This topic describes how the Mode source operates in broadband time-domain simulations, and how to significantly reduce the injection errors that can occur due to the mode mismatch.

## Introduction

The mode solver of the Mode source uses a frequency domain technique to calculate the modes of a structure. This technique is inherently single frequency and if the default source settings is used, the mode solver calculates the mode profiles at the center frequency of the source. For example, if the source range is 300-600 THz, the mode solver will calculate the modes at 450 THz. If the mode profile is relatively constant as a function of frequency, this works well. However, if the mode profile changes over the specified frequency range, there will be some reflection and scattering at the source injection plane. This can be understood in terms of the mode profile mismatch between the mode that actually exists at that frequency and the mode profile of the center frequency that is being injected. These errors will be most noticeable at the minimum and maximum frequencies where the mode mismatch is largest.

## Broadband Simulation Settings - FDTD

To significantly reduce the mode mismatch errors for broadband simulations, multi-frequency calculation can be enabled as shown on figure 2. Enabling this settings allows to specify the number of frequency points at which the mode profile will be calculated. The frequency points lie within the start-stop frequency range and they are located at the Chebyshev grid(rather than being linearly distributed within the start-stop range). The mode profiles at each frequency point can be displayed in the visualizer by clicking on the "Visualize Data" button as shown on figure 3.

Mode source settings |
Mode profiles calculated at the frequency points specified in the source settings
Actually injected frequency dependent fields in log scale |

## Number of Frequency Points, Accuracy and Performance Considerations

When setting up the number of frequency points for a broadband mode source simulations, it is important to realize that the frequency points do not represent the fields that are actually injected into the simulation. The injected, frequency dependent fields are obtained by interpolating the function at the specified frequency points during the simulation initialization. Since the points are located on Chebyshev grid, the interpolation can be very precise even with fairly small amount of points depending on the function smoothness. As a result, adding large number of frequency points can increase the simulation time without any considerable gains in accuracy. Therefore, it is recommended to start with a smaller number of frequency points and conduct convergence testing. An example to demonstrate this behavior is shown on the mode profiles above at the specified frequency points and the actual injected fields.

## Example

To demonstrate the multifrequency calculation feature, we will use an example of a copper wire with thin dielectric coating operating in broadband THz range. The theory tells us that the pulse propagates along the wire as a surface wave and produces a chirped signal. Additionally, since the dielectric coating causes the injected mode to be dispersive, the mode profile changes drastically with frequency. As such, this example can nicely demonstrate the advantages of multifrequency calculation feature for broadband simulations with mode source. The simulation setup is using three time monitors to record the time profile of the pulse right after the injection and after it propagates along 40 mm of the coated wire surface. The distance of 4 cm is chosen intentionally to allow us to compare the results with the reference paper that includes experimental and analytical results. The third monitor behind the source allows us to observe the amount of back-scattering.

To run the example, open the simulation file broadband_mode_source_thz_wire.fsp and run the associated script broadband_mode_source_thz_wire.lsf. The script will run one simulation in a single frequency mode and then it will repeat the same simulation with multifrequency calculation turned on and 15 frequency points. The movie shown in figure 5 represents the single frequency simulation. It can be clearly seen that there is considerable amount of back-scattering during the injection and the initial pulse propagation demonstrates interference due to the mode mismatch. As opposed to that, the movie in figure 6 shows that the multifrequency mode calculation significantly decreases the back scattering and the propagation along the wire surface produces nice clean chirped signal.

The advantage of the broadband mode source is also very clear when we compare the time profiles of the injected and propagated pulse. The multifrequency simulation results captured in figure 8 show that the time domain profile(Ey fields) of the injected pulse is cleaner and the dispersion creates a chirped signal that is very well aligned with the analytical and experimental results.

Single frequency mode source calculation |
Multifrequency mode source calculation |

Field of the injected and propagated pulse with single frequency calculation |
Field of the injected and propagated pulse with multi-frequency calculation |

## Broadband Simulation in varFDTD

The mode solver of the Mode source in varFDTD does not feature the multifrequency calculation option. Therefore, the behavior of the mode source is ideal for single frequency simulations, but creates a potential problem for broadband simulations. As noted above, the mode solver calculates the mode profile at the center frequency of the source range. The selected mode profile will then be injected over the entire frequency range of the source, which can result in injection and back-scattering errors. To avoid these errors in varFDTD, it might be necessary to use a smaller source bandwidth in order to minimize these problems. Unfortunately, if you need to collect broadband data, this means you will have to run multiple simulations, which will increase the overall simulation time.