# Finite-difference frequency-domain method

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The finite-difference frequency-domain (FDFD) method is a numerical solution method for problems usually in electromagnetism and sometimes in acoustics, based on finite-difference approximations of the derivative operators in the differential equation being solved.

While "FDFD" is a generic term describing all frequency-domain finite-difference methods, the title seems to mostly describe the method as applied to scattering problems. The method shares many similarities to the finite-difference time-domain (FDTD) method so much of the literature on FDTD can be directly applied. The method works by transforming Maxwell's equations (or other partial differential equation) for sources and fields at a constant frequency into matrix form $Ax = b$. The matrix A is derived from the wave equation operator, the column vector x contains the field components, and the column vector b describes the source. The method is capable of incorporating anisotropic materials, but off-diagonal components of the tensor require special treatment.

Strictly speaking, there are at least two categories of "frequency-domain" problems in electromagnetism.[1] One is to find the response to a current density J with a constant frequency ω, i.e. of the form $\mathbf{J}(\mathbf{x}) e^{i\omega t}$, or a similar time-harmonic source. This frequency-domain response problem leads to an $Ax = b$ system of linear equations as described above. An early description of a frequency-domain response FDTD method to solve scattering problems was published by Christ and Hartnagel (1987).[2] Another is to find the normal modes of a structure (e.g. a waveguide) in the absence of sources: in this case the frequency ω is itself a variable, and one obtains an eigenproblem $Ax = \lambda x$ (usually, the eigenvalue λ is ω2). An early description of an FDTD method to solve electromagnetic eigenproblems was published by Albani and Bernardi (1974).[3]

## Implementing the method

1. Use a Yee grid because it offers the following benefits: (1) it implicitly satisfies the zero divergence conditions to avoid spurious solutions, (2) it naturally handles physical boundary conditions, and (3) it provides a very elegant and compact way of approximating the curl equations with finite-differences.
2. Much of the literature on finite-difference time-domain (FDTD) methods applies to FDFD, particularly topics on how to represent materials and devices on a Yee grid.