the diffraction efficiency was preserved. The influence of storage life on the diffraction efficiency and surface profile of DCP gratings were also investigated. It was concluded that DCP gratings offer much better stability compared with the dichromated gelatin.     

Efficient coupling into polymer waveguides by gratings

Investigations of highly efficient grating couplers for polymer slab and strip waveguides fabricated by electron-beam lithography are reported. A maximum input efficiency of 67% is achieved. The electron-beam direct-writing technique allows one to replicate the original gratings into polymer substrates by embossing. An all-polymeric optical chip with efficient grating couplers is demonstrated. Waveguide grating couplers with blazed profile and variable grating depth are investigated. Thus, the intensity distribution of the outcoupled light is matched to a Gaussian-like profile. A focusing blazed grating that couples the light with an efficiency of 42% into a polymer strip waveguide is reported. A curvature correction of the grating lines allows one to improve the focusing properties.     

Boundary integral equation Neumann-to-Dirichlet map method for gratings in conical diffraction

Boundary integral equation methods for diffraction gratings are particularly suitable for gratings with complicated material interfaces but are difficult to implement due to the quasi-periodic Green's function and the singular integrals at the corners. In this paper, the boundary integral equation Neumann-to-Dirichlet map method for in-plane diffraction problems of gratings [Y. Wu and Y. Y. Lu, J. Opt. Soc. Am. A26, 2444 (2009)] is extended to conical diffraction problems. The method uses boundary integral equations to calculate the so-called Neumann-to-Dirichlet maps for homogeneous subdomains of the grating, so that the quasi-periodic Green's functions can be avoided. Since wave field components are coupled on material interfaces with the involvement of tangential derivatives, a least squares polynomial approximation technique is developed to evaluate tangential derivatives along these interfaces for conical diffraction problems. Numerical examples indicate that the method performs equally well for dielectric or metallic gratings.     

Analytic design and solutions for resonance domain diffractive optical elements

A model for designing and analyzing complicated surface relief diffractive elements in the resonance domain is developed. It is based on subdividing the complicated diffractive element into many highly efficient local diffraction gratings whose surface relief modulations can be effectively characterized as slanted volume gratings for which closed form analytic solutions exist. The model is illustrated by finding in the resonance domain the local period, effective slant angle, and groove depth at each location on an off-axis cylindrical diffractive lens.     

竖向成层介质中标量波传播问题的高精度人工边界条件

为了实现含竖向成层介质以及表面不规则地形场地中标量波传播问题的高效且高精度求解,该文基于连分式展开和扩展的一致边界,建立了一种频域下折线形高精度人工边界条件。通过在每个竖向地层内引入独立的斜角坐标变换,新的人工边界条件可以用于多起伏地表地形条件。新的折线形人工边界在频域下推导,仅含有连分式阶数一个待定实参数,用于调整计算精度,该参数不随外行波的频率和传播角度改变。人工边界条件可以与内域有限元方程无缝耦合,应用简单方便。由于新边界条件的高精度,内域尺寸可以取较小甚至可以直接将人工边界加在结构周围或者地表,从而极大提高计算效率。通过典型数值算例,将人工边界计算模型与有限元大模型的解进行了对比分析,验证了该文提出的折线形人工边界条件的有效性和高精度。     

二维金属光栅的矢量模式研究

用矢量模式场来描述二维金属光栅各层中的电磁场。由边界条件,推导出模式场方程组;借助反射透射系数阵递推算法,得出了方程组的解。这种矢量模式方法物理概念清晰且稳定收敛,可以处理任意偏振态平面波入射到二维亚波长金属光栅中的衍射问题。     

Characterization of Diffraction Gratings by use of a Tabletop Soft-X-Ray Laser

We have demonstrated the use of a high-repetition-rate 46.9-nm tabletop laser to characterize diffraction gratings designed for grazing-incidence operation in the soft-x-ray spectral region. The efficiencies for various diffraction orders were measured as a function of angle of incidence and compared with the results of model simulations. This measurement technique provides benchmarks with which to improve electromagnetic codes used in the design of soft-x-ray diffraction gratings. The results illustrate the potential of compact tabletop soft-x-ray lasers for use as a new tool for characterization of short-wavelength optics at the manufacturer's site.     

Off-Bragg analysis of the diffraction efficiency of transmission photorefractive holograms

Formulas for calculating the diffraction efficiency of gratings recorded in a photorefractive medium are given. The analysis uses coupled-wave theory for photorefractive hologram gratings and takes into account the photorefractive phase shift and fringe-bending effect. General solutions for diffracted (signal) and undiffracted (reference) waves are derived in a closed-form expression. By use of the derived formulas the diffraction efficiency for angle mismatch from the Bragg condition can easily be evaluated as compared with numerical-analysis methods. The diffraction efficiency is also quantified in terms of medium parameters and recording and reconstruction conditions.     

Two-fold mechanism of the interaction of incident light with a phase diffraction grating

It is suggested to use antireflection diffraction gratings for introducing light into the active regions of optoelectronic devices. The dependence of the diffraction efficiency on the grating groove depth, which controls the reflection from the optical surface and forms the diffraction order in the material, has been numerically modeled.     

光栅衍射特性的耦合波分析、计算与讨论

本文从麦克斯韦方程组及电磁场边界条件出发,推导了广泛应用于各类光栅衍射问题的矢量分析方法--严格的耦合波分析方法.针对光栅的衍射特性,编写了基于严格的耦合波分析方法的计算程序,并以TE模情形为例对光栅的衍射效率和收敛性作了数值计算.结果表明,当谐波数不断增加,即便对于厚光栅(d/λ>10)情形,光栅的衍射效率仍将收敛于某一确定值.     

Dynamic Control of Light Direction Enabled by Stimuli‐Responsive Liquid Crystal Gratings

The ability to control light direction with tailored precision via facile means is long‐desired in science and industry. With the advances in optics, a periodic structure called diffraction grating gains prominence and renders a more flexible control over light propagation when compared to prisms. Today, diffraction gratings are common components in wavelength division multiplexing devices, monochromators, lasers, spectrometers, media storage, beam steering, and many other applications. Next‐generation optical devices, however, demand nonmechanical, full and remote control, besides generating higher than 1D diffraction patterns with as few optical elements as possible. Liquid crystals (LCs) are great candidates for light control since they can form various patterns under different stimuli, including periodic structures capable of behaving as diffraction gratings. The characteristics of such gratings depend on several physical properties of the LCs such as film thickness, periodicity, and molecular orientation, all resulting from the internal constraints of the sample, and all of these are easily controllable. In this review, the authors summarize the research and development on stimuli‐controllable diffraction gratings and beam steering using LCs as the active optical materials. Dynamic gratings fabricated by applying external field forces or surface treatments and made of chiral and nonchiral LCs with and without polymer networks are described. LC gratings capable of switching under external stimuli such as light, electric and magnetic fields, heat, and chemical composition are discussed. The focus is on the materials, designs, applications, and future prospects of diffraction gratings using LC materials as active layers.     

On the use of bubble functions in the local buckling analysis of plate structures by the spline finite strip method

This paper augments bubble functions to the ordinary spline finite strip method in order to calculate the elastic local buckling coefficients of plates and plate structures. The results show that the use of bubble functions improves significantly the convergence of the spline finite strip method in terms of the strip subdivision, and therefore leads to smaller storage requirements for the global stiffness and stability matrices, and faster eigenvalue extraction. Benchmark numerical investigations are presented, including the study of plates with different boundary conditions under uniaxial and biaxial stresses, plates with different aspect ratios under shear, and a stiffened panel under combined shear and compression that has been studied elsewhere. These studies demonstrate that by implementation of the bubble functions, rapid convergence of the solution is obtained. The formulation is ideal for analysing local buckling under a variety of boundary and loading conditions. Copyright © 2000 John Wiley & Sons, Ltd.