Finite Difference Frequency Domain Solver (FDFD)

The Finite Difference Frequency Domain (FDFD) solver calculates the spatial distribution of electromagnetic fields at the target frequency by solving Maxwell's equations in the frequency domain. The basic principles of FDFD are outlined as follows.

• Frequency-domain Maxwell's Equations

  For source-free materials, Maxwell’s curl equations can be given by:

$\nabla \times \mathbf E = k_0 \mu_r \tilde{\mathbf H}$

$\nabla \times \mathbf H = k_0 \varepsilon_r \mathbf E$

• FDFD Mesh

  • Discretize space in Yee grids so that the electric and magnetic fields are staggered in space. The electric fields are located at the center of grid ridges while the magnetic fields are located at the center of grid surfaces.

  • The materials in FDTD are discretized based on 'Yee cell' grids. The material in each point of Yee grids is determined based on the structure at this point, so that the material distribution is discretized in Yee grids.

• Finite Difference Algorithm

  Based on Maxwell's equations in frequency domain, the y-axis magnetic field component can be discretized in the Yee cell and expressed as:

  $\frac{\partial E_x}{\partial z} - \frac{\partial E_z}{\partial x} = \mu_{yy} \tilde{H_y}$

  When linear expansion is done along the coordinate axis in space, the above equation can be transformed into the following form:

  ${\mathbf D_z^E E_x} - {\mathbf D_x^E E_z} = \mu_{yy} \tilde{H_y}$

  Likewise:

  ${\mathbf D_z^H \tilde{H_x}} - {\mathbf D_x^H \tilde{H_z}} = \varepsilon_{yy} E_y$

  Wherein, ${E_z}$, ${E_x}$, ${E_y}$, $\tilde{H_z}$, $\tilde{H_x}$ and $\tilde{H_y}$ are column vectors containing all electric and magnetic field components throughout the entire mesh; ${\mathbf D_z^E }$, ${\mathbf D_x^E}$, ${\mathbf D_z^H}$ and ${\mathbf D_x^H}$ are banded matrices used to calculate the spatial derivatives of the electromagnetic field components on the mesh; $\mu_{yy}$ and $\varepsilon_{yy}$ are the diagonal matrices containing the relative permeability and permittivity tensor components of the entire mesh along its central diagonal.

  Based on the equation above, Maxwell's equations at constant frequencies are transformed into matrix form: $Ax=b$.

  ${\mathbf A_E \mathbf E} = \mathbf f_E$

  The matrix $A$ is the wave matrix in physical space, while the column vector $x$ represents the electromagnetic field components to be resolved, and the column vector $b$ represents the source.

[1] Rumpf, Raymond C et al. “Rigorous electromagnetic analysis of volumetrically complex media using the slice absorption method.” Journal of the Optical Society of America. A, Optics, image science, and vision vol. 24,10 (2007): 3123-34.