Fast convergence with spectral volume integral equation for crossed block-shaped gratings with improved material interface conditions

For block-shaped dielectric gratings with two-dimensional periodicity, a spectral-domain volume integral equation is derived in which explicit Fourier factorization rules are employed. The Fourier factorization rules are derived from a projection-operator framework and enhance the numerical accuracy of the method, while maintaining a low computational complexity of O(NlogN) or better and a low memory demand of O(N).     

Group-theoretic approach to enhancing the Fourier modal method for crossed gratings with square symmetry

The Fourier modal method for crossed gratings with square symmetry is reformulated by use of a group-theoretic approach that we developed recently. In the new formulation, a crossed-grating problem is decomposed into six symmetrical basis problems whose field distributions are the symmetry modes of the grating. Then the symmetrical basis problems are solved with symmetry simplifications, whose solutions are superposed to get the solution of the original problem. Theoretical and numerical results show that when the grating is at some Littrow mountings, the computation efficiency can be improved effectively: The memory occupation is reduced by 3/4 and the computation time is reduced by a factor from 25.6 to 64 in different incident cases. Numerical examples are given to show the effectiveness of the new formulation.     

Computing shape parameter sensitivity of the field of one-dimensional surface-relief gratings by using an analytical approach based on RCWA

The rigorous coupled-wave analysis (RCWA) is a method to compute diffraction of a field by a given grating structure. Within various applications, such as metrology, it is important to know how the field reacts to small perturbations in the grating. This behavior can be expressed by the field derivatives with respect to a certain parameter. Approximations of these derivatives can be found by using finite differences where the field is computed for a neighboring value of the parameter, and the difference gives the derivative. Unfortunately, RCWA involves solving eigenvalue systems that are computationally expensive. Therefore, a faster alternative is given that computes the derivatives by straightforward differentiation of the relations within RCWA. Solving additional eigensystems is replaced by finding derivatives of eigenvalues and eigenvectors, which is less computationally expensive.     

Range-finding method using diffraction gratings

A model in geometric optics, along with some preliminary experimental results for a new range-finding method that exploits near-field diffraction phenomena found with plane gratings, is presented. Among the characteristics investigated is a magnification effect applicable to three-dimensional microscopy. A variety of embodiments of the method is disclosed, including an off-axis illumination model and a method of near-field focus compensation that takes advantage of the Scheimpflug condition.     

Group-theoretic approach to the enhancement of the Fourier modal method for crossed gratings: C2 symmetry case

A new formulation of the Fourier modal method for crossed gratings with symmetry considerations is established by using the group-theoretic approach that we have developed recently. Considering crossed gratings with the C2 symmetry (invariance after rotation about the normal of the mean grating plane through angle pi), we present in detail the construction of the new algorithm, illustrate the improved computation efficiency, and discuss its application. It is shown theoretically and numerically that when the grating is Littrow mounted and the truncated reciprocal lattice of the diffracted field also has the C2 symmetry, the maximum effective truncation number of the algorithm is doubled and the computation time is reduced by a factor of 4. The time saving factor is increased to 8 for the special case of normal incidence.     

Design of retrodiffraction gratings for polarization-insensitive and polarization-sensitive characteristics by using the Taguchi method

We present the design of retrodiffraction gratings that utilize total internal reflection (TIR) in a lamellar configuration to achieve high performance for both TE and TM polarized light and polarization-sensitive performance for gratings behaving as polarizer filters; the design was based on rigorous coupled wave analysis (RCWA) and the Taguchi method. The components can thus be fabricated from a single dielectric material and do not have to be coated with a metallic or dielectric film layer to enhance the reflectance. The effects of the structural and optical parameters of lamellar gratings were investigated, and the TIR gratings in a lamellar configuration were structurally and optically optimized in terms of the signal-to-noise ratio (S/N) and a statistical analysis of variance (ANOVA) of the refractive index, grating period, filling factor, and grating depth as control factors and the estimated efficiency by RCWA as a noise factor. For more accurate robustness, a two-step optimization process was used for each purpose. For TIR gratings designed to perform similarly for TE and TM incident polarization, the -1st-order efficiencies were estimated to be up to 92.0% and 88.5% for TE and TM polarization, respectively. Also, for the TIR gratings designed to achieve polarization-sensitive performance when behaving as a polarizer filters, the -1st-order diffraction efficiencies for TE and TM polarization were estimated to be up to 95.5% and 2.7%, respectively. From these analysis results, it was confirmed that the Taguchi method shows feasibility for an optimization approach to a technique for designing optical devices.     

Adaptive finite-element method for diffraction gratings

A second-order finite-element adaptive strategy with error control for one-dimensional grating problems is developed. The unbounded computational domain is truncated to a bounded one by a perfectly-matched-layer (PML) technique. The PML parameters, such as the thickness of the layer and the medium properties, are determined through sharp a posteriori error estimates. The adaptive finite-element method is expected to increase significantly the accuracy and efficiency of the discretization as well as reduce the computation cost. Numerical experiments are included to illustrate the competitiveness of the proposed adaptive method.     

Analysis of diffraction gratings by using an edge element method

Typically the grating problem is formulated for TE and TM polarizations by using, respectively, the electric and magnetic fields aligned with the grating wall and perpendicular to the plane of incidence, and this leads to a one-field-component problem. For some grating profiles such as metallic gratings with a triangular profile, the prediction of TM polarization by using a standard finite-element method experiences a slower convergence rate, and this reduces the accuracy of the computed results and also introduces a numerical polarization effect. This discrepancy cannot be seen as a simple numerical issue, since it has been observed for different types of numerical methods based on the classical formulation. Hence an alternative formulation is proposed, where the grating problem is modeled by taking the electric field as unknown for TM polarization. The application of this idea to both TE and TM polarizations leads to a two-field-component problem. The purpose of the paper is to propose an edge finite-element method to solve this wave problem. A comparison of the results of the proposed formulation and the classical formulation shows improvement and robustness in the new approach.     

Fourier-matching pseudospectral modal method for diffraction gratings

A Fourier-matching pseudospectral modal method [PSMM(f)] is developed for analyzing lamellar diffraction gratings or grating stacks. A Chebyshev pseudospectral method is first used to accurately calculate the eigenmodes of the grating layers, and then the Fourier coefficients are matched at the interfaces between the layers. Compared with an existing pseudospectral modal method based on point matching, the PSMM(f) is more robust and accurate. The method performs better than the standard Fourier modal method for gratings involving metals.     

Issues in convergence improvement for non-linear finite element programs

Systems of non-linear equations as they arise when analysing various physical phenomena and technological processes by the implicit Finite Element Method (FEM) are commonly solved by the Newton–Raphson method. The modelling of sheet metal forming processes is one example of highly non-linear problems where the iterative solution procedure can become very slow or diverge. This paper focuses on techniques to overcome these numerical difficulties. Several methods to generate initial guesses within the radius of convergence are proposed. Appropriate stopping criteria for the iterative procedure are discussed. A combination of various line search methods with the continuation method is proposed. The efficiency and robustness of these numerical procedures are compared based on a set of test examples. A particular form of line search was identified which allows the stable and efficient solution of highly non-linear sheet metal forming problems. Even though the present investigations were motivated by the application of the implicit FEM to the simulation of sheet metal forming processes, the findings are general enough to be applicable to a wide spectrum of non-linear FEM applications. © 1997 John Wiley & Sons, Ltd.     

Estimation of velocity vector angles using the directional cross-correlation method

A method for determining both velocity magnitude and angle in any direction is suggested. The method uses focusing along the velocity direction and cross-correlation for finding the correct velocity magnitude. The angle is found from beamforming directional signals in a number of directions and then selecting the angle with the highest normalized correlation between directional signals. The approach is investigated using Field II simulations and data from the experimental ultrasound scanner RASMUS and a circulating flow rig with a parabolic flow having a peak velocity of 0.3 m/s. A 7-MHz linear array transducer is used with a normal transmission of a focused ultrasound field. In the simulations the relative standard deviation of the velocity magnitude is between 0.7% and 7.7% for flow angles between 45 degrees and 90 degrees. The study showed that angle estimation by directional beamforming can be estimated with a high precision. The angle estimation performance is highly dependent on the choice of the time ktprf x Tprf (correlation time) between signals to correlate. One performance example is given with a fixed value of ktprf for all flow angles. The angle estimation on measured data for flow at 60 degrees to 90 degrees yields a probability of valid estimates between 68% and 98%. The optimal value of ktprf for each flow angle is found from a parameter study; with these values, the performance on simulated data yields angle estimates with no outlier estimates and with standard deviations below 2 degrees.     

Pseudospectral modal method for conical diffraction of gratings

For lamellar gratings and other layered periodic structures, the modal methods (including both analytic and numerical ones) are often the most efficient, since they avoid the discretization of one spatial variable. The pseudospectral modal method (PSMM) previously developed for in-plane diffraction problems of one-dimensional gratings achieves high accuracy for a small number of discretization points, and it outperforms most other modal methods. In this paper, an extension of the PSMM to conical diffraction problems is presented and implemented. Numerical examples are used to demonstrate the high accuracy and excellent convergence property of this method for both dielectric and metallic gratings.     

High order integral equation method for diffraction gratings

Conventional integral equation methods for diffraction gratings require lattice sum techniques to evaluate quasi-periodic Green's functions. The boundary integral equation Neumann-to-Dirichlet map (BIE-NtD) method in Wu and Lu [J. Opt. Soc. Am. A 26, 2444 (2009)], [J. Opt. Soc. Am. A 28, 1191 (2011)] is a recently developed integral equation method that avoids the quasi-periodic Green's functions and is relatively easy to implement. In this paper, we present a number of improvements for this method, including a revised formulation that is more stable numerically, and more accurate methods for computing tangential derivatives along material interfaces and for matching boundary conditions with the homogeneous top and bottom regions. Numerical examples indicate that the improved BIE-NtD map method achieves a high order of accuracy for in-plane and conical diffractions of dielectric gratings.     

High-order finite difference modal method for diffraction gratings

A new high-order finite difference modal method (FDMM) is developed for analyzing diffraction gratings in conical and classical mountings. The difference scheme is constructed by enforcing the internal interface conditions in each grating layer to high-order derivatives, and it gives a high order of accuracy for computing the eigenmodes of the grating layer. Between different layers, the interface conditions are implemented using a Fourier matching scheme and a point matching scheme. Compared with the standard Fourier modal method, the high-order FDMM is more efficient since the matrices in the discretized eigenvalue problems are sparse. Numerical examples are used to illustrate the performance of the method.     

相位扫描法制作变栅距光栅

提出了一种制作变栅距(VLS)光栅的相位扫描方法。该方法的主要装置包括一个用于控制刻划机运动的光栅干涉仪和一个相位扫描机构。如果调整光栅干涉仪,保证接收场中只有两条干涉条纹,然后改变用于对条纹进行计数的光电传感器的位置,就可以刻划出具有变栅距的刻槽。对光电式光栅刻划机的控制系统和结构都做了详细论述。按照上述方法成功刻划出了试验性的VLS光栅,它的最小栅距增量为0.33nm,并对在制作过程中产生的误差进行了讨论。采用测量衍射角的方法进行了栅距检测试验,由变栅距光栅和等栅距光栅作出的拟合曲线表明:相位扫描方法是加工具有亚纳米栅距增量的VLS光栅的有效方法,该方法对超精密定位也具有借鉴作用。     

Finite-number-of-periods holographic gratings with finite-width incident beams: analysis using the finite-difference frequency-domain method

The effects of finite number of periods (FNP) and finite incident beams on the diffraction efficiencies of holographic gratings are investigated by the finite-difference frequency-domain (FDFD) method. Gratings comprising 20, 15, 10, 5, and 3 periods illuminated by TE and TM incident light with various beam sizes are analyzed with the FDFD method and compared with the rigorous coupled-wave analysis (RCWA). Both unslanted and slanted gratings are treated in transmission as well as in reflection configurations. In general, the effect of the FNP is a decrease in the diffraction efficiency with a decrease in the number of periods of the grating. Similarly, a decrease in incident-beam width causes a decrease in the diffraction efficiency. Exceptions appear in off-Bragg incidence in which a smaller beam width could result in higher diffraction efficiency. For beam widths greater than 10 grating periods and for gratings with more than 20 periods in width, the diffraction efficiencies slowly converge to the values predicted by the RCWA (infinite incident beam and infinite-number-of-periods grating) for both TE and TM polarizations. Furthermore, the effects of FNP holographic gratings on their diffraction performance are found to be comparable to their counterparts of FNP surface-relief gratings.     

Pump efficiency improvement of a C-band tunable fiber laser using optical circulator and tunable fiber gratings

We propose and demonstrate a tunable fiber laser based on an optical circulator (OC) and two tunable fiber Bragg gratings (TFBGs). The OC acts as a pump power router to improve the pumping efficiency, and a 4% increase in overall conversion efficiency has been observed. The combined tuning spectra range of two TFBGs could cover the entire C-band spectrum from 1530 to 1560 nm. Stable laser output power above 10 dBm is obtained using 1.9 m of erbium-doped fiber and TFBGs with 50% reflectivity. With power equalization by using variable optical attenuators, the power variation is less than 0.1 dB in the whole C band with narrow linewidth of 0.05 nm. A signal-to-noise ratio of 60 dB and a continuous tuning resolution of 0.5 nm have been achieved. The TFBG-based tunable fiber laser can be a promising light source for WDM transmission and fiber sensor applications.     

New method for the fabrication of stratified gratings and their applications

Anew method for the preparation of stratified light-sensitive film is developed, and the stratified gratings (SG's) are recorded in the film. The sensitive layers on both sides of a dichromated cellulose triacetate film are produced simultaneously through chemical reaction and not with the conventional coating technique. Compared with SG's in other materials made with coating techniques, double-layer SG's in the film have, to my knowledge, the highest experimental diffraction efficiency (~54%) in addition to their having a simple recording optical system. The diffraction efficiency and the periodic Bragg selectivity of the SH in the film is given. Based on the SG's of the film, several beam splitters with 2, 3, 4, or 7 fan-outs and higher than 80% total diffraction efficiencies are realized experimentally. The advantages of this method as compared with others, such as the method based on volume holographic beam splitters, are explained.     

Modal method for classical diffraction by slanted lamellar gratings

We consider lamellar gratings made of dielectric or lossy materials used in classical diffraction mounts. We show how the modal diffraction formulation may be generalized to deal with slanted lamellar gratings and illustrate the accuracy and versatility of the new method through study of highly slanted gratings in a homogenization limit. We also comment on the completeness of the eigenmode basis and present tests enabling this completeness to be verified numerically.