3.4 Periodic Boundary Conditions

QuickWave enables periodic boundary conditions (PBC), which allow for analysing infinitely periodic structures. Direct FDTD modelling of the infinite periodic structure is impossible because of the finite computer resources. Approximation of the infinite problem by using a large number of the structure periods is possible but inefficient. PBC implemented in QW-3D software package allows reducing the model size to only one period of the structure and effectively analyse problems commonly occurring e.g. in optics where diffraction gratings as well as the other kinds of frequency selective surfaces (FSS) are of much interest. Fundamentals of the periodic FDTD algorithm implemented in QW-3D are described in reference [17] in User Guide 3D: Bibliography. According to them real and imaginary grids of electromagnetic components are defined. Calculation of both grids is performed concurrently and independently except for the periodic boundaries where these grids are coupled according to the phase coefficient e^j^y (y - Floquet phase shift per period). The imaginary parts of excitation parameters for periodic circuits are defined analogously as for typical 3D structures (real parts). It is worth noting at this point that in order to excite a pure travelling wave, excitation delay between the two grids has to be set to the quarter of the period (quadrature) at the frequency of interest.

The Floquet phase shift per period y is to be determined by the user in each direction in which the PBC are applied, and may be easily calculated from the following relations:

where L_i is a spatial period of the structure along i axis, c stands for the light velocity, j and q are the coordinates of the spherical coordinate system. According to the above equations the angle of incidence is frequency dependent if the phase shift per period y is fixed. In other words, pulse excitation of a plane wave may generate a set of plane waves with a variable angle of incidence. However, if a relatively narrow excitation bandwidth is set we may assume that deviation of the angle of incidence may be neglected.

Some examples of periodic boundary conditions applications:

The plane wave scaterrometry of periodic structures and evaluation of higher order modes (see User Guide 3D: Plane wave excitation scenario).
The analysis of eigenvalue problems. Eigenmodes usually refer to the resonating modes of the considered structure and each of these modes is associated with a specific wave vector, called eigenvalue. Many physical structures are periodic, like photonic crystals (PhC). The PBC can be used for the extraction of eigenmodes e.g. in PhC (see User Guide 3D: Plane TEM example).
Calculations of reflection characteristics of frequency selective surfaces (FSS) or absorbers with periodic resistive layer ([211], [213] in User Guide 3D: Bibliography).