RCWA(input::RCWAInput) -> RCWAOutput

Run a full RCWA simulation and return an RCWAOutput.

Arguments

• input::RCWAInput: Simulation parameters including the incoming wave, layer stack, and number of harmonics.

_get_block(FM, Nx, Ny, Δq, P)

Build the Toeplitz block for Y-harmonic difference Δq from the FFT FM of a pattern of native size (Nx, Ny).

Each entry is the exact Fourier coefficient of the piecewise-constant function defined by the pixel grid. This is the raw DFT coefficient multiplied by the rectangular-pixel transfer function h(k, N), which makes the result independent of the input resolution.

Stabilize eigenvectors of nearly-degenerate eigenvalue clusters.

When eigen() encounters degenerate or nearly-degenerate eigenvalues, it returns arbitrary eigenvectors within each degenerate subspace. This causes numerical issues downstream (ill-conditioned W⁻¹ in S-matrix boundary matching).

This function groups nearly-degenerate eigenvalues into clusters and, within each cluster, applies orthogonal Procrustes alignment to rotate the eigenvectors so they best align with the identity matrix (the free-space eigenvector basis).

function compute_bottom_scattering_matrix(free_space_modes, bottom_modes)

Compute the scattering matrix between bottom (transmissive) homogeneous medium and an empty medium. Equivalently, it may be viewed as a general scattering matrix in the special case where the thickness of the RCWA layer is zero.

Arguments

• free_space_modes::LayerModes: Modes associated to free space.
• bottom_modes::LayerModes: Modes associated to the bottom homogeneous medium.

function compute_global_scattering_matrix(
    modes, free_space_modes, wavelength, thicknesses
)

Compute the global scattering matrix associated to a stack of RCWA layers.

Arguments

• modes::AbstractVector{LayerModes}: Modes associated to each layer, including the top and bottom half-spaces as the first and last elements.
• free_space_modes::LayerModes: Modes associated to free space.
• wavelength::Real: Wavelength of the incoming light.
• thicknesses::AbstractVector{<:Real}: Thicknesses of each layer. The first and last elements must be Inf.

Compute eigenmodes analytically for a homogeneous layer.

In a homogeneous medium, all spatial harmonics decouple, so the eigenvectors are simply the identity matrix. The eigenvalues are determined by the dispersion relation: λn = sqrt(kxn² + ky_n² - εμ).

function compute_symmetric_scattering_matrix(
    layer_modes, free_space_modes, wavelength, layer_thickness
)

Compute the scattering matrix associated to an RCWA layer wedged between two empty media.

Arguments

• layer_modes::LayerModes: Modes associated to the RCWA layer.
• free_space_modes::LayerModes: Modes associated to free space.
• wavelength::Real: Wavelength of the incoming light.
• layer_thickness::Real: Thickness of the RCWA layer.

function compute_top_scattering_matrix(top_modes, free_space_modes)

Compute the scattering matrix between top (reflective) homogeneous medium and an empty medium. Equivalently, it may be viewed as a general scattering matrix in the special case where the thickness of the RCWA layer is zero.

Arguments

• top_modes::LayerModes: Modes associated to the top homogeneous medium.
• free_space_modes::LayerModes: Modes associated to free space.

convolve(layer, number_of_harmonics)

Apply a convolution to a given layer. Concretely, this amounts to taking the 2D Fourier transform, and subsequently reordering the elements in a Toeplitz block matrix form.

Arguments

• layer::Layer: The layer you want to convolve.
• number_of_harmonics::Tuple{<:Integer, <:Integer}: Number of harmonics in X and Y, respectively.

diffraction_efficiencies(output; polarization = :x) -> (DE_ref, DE_trn)

Power per diffraction order normalized to the incident power, returned as two P x Q real matrices. In the lossless case, sum(DE_ref) + sum(DE_trn) ≈ 1.

The power is computed from the z-component of the Poynting vector, which for a plane wave with transverse fields (Ex, Ey) and wave vector (kx, ky, kz) is:

Sz = Re(1/(kz* μ*)) x [(kz²+kx²)|Ex|² + 2 kx ky Re(Ex Ey*) + (ky²+kz²)|Ey|²]

function prepare_wave_vectors(
    source,
    top_medium,
    bottom_medium,
    period,
    number_of_harmonics
)

Prepare wave vectors in a format convenient for subsequent RCWA simulation steps.

Arguments

• source::Source: Incoming wave as described in a vacuum.
• top_medium::Layer: Homogeneous reflection region.
• bottom_medium::Layer: Homogeneous transmission region.
• period::Tuple{<:Real, <:Real}: Periodicity, or equivalently, size of your unit cells.
• number_of_harmonics::Tuple{<:Integer, <:Integer}: Number of harmonics to be used in your RCWA simulation.

reflection_coefficients(output; polarization = :x) -> (r_x, r_y)

Complex reflected field amplitudes per diffraction order, returned as two P x Q matrices (one per polarization component).

function star_product(A, B)

Redheffer star product, which may be regarded as the 'plumbing' of a coupled pair of scattering matrices.

Arguments

• A::AbstractScatteringMatrix
• B::AbstractScatteringMatrix

transmission_coefficients(output; polarization = :x) -> (t_x, t_y)

Complex transmitted field amplitudes per diffraction order, returned as two P x Q matrices (one per polarization component).

struct BTTB

Square block Toeplitz matrix of square Toeplitz blocks.

struct ConvolvedLayer

Layer in which the electric permittivity ε and magnetic permeability μ have been prepared for use in RCWA computation. Specifically, ε and μ have been Fourier transformed, and its Fourier coefficient have been rearranged in a convolution matrix format which is convenient for efficiently performing RCWA.

Properties

• conv_eps::BTTB: Convolved electric permittivity.
• conv_mu::BTTB: Convolved magnetic permeability.

struct Layer

Layer described in terms of complex electric permittivity and complex magnetic permeability.

Note that we use the negative sign convention where a wave propagating in the +z-direction is written as exp(-ikz). As such, the imaginary part of the complex permittivity is negative.

The input matrices use Cartesian convention: columns correspond to the X direction, rows to Y, with row 1 being the +Y edge. Internally, the matrices are stored with dim1 = X and dim2 = Y.

Properties

• eps::Matrix{ComplexF64}: Complex relative electric permittivity per pixel.
• mu::Matrix{ComplexF64}: Complex relative magnetic permeability per pixel.

function Layer(nk)

Describe a layer in terms of its complex refractive index.

This description may only be used for non-magnetic materials.

Note that we use the negative sign convention where a wave propagating in the +z-direction is written as exp(-ikz). As such, the imaginary part of the complex refractive index is negative.

Arguments

• nk::AbstractMatrix{<:Number}: Complex refractive index per pixel.

struct PreparedWaveVectors

Wave vectors presented in a shape convenient for subsequent simulation steps.

Properties

• waveVectorsX::Diagonal{ComplexF64}: Diagonal matrix of wave vectors in X.
• waveVectorsY::Diagonal{ComplexF64}: Diagonal matrix of wave vectors in Y.
• waveVectorsZReflection::Diagonal{ComplexF64}: Diagonal matrix of wave vectors in reflection region.
• waveVectorsZTransmission::Diagonal{ComplexF64}: Diagonal matrix of wave vectors in transmission region.

struct RCWAInput

All parameters needed to run an RCWA simulation.

Properties

• source::Source: Incoming wave specification (angles, wavelength).
• stack::Stack: Layer stack including top and bottom half-spaces.
• order::Tuple{Int64, Int64}: Maximum harmonic order (N, M). The simulation includes orders -N..N and -M..M, for a total of (2N+1)(2M+1) harmonics.

struct RCWAOutput

Result of an RCWA simulation. Holds the global scattering matrix together with all metadata needed to extract reflection/transmission coefficients and diffraction efficiencies.

Use reflection_coefficients, transmission_coefficients, and diffraction_efficiencies to query the result.

Properties

• input::RCWAInput: Input that was used to produce this output.
• waveVectors::PreparedWaveVectors: Wave vectors for all diffraction orders.
• modes::Vector{LayerModes}: Eigenmodes of each layer (including top and bottom half-spaces as the first and last elements).
• scatteringMatrix::ScatteringMatrix: Global scattering matrix of the stack.

struct ScatteringMatrix

Scattering matrix (S-matrix) describing the reflection and transmission of a periodic layer wedged in between two homogeneous media.

Properties

• S11::Matrix{ComplexF64}: Reflection in the top medium.
• S12::Matrix{ComplexF64}: Transmission from the bottom to the top medium.
• S21::Matrix{ComplexF64}: Transmission from the top to the bottom medium.
• S22::Matrix{ComplexF64}: Reflection in the bottom medium.

struct Source

Describes incoming wave in a vacuum.

Properties

• azimuthalAngle::Float64: Also called ϕ. Some people call it θ.
• elevationAngle::Float64: Also called θ. Some people call it ϕ.
• wavelength::Float64: Wavelength in whatever units you like so long as you're being consistent.

struct Stack

A stack of layers with associated thicknesses. The first and last layers are semi-infinite half-spaces (thickness = Inf) and must be homogeneous.

Properties

• layers::Vector{<:AbstractLayer}: All layers, including the top and bottom half-spaces as the first and last elements.
• thicknesses::Vector{Float64}: Thickness of each layer. The first and last elements must be Inf.
• period::Tuple{Float64, Float64}: Unit cell size in (X, Y).

struct SymmetricScatteringMatrix

Scattering matrix of an RCWA layer wedged between identical homogeneous media on both sides. By symmetry, the scattering matrix may be expressed as a block matrix S = [R T; T R], where R and T are reflection and transmission matrices, respectively.

Properties

• S11::Matrix{ComplexF64}: Reflection in either homogeneous medium.
• S12::Matrix{ComplexF64}: Transmission from either homogeneous medium into the periodic layer.