===Time Domain Simulation Of Periodic Structures===
[[Image:fdtd_perdiag_tn.png|thumb|350px|Diagram of a periodic structure illuminated by an obliquely incident plane wave in [[FDTD Module]]]]
A periodic structure is one that repeats itself infinitely in one, two or three directions. EM.Cube's [[FDTD Module]] allows you to simulate doubly periodic structures with periodicities along the X and Y directions. Many interesting structures such as frequency selective surfaces (FSS), electromagnetic band-gap (EBG) structures and metamaterial structures can be modeled using periodic geometries. In the case of an infinitely extended periodic structure, it is sufficient to analyze only a unit cell. In the FDTD method, this is accomplished by applying periodic boundary conditions (PBC) at the side walls of the computational domain. The application of the PBC is straightforward for the case of a normally incident plane wave source since the fields do not experience any delay as they travel across the unit cell. Obliquely incident plane waves, on the other hand, cause a time delay in the transverse plane. This delay requires knowledge of the future values of the fields at any time step.
A number of techniques have been proposed to solve this problem. EM.Cube uses a recently developed novel technique that is known as Direct Spectral FDTD or Constant Transverse Wavenumber method. In this technique, the components of the transverse (horizontal) wavenumber are kept constant in the direction of periodicity. This technique shows a significant advantage over the other methods for simulation of the incident illuminations close to the grazing angles.
[[Image:fdtd_perdiag_tn.png]]
Figure 1: Diagram of a periodic structure illuminated by an obliquely incident plane wave in [[FDTD Module]].
The figure above shows a doubly periodic structure with periods S<sub>x</sub> and S<sub>y</sub> along the X and Y directions, respectively. The computational domain is terminated with PBC in both X and Y directions. Along the positive and negative Z directions, it is terminated with CPML layers. Bear in mind that the PBC is also applied to the CPML layers. The computational domain is excited by a TM<sub>z</sub> or TE<sub>z</sub> plane wave incident at z = z<sub>0</sub>. The plane wave incidence angles are denoted by θ (elevation) and φ (azimuth) in the spherical coordinate system. The constant wavenumber components k<sub>x</sub> and k<sub>y</sub> in this case are defined as: