Grating theory: new equations in Fourier space leading to fast converging results for TM polarization

Modified rigorous coupled-wave analysis for multi-layer deformable gratings with arbitrary profiles and materials

In this paper, the rigorous coupled-wave analysis (RCWA) is extended for general multi-layer deformable gratings with arbitrary numbers of layers, surface profiles, layer offsets, and materials. The contribution from the offset between grating layers and/or due to the movement of the deformable grating layer is included in the expansion of the relative permittivity by the Fourier series, enabling the calculations of deformable gratings commonly used in many optical-based displacement sensing devices. The accuracy and efficiency of the extended RCWA are verified by a number of grating models. It is found that the numerical results are in excellent agreement with those from the finite element method, while the RCWA method costs only ∼1/10 in computation time when compared to its counterpart. Our approach can be used for fast calculation and optimization of multi-layer deformable gratings for optical displacement sensing applications.

VarRCWA: An Adaptive High-Order Rigorous Coupled Wave Analysis Method

Semianalytical methods, such as rigorous coupled wave analysis, have been pivotal in the numerical analysis of photonic structures. In comparison to other numerical methods, they have a much lower computational cost, especially for structures with constant cross-sectional shapes (such as metasurface units). However, when the cross-sectional shape varies even mildly (such as a taper), existing semianalytical methods suffer from high computational costs. We show that the existing methods can be viewed as a zeroth-order approximation with respect to the structure's cross-sectional variation. We derive a high-order perturbative expansion with respect to the cross-sectional variation. Based on this expansion, we propose a new semianalytical method that is fast to compute even in the presence of large cross-sectional shape variation. Furthermore, we design an algorithm that automatically discretizes the structure in a way that achieves a user-specified accuracy level while at the same time reducing the computational cost.

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.

Simple reformulation of the coordinate transformation method for gratings with a vertical facet or overhanging profile

We reformulate the coordinate transformation method (C method) for gratings with a vertical facet or overhanging profile (overhanging gratings), in which no tensor concept is involved, only the knowledge of elementary mathematics and Maxwell's equations in the rectangular coordinate system is used, and we provide a detailed recipe for programming. This formulation is easy to understand and implement. It adopts the strategy of a rotating coordinate system from Plumey et al. [J. Opt. Soc. Am. A14, 610 (1997)JOAOD60740-323210.1364/JOSAA.14.000610] and expresses it with the method of changing variables from Li et al. [Appl. Opt.38, 304 (1999)APOPAI0003-693510.1364/AO.38.000304]. We investigate several typical overhanging gratings by the reformulated C method, and we validate and compare the results with the Fourier modal method, which shows that it is superior, especially for metal deep smooth gratings. This reformulation can facilitate the research in light couplers for optical engineers.

Aperiodic differential method associated with FFF: an efficient electromagnetic computational tool for integrated optical waveguides modelization

A reformulation of the differential theory associated with fast Fourier factorization used for periodic diffractive structures is presented. The incorporation of a complex coordinate transformation in the propagation equations allows the modeling of semi-infinite open problems through an artificially periodized space. Hence, the outgoing wave conditions of an open structure must be satisfied. On the other hand, the excitation technique must be adjusted to adapt with guided structures. These modifications turn the differential theory into an aperiodic tool used with guided optical structure. Our method is verified through numerical results and comparisons with the aperiodic Fourier modal method showing enhanced convergence and accuracy, especially when complex-shaped photonic guided devices are considered.

Optimization of all-dielectric structures for color generation

In this work, we propose an inversion scheme to tailor the chromatic response of an all-dielectric structure. To this end, we couple, through a previously defined objective functional involving the concept of color difference, a forward solver with an optimization algorithm. The former is based on the differential method, whereas the latter is based on particle swarm optimization. The optimal geometrical parameters of the structure that generates a specific color are obtained through the solution of an approximation problem. We illustrate the performance of our inversion scheme through examples and discuss its limitations and potential applications.

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.

Local normal vector field formulation for periodic scattering problems formulated in the spectral domain

We present two adapted formulations, one tailored to isotropic media and one for general anisotropic media, of the normal vector field framework previously introduced to improve convergence near arbitrarily shaped material interfaces in spectral simulation methods for periodic scattering geometries. The adapted formulations enable the definition and generation of the normal vector fields to be confined to a region of prolongation that includes the material interfaces but is otherwise limited. This allows for a more flexible application of geometrical transformations like rotation and translation per scattering object in the unit cell. Moreover, these geometrical transformations enable a cut-and-connect strategy to compose general geometries from elementary building blocks. The entire framework gives rise to continuously parameterized geometries.

Improved multimodal method for the acoustic propagation in waveguides with a wall impedance and a uniform flow

We present an efficient multimodal method to describe the acoustic propagation in the presence of a uniform flow in a waveguide with locally a wall impedance treatment. The method relies on a variational formulation of the problem, which allows to derive a multimodal formulation within a rigorous mathematical framework, notably to properly account for the boundary conditions on the walls (being locally the Myers condition and the Neumann condition otherwise). Also, the method uses an enriched basis with respect to the usual cosine basis, able to absorb the less converging part of the modal series and thus, to improve the convergence of the method. Using the cosine basis, the modal method has a low convergence, 1/N, with N the order of truncation. Using the enriched basis, the improvement in the convergence is shown to depend on the Mach number, from 1/N5 to roughly 1/N1.5 for M=0 to M close to unity. The case of a continuously varying wall impedance is considered, and we discuss the limiting case of piecewise constant impedance, which defines pressure edge conditions at the impedance discontinuities.

Generalized method for retrieving effective parameters of anisotropic metamaterials

Electromagnetic or acoustic metamaterials can be described in terms of equivalent effective, in general anisotropic, media and several techniques exist to determine the effective permeability and permittivity (or effective mass density and bulk modulus in the context of acoustics). Among these techniques, retrieval methods use the measured reflection and transmission coefficients (or scattering coefficients) for waves incident on a metamaterial slab containing few unit cells. Until now, anisotropic effective slabs have been considered in the literature but they are limited to the case where one of the axes of anisotropy is aligned with the slab interface. We propose an extension to arbitrary orientations of the principal axes of anisotropy and oblique incidence. The retrieval method is illustrated in the electromagnetic case for layered media, and in the acoustic case for array of tilted elliptical particles.

Local transformation leading to an efficient Fourier modal method for perfectly conducting gratings

We present an efficient Fourier modal method for wave scattering by perfectly conducting gratings (in the two polarizations). The method uses a geometrical transformation, similar to the one used in the C-method, that transforms the grating surface into a flat surface, thus avoiding to question the Rayleigh hypothesis; also, the transformation only affects a bounded inner region that naturally matches the outer region; this allows applying a simple criterion to select the ingoing and outgoing waves. The method is shown to satisfy reciprocity and energy conservation, and it has an exponential rate of convergence for regular groove shapes. Besides, it is shown that the size of the inner region, where the solution is computed, can be reduced to the groove depth, that is, to the minimal computation domain.

Propagation in waveguides with varying cross section and curvature: a new light on the role of supplementary modes in multi-modal methods

We present an efficient multi-modal method to describe the acoustic propagation in waveguides with varying curvature and cross section. A key feature is the use of a flexible geometrical transformation to a virtual space in which the waveguide is straight and has unitary cross section. In this new space, the pressure field has to satisfy a modified wave equation and associated modified boundary conditions. These boundary conditions are in general not satisfied by the Neumann modes, used for the series representation of the field. Following previous work, an improved modal method (MM) is presented, by means of the use of two supplementary modes. Resulting increased convergences are exemplified by comparison with the classical MM. Next, the following question is addressed: when the boundary conditions are verified by the Neumann modes, does the use of supplementary modes improve or degrade the convergence of the computed solution? Surprisingly, although the supplementary modes degrade the behaviour of the solution at the walls, they improve the convergence of the wavefield and of the scattering coefficients. This sheds a new light on the role of the supplementary modes and opens the way for their use in a wide range of scattering problems.

Plane wave expansion method used to engineer photonic crystal sensors with high efficiency

A photonic crystal waveguide (PhC-WG) was reported to be usable as an optical sensor highly sensitive to various material parameters, which can be detected via changes in transmission through the PhC-WG caused by small changes of the refractive index of the medium filling its holes. To monitor these changes accurately, a precise optical model is required, for which the plane wave expansion (PWE) method is convenient. We here demonstrate the revision of the PWE method by employing the complex Fourier factorization approach, which enables the calculation of dispersion diagrams with fast convergence, i.e., with high precision in relatively short time. The PhC-WG is proposed as a line defect in a hexagonal array of cylindrical holes periodically arranged in bulk silicon, filled with a variable medium. The method of monitoring the refractive index changes is based on observing cutoff wavelengths in the PhC-WG dispersion diagrams. The PWE results are also compared with finite-difference time-domain calculations of transmittance carried out on a PhC-WG with finite dimensions.

Wave propagation through penetrable scatterers in a waveguide and through a penetrable grating

A multimodal method based on the admittance matrix is used to analyze wave propagation through scatterers of arbitrary shape. Two cases are considered: a waveguide containing scatterers, and the scattering of a plane wave at oblique incidence to an infinite periodic row of scatterers. In both cases, the problem reduces to a system of two sets of first-order differential equations for the modal components of the wavefield, similar to the system obtained in the rigorous coupled wave analysis. The system can be solved numerically using the admittance matrix, which leads to a stable numerical method, the basic properties of which are discussed (convergence, reciprocity, energy conservation). Alternatively, the admittance matrix can be used to get analytical results in the weak scattering approximation. This is done using the plane wave approximation, leading to a generalized version of the Webster equation and using a perturbative method to analyze the Wood anomalies and Fano resonances.

Computationally efficient finite-difference modal method for the solution of Maxwell's equations

In this work, a new implementation of the finite-difference (FD) modal method (FDMM) based on an iterative approach to calculate the eigenvalues and corresponding eigenfunctions of the Helmholtz equation is presented. Two relevant enhancements that significantly increase the speed and accuracy of the method are introduced. First of all, the solution of the complete eigenvalue problem is avoided in favor of finding only the meaningful part of eigenmodes by using iterative methods. Second, a multigrid algorithm and Richardson extrapolation are implemented. Simultaneous use of these techniques leads to an enhancement in terms of accuracy, which allows a simple method such as the FDMM with a typical three-point difference scheme to be significantly competitive with an analytical modal method.

Differential theory for anisotropic cylindrical objects with an arbitrary cross section

We extend the differential theory to anisotropic cylindrical structures with an arbitrary cross section. Two cases have to be distinguished. When the anisotropic cylinders do not contain the origin, the scattering matrix of the device is calculated from the extended differential theory with the help of the scattering matrix propagation algorithm. The fields outside the cylinders are described by Fourier-Bessel expansions. When the origin is located in one cylinder, the fields inside the cylinder are expressed from a semi-analytical theory related to a homogeneous anisotropic medium. In this second case, the formalism of the scattering matrix propagation algorithm is not exactly the same and requires suitable change. The numerical results are in good agreement with the ones obtained for the diffraction by one circular cylinder. The theory is then applied on the diffraction by an elliptical cylinder.

Optimizing and characterizing grating efficiency for a soft X-ray emission spectrometer

The efficiency of soft X-ray diffraction gratings is studied using measurements and calculations based on the differential method with the S-matrix propagation algorithm. New open-source software is introduced for efficiency modelling that accounts for arbitrary groove profiles, such as those based on atomic force microscopy (AFM) measurements; the software also exploits multi-core processors and high-performance computing resources for faster calculations. Insights from these calculations, including a new principle of optimal incidence angle, are used to design a soft X-ray emission spectrometer with high efficiency and high resolution for the REIXS beamline at the Canadian Light Source: a theoretical grating efficiency above 10% and resolving power E/ΔE > 2500 over the energy range from 100 eV to 1000 eV are achieved. The design also exploits an efficiency peak in the third diffraction order to provide a high-resolution mode offering E/ΔE > 14000 at 280 eV, and E/ΔE > 10000 at 710 eV, with theoretical grating efficiencies from 2% to 5%. The manufactured gratings are characterized using AFM measurements of the grooves and diffractometer measurements of the efficiency as a function of wavelength. The measured and theoretical efficiency spectra are compared, and the discrepancies are explained by accounting for real-world effects: groove geometry errors, oxidation and surface roughness. A curve-fitting process is used to invert the calculations to predict grating parameters that match the calculated and measured efficiency spectra; the predicted blaze angles are found to agree closely with the AFM estimates, and a method of characterizing grating parameters that are difficult or impossible to measure directly is suggested.

The electromagnetics of light transmission through subwavelength slits in metallic films

By numerically calculating the relevant electromagnetic fields and charge current densities, we show how local charges and currents near subwavelength structures govern light transmission through subwavelength apertures in a real metal film. The illumination of a single aperture generates surface waves; and in the case of slits, generates them with high efficiency and with a phase close to -π with respect to a reference standing wave established at the metal film front facet. This phase shift is due to the direction of induced charge currents running within the slit walls. The surface waves on the entrance facet interfere with the standing wave. This interference controls the profile of the transmission through slit pairs as a function of their separation. We compare the calculated transmission profile for a two-slit array to simple interference models and measurements [Phys. Rev. B 77(11), 115411 (2008)].

Fourier factorization with complex polarization bases in the plane-wave expansion method applied to two-dimensional photonic crystals

We demonstrate an enhancement of the plane wave expansion method treating two-dimensional photonic crystals by applying Fourier factorization with generally elliptic polarization bases. By studying three examples of periodically arranged cylindrical elements, we compare our approach to the classical Ho method in which the permittivity function is simply expanded without changing coordinates, and to the normal vector method using a normal-tangential polarization transform. The compared calculations clearly show that our approach yields the best convergence properties owing to the complete continuity of our distribution of polarization bases. The presented methodology enables us to study more general systems such as periodic elements with an arbitrary cross-section or devices such as photonic crystal waveguides.

Aperiodic Fourier modal method in contrast-field formulation for simulation of scattering from finite structures

This paper extends the area of application of the Fourier modal method (FMM) from periodic structures to aperiodic ones, in particular for plane-wave illumination at arbitrary angles. This is achieved by placing perfectly matched layers at the lateral sides of the computational domain and reformulating the governing equations in terms of a contrast field that does not contain the incoming field. As a result of the reformulation, the homogeneous system of second-order ordinary differential equations from the original FMM becomes non-homogeneous. Its solution is derived analytically and used in the established FMM framework. The technique is demonstrated on a simple problem of planar scattering of TE-polarized light by a single rectangular line.

Fourier modal method for relief gratings with oblique boundary conditions

Oblique boundary conditions are introduced in the Fourier modal method at each slice of the staircase decomposition of an arbitrary profile of a dielectric corrugation grating. The precision and convergence improvement are demonstrated by comparison with reference methods.

Formulation of the normal vector RCWA for symmetric crossed gratings in symmetric mountings

The utilization of symmetries in rigorous coupled wave analysis in order to reduce the matrix sizes and thus the computation time is an appropriate measure for an effective numerical implementation with no loss of accuracy. Another method to improve the convergence is the so-called normal vector method. This method is based on the transformation of the components of the electromagnetic fields in a lateral plane (or slice) from global coordinates into local normal and tangential coordinates relative to the boundaries between two different materials. In this way, the lateral boundary conditions of the electromagnetic field can be imposed more correctly as opposed to the traditional approximation introduced by Li, resulting in a better convergence in many cases. This paper shows how both methods can be combined to attain an optimum solution.

Analysis of semi-infinite periodic structures using a domain reduction technique

A new boundary condition is introduced to calculate the effective impedance matrix of semi-infinite periodic structures such as photonic crystals and metamaterials, which leads to a reduction of the solution space. The obtained effective impedance matrix allows one to relate a matrix to a PC, which includes all of its properties in terms of reflection from its interface. For one-dimensional photonic crystals or multilayer films, it is shown that a closed-form equation can be found for the effective impedance. For two-dimensional photonic crystals the impedance is obtained using the scattering matrices by solving a unilateral quadratic matrix equation. Several examples are outlined to validate the developed scheme. In the examples, the goal is mainly the computation of the reflection from a semi-infinite periodic structure when a plane wave illuminates its boundary.

Analyzing diffraction gratings by a boundary integral equation Neumann-to-Dirichlet map method

For analyzing diffraction gratings, a new method is developed based on dividing one period of the grating into homogeneous subdomains and computing the Neumann-to-Dirichlet (NtD) maps for these subdomains by boundary integral equations. For a subdomain, the NtD operator maps the normal derivative of the wave field to the wave field on its boundary. The integral operators used in this method are simple to approximate, since they involve only the standard Green's function of the Helmholtz equation in homogeneous media. The method retains the advantages of existing boundary integral equation methods for diffraction gratings but avoids the quasi-periodic Green's functions that are expensive to evaluate.

Chebyshev collocation Dirichlet-to-Neumann map method for diffraction gratings

For diffraction gratings with layered refractive index profiles, the Fourier modal method is widely used. However, it is quite expensive to calculate the eigenmodes for each layer, especially when the structure involves absorptive media. We develop an efficient method that avoids the eigenvalue problems based on the so-called Dirichlet-to-Neumann (DtN) map. For each layer, the DtN map is an operator that maps the wave field to its normal derivative on one period of the boundaries of the layer, and it is approximated by a matrix. An efficient procedure for computing the DtN map is developed based on a Chebyshev collocation method and a fourth-order finite difference method for discretizing the uniform and the periodic directions, respectively. The efficiency and accuracy of our method are illustrated by numerical examples.

Fourier factorization with complex polarization bases in modeling optics of discontinuous bi-periodic structures

The coupled wave theory dealing with optics of discontinuous two-dimensional (2D) periodic structures is reformulated by using Fourier factorization with complex polarization bases, which is a generalized implementation of the fast Fourier factorization rules. The modified approach yields considerably improved convergence properties, as shown on an example of a 2D quartz grating. The method can also be applied to the calculation of 2D photonic band structures or nonperiodic cylindrical devices, and can be generalized to elements with arbitrary cross-sections.

An enhanced contrast to detect bulk objects under arbitrary rough surfaces

We study a selective light scattering elimination procedure in the case of highly scattering rough surfaces. Contrary to the case of low scattering levels, the elimination parameters are shown to depend on the sample microstructure and to present rapid variations with the scattering angle. On the other hand, when the slope of the surface is moderated, we show that this parameters present smoother variations and little dependence to the microstructure, even when the roughness is high. These results allow an important selective reduction of the scattered light, with a basic experimental mounting and an analytical determination of the elimination parameters. Such selective scattering reduction is demonstrated by simulations and experiments and applied to the imaging of an object situated under a highly rough surface.

Longitudinal Legendre polynomial expansion of electromagnetic fields for analysis of arbitrary-shaped gratings

The Legendre polynomial expansion method (LPEM), which has been successfully applied to homogenous and longitudinally inhomogeneous gratings [J. Opt. Soc. Am. B24, 2676 (2007)], is now generalized for the efficient analysis of arbitrary-shaped surface relief gratings. The modulated region is cut into a few sufficiently thin arbitrary-shaped subgratings of equal spatial period, where electromagnetic field dependence is now smooth enough to be approximated by keeping fewer Legendre basis functions. The R-matrix propagation algorithm is then employed to match the Legendre polynomial expansions of the transverse electric and magnetic fields across the upper and lower interfaces of every slice. The proposed strategy then enhances the overall computational efficiency, reduces the required memory size, and permits the efficient study of arbitrary-shaped gratings. Here the rigorous approach is followed, and analytical formulas of the involved matrices are given.

Semi-analytical method for light interaction with 1D-periodic nanoplasmonic structures

We present a detailed description of a computationally efficient, semi-analytical method (SAM) to calculate the electomagnetic field distribution in a 1D-periodic, subwavelength-structured metal film placed between dielectric substrates. The method is roughly three orders of magnitude faster than the finite-element method (FEM). SAM is used to study the resonant transmission of light through nanoplasmonic structures, and to analyze the role of fundamental and higher-order Bloch surface plasmons in transmission enhancement. The method is also suitable for solving the eigenvalue problem and finding modes of the structure. Results obtained with SAM, FEM, and the finite-difference time-domain method show very good agreement for various parameters of the structure.

Pseudo-Fourier modal analysis of two-dimensional arbitrarily shaped grating structures

The pseudo-Fourier modal analysis of two-dimensional arbitrarily shaped grating structures is described. It is shown that the pseudo-Fourier modal analysis has an advantage of improved structure modeling over the conventional rigorous coupled-wave analysis. In the conventional rigorous coupled-wave analysis, grating structures are modeled by the staircase approximation, which is well known to have inherent significant errors under TM polarization. However, in the pseudo-Fourier modal analysis, such a limitation of the staircase approximation can be overcome through the smooth-structure modeling based on two-dimensional Fourier representation. The validity of the claim is proved with some comparative numerical results from the proposed pseudo-Fourier modal analysis and the conventional rigorous coupled-wave analysis.

Simplified optical scatterometry for periodic nanoarrays in the near-quasi-static limit

Scatterometry is now proven to be a very powerful technique for measurement of subwavelength periodic structures. However it requires heavy numerical calculations of the scattered optical waves from the structure. For periodic nanoarrays with feature size less than 100 nm, it is possible to simplify this using the Rytov near-quasi-static approximation valid for feature periods only few time less than the wavelength. The validity is investigated by way of comparison with exact numerical results obtained with the eigenfunctions approach. It is shown to be adequate for the determination of the structure parameters from the specularly reflected or transmitted waves and their polarization or ellipsometric properties. The validity of this approach is applied to lamellar nanoscale grating photoresist lines on Si substrate. The high sensitivity of the signals to the structure parameters is demonstrated using wavelengths of only few times the period.

Wave characteristics in gratings by linear superposition of retarded fields

We present a formalism for the wave characteristics in gratings and periodic dielectrics based on the linear superposition of retarded fields. The idea is based on the physical picture that an incident field affects the charges in the material forming the gratings and hence leads to oscillating current and charge densities, which in turn generate more fields via the retarded potential. A set of self-consistent equations for the electric field and current and charge densities is derived. Expressions for the electric field everywhere, including the reflected and transmitted fields, are derived. The formalism is then applied to the calculation of diffraction efficiency so as to illustrate its application and to establish its validity by comparing results with the rigorous coupled-wave method. We further generalize the formalism to include possible anisotropy and nonlinearity in the response of the material forming the grating.

Simulation of two-dimensional Kerr photonic crystals via fast Fourier factorization

We present an adaptation of the fast Fourier factorization method to the simulation of two-dimensional (2D) photonic crystals with a third-order nonlinearity. The algorithm and its performance are detailed and illustrated via the simulation of a Kerr 2D photonic crystal. A change in the transmission spectrum at high intensity is observed. We explain why the change does not reduce to a translation (redshift) but rather consists in a deformation and why one side of the bandgap is more suited to a switching application than the other one.

Efficient finite-element, Green's function approach for critical-dimension metrology of three-dimensional gratings on multilayer films

We present an efficient method for calculating the reflectivity of three-dimensional gratings on multilayer films based on a finite-element, Green's function approach. Our method scales as NlogN, where N is the number of plane waves used in the expansion. Therefore, it is much more efficient than the commonly adopted rigorous-coupled-wave analysis (RCWA), which scales as N3. We demonstrate the effectiveness of this method by applying it to a two-dimensional periodic array of contact holes on a multilayer film. We find that our Green's function approach is about one order of magnitude faster than the RCWA approach when applied to typical contact holes considered in industry. For most cases, this method is efficient enough for application as a realtime, critical-dimension metrology tool.

Study of the differential theory of lamellar gratings made of highly conducting materials

Differential theory is said to be difficult to apply to surface-relief gratings made of metals with very high conductivity even though the formulation follows Li's Fourier factorization rules. Recently, Popov et al. [J. Opt. Soc. Am. 21, 199 (2004)] pointed out this difficulty and explained that its origin is related to the inversion of Toeplitz matrices constructed by the permittivity distribution inside the groove region. The current paper provides information about the differential theory for highly conducting gratings and considers the numerical instability problems. A stable calculation for lossless gratings is described, based on the extrapolation technique with the assumption of small losses.

Light diffraction by a three-dimensional object: differential theory

The differential theory of diffraction of light by an arbitrary object described in spherical coordinates is developed. Expanding the fields on the basis of vector spherical harmonics, we reduce the Maxwell equations to an infinite first-order differential set. In view of the truncation required for numerical integration, correct factorization rules are derived to express the components of D in terms of the components of E, a process that extends the fast Fourier factorization to the basis of vector spherical harmonics. Numerical overflows and instabilities are avoided through the use of the S-matrix propagation algorithm for carrying out the numerical integration. The method can analyze any shape and/or material, dielectric or conducting. It is particularly simple when applied to rotationally symmetric objects.

Fast iterative, coupled-integral-equation technique for inhomogeneous profiled and periodic slabs

A fast coupled-integral-equation (CIE) technique is developed to compute the plane-TE-wave scattering by a wide class of periodic 2D inhomogeneous structures with curvilinear boundaries, which includes finite-thickness relief and rod gratings made of homogeneous material as special cases. The CIEs in the spectral domain are derived from the standard volume electric field integral equation. The kernel of the CIEs is of Picard type and offers therefore the possibility of deriving recursions, which allow the computation of the convolution integrals occurring in the CIEs with linear amounts of arithmetic complexity and memory. To utilize this advantage, the CIEs are solved iteratively. We apply the biconjugate gradient stabilized method. To make the iterative solution process faster, an efficient preconditioning operator (PO) is proposed that is based on a formal analytical inversion of the CIEs. The application of the PO also takes only linear complexity and memory. Numerical studies are carried out to demonstrate the potential and flexibility of the CIE technique proposed. Though the best efficiency and accuracy are observed at either low permittivity contrast or high conductivity, the technique can be used in a wide range of variation of material parameters of the structures including when they contain components made of both dielectrics with high permittivity and typical metals.

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.

Numerical study on the spectroscopic ellipsometry of lamellar gratings made of lossless dielectric materials

The spectroscopic ellipsometry of lamellar gratings made of lossless dielectric materials is studied numerically by using the rigorous coupled-wave method with the use of Li's Fourier factorization rules [J. Opt. Soc. Am. A 13, 1870 (1996)], which are known to improve the convergence on the analyses of metallic gratings. Numerical results show that the calculation method also provides fast convergence on lossless gratings, and accurate values of the ellipsometric angles are obtained in very short computation times. Moreover, estimation of grating parameters is investigated by using a cost function defined by the average distance on the Poincaré sphere, and it is shown that the computation required for accurate estimation is possible in reasonable computation time.

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.

Analysis and design of a concave diffraction grating with total-internal-reflection facets by a hybrid diffraction method

A novel hybrid diffraction method is introduced to simulate the diffraction and imaging of a planar-integrated concave grating that has total internal reflection (TIR) facets. The Kirchhoff-Huygens diffraction formula is adopted to simulate the propagation of the lightwave field in the free-propagation region, and a rigorous coupled-wave analysis is used to calculate the polarization-dependent diffraction by the grating. The hybrid diffraction method can be used to analyze accurately the imaging properties as well as the polarization-dependent diffraction characteristics of a concave grating. The dependence of several merit parameters of a concave grating with TIR facets on its basic geometric parameters is studied. Compared with one with metallic echelle facets, a concave grating with TIR facets shows a much lower polarization-dependent loss. Since more performance specifications can be considered in the design of a concave grating than with the conventional scalar method, design error can be reduced greatly with the present hybrid diffraction method.

Differential theory: application to highly conducting gratings

The recently developed fast Fourier factorization method, which has greatly improved the application range of the differential theory of gratings, suffers from numerical instability when applied to metallic gratings with very low losses. This occurs when the real part of the refractive index is small, in particular, smaller than 0.1-0.2, for example, when silver and gold gratings are analyzed in the infrared region. This failure can be attributed to Li's "inverse rule" [L. Li, J. Opt. Soc. Am. A 13, 1870 (1996)] as shown by studying the condition number of matrices that have to be inverted. Two ways of overcoming the difficulty are explored: first, an additional truncation of the matrices containing the coefficients of the differential system, which reduces the numerical problems in some cases, and second, an introduction of lossier material inside the bulk, thus leaving only a thin layer of the highly conducting metal. If the layer is sufficiently thick, this does not change the optical properties of the system but significantly improves the convergence of the differential theory, including the rigorous coupled-wave method, for various types of grating profiles.

Theoretical study of photonic band gaps in woodpile crystals

We investigate numerically the existence of photonic band gaps in woodpile crystals. We present a numerical method specifically developed to solve Maxwell's equations in such photonic structures. It is based upon a rigorous mathematical formulation and leads to a considerable improvement of the convergence speed as compared to other existing numerical methods. We tested our method by comparing the calculated reflectivity with measurements on an actual sample, i.e., a silicon woodpile photonic crystal designed for 1.5 microm wavelength. Excellent agreement is obtained, provided the main structural imperfections of the sample are taken into account. We show that the existence of photonic band gaps in woodpile crystals requires an index contrast higher than 2.05 +/- 0.01. The effects of imperfections of such structures with an index contrast equal to 2.25 are also investigated. Thus, the relative band gap width falls from 3.5% to 2.2% with structurals imperfection similar to those of the sample.

Comparative study of the modeling of three-dimensional photonic bandgap structures

A comparative study of theoretical models of different three-dimensional photonic bandgap (3D-PBG) structures has been performed, taking into account instability and convergence problems. Some rules for solving these problems and for reducing the computational time by finding symmetries in the structures are also explained. Finally, some applications produced by defects in 3D structures are shown by studying the creation of a complete bandgap in one of them and the variation of partial bandgaps in several 3D-PBG structures when several parameters of the defects, such as the number of layers stacked at each side of the defect, its thickness, and the real and imaginary parts of its index of refraction, are changed.

Numerical integration schemes used on the differential theory for anisotropic gratings

Several formulations of the differential theory for anisotropic gratings are investigated numerically. Conventional formulations and recent formulations based on Li's Fourier factorization rules are applied to a sinusoidal-profiled grating made of an anisotropic and conducting material. For both types of formulation, the numerical results of the differential and the rigorous coupled-wave methods are presented, and only the differential method based on Li's Fourier factorization rules provides a reliable convergence. Moreover, several numerical integration schemes used on the differential method are examined, and the advantage of the implicit integration schemes is shown.

From scattering or impedance matrices to Bloch modes of photonic crystals

The dispersion relation of Bloch waves is derived from the properties of a single grating layer. A straightforward way to impose the Bloch condition leads to the calculation of the eigenvalues of the transfer matrix through the single grating layer. Unfortunately, the transfer-matrix algorithm is known to be unstable as a result of the growing evanescent waves. This problem appears again in the calculation of the eigenvalues, making unusable the transfer matrix in numerous practical problems. We propose two different algorithms to circumvent this problem. The first one takes advantage of scattering matrices, while the second one takes advantage of impedance matrices. Numerical evidence of the efficiency of the algorithms is given. Dispersion diagrams of simple cubic and woodpile photonic crystals are obtained by using, respectively, the scattering and impedance matrices.

Differential theory of gratings made of anisotropic materials

Arbitrary profiled gratings made with anisotropic materials are discussed; the anisotropic character concerns electric and/or magnetic properties. Our aim is to avoid the use of the staircase approximation of the profile, whose convergence is questionable. A coupled first-order differential-equation set is derived by taking into account Li's remarks about Fourier factorization [J. Opt. Soc. Am. A 13, 1870 (1996)], but the present formulation shows that, in return for a convenient form of the differential system, it is possible to use only the intuitive Laurent rule. Our method, when applied to the simpler case of isotropic gratings, is shown to be consistent with that of previous studies. Moreover, from the numerical point of view, the convergence of our formulation for an anisotropic grating is faster than that of the conventional differential method.

Staircase approximation validity for arbitrary-shaped gratings

An electromagnetic study of the staircase approximation of arbitrary shaped gratings is conducted with three different grating theories. Numerical results on a deep aluminum sinusoidal grating show that the staircase approximation introduces sharp maxima in the local field map close to the edges of the profile. These maxima are especially pronounced in TM polarization and do not exist with the original sinusoidal profile. Their existence is not an algorithmic artifact, since they are found with different grating theories and numerical implementations. Since the number of the maxima increases with the number of the slices, a greater number of Fourier components is required to correctly represent the electromagnetic field, and thus a worsening of the convergence rate is observed. The study of the local field map provides an understanding of why methods that do not use the staircase approximation (e.g., the differential theory) converge faster than methods that use it. As a consequence, a 1% accuracy in the efficiencies of a deep sinusoidal metallic grating is obtained 30 times faster when the differential theory is used in comparison with the use of the rigorous coupled-wave theory. A theoretical analysis is proposed in the limit when the number of slices tends to infinity, which shows that even in that case the staircase approximation is not well suited to describe the real profile.