Staircase approximation validity for arbitrary-shaped gratings

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.

Two-dimensional guided-mode resonance gratings with an etch-stop layer and high tolerance to fabrication errors

Guided-mode resonance (GMR) bandpass filters have many important applications. The tolerance of fabrication errors that easily cause the transmission wavelength to shift has been well studied for one-dimensional (1D) anisotropic GMR gratings. However, the tolerance of two-dimensional (2D) GMR gratings, especially for different design architectures, has rarely been explored, which prevents the achievement of a high-tolerance unpolarized design. Here, GMR filters with common 2D zero-contrast gratings (ZCGs) were first investigated to reveal their differences from 1D gratings in fabrication tolerance. We demonstrated that 2D ZCGs are highly sensitive to errors in the grating linewidth against the case of 1D gratings, and the linewidth orthogonal to a certain polarization direction has much more influence than that parallel to the polarization. By analyzing the electromagnetic fields, we found that there was an obvious field enhancement inside the gratings, which could have a strong effect on the modes in the waveguide layer through the field overlap. Therefore, we proposed the introduction of an etch-stop (ES) layer between the gratings and the waveguide-layer, which can effectively suppress the interaction between the gratings and modal evanescent fields, resulting in 4-fold increased tolerance to the errors in the grating linewidth. Finally, the proposed etch-stop ZCGs (ES-ZCGs) GMR filters were experimentally fabricated to verify the error robustness.

Numerical study of the thermally adaptive emissivity of VO2-polymer nanostructured coatings

The emissivity of an opal photonic crystal loaded with thermochromic VO₂ nanoparticles is studied through optical calculations, highlighting the influence of the structure by comparison with a homogenized model. Parameters are first set to maximize the structure influence on material emissivity. Then, a full study of the influence of the VO₂ concentration is made to identify, on one hand, cases with the highest structure impact, and on the other hand, interesting cases for applications such as energy-efficient coatings for buildings, satellites, and camouflage applications.

Electrically Tuneable Optical Diffraction Gratings Based on a Polymer Scaffold Filled with a Nematic Liquid Crystal

We present an experimental and theoretical investigation of the optical diffractive properties of electrically tuneable optical transmission gratings assembled as stacks of periodic slices from a conventional nematic liquid crystal (E7) and a standard photoresist polymer (SU-8). The external electric field causes a twist-type reorientation of the LC molecules toward a perpendicular direction with respect to initial orientation. The associated field-induced modification of the director field is determined numerically and analytically by minimization of the Landau–de Gennes free energy. The optical diffraction properties of the associated periodically modulated structure are calculated numerically on the basis of rigorous coupled-wave analysis (RCWA). A comparison of experimental and theoretical results suggests that polymer slices provoke planar surface anchoring of the LC molecules with the inhomogeneous surface anchoring energy varying in the range 5–20 μJ/m². The investigated structures provide a versatile approach to fabricating LC-polymer-based electrically tuneable diffractive optical elements (DOEs).

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.

Coordinate transformation method for modeling three-dimensional photonic structures with curved boundaries

The coordinate transformation method (C method) is a powerful tool for modeling photonic structures with curved boundaries of discontinuities. As a modal method upon the Fourier basis, the C method has superior computational efficiency and rich physical intuitiveness compared to other full-wave numerical methods. But presently the C method is limited to two-dimensional (2D) structures if the boundaries between adjacent z-invariant layers are of generally different profiles [with (x,y,z) being the Cartesian coordinate]. Here we report a nontrivial extension of the C method to the general case of three-dimensional (3D) structures with curved boundaries of different profiles between adjacent layers. This extension drastically enlarges the applicability of the C method to the various interesting structures in nanophotonics and plasmonics. The extended 3D-C method adopts a hybrid coordinate transformation which includes not only the z-direction coordinate transformation in the classical C method but also the x- and y-direction matched coordinates adopted in the Fourier modal method (FMM), so as to exactly model the curved boundaries in all the three directions. The method also incorporates the perfectly matched layers (PMLs) for aperiodic structures and the adaptive spatial resolution (ASR) for enhancing the convergence. A modified numerically-stable scattering-matrix algorithm is proposed for solving the equations of boundary condition between adjacent z-invariant layers, which are derived via a transformation of the full 3D covariant field-components between the different curvilinear coordinate systems defined by the different-profile top and bottom boundaries of each layer. The validity of the extended 3D-C method is tested with several numerical examples.

Comparison of absorption simulation in semiconductor nanowire and nanocone arrays with the Fourier modal method, the finite element method, and the finite-difference time-domain method

For the design of nanostructured semiconductor solar cells and photodetectors, optics modelling can be a useful tool that reduces the need of time-consuming and costly prototyping. We compare the performance of three of the most popular numerical simulation methods for nanostructure arrays: the Fourier modal method (FMM), the finite element method (FEM) and the finite-difference time-domain (FDTD) method. The difference between the methods in computational time can be three orders of magnitude or more for a given system. The preferential method depends on the geometry of the nanostructures, the accuracy needed from the simulations, whether we are interested in the total, volume-integrated absorption or spatially resolved absorption, and whether we are interested in broadband or narrowband response. Based on our benchmarking results, we provide guidance on how to choose the method.

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.

Demonstration of a dual-channel two-dimensional reflection grating filter

A dual-channel two-dimensional (2D) reflection grating filter operating around the 1.55 µm wavelength region is demonstrated, exhibiting dual-channel reflection peaks at 1.492 µm and 1.647 µm. The sidebands intrinsic to this kind of grating are suppressed by appropriately designed antireflective thin films, and this can be proved by equivalent medium theory. Using the modal analysis method, the excitation modes of the dual-channel reflection peaks are determined to be the TM0 (1.490 µm) and TE0 (1.638 µm) modes. The estimated relative errors in the wavelength determination of these modes are less than 1%. This is found to be in accord with analyses of the reflectivity spectra and electromagnetic fields. The dual-channel reflection peaks are sensitive to the background refractive index and may be useful in biosensing applications.

Physical-optics propagation through curved surfaces

Curved surfaces are the basic elements of various optical components and systems such as microscopy systems, diffractive optical elements, freeform components, microlens arrays, etc. In order to model the propagation through curved surfaces fully vectorially and fast, the local plane interface approximation (LPIA) [Appl. Opt.39, 3304 (2000)APOPAI0003-693510.1364/AO.39.003304] is often used. However, the evaluation of the validity and accuracy of this method has, to our knowledge, not yet been fully addressed in the literature. In this work, we compare the field on the curved surface obtained by LPIA with that obtained with the finite element method. We find it is highly accurate even when the size of the curved surface is on the scale of micrometers. We further evaluate the limitation of LPIA in the cases of multi-reflection/transmission and internal resonance.

Direct S-matrix calculation for diffractive structures and metasurfaces

The paper presents a derivation of analytical components of S matrices for arbitrary planar diffractive structures and metasurfaces in the Fourier domain. The attained general formulas for S-matrix components can be applied within both formulations in the Cartesian and curvilinear metric. A numerical method based on these results can benefit from all previous improvements of the Fourier domain methods. In addition, we provide expressions for S-matrix calculation in the case of periodically corrugated layers of two-dimensional materials, which are valid for arbitrary corrugation depth-to-period ratios. As an example, the derived equations are used to simulate resonant grating excitation of graphene plasmons and the impact of a silica interlayer on corresponding reflection curves.

Treatment of nonconvergence of the Fourier modal method and C method arising from field hypersingularities at lossless metal-dielectric arbitrary-angle edges

Here, we report perturbative approaches to overcome the recently reported nonconvergence of the Fourier modal method (FMM) and the coordinate transformation method (C method) caused by the field hypersingularities (also called irregular field singularities) at lossless metal-dielectric arbitrary-angle edges. For the example of triangular gratings, we replace the sharp edge with a rounded edge to remove the hypersingularities at the edge. With such profile perturbations, we observe the convergence of the C method. The converged values of the diffraction efficiency are tested by the finite element method. However, with the radius of the rounded edge approaching zero, the converged values of the diffraction efficiency cannot approach a fixed value. For the example of parallelogram gratings, we smooth the sharp lamellar boundaries with a medium having a gradually varied refractive index to remove the hypersingularities. With the decrease of the width of the perturbative medium, the converged values of the diffraction efficiency can approach a fixed value for some numerical examples but cannot for other examples. For parallelogram gratings with a period much smaller than the wavelength, we surprisingly find that the FMM tends to converge despite the existence of hypersingularities, and the converged value consists well with the theoretical value given by the effective medium theory.

Fourier method for modeling slanted lamellar gratings of arbitrary end-surface shapes in conical mounting

An efficient modal method for numerically modeling slanted lamellar gratings of isotropic dielectric or metallic media in conical mounting is presented. No restrictions are imposed on the slant angle and the length of the lamellae. The end surface of the lamellae can be arbitrary, subject to certain restrictions. An oblique coordinate system that is adapted to the slanted lamella sidewalls allows the most efficient way of representing and manipulating the electromagnetic fields. A translational coordinate system that is based on the oblique Cartesian coordinate system adapts to the end-surface profile of the lamellae, so that the latter can be handled simply and easily. Moreover, two matrix eigenvalue problems of size 2N × 2N, one for each fundamental polarization of the electromagnetic fields in the periodic lamellar structure, where N is the matrix truncation number, are derived to replace the 4N × 4N eigenvalue problem that has been used in the literature. The core idea leading to this success is the polarization decomposition of the electromagnetic fields inside the periodic lamellar region when the fields are expressed in the oblique translational coordinate system.

Modal method for the 2D wave propagation in heterogeneous anisotropic media

A multimodal method based on a generalization of the admittance matrix is used to analyze wave propagation in heterogeneous two-dimensional anisotropic media. The heterogeneity of the medium can be due to the presence of anisotropic inclusions with arbitrary shapes, to a succession of anisotropic media with complex interfaces between them, or both. Using a modal expansion of the wave field, the problem is reduced to a system of two sets of first-order differential equations for the modal components of the field, similar to the system obtained in the rigorous coupled wave analysis. The system is solved numerically, using the admittance matrix, which leads to a stable numerical method, the basic properties of which are discussed. The convergence of the method is discussed, considering arrays of anisotropic inclusions with complex shapes, which tend to show that Li's rules are not concerned within our approach. The method is validated by comparison with a subwavelength layered structure presenting an effective anisotropy at the wave scale.

Reflective properties of randomly rough surfaces under large incidence angles

The reflective properties of randomly rough surfaces at large incidence angles have been reported due to their potential applications in some of the radiative heat transfer research areas. The main purpose of this work is to investigate the formation mechanism of the specular reflection peak of rough surfaces at large incidence angles. The bidirectional reflectance distribution function (BRDF) of rough aluminum surfaces with different roughnesses at different incident angles is measured by a three-axis automated scatterometer. This study used a validated and accurate computational model, the rigorous coupled-wave analysis (RCWA) method, to compare and analyze the measurement BRDF results. It is found that the RCWA results show the same trend of specular peak as the measurement. This paper mainly focuses on the relative roughness at the range of 0.16<σ/λ<5.35. As the relative roughness decreases, the specular peak enhancement dramatically increases and the scattering region significantly reduces, especially under large incidence angles. The RCWA and the Rayleigh criterion results have been compared, showing that the relative error of the total integrated scatter increases as the roughness of the surface increases at large incidence angles. In addition, the zero-order diffractive power calculated by RCWA and the reflectance calculated by Fresnel equations are compared. The comparison shows that the relative error declines sharply when the incident angle is large and the roughness is small.

Numerical Modeling of Sub-Wavelength Anti-Reflective Structures for Solar Module Applications

This paper reviews the current progress in mathematical modeling of anti-reflective subwavelength structures. Methods covered include effective medium theory (EMT), finite-difference time-domain (FDTD), transfer matrix method (TMM), the Fourier modal method (FMM)/rigorous coupled-wave analysis (RCWA) and the finite element method (FEM). Time-based solutions to Maxwell's equations, such as FDTD, have the benefits of calculating reflectance for multiple wavelengths of light per simulation, but are computationally intensive. Space-discretized methods such as FDTD and FEM output field strength results over the whole geometry and are capable of modeling arbitrary shapes. Frequency-based solutions such as RCWA/FMM and FEM model one wavelength per simulation and are thus able to handle dispersion for regular geometries. Analytical approaches such as TMM are appropriate for very simple thin films. Initial disadvantages such as neglect of dispersion (FDTD), inaccuracy in TM polarization (RCWA), inability to model aperiodic gratings (RCWA), and inaccuracy with metallic materials (FDTD) have been overcome by most modern software. All rigorous numerical methods have accurately predicted the broadband reflection of ideal, graded-index anti-reflective subwavelength structures; ideal structures are tapered nanostructures with periods smaller than the wavelengths of light of interest and lengths that are at least a large portion of the wavelengths considered.

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.

Coordinate transformation formulation of electromagnetic scattering from imperfectly periodic surfaces

This paper considers the electromagnetic scattering problem of periodically corrugated surface with local imperfection of structural periodicity, and presents a formulation based on the coordinate transformation method (C-method). The C-method is originally developed to analyze the plane-wave scattering from perfectly periodic structures, and uses the pseudo-periodic property of the fields. The fields in imperfectly periodic structures are not pseudo-periodic and the C-method cannot be directly applied. This paper introduces the pseudo-periodic Fourier transform to convert the fields in imperfectly periodic structures to pseudo-periodic ones, and the C-method becomes then applicable.

Analysis of two-dimensional photonic band gaps of any rod shape and conductivity using a conical-integral-equation method

The conical-boundary-integral-equation method has been proposed for calculation of the sensitive optical response of two-dimensional photonic band gaps (PBGs), including dielectric, absorbing, and high-conductive rods of various shapes working in any wavelength range. It is possible to determine the diffracted field by computing the scattering matrices separately for any grating boundary profile. The computation of the matrices is based on the solution of a 2×2 system of singular integral equations at each interface between two different materials. The advantage of our integral formulation is that the discretization of the integral equation system and the factorization of the discrete matrices, which takes the majority of the computing time, are carried out only once for a boundary. It turns out that a small number of collocation points per boundary combined with a high convergence rate can provide an adequate description of the dependence on diffracted energy of very different PBGs illuminated at arbitrary incident and polarization angles. The numerical results presented describe the significant impact of rod shape on diffraction in PBGs supporting polariton-plasmon excitation, particularly in the vicinity of resonances and at high filling ratios. The diffracted energy response calculated vs the array cell geometry parameters was found to vary from a few up to a few hundred percent. The influence of other types of anomalies (i.e., waveguide anomalies, cavity modes, Fabry-Perot and Bragg resonances, Rayleigh orders, etc.), conductivity, and polarization states on the optical response is demonstrated.

Application of the three-dimensional aperiodic Fourier modal method using arc elements in curvilinear coordinates

This paper deals with a full vectorial generalization of the aperiodic Fourier modal method (AFMM) in cylindrical coordinates. The goal is to predict some key characteristics such as the bending losses of waveguides having an arbitrary distribution of the transverse refractive index. After a description of the method, we compare the results of the cylindrical coordinates AFMM with simulations by the finite-difference time-domain (FDTD) method performed on an S-bend structure made by a 500 nm × 200 nm silicon core (n=3.48) in silica (n=1.44) at a wavelength λ=1550 nm, the bending radius varying from 0.5 up to 2 μm. The FDTD and AFMM results show differences comparable to the variations obtained by changing the parameters of the FDTD simulations.

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.

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.

All-purpose finite element formulation for arbitrarily shaped crossed-gratings embedded in a multilayered stack

We propose a novel formulation of the finite element method adapted to the calculation of the vector field diffracted by an arbitrarily shaped crossed-grating embedded in a multilayered stack and illuminated by an arbitrarily polarized plane wave under oblique incidence. A complete energy balance (transmitted and reflected diffraction efficiencies and losses) is deduced from field maps. The accuracy of the proposed formulation has been tested using classical cases computed with independent methods. Moreover, to illustrate the independence of our method with respect to the shape of the diffractive object, we present the global energy balance resulting from the diffraction of a plane wave by a lossy thin torus crossed-grating. Finally, computation time and convergence as a function of the mesh refinement are discussed. As far as integrated energy values are concerned, the presented method shows a remarkable convergence even for coarse meshes.

Polarization-dependent GaN surface grating reflector for short wavelength applications

This study proposes a one-dimensional sub-wavelength grating structure on GaN surface which behaves as a reflector for transverse-electric polarized light in the blue wavelength range. The rigorous coupled-wave analysis method was used to analyze the effects of various structural parameters on the reflectance spectra of the grating. Based on the optimal design, a GaN surface grating reflector (SGR) was fabricated using holographic lithography and dry etching processes. It showed reflectance that exceeded 90% over a 60-nm bandwidth. The obtained experimental results were in good agreement with simulated ones. The SGR has an advantage of structural simplicity, which should greatly facilitate the fabrication and integration of high reflectors on GaN-based short-wavelength photonic devices.

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.

Demonstration of a cavity coupler based on a resonant waveguide grating

Thermal noise in multilayer optical coatings may not only limit the sensitivity of future gravitational wave detectors in their most sensitive frequency band but is also a major impediment for experiments that aim to reach the standard quantum limit or to cool mechanical systems to their quantum ground state. Here, we present the experimental realization and characterization of a cavity coupler, which is based on a surface relief guided ode resonant grating. Since the required thickness of the dielectric coating is dramatically decreased compared to conventional mirrors, it is expected to provide low mechanical loss and, thus, low thermal noise. The cavity coupler was incorporated into a Fabry-Perot resonator together with a conventional high quality mirror. The finesse of this cavity was measured to be F = 657, which corresponds to a coupler reflectivity of R = 99.08 %.

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.

Hybridization of electromagnetic numerical methods through the G-matrix algorithm

For the sake of numerical performance, we hybridize two common approaches often used in electromagnetic computations, namely the finite-element method and the aperiodic Fourier modal method. To that end, we propose an extension of the classical S-matrix formalism to numerical situations, which requires handling different mathematical representations of the electromagnetic fields. As shown with a three-dimensional example, the proposed G-matrix formalism is stable and allows for an enhanced performance in terms of numerical accuracy and efficiency.

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.

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.

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.

Pseudo-Fourier modal analysis on dielectric slabs with arbitrary longitudinal permittivity and permeability profiles

A pseudo-Fourier modal analysis method for analyzing finite-sized dielectric slabs with arbitrary longitudinal permittivity and permeability profiles is proposed. In the proposed method, the permittivity and permeability profiles are represented by the Fourier expansion without using the conventional staircase approximation. The total electromagnetic field distribution inside a dielectric slab is a linear superposition of extracted pseudo-Fourier eigenmodes with specific coupling coefficients selected to satisfy given boundary conditions. The proposed pseudo-Fourier modal analysis method shows excellent agreement with the conventional rigorous coupled-wave analysis with the S-matrix method.

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.

Comparing the Fourier modal method with the C method: analysis of conducting multilevel gratings in TM polarization

The coordinate transformation method (C method) with adaptive spatial resolution and the Fourier modal method (FMM) are compared in the case of conducting discontinuous multilevel gratings in TM polarization. A procedure permitting analysis of such gratings more efficiently with the C method than with the FMM is presented. The C method is observed to converge more rapidly than the FMM, whose instabilities are shown to harm the convergence in the aforementioned case.