Modal spectral element method with modified Legendre polynomials to analyze binary crossed gratings.

In a previous paper, a modal spectral element method (SEM), the originality of which comes from the use of a hierarchical basis built with modified Legendre polynomials, was shown to be very powerful for the analysis of lamellar gratings. In this work, keeping the same ingredients, the method has been extended to the general case of binary crossed gratings. The geometric versatility of the SEM is illustrated with gratings whose patterns are not aligned with the boundaries of the elementary cell. The method is validated by a comparison to the Fourier modal method (FMM) in the case of anisotropic crossed gratings and with the FMM with adaptive spatial resolution in the case of a square-hole array in a silver film.

Modal analysis of diffraction by snake gratings using a tensor product of pseudo-periodic functions and Legendre polynomials.

The problem of diffraction by snake gratings is presented and formulated as an eigenvalue eigenvector problem. A numerical solution is obtained thanks to the method of moments where a tensor product of pseudo-periodic functions and Legendre polynomials is used as expansion and test functions. The method is validated by comparison with the usual Fourier modal method (FMM) as applied to crossed gratings. Our method is shown to be more efficient than the FMM in the case of metallic gratings.

Domain Decomposition Spectral Method Applied to Modal Method: Direct and Inverse Spectral Transforms.

We introduce a Domain Decomposition Spectral Method (DDSM) as a solution for Maxwell's equations in the frequency domain. It will be illustrated in the framework of the Aperiodic Fourier Modal Method (AFMM). This method may be applied to compute the electromagnetic field diffracted by a large-scale surface under any kind of incident excitation. In the proposed approach, a large-size surface is decomposed into square sub-cells, and a projector, linking the set of eigenvectors of the large-scale problem to those of the small-size sub-cells, is defined. This projector allows one to associate univocally the spectrum of any electromagnetic field of a problem stated on the large-size domain with its footprint on the small-scale problem eigenfunctions. This approach is suitable for parallel computing, since the spectrum of the electromagnetic field is computed on each sub-cell independently from the others. In order to demonstrate the method's ability, to simulate both near and far fields of a full three-dimensional (3D) structure, we apply it to design large area diffractive metalenses with a conventional personal computer.

Spectral element method with modified Legendre polynomials for modal analysis of lamellar gratings.

We report on the derivation of a spectral element method whose originality comes from the use of a hierarchical basis built with modified Legendre polynomials. We restrict our work to TM polarization, which is the most challenging. The validation and convergence are carefully checked for metallic dielectric gratings. The method is shown to be highly efficient and remains stable for huge truncation numbers. All the necessary information is given so that non-specialists can implement the method.

Matched coordinates for the analysis of 1D gratings.

The Fourier modal method (FMM) is certainly one of the most popular and general methods for the modeling of diffraction gratings. However, for non-lamellar gratings it is associated with a staircase approximation of the profile, leading to poor convergence rate for metallic gratings in TM polarization. One way to overcome this weakness of the FMM is the use of the fast Fourier factorization (FFF) first derived for the differential method. That approach relies on the definition of normal and tangential vectors to the profile. Instead, we introduce a coordinate system that matches laterally the profile and solve the covariant Maxwell's equations in the new coordinate system, hence the name matched coordinate method (MCM). Comparison of efficiencies computed with MCM with other data from the literature validates the method.

Polynomial modal analysis of slanted lamellar gratings.

The problem of diffraction by slanted lamellar dielectric and metallic gratings in classical mounting is formulated as an eigenvalue eigenvector problem. The numerical solution is obtained by using the moment method with Legendre polynomials as expansion and test functions, which allows us to enforce in an exact manner the boundary conditions which determine the eigensolutions. Our method is successfully validated by comparison with other methods including in the case of highly slanted gratings.

Polynomial modal analysis of lamellar diffraction gratings in conical mounting.

An efficient numerical modal method for modeling a lamellar grating in conical mounting is presented. Within each region of the grating, the electromagnetic field is expanded onto Legendre polynomials, which allows us to enforce in an exact manner the boundary conditions that determine the eigensolutions. Our code is successfully validated by comparison with results obtained with the analytical modal method.

Efficient implementation of B-spline modal method for lamellar gratings.

The B-spline modal method (BMM) as applied to lamellar gratings analysis is revisited, and a new implementation is presented. The main difference with our previous work is that we now take into account discontinuities by putting a spline with a degenerate knot on them. Our new approach is compared with other implementations of the BMM and is shown to be superior in terms of numerical convergence.

Profile reconstruction of periodic interface.

The reconstruction problem for periodic (arbitrary profiled within a period) boundary between two homogeneous media is considered. Our approach to the solution of the inverse problem is based on the Tikhonov regularization technique, which requires successive selection of the boundaries on the basis of multiple solutions of the direct problem of wave diffraction by the candidate boundaries. The analytical numerical C method has been chosen as a simple but rather efficient tool for the direct problem solving. The scheme for numerical tests of algorithms and criteria for reconstruction accuracy have been suggested and verified. Results of numerical experiments that prove the validity of the approach are presented.

Reformulation of the modal method based on Gegenbauer polynomial expansion for cylindrical structures with arbitrary cross sections.

The work presented here focuses on the numerical modeling of cylindrical structure eigenmodes with an arbitrary cross section using Gegenbauer polynomials. The new eigenvalue equation leads to considerable reduction in computation time compared to the previous formulation. The main idea of this new formulation involves considering that the numerical scheme can be partially separated into two independent parts and the size of the eigenvalue matrix equation may be reduced by a factor of 2. We show that the ratio of the computation times between the first and current versions follows a linear relation with respect to the number of polynomials.

Mode solver based on Gegenbauer polynomial expansion for cylindrical structures with arbitrary cross sections.

We present a modal method for the computation of eigenmodes of cylindrical structures with arbitrary cross sections. These modes are found as eigenvectors of a matrix eigenvalue equation that is obtained by introducing a new coordinate system that takes into account the profile of the cross section. We show that the use of Hertz potentials is suitable for the derivation of this eigenvalue equation and that the modal method based on Gegenbauer expansion (MMGE) is an efficient tool for the numerical solution of this equation. Results are successfully compared for both perfectly conducting and dielectric structures. A complex coordinate version of the MMGE is introduced to solve the dielectric case.

Time-and-frequency domains approach to data processing in multiwavelength optical scatterometry of dielectric gratings.

This paper focuses on scatterometry problems arising in lithography production of periodic gratings. Namely, the paper introduces a theoretical and numerical-modeling-oriented approach to scatterometry problems and discusses its capabilities. The approach allows for reliable detection of deviations in gratings' critical dimensions (CDs) during the manufacturing process. The core of the approach is the one-to-one correspondence between the electromagnetic (EM) characteristics and the geometric/material properties of gratings. The approach is based on highly accurate solutions of initial boundary-value problems describing EM waves' interaction on periodic gratings. The advantage of the approach is the ability to perform simultaneously and interactively both in frequency and time domains under conditions of possible resonant scattering of EM waves by infinite or finite gratings. This allows a detection of CDs for a wide range of gratings, and, thus is beneficial for the applied scatterometry.

Fourier-matching pseudospectral modal method for diffraction gratings: comment.

Recently two variants of a pseudospectral modal method were developed for analyzing lamellar diffraction gratings: [J. Lightwave Technol. 27, 5151 (2009)] and [J. Opt. Soc. Am. A 28, 613 (2011)]. Both of them divide the computational domain into nonoverlapping subdomains and replace the spatial derivative in the Helmoltz equation by a differentiation matrix at the Chebyshev collocation points. The authors of the second reference claim that their method is more robust and accurate because they match the Fourier coefficient at the interfaces between the layers and drop some computed eigenmodes. We challenge these two ideas. Instead, we numerically demonstrate that by keeping all computed eigenmodes and by also numerically computing eigenmodes in homogeneous regions, the pseudospectral method performs better.

Field singularities at lossless metal-dielectric right-angle edges and their ramifications to the numerical modeling of gratings.

We mathematically prove and numerically demonstrate that the source of the convergence problem of the analytical modal method and the Fourier modal method for modeling some lossless metal-dielectric lamellar gratings in TM polarization recently reported by Gundu and Mafi [J. Opt. Soc. Am. A 27, 1694 (2010)] is the existence of irregular field singularities at the edges of the grating grooves. We show that Fourier series are incapable of representing the transverse electric field components in the vicinity of an edge of irregular field singularity; therefore, any method, not necessarily of modal type, using Fourier series in this way is doomed to fail. A set of precise and simple criteria is given with which, given a lamellar grating, one can predict whether the conventional implementation of a modal method of any kind will converge without running a convergence test.

Derivation of plasmonic resonances in the Fourier modal method with adaptive spatial resolution and matched coordinates.

A very stable approach for finding optical resonances is to solve an eigenvalue equation that evolves from the linearization of the inverse scattering matrix. In this paper, we show how to use this approach in the Fourier modal method so that advanced coordinate transformation methods such as adaptive spatial resolution and matched coordinates can be included. Furthermore, we present a way that accelerates the computation of the inverse scattering matrix tremendously and allows the derivation of the resonant field distribution inside the structure efficiently.

Perturbation method for the Rigorous Coupled Wave Analysis of grating diffraction.

The perturbation method is combined with the Rigorous CoupledWave Analysis (RCWA) to enhance its computational speed. In the original RCWA, a grating is approximated by a stack of lamellar gratings and the number of eigenvalue systems to be solved is equal to the number of subgratings. The perturbation method allows to derive the eigensolutions in many layers from the computed eigensolutions of a reference layer provided that the optical and geometrical parameters of these layers differ only slightly. A trapezoidal grating is considered to evaluate the performance of the method.

Light wheel buildup using a backward surface mode.

When a guided mode is excited in a dielectric slab coupled to a backward surface wave at the interface between a dielectric and a left-handed medium, light is confined in the structure: this is a light wheel. Complex plane analysis of the dispersion relation and coupled-mode formalism give deep insight into the physics of this phenomenon (lateral confinement and the presence of a dark zone).

Modal analysis of lamellar gratings using the moment method with subsectional basis and adaptive spatial resolution.

We formulate the problem of diffraction by a one-dimensional lamellar grating as an eigenvalue problem in which adaptive spatial resolution is introduced thanks to a new coordinate system that takes into account the permittivity profile function. We use the moment method with triangle functions as expansion functions and pulses as test functions. Our method is successfully compared with the Fourier modal method and the frequency domain finite difference method.

Matched coordinates and adaptive spatial resolution in the Fourier modal method.

Several improvements have been introduced for the Fourier modal method in the last fifteen years. Among those, the formulation of the correct factorization rules and adaptive spatial resolution have been crucial steps towards a fast converging scheme, but an application to arbitrary two-dimensional shapes is quite complicated.We present a generalization of the scheme for non-trivial planar geometries using a covariant formulation of Maxwell's equations and a matched coordinate system aligned along the interfaces of the structure that can be easily combined with adaptive spatial resolution. In addition, a symmetric application of Fourier factorization is discussed.

Recognition of diffraction-grating profile using a neural network classifier in optical scatterometry.

Optical scatterometry has been given much credit during the past few years in the semiconductor industry. The geometry of an optical diffracted structure is deduced from the scattered intensity by solving an inverse problem. This step always requires a previously defined geometrical model. We develop an artificial neural network classifier whose purpose is to identify the structural geometry of a diffraction grating from its measured ellipsometric signature. This will take place before the characterization stage. Two types of geometry will be treated: sinusoidal and symmetric trapezoidal. Experimental results are performed on two manufactured samples: a sinusoidal photoresist grating deposited on a glass substrate and a trapezoidal grating etched on a SiO2 substrate with periods of 2 microm and 0.565 microm, respectively.