Plane wave illumination of the infinitely periodic structure
The present example considers a plane wave illumination of the infinitely periodic structure.
Analyzing plane wave illumination of the infinitely periodic structure problems occur in optics where diffraction gratings as well as the other kinds of frequency selective surfaces are growing interest. 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 grating periods is possible but inefficient.
Plane wave illumination of the infinitely periodic structure.
Simulation model considers a cubic volume of free space, with absorbing exterior boundaries on the top and the bottom (marked as blue rectangles), a surface for near-to-far field transformation (NTF box, marked as reen box) a surface for plane wave excitation (PLW box, marked as red box).
Plane wave illumination of the infinitely periodic structure project in QW-Editor.
Periodic boundary conditions (PBC) implemented in QuickWave software package allows reducing the model size to only one period of the structure.
We impose PBC along X and Y assuming infinite extension in these directions. As a consequence, both plane wave and NTF boxes stick to the lateral boundaries of the scenario. The type of the circuit is set to 3DP with periodicity in X and Y direction. It means that periodic boundary conditions are located at the lateral boundaries of the scenario and force a particular Floquet Phase shift per period.
Circuit dialogue with PBC settings.
It is advised to refer to Free space incident wave for detailed discussion regarding free space excitation with a plane wave or Gaussian beam.
Since we impose periodicity only along X and Y directions, phase shift shift per period in z direction is not needed. The angle of incidence is frequency dependent if the phase shift per period is fixed. In other words, pulse excitation in plane wave box 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.
Configuration dialogue of a plane wave box excitation (free space incident wave).
The excitation spectrum is around 10 GHz. The plane wave illumination angle indeed has been set to Phi=0 degrees, Theta=135 degrees. However, since we have set PBC along X and Y directions, only Z-directed walls of the PLW box will be used for exciting the wave. The -X, +X, -Y and +Y walls of PLW are deactivated.

According to the fundamentals of the periodic FDTD algorithm described in M.Celuch-Marcysiak, W.K.Gwarek, "Spatially looped algorithms for time-domain analysis of periodic structures", IEEE Trans. Microwave Theory Tech., vol.MTT-43, No.4, April 1995, pp.860-865, and implemented in QuickWave, real and imaginary grids of electromagnetic components are defined. Calculation of both grids is performed independently except for the periodic boundaries where these grids are coupled according to the phase coefficient e. In our example, amplitude has been set to both real and imaginary grids. Moreover, in order to excite a pure travelling plane wave, excitation delay between the two grids has to be set to the quarter of the period (quadrature) at the frequency of our interest, i.e. 10 GHz. In this case delay of the imaginary grid excitation amounts to 0.025 ns.

The radiation pattern calculation is set at 10 GHz.
Near To Far postprocessing configuration dialogue.
We are interested in calculating the 2D scattering patterns for the considered scenario. The scattering pattern will be calculated versus Theta angle varying between 0 and 360 degrees with a step of 1 degrees. They will be calculated with a constant Phi angle equal to 0 degrees. The definition of the angles is explained in the lower right part of the 2D Radiation Patterns configuration dialogue. Note that this definition depends on the choice of the reference axis. The angle Theta is always counted from the reference axis (Z in the considered example). The angle Phi is always counted around it.
2D Radiation Patterns configuration dialogue.
Pickup Walls dialogue.
Only one wall of the NTF box will take part in NTF calculations. This is +Z wall located just below the active PLW wall, and flux of power is incoming through this wall. Therefore the value of radiated (outgoing) power shown in the Results window status is negative: Pr=-5967.1953 [W].
2D scattering patterns calculated at 10 GHz.
There is an incident lobe indicated around Theta=135 degrees, which is consistent with the settings of Theta plane wave illumination angle. Since there is no scattering body inside the considered volume we observe nulls at all of the possible diffraction orders indicated with the vertical dashed lines.
Ey field component of plane wave excitation (Phi=0 degrees, Theta=135 degrees and Polarisation=90 degrees) at 10 GHz.
Time dependent Ey field distribution of plane wave excitation (Phi=0 degrees, Theta=135 degrees and Polarisation=90 degrees) at 10 GHz.