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Fig. 1. General view of the structure
Plane wave illumination of the infinitely periodic structure
In some applications it is desirable to analyze plane wave illumination of the infinitely periodic structure. Such 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.We present a novel approach implemented in QW-3D software package which allows reducing the model size to only one period of the structure. Fundamentals of the periodic FDTD algorithm are described in reference .
Fig. 2. Antenna results window
The type of the circuit is set for 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 Y.
The excitation spectrum is around 10 GHz. The plane wave illumination angle indeed has been set to (j, Q) = (00, 1350). However, since we have set PBC along x and y directions, PLW walls need to be set only in the Z direction.
According to the fundamentals of the periodic FDTD algorithm described in  and implemented in QW-3D, 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 ejY. 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.
After simulation converges we may look at the radiation pattern in the XZ-plane to see that there is an incident lobe indicated around Q = 1350 (see Fig. 2). 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. Due to the finite size of NTF wall, where the Fourier transform is performed, radiation pattern is somewhat broadened, rather than the Dirac delta. Nevertheless, as it may be seen in the Fig. 2, nulls of the broadened incident beam are located within The numerical accuracy at all of the diffraction orders. In general, it may be proven that if NTF is processed over the integer multiple of spatial periods of the structure, all the scattered beams do not disturb one another in the NTF radiation pattern. This it is advantageous because we can use NTF in a periodic scenario that is often electrically short and easily extract the reflection coefficient for any diffraction order.
Now the excitation is changed to the sinusoidal at 10 GHz. The beam is excited at the top PLW and suppressed at the bottom wall. There is no field outside this area in the so-called scattering field region. Periodic boundary conditions imposed at the lateral walls support the propagation of the plane wave at the chosen angle and suppress at the other angles.
We consider now the PLW illumination of the regularly arranged 10 mm x 10 mm rectangular metal patches (see Fig. 4). Principal period is 20 mm in both directions.
Fig. 3. Ey polarised plane wave propagation in the XZ plane for 10 GHz
Eigenmodes in photonic crystals (PhC)
Another wide scope of applications for PBC (Periodic Boundary Conditions) is related to 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). We will show how to take the advantage of PBC for the extraction of eigenmodes in PhC consisting of the infinite lattice of dielectric GaAs rods of radius r = 1 mm and lattice constant a = 10 mm.
Fig. 6. Infinite rectangular lattice of GaAs rods (r = 1 um, a = 10 um) and its model
The model consists of one period of the lattice. The aim is to find TM modes with Ez polarization that satisfy the phase shift per period Yx = Yy = 0 rad but with no phase variation along the rods (perpendicular to the picture). Hence, the model may be reduced to one FDTD layer with electric boundary conditions at the top and at the bottom. The structure is excited with a wideband delta signal to perform a Fourier transform of the injected electric current. Consequently, we will observe several resonances indicating eigenmodes that satisfy the imposed phase shift per period.
Fig. 7. Spectrum of the injected electric current
Fig. 8. Envelope of the electric and magnetic components at f = 18.34 THz
Let us consider the same scenario but with a sinusoidal excitation at f = 18.34 THz which is the first resonance highlighted in the Fig. 7. Fig. 8 shows envelopes of the electric and magnetic field components, respectively. As expected, there is no phase shift over the period of the structure along X and Y axes.
Follow the same procedure to collect the data for several phase shifts, we will obtain a so called photonic bandgap (PBG) diagram in the first Brillouin zone. There is a bandgap within the range a/l = 0.42 - 0.49 (f = 12.6 - 14.7 THz). It means that no TM mode can propagate in such a 2D lattice within this frequency range. In other words, such a Structure may be treated as a very good shield for TM polarization incident at any angle within the bandgap range.
Fig. 9. PBG diagram in the first Brillouin zone of the considered example
Fig. 4. General view of the structure
Fig. 5 presents the radiation pattern obtained for this scenario. We may observe the 0th diffraction order beam at the 450. The higher reflection orders neighbouring to the 0th order are not disturbed at all (see the nulls indicated with dashed lines). Moreover, according to the following relation:
ml / L = sin (Qinc) + sin (Qm)
where Qinc is an illumination angle, Qm is the mth diffraction order and L stands for the spatial period, we may also expect the presence of the 1st diffraction order at Q1 = -52.40 and indeed there is another beam indicated at the Radiation pattern (see Fig. 5).
Fig. 5. Antenna results window
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