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Rigorous coupled-wave analysis

Rigorous coupled-wave analysis (RCWA), also known as the Fourier modal method (FMM), is a semi-analytical numerical method in computational electromagnetics for modeling the diffraction of electromagnetic waves by periodic dielectric structures, such as planar gratings bounded by different media.^[1] It expands the electromagnetic fields into plane-wave components using Fourier series and solves the resulting coupled differential equations to compute exact diffraction efficiencies for both transmission and reflection configurations.^[2] Originally formulated for gratings with arbitrary fringe orientations, including slanted profiles, RCWA accounts for all diffraction orders without approximations beyond the periodic boundary conditions.^[1] Introduced in 1981 by M. G. Moharam and T. K. Gaylord, RCWA addressed limitations in earlier approximate theories like modal methods and scalar coupled-wave approaches, which often failed for deep or slanted gratings by providing rigorous solutions that converge accurately with increasing numbers of spatial harmonics.^[1] Subsequent refinements, such as those in 1995 by M. G. Moharam and colleagues, enhanced numerical stability and efficiency for binary gratings under TE and TM polarizations, including conical diffraction cases, by reformulating the eigenvalue problems to avoid underflow and overflow issues in computations.^[2] These improvements exploit grating symmetries to reduce computational demands by up to an order of magnitude while maintaining high precision for structures with grating periods comparable to or smaller than the wavelength.^[2] RCWA has become a cornerstone in photonics design due to its balance of accuracy and speed compared to full-wave methods like finite-difference time-domain (FDTD) simulations, particularly for multilayer periodic media where it enables rapid optimization of parameters like grating depth and profile.^[3] Key applications include diffractive optical elements, such as holograms and photonic crystals, where it predicts scattering from subwavelength structures; surface plasmon polaritons in grating-based sensors; and metasurfaces for beam steering and polarization control.^[4] In recent extensions, RCWA has been adapted for non-optical domains, including acoustic wave propagation in periodic media for ultrasound transducer design.^[5] Its versatility supports both 1D and 2D gratings, with formulations for stratified structures reducing dimensionality for efficient near- and far-field analysis.^[6]

Introduction

Definition and overview

Rigorous coupled-wave analysis (RCWA), also known as the Fourier modal method (FMM), is a semi-analytical computational technique for solving Maxwell's equations to model electromagnetic wave scattering and diffraction in periodic dielectric structures, such as optical gratings, without scalar approximations that neglect polarization effects.^[7] This full-vectorial approach accounts for both transverse electric (TE) and transverse magnetic (TM) polarizations, enabling precise predictions of diffraction efficiencies and field distributions in structures with subwavelength periodicity.^[7] Originally formulated by Moharam and Gaylord in 1981 for planar grating diffraction, RCWA divides the periodic structure into homogeneous layers parallel to the plane of periodicity. Within each layer, the electromagnetic fields and relative permittivity are expanded as Fourier series, leveraging Floquet theory to represent the quasi-periodic nature of the solutions. This expansion transforms Maxwell's equations into a system of matrix ordinary differential equations along the direction of stratification, from which modal solutions are derived and propagated layer-by-layer using transfer or scattering matrices.^[8] Key advantages of RCWA include its rigor in handling vectorial effects and material anisotropies, computational efficiency for periodic geometries due to the semi-analytical reduction of partial differential equations to algebraic systems or ordinary differential equations, and suitability for multilayered configurations where numerical stability can be ensured through proper formulation.^[7]^[8]

Historical development

The rigorous coupled-wave analysis (RCWA) method originated in 1981 with the seminal work of M. G. Moharam and T. K. Gaylord, who introduced a framework for computing diffraction efficiencies in planar periodic gratings by deriving coupled-wave equations from Maxwell's equations in stratified media using Fourier expansions. This approach provided a numerically efficient alternative to integral methods, enabling accurate predictions of electromagnetic wave interactions with periodic structures for both TE and TM polarizations. In the 1990s, key advancements addressed convergence issues and expanded the method's scope to more general geometries. Moharam and collaborators formulated a stable implementation in 1995 that incorporated the enhanced transmittance matrix approach, ensuring numerical robustness for multilayer surface-relief gratings and extending calculations to conical diffraction under oblique incidence.^[9] Lifeng Li contributed significantly in 1996 by developing the S-matrix algorithm, a recursive matrix method for layered diffraction gratings that enhanced computational stability and efficiency by avoiding ill-conditioned matrices in forward and backward propagations.^[10] That same year, Li's formulation on Fourier series for discontinuous periodic structures introduced the Fourier factorization technique, which improved convergence for sharp-edged profiles by correctly factoring the permittivity at interfaces and reducing Gibbs oscillations in expansions.^[11] The 2000s saw refinements aimed at broader applicability and better handling of non-ideal boundaries. In 2001, Evgeny Popov advanced the method through a Fourier-space reformulation of Maxwell's equations for fast-converging solutions in arbitrary shaped, periodic, anisotropic media.^[12] Building on earlier stability techniques, Jean-Paul Hugonin and Philippe Lalanne further optimized the enhanced transmittance matrix in their 2005 grating analysis frameworks, improving numerical conditioning for complex multilayer systems and enabling reliable simulations of high-contrast dielectric interfaces. Post-2010 developments have focused on adaptive and high-performance variants for emerging photonic applications. In 2022, Ziwei Zhu and Changxi Zheng proposed VarRCWA, an adaptive high-order RCWA that dynamically varies the expansion basis to handle large cross-sectional changes in waveguides and metasurfaces, achieving faster convergence and higher accuracy compared to standard implementations for structures like photonic crystal fibers.^[13] In 2023, Lin Wang and colleagues introduced the global NV-ETM RCWA, merging normal vector fields with enhanced transmittance matrices to streamline the optimization of multilayer metasurface absorbers, particularly for broadband radar stealth, reducing computation time by orders of magnitude while maintaining rigorous accuracy.^[14] More recent works, such as a 2024 study on RCWA for optical force computations in periodic structures and a 2025 enhancement using adaptive spatial resolution for complex arrays, continue to expand its applications.^[15]^[16]

Fundamental Principles

Periodic structures and Floquet theory

Periodic structures in electromagnetics refer to media characterized by infinite repetition of a unit cell in one or more spatial directions, such as one-dimensional (1D) diffraction gratings or two-dimensional (2D) photonic crystals.^[17] This periodicity imposes translational invariance, leading to wave solutions in the form of Bloch waves that propagate through the structure without scattering at unit cell boundaries.^[18] Examples include dielectric gratings used in optical devices, where the permittivity varies periodically, enabling control over light diffraction and filtering.^[19] Floquet's theorem, originally developed for linear differential equations with periodic coefficients, states that the electromagnetic fields in such structures can be expressed as the product of a periodic function with the same periodicity as the structure and a phase factor \exp(i \boldsymbol{\beta} \cdot \mathbf{r}), where \boldsymbol{\beta} is the Bloch wavevector confined to the first Brillouin zone. Mathematically, for a field component U(\mathbf{r}),

U(\mathbf{r}) = p(\mathbf{r}) \exp(i \boldsymbol{\beta} \cdot \mathbf{r}),

where p(\mathbf{r} + \mathbf{d}) = p(\mathbf{r}) for any lattice vector \mathbf{d}.^[20] This decomposition implies that the fields consist of discrete diffraction orders or space harmonics, each with parallel wavevectors \mathbf{k}_{\parallel, lm} = \boldsymbol{\beta} + l \mathbf{b}_x + m \mathbf{b}_y, where \mathbf{b}_x and \mathbf{b}_y (if applicable) are the reciprocal lattice vectors in the periodic in-plane directions, and the perpendicular (z) components k_{z, lm} determined by the material dispersion relation.$$] ^[20] Historically, Floquet's theorem was introduced by Gaston Floquet in 1883 for one-dimensional periodic systems, later extended by Felix Bloch in 1928 to three dimensions for electron waves in crystals, and applied to electromagnetic gratings in optics during the 1960s as part of coupled-wave theories for diffraction analysis.^[19] In scattering scenarios, an incident plane wave on a periodic structure excites a discrete set of forward- and backward-propagating modes, along with evanescent modes, governed by the Floquet condition that ensures phase matching across periods.^[20] Only a finite number of these modes are propagating, depending on the wavelength and grating period, while the rest decay exponentially, allowing efficient numerical modeling by truncating higher-order terms.^[20] This contrasts with non-periodic structures, where scattering involves a continuous spectrum of modes requiring an infinite, non-discrete basis for expansion, complicating analysis and computation.^[20] The modal discreteness in periodic cases thus enables Bloch wave solutions that capture band structures and forbidden gaps, essential for designing photonic devices like filters and waveguides.^[19]

Maxwell's equations in stratified media

In the context of rigorous coupled-wave analysis (RCWA), stratified media are modeled as a stack of parallel layers, each invariant along the z-direction but featuring a permittivity \epsilon(x, y, z) that varies periodically in the x-y plane within the layer.^[20] This assumption simplifies the electromagnetic problem by treating the structure as translationally invariant in z within each layer, allowing the z-dependence to be isolated while the in-plane periodicity is handled separately.^[21] Such media are common in applications like diffraction gratings and photonic crystals, where the layering enables an invariant embedding approach for propagation.^[20] The starting point for RCWA is the time-harmonic form of Maxwell's equations, assuming fields vary as e^{-i\omega t} and a non-magnetic medium with permeability \mu = 1. These equations are: [ \nabla \times \mathbf{E} = i \omega \mu_0 \mathbf{H},

\nabla \times \mathbf{H} = -i \omega \epsilon_0 \epsilon(x, y, z) \mathbf{E},

where $\mathbf{E}$ and $\mathbf{H}$ are the electric and magnetic field vectors, $\omega$ is the angular frequency, $\epsilon_0$ and $\mu_0$ are the vacuum permittivity and permeability, and $\epsilon(x, y, z)$ is the relative permittivity.[](https://doi.org/10.1364/JOSA.71.000811) This convention ensures the equations capture the vectorial nature of the fields without loss of generality for isotropic media.[](https://doi.org/10.1364/JOSAA.12.001068) Within each stratified layer, the translational invariance in z reduces the partial differential equations to ordinary differential equations (ODEs) in z after handling in-plane (x-y) derivatives, which exploit the periodicity. The fields are described using tangential components—specifically, the in-plane [electric field](/page/Electric_field) elements $E_x$ and $E_y$, and the in-plane [magnetic field](/page/Magnetic_field) elements $H_x$ and $H_y$—to form a [state vector](/page/State_vector) $\Psi(z) = [E_x, E_y, H_x, H_y]^T$. This choice of tangential fields ensures [continuity](/page/Continuity) across layer interfaces and facilitates boundary matching.[](https://doi.org/10.1364/JOSA.71.000811) The propagation through the layer is then governed by the invariant embedding equation:

\frac{d}{dz} \Psi(z) = i K(z) \Psi(z),

where $K(z)$ is the dynamical matrix incorporating the material properties and wave parameters.[](https://doi.org/10.1364/JOSAA.12.001068) This first-order ODE system describes the evolution of the fields along z, with solutions involving exponentials of the matrix $K(z)$.[](https://doi.org/10.1364/JOSA.71.000811) The vectorial formulation inherently couples transverse electric (TE) and transverse magnetic (TM) polarizations, particularly under oblique incidence, due to the off-diagonal terms in $K(z)$ arising from the periodic permittivity contrast.[](https://doi.org/10.1364/JOSAA.12.001068) This coupling distinguishes RCWA from scalar approximations and ensures accurate modeling of cross-polarization effects in stratified periodic structures.[](https://doi.org/10.1364/JOSA.71.000811) ## Mathematical Formulation ### Field expansions in Fourier space In rigorous coupled-wave analysis (RCWA), the electromagnetic fields in each homogeneous layer of a stratified periodic structure are expressed using [Fourier](/page/Fourier) expansions to exploit the inherent periodicity in the lateral (x and y) directions, while treating the propagation direction ([z](/page/Z)) analytically. This approach transforms the [continuous wave](/page/Continuous_wave) equation into a [discrete](/page/Discrete) [modal](/page/Modal) representation. For a general structure with two-dimensional periodicity of periods $\Lambda_x$ and $\Lambda_y$, the tangential components of the electric and magnetic fields—specifically $E_x$, $E_y$, $H_x$, and $H_y$—are expanded as double [Fourier series](/page/Fourier_series):

E_x(x, y, z) = \sum_{m=-\infty}^{\infty} \sum_{n=-\infty}^{\infty} E_{x,mn}(z) \exp\left[i (k_{x,m} x + k_{y,n} y)\right],

with analogous forms for the other components, where the diffraction-order [wave vector](/page/Wave_vector)s are defined as $k_{x,m} = k_x + 2\pi m / \Lambda_x$ and $k_{y,n} = k_y + 2\pi n / \Lambda_y$. Here, $k_x$ and $k_y$ are the incident [wave vector](/page/Wave_vector) components in the x and y directions, respectively. In numerical implementations, the infinite sums are truncated to a finite number of orders, typically $2N+1 \times 2N+1$ terms for [convergence](/page/Convergence).[](https://opg.optica.org/josaa/abstract.cfm?URI=josaa-12-5-1068)[](https://empossible.net/wp-content/uploads/2019/08/Lecture-7a-RCWA-Formulation.pdf) The [relative permittivity](/page/Relative_permittivity) $\epsilon_r(x, y)$ within each layer, which encodes the periodic material inhomogeneity, is similarly expanded in a double [Fourier series](/page/Fourier_series):

\epsilon_r(x, y) = \sum_{m=-\infty}^{\infty} \sum_{n=-\infty}^{\infty} \epsilon_{mn} \exp\left[i (k_{x,m} x + k_{y,n} y)\right],

where the coefficients $\epsilon_{mn}$ are computed from the spatial profile of the [permittivity](/page/Permittivity), often forming a block-Toeplitz [matrix](/page/Matrix) in the discrete formulation. The [state vector](/page/State_vector) $\Psi(z)$ collects the [Fourier](/page/Fourier) coefficients of all tangential [field](/page/Field) components across the [diffraction](/page/Diffraction) orders: $\Psi(z) = [\{E_{x,mn}(z)\}, \{E_{y,mn}(z)\}, \{H_{x,mn}(z)\}, \{H_{y,mn}(z)\}]^T$, resulting in a [vector](/page/Vector) of [dimension](/page/Dimension) $4(2N+1)^2$. This [vector](/page/Vector) fully describes the transverse [field](/page/Field) state at any z within the layer.[](https://opg.optica.org/josaa/abstract.cfm?URI=josaa-12-5-1068)[](https://empossible.net/wp-content/uploads/2019/08/Lecture-7a-RCWA-Formulation.pdf) Substituting these expansions into [Maxwell's](/page/Maxwell's) curl equations for stratified media yields a system of first-order ordinary differential equations for the coefficients in $\Psi(z)$. In matrix form, this becomes $\frac{d}{dz} \Psi(z) = i K \Psi(z)$, where $K$ is the dynamical matrix composed of blocks representing free-space propagation (diagonal terms involving $\kappa_{mn} = \sqrt{k_0^2 \epsilon \mu - k_{x,m}^2 - k_{y,n}^2}$, with $k_0 = 2\pi / \lambda$) and off-diagonal coupling terms derived from the Fourier coefficients of $\epsilon_r$. Solving this eigenvalue problem, $K \mathbf{v}_l = \gamma_l \mathbf{v}_l$, provides the propagation constants $\gamma_l$ and eigenvectors (modal vectors) $\mathbf{v}_l$, allowing the field in the layer to be reconstructed as a superposition $\Psi(z) = \sum_l a_l \mathbf{v}_l \exp(i \gamma_l z)$.[](https://opg.optica.org/josaa/abstract.cfm?URI=josaa-12-5-1068)[](https://empossible.net/wp-content/uploads/2019/08/Lecture-7a-RCWA-Formulation.pdf) For the one-dimensional case, relevant to planar gratings periodic only in the x-direction ($\Lambda_y \to \infty$, so $k_y$ is fixed and $n=0$), the expansions reduce to single sums over m, and the [state vector](/page/State_vector) simplifies (e.g., to two components per [polarization](/page/Polarization) for [TE](/page/TE) or TM incidence). The dynamical [matrix](/page/Matrix) $K$ then features explicit blocks for propagation, such as

K = \begin{pmatrix} 0 & P \ Q & 0 \end{pmatrix},

where $P$ and $Q$ incorporate diagonal free-space terms (e.g., $\kappa_m / k_0$) and coupling via the [permittivity](/page/Permittivity) Fourier coefficients $\epsilon_m$, forming the core algebraic system that discretizes the problem for efficient [computation](/page/Computation). This formulation, originally developed for E-[polarization](/page/Polarization), extends straightforwardly to general cases while preserving the modal eigenvalue structure.[](https://www.researchgate.net/publication/239664514_Rigorous_coupled-wave_analysis_of_planar-grating_diffraction) ### Fourier factorization technique In rigorous coupled-wave analysis (RCWA) of discontinuous periodic structures, the standard [Fourier](/page/Fourier) expansion of the [inverse](/page/Inverse) permittivity $1/\varepsilon$ exhibits the [Gibbs phenomenon](/page/Gibbs_phenomenon) at [dielectric](/page/Dielectric) interfaces, resulting in slow convergence rates, especially when formulating Ampere's law for TM-polarized waves.[](https://opg.optica.org/josaa/fulltext.cfm?uri=josaa-13-9-1870) To mitigate this, Li developed [factorization](/page/Factorization) rules in 1996 for 1D periodic structures, which ensure accurate handling of products involving discontinuous functions by distinguishing between Laurent's rule for non-concurrent discontinuities and the [inverse](/page/Inverse) rule for cases where the product remains continuous despite jumps in $\varepsilon(x)$.[](https://opg.optica.org/josaa/fulltext.cfm?uri=josaa-13-9-1870) For a [binary](/page/Binary) [dielectric](/page/Dielectric) profile, $\varepsilon(x)$ is factorized into a reference value $\varepsilon_0$ (typically the average [permittivity](/page/Permittivity)) plus a discontinuous correction $\varepsilon(x) - \varepsilon_0$, with the expansion of $1/\varepsilon$ employing forward or backward differences to approximate stair-like interfaces and maintain [uniform convergence](/page/Uniform_convergence).[](https://opg.optica.org/josaa/fulltext.cfm?uri=josaa-13-9-1870) These rules dictate that [Fourier](/page/Fourier) coefficients of $1/\varepsilon$ use one-sided values at discontinuities—specifically, the [permittivity](/page/Permittivity) from the side corresponding to the direction of field differentiation—to avoid artificial oscillations. The resulting Fourier series for the inverse permittivity is

\frac{1}{\varepsilon(x)} \approx \sum_m c_m \exp(i k_m x),

where the coefficients $c_m$ (for $m \neq 0$) are computed via the inverse [discrete Fourier transform](/page/Discrete_Fourier_transform) of the factored $1/\varepsilon$, incorporating the difference in reciprocal [permittivities](/page/Permittivity) scaled by the profile's [geometry](/page/Geometry), such as $\frac{1/\varepsilon_1 - 1/\varepsilon_2}{i 2\pi m} (1 - e^{-i 2\pi m f})$ for a rectangular [grating](/page/Grating) with fill factor $f$, $\varepsilon_1$, and $\varepsilon_2$.[](https://opg.optica.org/josaa/fulltext.cfm?uri=josaa-13-9-1870) The zeroth-order coefficient is the spatial average $c_0 = f/\varepsilon_1 + (1-f)/\varepsilon_2$. The fast Fourier factorization (FFF) method extends these rules for efficient numerical implementation, approximating discontinuous $\varepsilon$ profiles with smoothed transitions (e.g., via [convolution](/page/Convolution) with a narrow [kernel](/page/Kernel)) or deriving exact analytical expressions for rectangular cases, which allows rapid computation without full matrix inversion.[](https://www.sciencedirect.com/science/article/abs/pii/S0030401801013098) This approach is particularly effective for slanted or lamellar gratings, reducing computational cost while preserving accuracy. For two-dimensional (2D) structures, Li's rules are generalized by decomposing $\varepsilon(x,y)$ into separable $x$- and $y$-dependent factors and applying the 1D rules independently to each, forming block-diagonal matrices for the expansions.[](https://iopscience.iop.org/article/10.1088/1464-4258/5/4/307) However, crossed gratings present challenges due to concurrent discontinuities in both directions, requiring careful [polarization](/page/Polarization) basis selection to avoid non-uniform convergence.[](https://iopscience.iop.org/article/10.1088/1464-4258/5/4/307) Overall, these techniques transform the [convergence](/page/Convergence) behavior of RCWA from algebraic $O(1/N)$ (with truncation order $N$) to near-exponential for profiles amenable to proper [factorization](/page/Factorization), enabling reliable simulations of [complex](/page/Complex) stratified media with fewer harmonics.[](https://opg.optica.org/josaa/fulltext.cfm?uri=josaa-13-9-1870) ## Layered Structure Analysis ### Coupled-wave equations In rigorous coupled-wave analysis (RCWA), the propagation of electromagnetic waves within each stratified layer of a periodic structure is governed by a set of first-order differential equations derived from [Maxwell's equations](/page/Maxwell's_equations). The tangential components of the electric and [magnetic fields](/page/The_Magnetic_Fields) at a fixed z-position are expanded in a [Fourier series](/page/Fourier_series) using the Floquet-Bloch basis to account for the lateral periodicity:

\mathbf{E}\parallel(x, y, z) = \sum{g} \mathbf{E}{g}(z) \exp(i \mathbf{k}{\parallel, g} \cdot \mathbf{r}_\parallel),

\mathbf{H}\parallel(x, y, z) = \sum{g} \mathbf{H}{g}(z) \exp(i \mathbf{k}{\parallel, g} \cdot \mathbf{r}_\parallel),

where $\mathbf{r}_\parallel = (x, y)$, $\mathbf{k}_{\parallel, g} = \mathbf{k}_{\parallel, 0} + g \mathbf{G}$ with $\mathbf{G}$ the [reciprocal lattice](/page/Reciprocal_lattice) vector, and the sum is truncated to $N$ diffraction orders on each side for numerical [computation](/page/Computation), yielding 2N+1 terms total. Substituting these expansions into [Maxwell's](/page/Maxwell's) curl equations within a layer where the [permittivity](/page/Permittivity) $\varepsilon(x, y)$ is periodic but independent of z (i.e., stratified media) results in a coupled system for the coefficient vectors. The [state vector](/page/State_vector) $\Psi(z) = [\mathbf{E}_x(z), \mathbf{E}_y(z), \mathbf{H}_x(z), \mathbf{H}_y(z)]^T$, each a (2N+1)-dimensional vector of [Fourier](/page/Fourier) amplitudes, satisfies the matrix [differential equation](/page/Differential_equation)

\frac{d}{dz} \Psi(z) = i k_0 M(\varepsilon) \Psi(z),

where $k_0 = \omega / c$ is the free-space [wavenumber](/page/Wavenumber) and $M(\varepsilon)$ is a sparse [4(2N+1)] × [4(2N+1)] block matrix (for 1D periodicity with general polarization) constructed from diagonal matrices of the lateral wavevectors $\mathbf{k}_{\parallel, g}$ and convolution matrices representing the [Fourier](/page/Fourier) coefficients $\varepsilon_{pq}$ of the [permittivity](/page/Permittivity). The off-diagonal elements of these convolution matrices, arising from the non-constant [Fourier](/page/Fourier) components of $\varepsilon$ (i.e., $\varepsilon_{pq}$ for $p \neq q$), introduce the mode coupling that mixes the diffraction orders, enabling rigorous treatment of multiple scattering within the layer.[](https://opg.optica.org/josaa/abstract.cfm?uri=josaa-12-5-1068) Since the layer is homogeneous in z, the matrix $M(\varepsilon)$ is constant, allowing an exact [modal](/page/Modal) solution via eigenvalue decomposition. The eigenvalues $\gamma_l$ (with $l = 1, \dots, 4(2N+1)$) represent the z-directed [propagation](/page/Propagation) constants ([complex](/page/Complex) for evanescent modes), and the corresponding eigenvectors $\mathbf{u}_l$ describe the [modal](/page/Modal) [field](/page/Field) profiles across the Fourier orders. The general solution is then a superposition of forward- and backward-propagating modes:

\Psi(z) = \sum_{l=1}^{4(2N+1)} \left[ a_l^+ \exp(i \gamma_l z) + a_l^- \exp(-i \gamma_l z) \right] \mathbf{u}_l,

where the coefficients $a_l^\pm$ are determined by boundary conditions at layer interfaces. This [modal](/page/Modal) form captures the dispersive nature of wave [propagation](/page/Propagation) in the periodic medium, with [coupling](/page/Coupling) strength dictated by the $\varepsilon_{pq}$ terms that redistribute energy among orders.[](https://opg.optica.org/josaa/abstract.cfm?uri=josaa-12-5-1068) For a one-dimensional grating periodic in x (e.g., $\mathbf{G} = (2\pi / \Lambda, 0)$), the formulation simplifies significantly, particularly for TE polarization where the electric field is $\mathbf{E} = (0, E_y, 0)$ and the relevant tangential fields are $E_y$ and $H_x$. The system reduces to a 2(2N+1) × 2(2N+1) matrix equation for the vectors of $E_y$ and $H_x$ Fourier coefficients, with the matrix blocks incorporating the permittivity [Fourier series](/page/Fourier_series) $\varepsilon(x) = \sum_m \varepsilon_m \exp(i m G x)$. The perpendicular wavevector components for each order are $k_{z,n} = \sqrt{k_0^2 \varepsilon_0 - k_{x,n}^2}$, where $\varepsilon_0$ is the average [permittivity](/page/Permittivity) in the layer, and $k_{x,n} = k_{x,0} + n G$ with $G = 2\pi / \Lambda$. For [oblique](/page/Oblique) incidence, the incident Bloch [momentum](/page/Momentum) $\beta = k_{x,0} = k_0 \sin \theta$ (with $\theta$ the angle from the z-axis) is incorporated directly into the zeroth-order wavevector, shifting all $k_{x,n}$ and ensuring phase-matching with the incident [plane wave](/page/Plane_wave). This TE case exemplifies the coupling mechanism, as the off-diagonal $\varepsilon_m$ (m ≠ 0) terms in the permittivity matrix generate inter-order interactions in the eigenvalue problem for $\gamma_l$.[](https://opg.optica.org/josaa/abstract.cfm?uri=josaa-12-5-1068) ### Interface and boundary conditions In rigorous coupled-wave analysis (RCWA), the continuity of the tangential components of the [electric field](/page/Electric_field) $\mathbf{E}_\parallel$ and [magnetic field](/page/Magnetic_field) $\mathbf{H}_\parallel$ is enforced at each [interface](/page/Interface) between layers to satisfy Maxwell's boundary conditions. This ensures that the electromagnetic fields remain well-defined across discontinuities in material properties, such as [permittivity](/page/Permittivity) $\epsilon$ or permeability $\mu$. The fields are represented using a [state vector](/page/State_vector) $\Psi(z) = [E_x, E_y, H_x, H_y]^T$, where the [continuity](/page/Continuity) condition at an [interface](/page/Interface) located at $z = z_i$ yields $\Psi(z_i^+) = T \Psi(z_i^-)$, with $T$ being the [interface transfer matrix](/page/Transfer_matrix) that relates the field expansions on either side. For layers with identical material properties, $T$ simplifies to the [identity matrix](/page/Identity_matrix); otherwise, it incorporates adjustments for changes in the [Fourier](/page/Fourier) mode basis due to differing periodic [permittivities](/page/Permittivity). The matrix $T$ is constructed from the [Fourier](/page/Fourier) coefficients of the [permittivity](/page/Permittivity) in the adjacent layers to maintain [continuity](/page/Continuity) in the expanded basis.[](https://opg.optica.org/josaa/abstract.cfm?uri=josaa-12-5-1068) In standard RCWA for isotropic media, the formulation decouples for TE and TM polarizations, allowing the use of scattering matrices to propagate forward and backward [waves](/page/Waves) while enforcing tangential [continuity](/page/Continuity). At the top and bottom boundaries of the stratified structure, [radiation](/page/Radiation) conditions are applied to model semi-infinite homogeneous regions, stipulating that only outgoing [waves](/page/Waves) propagate away from the device—no incoming [waves](/page/Waves) from infinity. For a structure of finite thickness, the field expansions in these exterior regions are matched to the incident and reflected (or transmitted) plane [waves](/page/Waves), typically using a reduced set of modes corresponding to the propagating and evanescent fields as needed. This matching is achieved by setting the backward amplitudes to zero in the superstrate (for [reflection](/page/Reflection) problems) or forward amplitudes to zero in the [substrate](/page/Substrate) (for [transmission](/page/Transmission)), directly linking the layer interface solutions to observable [diffraction](/page/Diffraction) efficiencies. In terms of mode amplitudes, the interface conditions translate to relations between forward ($a_j^+$) and backward ($a_j^-$) [waves](/page/Waves) across layers, such as $a_{j+1}^+ = P_j a_j^+ + Q_j a_j^-$, where $P_j$ and $Q_j$ are matrices derived from the continuity enforcement and layer propagation factors. These connect the intra-layer coupled-wave solutions from the previous layer to the next, forming the basis for global field matching without delving into full [scattering](/page/Scattering) matrix [algebra](/page/Algebra).[](https://opg.optica.org/josaa/abstract.cfm?uri=josaa-12-5-1068) ## Numerical Aspects ### Scattering matrix methods In rigorous coupled-wave analysis (RCWA) for stratified periodic structures, the scattering matrix method provides a robust framework for propagating electromagnetic fields through multiple layers by relating the amplitudes of incoming and outgoing waves at interfaces. The scattering matrix $ S $ for a layer connects the outgoing wave amplitudes on both sides to the incoming ones, typically expressed as

\begin{pmatrix} \mathbf{b}^+ \ \mathbf{a}^- \end{pmatrix} = S \begin{pmatrix} \mathbf{a}^+ \ \mathbf{b}^- \end{pmatrix},

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Formulation for stable and efficient implementation of the rigorous ...
Crossref . 2. M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar grating diffraction—E-mode polarization and losses,” J. Opt. Soc. Am.
[8]
[PDF] Formulation of Rigorous Coupled‐Wave Analysis (RCWA)
In RCWA, it is only necessary to construct the cross section of the unit cells. Thickness is conveyed elsewhere. 5. 6. Page 4. 4/ ...
[9]
Stable implementation of the rigorous coupled-wave analysis for ...
A formulation for the implementation of the RCWA for surface-relief dielectric gratings incorporating the developed enhanced transmittance approach is presented ...
[10]
https://opg.optica.org/josaa/abstract.cfm?uri=josaa-13-5-1024
[11]
https://opg.optica.org/josaa/abstract.cfm?uri=josaa-13-9-1870
[12]
https://opg.optica.org/josaa/abstract.cfm?uri=josaa-18-11-2886
[13]
An Adaptive High-Order Rigorous Coupled Wave Analysis Method
Sep 14, 2022 · Semianalytical methods, such as rigorous coupled wave analysis, have been pivotal in the numerical analysis of photonic structures.
[14]
Global rigorous coupled wave analysis for design of multilayer ...
Nov 13, 2023 · In this work, a novel Global NV-ETM RCWA method is proposed to accelerate the optimization of the periodic stepped radar absorbing structure.
[15]
[PDF] FDTD Modeling of Periodic Structures: A Review - arXiv
FDTD Modeling of Periodic Structures: A Review ... Each mode independently satisfies the Floquet condition (equation (1)). Floquet's theorem may likewise describe ...
[16]
Periodic Structures - Bozzi - Wiley Online Library
Apr 15, 2005 · The basic theory of the propagation of electromagnetic fields in periodic structures is reported, and includes Floquet's theorem and Brillouin ...
[17]
[PDF] Analysis and Applications of Optical Diffraction by Gratings
The analysis of diffraction by gratings has a long and interesting history. Since 1930, there have been over 400 scientific papers on the subject of grating ...Missing: theorem | Show results with:theorem
[18]
https://onlinelibrary.wiley.com/doi/abs/10.1002/0471654507.eme306
[19]
http://users.ntua.gr/eglytsis/IO/Gaylord_1985.pdf
[20]
Rigorous coupled-wave analysis of planar-grating diffraction
Aug 9, 2025 · M. G. Moharam · T. K. Gaylord · R. Magnusson. Four alternative criteria for the occurence of Bragg regime diffraction by phase gratings are ...Missing: definition | Show results with:definition
[21]
Use of Fourier series in the analysis of discontinuous periodic ...
Use of Fourier series in the analysis of discontinuous periodic structures · 1. From the basic Maxwell equations, derive the coupled first-order equations or the ...
[22]
Fast Fourier factorization method applied to modal analysis of ...
We describe S4, a free implementation of the Fourier modal method (FMM), which has also been commonly referred to as rigorous coupled wave analysis (RCWA), for ...
[23]
Fourier modal method for crossed anisotropic gratings with arbitrary ...
Apr 30, 2003 · A Fourier modal method for analysing crossed anisotropic gratings is presented without any restrictions on the permittivity and permeability tensors.
[24]
https://opg.optica.org/josaa/abstract.cfm?uri=josaa-24-8-2313
[25]
[PDF] RCWA Extras | EMPossible
The enhanced transmittance matrix (ETM) method was the first technique applied to RCWA that fixed the instability. ETM is much faster than scattering ...<|control11|><|separator|>
[26]
Analysis of the Rigorous Coupled Wave Approach for p-polarized ...
We study the convergence properties of the two-dimensional Rigorous Coupled Wave Approach (RCWA) for p -polarized monochromatic incident light.
[27]
Efficient solution of Maxwell's equations for geometries with ...
Oct 15, 2012 · Thus, for M layers the total computational cost is O ( N 3 M ) . Similarly the memory requirements can be estimated to ...
[28]
[PDF] Validation and characterization of algorithms and software for ...
RCWA optimized. FDTD optimized. FDTD optimized. (c). (e). 93.2% (RCWA). 93.8% (FDTD). 96.8% (RCWA). 95.0% (FDTD). 83.0% (RCWA). 84.3% (FDTD). 93.7% (RCWA). 92.5 ...
[29]
Rigorous Coupled-Wave Analysis of Multilayered Grating Structures
Aug 5, 2025 · When modeling one-dimensional gratings using RCWA there are significant differences between TE and TM polarized light (see Figure 8). It has ...
[30]
Adaptive perfectly matched layer for Wood's anomalies in diffraction ...
Dec 4, 2012 · In this paper, we have proposed an adaptive PML that can treat rigorously Wood's anomalies in numerical analysis of diffraction gratings. It is ...
[31]
[PDF] Gratings: Theory and Numeric Applications | HAL
Feb 6, 2013 · Periodic structures in optics can serve for the photons in the same ... periodic functions and must themselves be Floquet- or quasi-periodic.
[32]
Volume Holographic Grating - Ansys Optics
In this article, we demonstrate how to simulate Volume Holographic Gratings (VHG) with the RCWA solver.Overview · Run and Results · Updating the Model with Your...
[33]
Volume holographic phase-retardation elements
We introduce phase variations of volume gratings, using the results of rigorous coupled-wave analysis (RCWA). Phase variations near the cutoff period are also ...
[34]
In situ spectroscopic ellipsometry and rigorous coupled wave ...
Sep 30, 2022 · Rigorous coupled wave analysis (RCWA) is conducted on in situ spectroscopic ellipsometry data to understand profile evolution during film ...
[35]
Accurate near-field calculation in the rigorous coupled-wave ...
Oct 29, 2015 · method is the rigorous coupled-wave analysis (RCWA) [14–. 16]. It ... [15] Moharam M G and Gaylord T K 1986 J. Opt. Soc. Am. A 3 · 1780–7.
[36]
Semiconductor - n&k Technology Inc. Metrology System ...
Critical dimensions (CD) of gratings of resist lines were obtained from RCWA modeling of optical spectra. An i-line resist with lines of larger spacing and a UV ...
[37]
On the complexity of aperiodic Fourier modal methods for finite ...
Aperiodicity is achieved by using perfectly matched layers (PMLs) [39] on the vertical boundaries in order to annihilate the periodic boundary condition.
[38]
https://opg.optica.org/oe/abstract.cfm?uri=oe-26-26-34622
[39]
[PDF] Scattering from finite structures : an extended Fourier modal method
Jan 1, 2011 · The choice of the S-matrix approach over the enhanced transmittance matrix approach ... Hugonin and Philippe Lalanne. Perfectly matched ...
[40]
Electromagnetic beam diffraction by a finite strip grating
We study the diffraction of an electromagnetic beam by a finite strip grating. The fields are expressed in terms of their angular spectra.
[41]
(PDF) Perfectly matched layers as nonlinear coordinate transforms
Aug 9, 2025 · Perfectly matched layers as nonlinear coordinate transforms: a ... RETICOLO implements the rigorous coupled wave analysis (RCWA) for 1D ...
[42]
Hybrid finite-element/rigorous coupled wave analysis for scattering ...
An adaptive PML approach [30] or other absorbing boundary conditions (ABCs) can be applied to address this issue. Fig. 12. -- PML distance (tair) effect on ...
[43]
Aperiodic Fourier modal method in contrast-field formulation for ...
The Fourier modal method (FMM), also referred to as rigorous coupled-wave analysis (RCWA), has a well established position in the field of rigorous diffraction ...
[44]
Efficient wave optics modeling of nanowire solar cells using rigorous ...
We investigate the accuracy of rigorous coupled-wave analysis (RCWA) for near-field computations within cylindrical GaAs nanowire solar cells and discover ...
[45]
Trench geometry and resist profiles from modeling of polarized ...
Profiles of trenches in silicon were calculated by fitting measured polarized reflectance spectra with theoretically calculated RCWA (rigorous coupled wave ...
[46]
Physics‐Informed Machine Learning for Inverse Design of Optical ...
Oct 11, 2023 · To expedite the computation for the inverse design of metasurfaces, a physics-informed deep learning (DL) framework is used. A tandem DL ...
[47]
Probabilistic inverse design of metasurfaces using mixture density ...
Mixture density neural networks (MDN) are used to design metasurfaces by predicting probability distributions of optimal structures, avoiding local minima.
[48]
TORCWA: GPU-accelerated Fourier modal method and gradient ...
First, TORCWA is a GPU-accelerated RCWA simulation. It performs simulation significantly faster with almost similar accuracy than CPU operation. In addition, it ...Missing: unfolded quasi-
[49]
Materials in rcwa
Lossy materials are represented with a positive imaginary refractive index (i.e. 2 + 0.1j . Materials with gain are represented by a negative imaginary ...Missing: challenges active loss<|control11|><|separator|>
[50]
Hybrid Model for Spectral Analysis of Multilayered Structures with ...
Apr 25, 2025 · Optimization of numerical algorithms showed that the combination of RCWA and FDTD methods in a hybrid format provides a balance between accuracy ...Missing: extreme anisotropy active
[51]
Topological photonics in nanoscaled systems with far-field radiation ...
Aug 25, 2025 · Topology is a powerful framework for controlling and manipulating light, minimizing detrimental perturbations on the photonic properties.