Bloch’s Boundary Condition

Compatibility:

Bloch’s theorem tells us that for periodic structure, field components have the
following properties:

FDTD - Equation 60

where R is the lattice vector, is the wave vector. Equation 60 is the boundary
condition used in the OptiFDTD band solver.

Initial excitation

Unlike conventional FDTD simulations where time domain excitation is continuous
wave and lasts on only some time steps, the FDTD band solver sets the initial value
in the first time step, and this initial field is expected to have non-zero projection on
the normal modes we are interested in.

Example: At t = 0 :

FDTD - Equation 61, 62 and 63

fdtdis chosen as the unit vector along the Cartesian system coordinator.

This keeps fdtd

FDTD - Figure 19 Flow chart of FDTD band solver

Figure 19: Flow chart of FDTD band solver

Note:

  • In OptiFDTD, the FDTD band solver only deals with the 2D square lattice and 2D hexagonal lattice.
  • In OptiFDTD, the band solver only accept lossless Isotropic and Anisotropic materials.

Unit cell and Brillouin zone

The following figures show the unit cell and Brillouin zone in the OptiFDTD
2D FDTD band solver.

FDTD - Figure 20 Unit cell and associated Brillouin zone for square lattice

Figure 20: Unit cell and associated Brillouin zone for square lattice

FDTD - Figure 21 Unit cell and Brillouin zone for Hexagonal lattice

Figure 21: Unit cell and Brillouin zone for Hexagonal lattice