where the new field distributions are computed from the old ones at discrete times.
The time step between computation rounds has to be small enough to make the simulation stable; the smaller the cell size is the shorter the time step has to be.
The stability criteria are discussed in more detail in [98, 99].
Absorbing layer (PML) Metal grating
Line of excitation Dielectric film Sampling line 1
Sampling line 2
Ä
z' y' x'
Figure 3.5: Two-dimensional FDTD simulation domain.
The cell size has to be kept small enough to model the hologram structure with a sufficient accuracy and to minimize the numerical dispersion. The numerical dis- persion arises from the discretization of the simulation domain and causes variation of the numerical phase velocity with wave propagation angle [98]. The numerical phase velocity is always smaller than the physical phase velocity of the wave. If the cell dimensions are 0.1λ, the maximum phase velocity error relative to the physical phase velocity is –1.3 %, and it reduces to –0.3 % if the cell dimensions are 0.05λ[98].
A non-uniform grid is used to model the hologram structure accurately near the slot edges [100]. Smaller cells are used near the slot edges than elsewhere. The cells are gradually narrowed in the x direction towards the slot edges. By this way, the locations of the slot edges and the field in the vicinity of the edges are modelled more accurately. The use of larger cells further away from the slot edges helps to keep the computational size of the simulation reasonable. In the z direction, the cell width is fixed. The simulation domain is very wide in the x direction (up to hundreds of thousands of FDTD cells) but quite narrow along thez axis (approximately one hundred cells). The cell dimensions vary typically between 0.02–0.05λ, where λ is the wavelength in free space. If the wavelength is given in Mylar (²r = 3.35) that is typically used as the hologram substrate, the variation of the cell dimensions is approximately 0.04–0.09λ.
In the two-dimensional FDTD simulation, two different polarizations are studied separately; the electric field is either parallel with the slots (y-directed) or perpen- dicular to them (tangential to the (x, z)-plane). The magnetic field is perpendicular to the electric field. In the first case (from now on called the vertical polarization), the electromagnetic field has components Ey, Hx, and Hz. In the latter case (the
horizontal polarization), the components are Hy, Ex, andEz. Figure 3.6 shows the FDTD cells at the vertical and horizontal polarization. The cross-polarization anal- ysis is not possible in the two-dimensional simulations, where the curvature of the slots, and thus the source of the cross polarization is omitted; a special method to analyze cross polarization is presented in [101].
E
+
yHz Hz
Hx
Hx
Vertical polarization
H
+
yEz Ez
Ex
Ex
Horizontal polarization
Figure 3.6: FDTD cells in the vertical and horizontal simulation.
The simulation domain is surrounded with an absorbing layer that serves to simulate the radiating free-space condition. The perfectly matched layer (PML) has been used for this purpose due to its very low reflectivity [98, 102].
3.3.2 Excitation
The sinusoidal excitation is placed a few cells away from the hologram structure (see Figure 3.5). The excitation is a so-called soft source, through which the reflected field can pass undisturbed [98]. At the line of excitation, the complex incident field is added to the computed field. At the vertical polarization, the incident electric field Ein is added to the electric field Ey. At the horizontal polarization, the equivalent incident magnetic field Hin = Ein/η0 is added to the magnetic field Hy, which induces the electric field components Ex and Ez. The wave impedance of vacuum η0 is 376.73 Ω.
Non-oscillating dc currents may be induced in the FDTD simulation, if the model has highly conductive current paths and the excitation is not turned on correctly [103, 104]. The hologram metal grating structure, which is modelled as a perfect electric conductor (PEC), is found to induce dc currents at the horizontal polarization. The dc-currents produce a stationary field, which can lead to interpretation errors in the post-processing of the simulation data. To avoid that, the dc-field components have to be filtered out from the simulation data. The filtering method has been used in earlier simulations. In this thesis work, however, an optimal ramp function has been used to turn the excitation smoothly on, which eliminates dc currents. It has
been found that a raised cosine function is optimal for this purpose [104]. When the excitation is turned on slowly during the first 4.5 sine wave cycles, no dc-field components can be observed in the simulation data.
3.3.3 Physical optics
The transmitted field in the aperture of the hologram is recorded along the sampling line (see Figure 3.5). In the post-processing, the time-domain data are transformed to the complex data, i.e., the amplitude and phase are resolved. From these data, the field further away from the hologram is calculated by using physical optics [63, 99].
The physical optics transformation is done on the same (x, z)-plane where the FDTD simulation is carried out.
In some simulations, the reflected field is the desired one [III,IV]. In such a case, the field in front of the hologram is recorded (see Figure 3.5). This is the total field including the reflected field and the field radiated from the excitation line towards the–z axis. The reflected field can be determined by subtracting the excitation field from the total field. In this thesis work, the simulation method is extended in such a way that it can be used to calculate the reflected field, as well as the field over the full horizontal angle range [III,IV].