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Finite-difference frequency-domain method

The finite-difference frequency-domain (FDFD) method is a numerical technique for solving time-harmonic Maxwell's equations in the frequency domain, primarily applied to electromagnetics problems such as wave propagation, scattering, and resonance.^[1] It discretizes the curl equations—∇ × E = -iωμH and ∇ × H = iωεE + J—using central finite differences on a staggered Yee grid, where electric and magnetic field components are offset to achieve second-order accuracy and maintain charge conservation.^[1] This approach transforms the partial differential equations into a sparse linear system Ax = b, solved iteratively (e.g., via BiCGSTAB or GMRES) for steady-state fields at a specified frequency ω, avoiding the temporal evolution required in time-domain methods.^[1]^[2] FDFD implementations typically incorporate perfectly matched layers (PML), such as uniaxial or convolutional PML, as absorbing boundary conditions to truncate computational domains while minimizing reflections from artificial boundaries.^[2] The method handles material inhomogeneities, dispersion, and anisotropy by incorporating permittivity tensors ε and permeability μ directly into the discretization, with subpixel smoothing techniques applied to approximate curved interfaces and reduce staircasing errors.^[1] For eigenmode problems, such as waveguide modes, FDFD formulates a generalized eigenvalue equation using transverse field components, enabling analysis of propagation constants and field profiles.^[3] Key applications of FDFD span photonics and electromagnetics, including the simulation of guided modes in optical waveguides, slab structures, and surface plasmon polaritons, as well as scattering from metallic nanostructures like bowtie antennas and 90° bends in plasmonic waveguides.^[2]^[1] It is also employed in electromagnetic induction tomography for imaging conductive subsurface features and in analyzing anisotropic devices, such as proton-exchanged LiNbO₃ waveguides, where full-vector modes reveal asymmetric field distributions due to arbitrary permittivity tensors.^[4]^[3] These capabilities make FDFD versatile for designing photonic band-gap structures, transmission lines, and active devices requiring multi-frequency parameter sweeps.^[2] Compared to the finite-difference time-domain (FDTD) method, FDFD provides direct access to frequency-specific steady-state solutions without Courant-Friedrichs-Lewy (CFL) time-step restrictions, offering efficiency for narrowband analyses or highly resonant systems where FDTD simulations may require excessively long run times.^[1] Relative to finite-element methods (FEM), FDFD benefits from a structured, banded system matrix that facilitates faster iterative solvers in large 3D problems, though it may require more grid points for irregular geometries unless enhanced with smoothing.^[1] Overall, FDFD's simplicity, scalability to millions of unknowns, and compatibility with parallel computing position it as a powerful tool for rigorous electromagnetic modeling.^[2]^[4]

Fundamentals

Definition and Principles

The finite-difference frequency-domain (FDFD) method is a numerical technique that approximates differential equations in the frequency domain using finite differences, primarily applied to linear, time-harmonic problems in electromagnetics. It discretizes Maxwell's equations in their phasor form to compute steady-state electromagnetic fields at specific frequencies, making it suitable for analyzing wave propagation, scattering, and resonance in structures such as antennas, photonic devices, and metamaterials.^[5]^[6] At its core, the FDFD method solves the phasor representations of wave equations—derived from time-harmonic assumptions where fields vary as e^{j\omega t}—directly at discrete frequencies without requiring time evolution. This contrasts with time-domain methods like the finite-difference time-domain (FDTD) approach, which simulate transient behaviors through iterative time-stepping and often necessitate Fourier transforms to obtain frequency-specific results; FDFD bypasses these steps, enabling straightforward computation of responses at targeted frequencies.^[5]^[2] Key advantages of FDFD include enhanced computational efficiency for narrowband analyses or single-frequency evaluations, where simulating broad spectra would be inefficient in time-domain methods, and its inherent stability for problems involving high-Q resonators or periodic steady-state behaviors. It excels in scenarios with dispersive media or where parallel computing can accelerate matrix solutions, though it may require larger memory for 3D problems compared to time-domain alternatives.^[5]^[6] The basic workflow begins with grid-based spatial discretization, typically using a Yee-like staggered grid to position electric and magnetic field components, followed by the assembly of a sparse matrix system representing the discretized equations. This linear system is then solved for the field values using direct or iterative solvers, yielding the frequency-domain solutions for post-processing such as power flow or far-field patterns.^[5]^[2]

Historical Background

The finite-difference frequency-domain (FDFD) method originated in the late 1970s and early 1980s as an extension of finite-difference techniques originally developed for solving partial differential equations in heat transfer and fluid dynamics to the field of computational electromagnetics. These early efforts focused on solving Maxwell's equations in the frequency domain for resonant structures, providing an alternative to emerging time-domain approaches by directly addressing steady-state problems at specific frequencies. The method's foundations were laid through discrete approximations of curl equations on structured grids, enabling efficient eigenvalue computations for cavities and waveguides.^[7] A seminal contribution came from T. Weiland, who in 1982 introduced a numerical finite-difference approach for solving Maxwell's equations in static, resonant, and transient contexts within cylindrically symmetric cavities, marking one of the first comprehensive formulations for frequency-domain electromagnetic simulations.^[8] Weiland extended this to three-dimensional eigenvalue problems in 1985, demonstrating unique numerical solutions for arbitrary-shaped cavities and establishing the method's viability for complex geometries in accelerator physics and microwave engineering. Early applications in the 1980s targeted waveguide problems, with influences from groups advancing frequency-domain extensions alongside time-domain methods, such as those led by A. Taflove in related finite-difference electromagnetics. In 1987, A. Christ and H. L. Hartnagel applied the FDFD method to three-dimensional shielded microwave structures, enabling analysis of device embedding and circuit integration, which broadened its use beyond cavities to practical microwave components.^[9] The 1990s saw advancements in two- and three-dimensional formulations for microwave circuits, with refinements in grid discretization and solver efficiency for scattering and propagation problems. By the 2000s, the method evolved from primarily scalar formulations to fully vectorial ones, incorporating full Maxwell curl equations for anisotropic media and complex boundaries. Integration with mode-matching techniques further enhanced its capability for hybrid waveguide discontinuities, as demonstrated in analyses of substrate-integrated waveguides.^[10] This evolution culminated in the incorporation of FDFD principles into commercial software by the early 2000s, notably through the Finite Integration Technique (FIT) in tools like CST Microwave Studio, which stemmed from Weiland's foundational work and facilitated widespread adoption in industry for electromagnetic design.

Mathematical Foundation

Governing Equations

The finite-difference frequency-domain (FDFD) method numerically solves the time-harmonic forms of Maxwell's equations for electromagnetic problems and the Helmholtz equation for acoustic problems, assuming fields vary as e^{i\omega t} where \omega is the angular frequency.^[1] In the electromagnetic case, the governing equations are the curl forms of Maxwell's equations in phasor notation:

\nabla \times \mathbf{E} = -i \omega \mu \mathbf{H}

\nabla \times \mathbf{H} = i \omega \epsilon \mathbf{E} + \mathbf{J}

Here, \mathbf{E} and \mathbf{H} are the electric and magnetic field phasors, \mu is the magnetic permeability, \epsilon is the electric permittivity, and \mathbf{J} is the impressed current density source term.^[1] Material properties in lossy media are incorporated via complex-valued permittivity \epsilon = \epsilon_r - i \frac{\sigma}{\omega} (where \epsilon_r is the real relative permittivity and \sigma is conductivity) and permeability \mu = \mu_r - i \frac{\sigma_m}{\omega} (with \mu_r the real relative permeability and \sigma_m magnetic loss), which account for absorption and dispersion effects.^[11] The source term \mathbf{J} represents impressed currents or fields and appears as a right-hand-side vector in the discretized equation system.^[1] For the acoustic case, the governing equation is the scalar time-harmonic wave equation, or Helmholtz equation:

\nabla^2 p + \left( \frac{\omega}{c} \right)^2 p = 0

where p is the acoustic pressure phasor and c is the speed of sound.^[12] In lossy acoustic media, the wavenumber term \left( \frac{\omega}{c} \right)^2 becomes complex to include attenuation, often via a complex bulk modulus K or density \rho such that c^2 = K / \rho.^[12] Sources, such as monopolar pressure injections, are included as a right-hand-side term S in the equation, e.g., \nabla^2 p + k^2 p = S.^[12]

Finite-Difference Discretization

In the finite-difference frequency-domain (FDFD) method, the spatial domain is discretized using a Yee-like staggered grid, where electric field components are positioned at the edges of the grid cells and magnetic field components at the faces, ensuring accurate representation of vector fields in Cartesian coordinates. This arrangement, originally developed for time-domain simulations but adapted for frequency-domain problems, facilitates the approximation of curl operators in Maxwell's equations by naturally incorporating the geometric staggering of fields. Uniform Cartesian meshes are commonly employed for simplicity, though non-uniform grids can be used to refine resolution in regions of interest, such as near material interfaces, while maintaining overall computational efficiency.^[5]^[2] The discretization of differential operators relies on central finite-difference schemes, which approximate spatial derivatives with second-order accuracy. For a scalar field \phi, the second derivative along the x-direction is given by

\frac{\partial^2 \phi}{\partial x^2} \approx \frac{\phi_{i+1} - 2\phi_i + \phi_{i-1}}{\Delta x^2},

where \Delta x is the grid spacing and indices denote grid points. In two- and three-dimensional cases, mixed partial derivatives, such as \partial^2 / \partial x \partial y, are handled similarly using central differences on the staggered components, for example, by averaging appropriately offset field values to preserve the structure of the vector wave equation. This approach extends to the full curl equations, where first derivatives are approximated as (f_{i+1/2} - f_{i-1/2}) / \Delta x, leveraging the half-integer staggering of E and H fields.^[2]^[13]^[5] For the complete vector formulation of Maxwell's equations in the frequency domain, the discretized fields are assembled into a global vector \mathbf{\phi} containing all E and H components across the grid, leading to a linear system A \mathbf{\phi} = \mathbf{b}. Here, A is a large, sparse matrix representing the discretized differential operator, typically with 7 to 13 non-zero entries per row in 2D and 3D, respectively, derived from edge- or node-based placements that align with the Yee grid. The right-hand side \mathbf{b} incorporates sources, and the matrix A combines the curl operators with material properties \mu and \epsilon, and frequency-dependent terms -i \omega \mu and i \omega \epsilon. This formulation arises directly from substituting the finite-difference approximations into the frequency-domain curl equations, resulting in a complex, non-Hermitian system amenable to iterative solvers.^[2]^[5]^[13] The central finite-difference scheme employed in FDFD achieves second-order accuracy, with truncation errors scaling as O(\Delta x^2) for uniform grids, as the leading error terms in the Taylor expansion of the difference operators are quadratic in the grid size. Error analysis confirms this convergence rate, demonstrated through numerical convergence studies where the solution error decreases quadratically with refinement of \Delta x, provided the grid resolves the wavelength adequately (typically \Delta x < \lambda / 10). Higher-order schemes exist but increase matrix bandwidth and complexity, making second-order central differences the standard for balancing accuracy and efficiency in most FDFD implementations.^[5]^[2]^[13]

Numerical Implementation

Solution Methods

The finite-difference frequency-domain (FDFD) method discretizes Maxwell's curl equations on a Yee grid, leading to a large sparse linear system that must be solved for the electric or magnetic fields at a given frequency. The discretization leads to a large sparse linear system A x = b, where A is assembled from the finite-difference approximations to the curl operators scaled by material properties (inverse permeability and permittivity), x is the vector of field components, and b incorporates the excitation sources. The matrix A is block-structured and sparse, with off-diagonal blocks representing derivative terms and diagonal blocks including material responses.^[5]^[14] For smaller problems, such as two-dimensional (2D) simulations, direct solvers like LU decomposition or sparse Cholesky factorization can be employed to obtain exact solutions of \mathbf{A} \mathbf{x} = \mathbf{b}. These methods factorize the matrix into lower and upper triangular components (or a single lower triangular for Cholesky in symmetric positive-definite cases), enabling efficient forward and back substitution. However, their cubic O(N^3) time complexity, where N is the number of grid points, renders them impractical for large three-dimensional (3D) problems, where N can exceed millions due to fine grid resolutions needed for wavelength-scale features.^[5]^[15] Iterative solvers are preferred for large-scale 3D FDFD systems, as they approximate the solution through successive matrix-vector multiplications without full factorization. Common choices include the conjugate gradient (CG) method for symmetric systems, the generalized minimal residual (GMRES) method for nonsymmetric cases, and the biconjugate gradient stabilized (BiCGSTAB) method, which offers robust convergence for indefinite matrices typical in lossy or dispersive media. To accelerate convergence and mitigate ill-conditioning from the discretized curls, preconditioners such as incomplete LU (ILU) factorization or multigrid methods are applied, reducing the effective condition number and iteration count from hundreds to tens. For instance, multigrid preconditioning leverages coarse-grid corrections to handle low-frequency errors efficiently. Recent advances include parallel overlapping domain decomposition methods for improved scalability in large-scale simulations and AI-augmented approaches for faster solving of FDFD systems.^[5]^[16]^[17]^[18] Parallelization techniques are essential to handle the computational demands of 3D FDFD, particularly for iterative solvers on high-performance computing platforms. Domain decomposition methods partition the computational grid into overlapping or non-overlapping subdomains, solving local systems in parallel and exchanging boundary data via message-passing interfaces like MPI, which can yield near-linear speedups with increasing core counts while reducing per-processor memory usage. GPU acceleration further enhances performance by mapping matrix-vector products to thousands of parallel threads using CUDA, achieving speedups of over 20× compared to single CPU implementations for problems with up to 10^7 unknowns. These approaches enable simulations of complex structures that would otherwise be infeasible on serial hardware.^[5]^[19]^[20]

Boundary Conditions and Absorbing Layers

In finite-difference frequency-domain (FDFD) simulations, artificial boundaries are essential for modeling open or unbounded domains while minimizing spurious reflections that can distort the solution. These boundaries truncate the computational grid, and appropriate conditions ensure that outgoing waves are absorbed or correctly propagated without re-entering the domain of interest. Common techniques in FDFD include perfectly matched layers (PML), absorbing boundary conditions (ABC) such as Mur's approximations, and periodic boundaries for structures with translational symmetry. Perfectly matched layers (PML) provide highly effective absorption for open boundaries in FDFD by adapting Berenger's original time-domain formulation to the frequency domain through complex coordinate stretching. This approach modifies Maxwell's equations by replacing spatial coordinates x with a complex stretched version \tilde{x} = x + i \int \frac{\sigma(\omega)}{\omega \epsilon_0} dx, where \sigma(\omega) is a frequency-dependent conductivity profile that increases toward the outer edge of the PML region, ensuring zero reflection for plane waves at any incidence angle and frequency. The resulting anisotropic, lossy medium absorbs waves without altering the impedance match at the interface, and in FDFD, it leads to a well-conditioned sparse matrix when implemented via stretched-coordinate PML (SC-PML). Mur's absorbing boundary conditions (ABC) offer a simpler alternative for truncating the domain, approximating the one-way wave equation for outgoing waves at the boundaries. The first-order Mur's ABC enforces the condition \frac{\partial E}{\partial n} + j k E = 0 on the boundary normal n, where k = \omega / c is the wavenumber, effectively simulating the Sommerfeld radiation condition for normal incidence. Higher-order versions, such as the second-order approximation, incorporate transverse derivatives for better accuracy at oblique angles, derived from a Taylor expansion of the square-root operator in the wave equation, reducing reflections to below -40 dB for angles up to 45 degrees in typical setups. In FDFD, these conditions are applied directly to the discretized fields at the grid edges.^[21] For periodic structures like photonic crystals, Bloch boundary conditions enforce phase continuity across the unit cell edges, modeling infinite periodicity without absorbing layers. The fields satisfy \mathbf{E}(\mathbf{r} + \mathbf{R}_l) = \mathbf{E}(\mathbf{r}) e^{i \mathbf{k} \cdot \mathbf{R}_l}, where \mathbf{R}_l is a lattice vector and \mathbf{k} is the Bloch wavevector, allowing computation over a single unit cell while capturing band structures. This phase shift is applied to the tangential field components at opposite boundaries, enabling efficient eigenmode extraction. Implementation of these boundaries in FDFD involves modifying the sparse system matrix \mathbf{A} \mathbf{x} = \mathbf{b} derived from the discretized curl equations, where \mathbf{x} represents the field vector. For PML, the stretching factors s_i = 1 + i \sigma_i / (\omega \epsilon_0) (with i = x, y, z) are incorporated into the derivative operators, altering the stencil coefficients near the PML interface and introducing auxiliary diagonal scaling matrices for improved numerical stability. Mur's ABC replaces the standard central-difference stencils at the boundaries with one-sided approximations that include the j k term, effectively modifying the matrix rows corresponding to boundary nodes. Periodic boundaries adjust the off-diagonal connections between edge nodes by multiplying with the phase factor e^{i k_q a_q}, where a_q is the cell size, without adding auxiliary variables. These modifications preserve the sparsity and structure of the matrix, facilitating solution via iterative methods like GMRES.^[21]

Comparisons with Other Methods

Versus Finite-Difference Time-Domain (FDTD)

The finite-difference frequency-domain (FDFD) method and the finite-difference time-domain (FDTD) method both discretize Maxwell's equations on structured grids but operate in fundamentally different domains. FDFD solves the steady-state response at a fixed angular frequency ω, making it inherently narrowband and suitable for problems where the response at specific frequencies is required.^[1] In contrast, FDTD simulates the broadband time evolution of electromagnetic fields through explicit time-stepping, capturing transient behaviors and enabling frequency-domain results via Fourier transforms.^[22] Efficiency trade-offs between the methods depend on the problem's spectral requirements. FDFD is generally more efficient for single-frequency or narrowband analyses, as it avoids the lengthy time iterations needed in FDTD to reach steady state, particularly in simulations involving metallic or dispersive materials where FDTD's uniform time step is constrained.^[1] Conversely, FDTD excels in transient or wideband scenarios, where a single simulation yields responses across a broad spectrum, though full frequency sweeps in FDTD can be comparable in computation time to multiple FDFD runs for equivalent grid sizes (e.g., 18.6 s for FDTD versus 25.6 s for FDFD in a 192-cell problem).^[22] Regarding stability and accuracy, FDFD employs a matrix-based formulation that provides unconditional stability, free from the Courant-Friedrichs-Lewy (CFL) condition limiting FDTD's time step (Δt ≤ Δx/c).^[1] This allows FDFD to handle fine grids without stability penalties, though it requires iterative solvers like BiCGSTAB or GMRES for large systems. FDTD's explicit scheme, while intuitive, can introduce dispersion errors in dispersive media and struggles with mode degeneracy in eigenvalue problems, such as photonic crystal band diagrams, where FDFD better resolves closely spaced modes.^[23] FDFD also supports exact absorbing boundary conditions, reducing non-physical reflections compared to FDTD's perfectly matched layers (PMLs), which may exhibit errors at low or high frequencies.^[22] In terms of memory and computation, both methods scale as O(N) with grid size N, but FDFD's sparse matrix storage and iterative solution enable lower memory usage (e.g., 0.6 GB versus potentially higher for FDTD's time history in large 100³-cell problems).^[1] However, FDTD often requires fewer overall operations for broadband cases due to its marching-in-time nature, whereas FDFD benefits from preconditioning (e.g., Jacobi or shifted complex PML) to reduce solve times from hours to minutes on multi-core systems.^[1] For photonic crystal analyses, FDTD can be faster per frequency point, but FDFD's accuracy in handling losses and multiple roots makes it preferable for precise eigenmode computations.^[23]

Versus Finite Element Method (FEM)

The finite-difference frequency-domain (FDFD) method and the finite element method (FEM) both solve Maxwell's equations in the frequency domain for electromagnetic problems, but they differ fundamentally in their discretization approaches. FDFD employs uniform Cartesian grids to approximate spatial derivatives directly through finite-difference stencils, leading to a straightforward replacement of differential operators in the governing equations. In contrast, FEM uses unstructured meshes composed of elements (e.g., triangles or tetrahedra) and relies on weak variational formulations, such as the Galerkin method, to integrate the equations over each element using basis functions.^[24]^[25] Regarding flexibility, FDFD is simpler to implement for problems with regular geometries, such as layered media or periodic structures, due to its structured grid that aligns naturally with orthogonal coordinates. However, this uniformity introduces staircasing errors when approximating curved or irregular boundaries, requiring finer grids to mitigate inaccuracies. FEM, on the other hand, excels in handling complex shapes and adaptive meshing, allowing refined resolution in regions of high field gradients while coarsening elsewhere, which makes it more suitable for arbitrary geometries like scatterers with intricate features.^[13]^[25] In terms of computational aspects, FDFD generates sparse matrices from local finite-difference stencils, enabling faster assembly times and efficient storage, particularly beneficial for large-scale simulations where only nearby grid points contribute to each equation. This results in lower preprocessing overhead compared to FEM, whose matrices are denser due to element connectivity across the mesh, often demanding more intensive mesh generation and higher memory usage during assembly. Nonetheless, both methods lead to large linear systems that require iterative solvers for solution, with FDFD's structured sparsity sometimes allowing for optimized preconditioners.^[24]^[13] For accuracy, both FDFD and FEM achieve second-order convergence on uniform or refined discretizations, but FEM generally provides superior performance in inhomogeneous media through higher-order shape functions that better capture material interfaces and field variations within elements. FDFD's accuracy can be limited by numerical dispersion and staircasing in non-Cartesian setups, though these can be alleviated with subgridding or higher-order differences at added cost.^[25]^[13]

Specialized Formulations

Susceptance Element Equivalent Circuit

The susceptance element equivalent circuit (SEEC) model provides a lumped-element interpretation of the finite-difference frequency-domain (FDFD) method, representing discretized electromagnetic cells as an equivalent electrical network for frequency-domain analysis. In this approach, Maxwell's equations are discretized on a Yee grid, mapping electric field components to nodal voltages and magnetic field components to branch currents, which allows the formulation of a circuit-like system amenable to standard circuit solvers such as SPICE.^[26] The model emphasizes susceptance elements, where capacitive susceptance B = \omega C arises from the grid's effective capacitance, capturing the reactive behavior of dielectric regions, while inductive elements account for magnetic permeability effects.^[26] Derivation of the SEEC begins from the 2D transverse magnetic (TM) formulation of FDFD, a scalar wave equation for the electric field, discretized to yield a nodal analysis equation of the form \Gamma_C V = B I + j \omega C V, where V represents nodal voltages (electric fields), I are excitation currents (related to sources), \Gamma_C is the nodal capacitance matrix incorporating grid geometry, B is the susceptance matrix from inter-node couplings, and C includes material susceptances.^[26] Each grid node is interpreted as a parallel LC circuit, with the capacitive branch derived from spatial discretization steps (e.g., impedance Z_1 = \Delta x / (j \omega \epsilon) for x-directed edges) and inductive susceptance from magnetic flux linkages via current-controlled current sources.^[26] This rearrangement transforms the second-order partial differential equations into a circuit admittance matrix, enabling the solution of field distributions through circuit theory.^[27] In usage, the SEEC simplifies the analysis of periodic structures, such as power-ground planes or photonic crystals, by relating electromagnetic field solutions to circuit parameters like total impedance Z = 1 / (j \omega C + 1 / (j \omega L)), where C and L are effective per-cell capacitance and inductance from the grid.^[26] For microwave filters, it facilitates the extraction of S-parameters by defining ports at excitation nodes, allowing direct computation of transmission and reflection coefficients without full matrix inversion at each frequency.^[26] This circuit analogy builds on the finite-difference discretization to model wave propagation as current-voltage interactions in a network.^[27] The primary advantages of the SEEC lie in its intuitiveness for microwave engineers familiar with circuit design, as it bridges electromagnetic simulation with traditional lumped-element tools, and its support for hybrid simulations where FDFD regions interface with active devices or SPICE models.^[26] By reducing the problem to a sparse circuit matrix, it achieves memory efficiency (e.g., simulating a 15 mm × 15 mm structure in milliseconds per frequency point versus seconds for direct methods) and enables advanced techniques like model-order reduction for broadband analysis.^[28]

Eigenmode Analysis

The finite-difference frequency-domain (FDFD) method is adapted for eigenmode analysis by solving the source-free Maxwell's equations in the frequency domain, leading to a generalized eigenvalue problem after discretization on a Yee grid. The governing equation, derived from the curl-curl form \nabla \times (\mu^{-1} \nabla \times \mathbf{E}) - \omega^2 \epsilon \mathbf{E} = 0, is discretized to yield the matrix form A \phi = \lambda B \phi, where A represents the discretized curl-curl operator, B is a diagonal mass matrix incorporating material properties like permittivity \epsilon and permeability \mu, \phi is the eigenvector corresponding to the electric field components, and \lambda = \omega^2 is the eigenvalue related to the squared resonant frequency.^[29] This formulation allows computation of resonant modes in cavities or waveguides by targeting eigenvalues near a specified shift \sigma, transforming the problem to (A - \sigma B)^{-1} B \phi = \frac{1}{\lambda - \sigma} \phi.^[29] For solving this sparse generalized eigenvalue problem, particularly in large-scale structures, the shift-and-invert Arnoldi method is commonly employed, which iteratively builds a Krylov subspace to approximate interior eigenvalues efficiently, often with preconditioning via fast transforms to reduce computational cost from O(N^2) to O(N \log N).^[29] For denser matrices or when full eigenspectra are needed, the QZ algorithm can be applied to compute all eigenvalues and eigenvectors by reducing the pair (A, B) to a generalized Schur form, though it is less scalable for very large systems. In lossy cavities, where material parameters introduce damping, the eigenvalues become complex (\lambda = \lambda_r + i \lambda_i), reflecting attenuation; the corresponding complex frequencies \omega = \sqrt{\lambda} enable extraction of mode decay rates.^[30] Eigenmode analysis in FDFD is applied to compute cutoff frequencies in waveguides, obtained as \omega_c = \sqrt{\lambda} for propagating modes where \lambda > k_z^2 (with k_z the propagation constant), and quality factors Q in resonators via Q = -\operatorname{Re}(\omega) / (2 \operatorname{Im}(\omega)), quantifying energy loss relative to stored energy.^[30] For example, in benchmark photonic crystal cavities, FDFD yields Q values on the order of $10^4 to $10^6, establishing the method's accuracy for high-Q modes without exhaustive time-domain simulations.^[30] In 2.5D extensions for quasi-3D structures invariant along the z-direction, fields are assumed as \mathbf{E}(x,y,z) = \mathbf{e}(x,y) e^{-j \beta z} (with \beta the propagation constant), reducing the 3D problem to a 2D eigenvalue equation (\omega^2 \varepsilon \mu + \partial_x^2 + \partial_y^2) \mathbf{e} = \beta^2 \mathbf{e}, discretized on a transverse Yee grid to form a sparse matrix for TE or TM modes.^[13] This approach efficiently computes dispersion relations and cutoff frequencies for guided-wave structures like rectangular waveguides, matching analytical results within 1% error for grids of 20-30 cells per dimension.^[13]

Applications

Electromagnetic Simulations

The finite-difference frequency-domain (FDFD) method plays a significant role in antenna analysis by enabling the computation of radiation patterns and input impedance at discrete frequencies for various antenna geometries, including patch and wire types. This approach discretizes Maxwell's equations on a Yee grid in the frequency domain, allowing for efficient evaluation of antenna performance under steady-state conditions without the need for time-stepping. For wire antennas, FDFD formulations have been used to assess input impedances in the presence of nearby scatterers, providing insights into mutual coupling effects that are critical for array designs.^[2] In the realm of microwave circuits, FDFD is extensively applied to model components such as filters, couplers, and discontinuities through 2D and 3D simulations, with a primary output being the extraction of scattering parameters (S-parameters) for broadband characterization. The method's ability to handle inhomogeneous media and complex geometries makes it suitable for full-wave analysis of planar and volumetric structures. A notable implementation involves combining FDFD with asymptotic waveform evaluation to compute S-parameters for 3D microwave devices like irises and resonators.^[31] For transmission lines, 2D FDFD variants have been employed in analysis of propagation characteristics.^[32] Propagation modeling using FDFD is particularly valuable for urban radiowave prediction, where 2.5D formulations approximate 3D effects to generate site-specific coverage maps for wireless communication systems. By solving the frequency-domain wave equation over terrain and building geometries, the method accounts for diffraction, reflection, and shadowing in dense environments. A dedicated FDFD propagation model for complex urban scenarios has shown path loss predictions with RMS error of 5.6 dB compared to measured values at 2.16 GHz, outperforming empirical models in accuracy for base station planning.^[33] To address large-scale problems, FDFD is often integrated in hybrid schemes with the method of moments (MoM), leveraging FDFD's volumetric discretization for interior regions and MoM's surface integral for exterior radiation. This combination mitigates memory demands in open-domain simulations. An iterative multi-region algorithm pairing FDFD and MoM has been implemented for electromagnetic scattering from composite structures, achieving runtime reductions of approximately a factor of 2.^[34]

Acoustic Problems

The finite-difference frequency-domain (FDFD) method adapts the scalar Helmholtz equation to model acoustic wave propagation, discretizing the governing partial differential equation on a structured grid to solve for pressure fields at specific frequencies. This approach is particularly suited for applications like room acoustics, where it enables the computation of transfer functions between source and receiver locations by assembling a sparse linear system that captures wave interactions within enclosed spaces. In muffler design, the method can simulate sound attenuation in duct-like geometries by modeling wave propagation and attenuation, predicting transmission loss across tonal frequencies.^[35] A representative example involves frequency-domain analysis of acoustic scattering from obstacles, such as in ocean or room environments, where FDFD resolves near-field interactions and mode coupling around irregular scatterers on radially symmetric grids. This yields accurate intensity distributions compared to parabolic approximations, with errors minimized beyond one wavelength from the source. For urban noise propagation or architectural design, such simulations quantify diffraction effects from barriers, aiding in the optimization of sound mitigation structures.^[36] Impedance boundary conditions in FDFD formulations account for energy absorption at surfaces, essential for modeling porous materials like acoustic liners or building facades. These conditions relate pressure and normal velocity at grid boundaries using complex impedance values derived from material properties.^[37] In vibroacoustics, FDFD couples acoustic pressure solutions with structural displacement fields to predict noise radiation from vibrating components, such as panels in vehicles or building envelopes. The method solves the Helmholtz equation alongside elastic wave equations on a shared grid, capturing fluid-structure interactions that drive interior noise levels from external vibrations. Compared to time-domain methods, FDFD offers advantages in directly obtaining steady-state frequency responses, which is efficient for tonal noise sources like machinery hum or harmonic excitations in buildings. This eliminates the need for Fourier transforms on broadband simulations, reducing computational overhead for narrowband analyses while maintaining stability with perfectly matched layers for open domains.^[38]

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[13]
[PDF] The Finite-Difference Frequency-Domain Method - hade.ch
Abstract—The finite-difference frequency-domain (FDFD) method is introduced and implemented in 1D, 2D and 2.5D for both driven mode and eigenmode analysis.
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[PDF] Formulation of Finite‐Difference Frequency‐Domain - EMPossible
Aug 24, 2019 · Matrix Wave Equations. 3D FDFD in Block Matrix Form. Slide 18 e e e e e e h h h h h h y z z y xx x. z x. x z yy y x y y x zz z y z z y xx x z x.Missing: ZJ = V
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[PDF] effect of choice of boundary condition on the - arXiv
Feb 28, 2022 · The choice of either the periodic or the Dirichlet boundary conditions outside the PML regions does not affect their absorption properties.
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An iterative solver for the finite‐difference frequency‐domain (FDFD ...
Sep 21, 2010 · A stable iterative solver for the simulation of optical waves in metals using finite difference frequency domain (FDFD) method is presented.Missing: methods | Show results with:methods
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https://arxiv.org/abs/2509.20801
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Graphics processor unit (GPU) acceleration of finite-difference ...
Aug 9, 2025 · This letter presents a parallel frequency-domain finite-difference (FDFD) algorithm based on multi-graphic processing unit (GPU) applied to ...
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The finite difference frequency domain method for the eigenanalysis ...
May 17, 2016 · The finite difference frequency domain method ... Mur's Absorbing Boundary Conditions (ABC) will be introduced within the FDFD formulation.
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[PDF] Preconditioned finite-difference frequency-domain for ... - HAL-ENAC
Sep 23, 2014 · FDTD when full frequency sweeps are required, indicating that for narrow-band applications FDFD will outperform FDTD. Moreover, general ...
[21]
[PDF] FDFD and FDTD Analysis of Photonic Crystals and Loss Effect on ...
FDTD and FDFD methods have high capability of electromagnetic calculation and propagation modes ... 3.1 Comparison of FDFD and FDTD for Band Diagram Calculation.
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### Summary of Comparisons: Finite-Difference Method (FDM) vs. Finite-Element Method (FEM) for Electromagnetic Simulations
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[PDF] A review on Computational Electromagnetics Methods
The mesh free method has several benefits over FEM and FDM such as Overlapping domain gives much support and gives good approximation.
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[PDF] Accelerated Frequency Domain Analysis by Susceptance-Element ...
... equivalent frequency-domain three-dimensional electromagnetic simulation for package structures is proposed. A robust and passive susceptance-element based ...
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[PDF] 3-D FDFD Non-conformal Domain Decomposition Method for ...
INTRODUCTION. 3-D finite-difference frequency-domain (FDFD) method based on the susceptance element equivalent circuit (SEEC) [1] has been presented for ...
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Fast memory-efficient full-wave 3D simulation of power planes ...
-wave 3D simulation of power planes | A circuit-equivalent ... susceptance element based equivalent circuit. The ... finite difference frequency domain method.
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[PDF] An Efficiently Preconditioned Eigenanalysis Of ... - Scholars' Mine
eigenvalue problem. ... Index Terms—Fast discrete sine and cosine transforms, finite-difference frequency domain (FDFD), inhomogeneously ... curl-curl operator to ...
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A 2-D Finite Difference Frequency Domain (FDFD) Eigenvalue ...
A 2-D FDFD eigenvalue method in orthogonal curvilinear coordinates is presented in this paper. It is based on the combination of Finite Difference Frequency ...
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Benchmarking five numerical simulation techniques for computing ...
... Q factor is deduced from the real and imaginary parts of the complex eigenfrequency as Q = −Re(ω)/(2Im(ω)) [11]. The resonance wavelength λ is given by λ ...
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Reconfigurable graphene-gold-based microstrip patch antenna
The method used to implement the reconfigurable graphene-gold-based microstrip patch antenna is the finite difference frequency domain (FDFD). In conclusion ...
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S-parameter extraction of 3-D microwave structures by using 3D ...
We extend the asymptotic waveform evaluation (AWE) to the calculation of S-parameters using FDFD method. In the AWE technique, the electric field at the input ...
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A Compact Finite-Difference Frequency-Domain Method for Analysis ...
Oct 1, 2019 · A compact finite-difference frequencydomain method (FDFD) is presented to analyze the propagation characteristics of microwave transmission ...Missing: seminal | Show results with:seminal<|control11|><|separator|>
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Application of the Finite-Difference Frequency-Domain (FDFD ...
The present work proposes a Finite-Difference Frequency-Domain (FDFD) propagation method for radiowave prediction in dense urban environments. The main goal is ...
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[PDF] the iterative multi-region algorithm using a hybrid finite difference ...
This paper presents a hybrid technique, which combines the desirable features of two different numerical methods, finite difference frequency domain (FDFD) and ...
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**Summary of Finite Difference Discretization of Scalar Helmholtz Equation for Acoustics**
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A Finite Difference Solution To The Helmholtz Equation In a Radially ...
Aug 9, 2025 · A finite difference (FD) method is developed and analyzed for the Helmholtz equation in a radially symmetric waveguide.
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https://pubs.aip.org/asa/jasa/article/133/1/EL1/826367/Bayesian-based-estimation-of-acoustic-surface
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Finite-Difference Frequency-Domain Scheme for Sound Scattering ...
Sep 18, 2023 · Firstly, the governing equation for flow–sound interaction is given based on the perturbation theory, and the FDFD program is built.