# Simulation time and Frequency domain monitors

The objective of this section is to explain the importance of simulation time for frequency domain monitors. If simulations are not run long enough for the electromagnetic fields to decay, the frequency domain monitor data may not be accurate. The ring resonator file is provided here for FDTD Solutions, for a similar setup in MODE Solutions, see Ring resonator.

Solvers
FDTD
VarFDTD

Associated files
usr_ring.fsp (253.6 KB)

Apodization

First, we will investigate the effect of simulation time on the frequency domain monitor data for the ring resonator taken from the getting started examples. The ring resonator geometry is seen in the picture above. A mode source is injected on the left hand side of the top waveguide. At certain wavelengths, the light builds up in intensity over multiple round-trips through the closed loop due to constructive interference. Radiation at wavelengths that do not coincide with the resonant wavelengths of the loop is attenuated by destructive interference. The resonant light can be coupled out through the bottom waveguide.

In this example, we will look at the field intensity at the output of the bottom waveguide as a function of time and frequency.

The first plot below shows the intensity of the Ez as a function of time. You can re-produce this plot by running usr_ring.fsp and plotting the Ez intensity from the time-drop monitor. Note that the simulation runs to 1500fs. The second plot shows the |Ez|^2 at the same point as a function of frequency. This plot can be re-produced by plotting the Intensity vs frequency/wavelength of Ez from the drop monitor. The drop monitor is a frequency domain power monitor.

The plots below can be created by setting the simulation time in usr_ring.fsp to 700fs, running the simulation and plotting the same data as above. Note the oscillations (which are circled in red) in the frequency domain plot. The oscillations are an artifact that arises because the simulation was not run long enough, as explained next.

Since FDTD is a time domain technique, the fields are calculated as a function of time. A Fourier transform gives the fields as a function of frequency.

$E(\omega ) = \int_{0}^{\infty} e^{-i\omega t} E(t) dt$

The frequency monitors calculate this integral up to the time when the simulation stops. This is the same as assuming that the fields are 0 beyond the end of the simulation. Mathematically, this is equivalent to multiplying the E(t) with a top hat function that starts at t = 0 and ends at the simulation time (Tsim).

$E_{sim}(t) = E(t) [H(0) - H(T_{sim})]$

where H(t) is the Heaviside function. It is possible to obtain the effect of a finite simulation time on frequency monitor data from basic properties of Fourier transforms. Below, we can see that E(w) calculated by the frequency monitor is equal to the true E(w) convolved with a sinc function.

$E_{sim}(\omega) = \int_{0}^{\infty} e^{-i\omega t} E(t) [H(0) - H(T_{sim})] dt$
$= E_{true} (\omega) * \int_{0}^{\infty} e^{-i\omega t} [H(0) - H(T_{sim})] dt$
$= E_{true} (\omega) * T_{sim} sinc (\omega T_{sim})$

To obtain the true E(w) from the frequency monitor, we would convolve E(w) with a delta function rather than a sinc function. The sinc function causes frequency mixing that reduces the accuracy of the monitor data.

The Fourier transforms of the top hat functions corresponding to the above simulations are shown below. Notice that the sinc function for the 1500fs simulation is much narrower. A narrow sinc function is preferable to a wide sinc function because it is closer to the ideal response of a delta function.

In the image below, the blue line E(w) from the simulation that ran for 1500fs. 1500fs is long enough to give a narrow sinc function that is a good approximation of a delta function for this simulation. Therefore, the result is very close to the true response of the system.

The red line shows the result of taking E(w) from the 1500fs simulation and convolving it with the Fourier Transform of the top hat function that has a width of 700fs. This result is very similar to the frequency domain plot from the simulation that ran for 700fs.

To conclude, the oscillations in the frequency domain data are caused by early termination of the simulation. These oscillations are the result of the convolution between the true response response of the system and the sinc function that comes from stopping the simulation too early. The convolution causes frequency mixing that reduces the accuracy of the simulation.

To avoid this problem, the simulation should be run long enough for the time domain fields to decay. This will ensure accurate frequency domain results. For simulations that decay very slowly, monitor Apodization can be used to minimize these problems.

Note: Auto shutoff

By default, simulations will automatically shut off when the total energy in the simulation volume drops to a small fraction of the maximum energy injected. The auto-shutoff was developed in order to make sure the simulations run long enough so that in most cases frequency domain data will be accurate.

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