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/* Waveform, Bandwidth & Stability */
The FDTD method provides a wideband simulation of your physical structure. Frequency domain techniques often require a tedious frequency sweep to calculate the port characteristics (S/Y/Z parameters). By contrast, EM.Cube's [[FDTD Module]] performs a discrete Fourier transform (DFT) of the time domain data to calculate these characteristics at the end of a single FDTD simulation run. In order to produce sufficient spectral information, an appropriate wideband temporal waveform is needed to excite the physical structure. The general form of EM.Cube's default excitation waveform is a Modulated Gaussian Pulse given by:
:<math> E_j^{inc}(r,t)=E_0(r) \exp \left(-\frac{(t-t_0)^2}{\tau ^2} \right) \cos \left(2\pi f_0 (t-t_0)-\Phi \right),\quad j=x,y,z </math><!-- [[Image:FDTD62.png]]-->
where f<sub>0</sub> is the center frequency, t<sub>0</sub> is the time delay, t is the Gaussian pulse width, and F is a constant phase. In the limits, the above waveform can be reduced either to a simple Gaussian pulse :
:<math> E_j^{inc}(r,t) = E_0(r) \exp \left(-\frac{(t-t_0)^2}{\tau ^2} \right),\quad j=x,y,z </math><!-- [[Image:FDTD89.png]]-->
or to a continuous, single-tone, sinusoidal waveform with a frequency of f<sub>0</sub><nowiki>:</nowiki>
:<math> E_j^{inc}(r,t) = E_{0} (r)\cos (2\pi f_0 (t-t_0)-\Phi),\quad j=x,y,z </math><!-- [[Image:FDTD90.png]]-->
The choice of the waveform, its bandwidth and time delay are important for the convergence behavior of the FDTD time marching loop. By default, EM.Cube uses a modulated Gaussian waveform with optimal parameters: t = 0.966/Δf and t<sub>0</sub> = 4.5t, where Δf is the specified bandwidth of the simulation. The time delay t<sub>0</sub> is chosen so that the temporal waveform has an almost zero value at t = 0. Of the above waveforms, modulated Gaussian and sinusoidal waveforms are band pass with no DC content, while the Gaussian pulse is low pass with a frequency spectrum that is concentrated around f = 0. In a typical FDTD simulation, you set a center frequency for the structure of interest and then specify a bandwidth around this center frequency. These together determine the lowest and highest spectral contents of your FDTD waveform. Note that setting a bandwidth equal to 2f<sub>0</sub> sets the lowest frequency to DC (f<sub>min</sub> = 0), which you may want to avoid in certain applications. On the other hand, using a Gaussian pulse waveform, you do want to set Δf = 2f<sub>0</sub>. In contrast to the wideband, exponentially decaying, Gaussian pulse and modulated Gaussian waveforms, the sinusoidal waveform is extremely narrowband and single-frequency indeed. It does not decay over time and continues to oscillate indefinitely after reaching a steady state.
Another issue of concern in an FDTD simulation is the numerical stability of the time marching scheme. You can set the mesh grid cell size to any fraction of a wavelength. Normally, you would expect to get better and more accurate results if you increase the mesh resolution. However, the time step is inversely proportional to the maximum grid cell size in order to satisfy the Courant-Friedrichs-Levy (CFL) stability condition:
:<math> \Delta t \le \frac{K_{CFL}} {c\sqrt{\left(\dfrac{1}{\Delta x_{min}} \right)^2 + \left(\dfrac{1}{\Delta y_{min}} \right)^2 + \left(\dfrac{1}{\Delta z_{min}} \right)^2 } } </math><!-- [[Image:FDTD91(1).png]]-->
where c is the speed of light, and K<sub>CFL</sub> is a constant. EM.Cube uses a default value of K<sub>CFL</sub> = 0.9. For a uniform grid with equal cell dimensions along the X, Y and Z directions, i.e. Δx = Δy = Δz = Δ, and the CFL condition reduces to:
:<math> \Delta t \le K_{CFL} \frac{\Delta}{\sqrt{3}c}</math><!-- [[Image:FDTD92.png]]-->
As can be seen from the above criterion, a high resolution mesh requires a smaller time step. Since you need to let the fields in the computational domain fully evolve over time, a smaller time step will require a larger number of time steps to achieve convergence. EM.Cube automatically chooses a time step that satisfies the CFL condition.
