In this example, we use the S-Parameter extraction method to calculate the group delay in a waveguide ring resonator. The same methodology can be applied to calculate the group delay of any passive device.
Simulation setup
The ring_group_delay_fdtd.fsp and ring_group_delay_mode.lms files are for simulation using FDTD and MODE respectively. Each file contains a single straight silicon waveguide coupled to a ring resonator. These examples are based on the ring resonator in the FDTD and MODE getting started example section except that there is only a single waveguide, making it a 2-port rather than a 4-port device. The gap between waveguides, simulation time and source bandwidth have also been modified to work for this example.
This example is particularly interesting for an analysis of group delay because this ring modulator is an all-pass filter. In the ideal device, without any loss or reflection mechanisms, there should be 100% transmission of the incident power. The effect of the wavelength dependent ring resonances will therefore be seen in the group delay. Indeed, we can expect to see peaks in the group delay for resonant wavelengths of the ring, since the light will stay trapped in the ring for a long time.
Results
In the examples below, the Hz-field data is extracted from a time monitor and a Fourier transform is performed. This approach is valid for single mode waveguides and the H field was selected because we are using a TE-like mode. For multi-mode waveguides, modal decomposition would be necessary. The phase information is extracted after the Fourier transform and subsequently the derivative is taken to generate the group delay plot. The plots below are obtained from the above associated files with FDTD and MODE separately. 3D FDTD has more accurate results but slower in simulation speeds than 2.5D Propagator. Therefore, MODE allows a longer simulation time and a broader bandwidth with a faster simulation speed in this comparison. The FDTD plots are based on a higher loss (reduced gap distance) configuration just to increase simulation speed for demonstration purposes.
3D FDTD Simulation: Gap: 0.02 um Simulation time: 1.5 ps wavelength range: 1.5 - 1.6 um |
2.5D MODE Propagator: Gap: 0.1 um Simulation time: 12 ps wavelength range: 1.4 - 1.7 um |
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Hz vs time |
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phase vs wavelength |
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group delay vs wavelength |
* note that the X axis of the left/right hand figures is different
The resonant wavelengths can stay in the ring for many cycles, compared to those wavelengths that do not resonate in the ring. The group delay plots show this feature and indicate the position of the peaks. There are some losses in the system, especially at the coupling point. In each cycle, light from the ring can be scattered at the coupling point either in a backwards direction or into radiation modes. This explains why the transmission is less than 100% at resonant wavelengths at the "through" port.
The simulation time is an important factor for these simulations, especially for shorter wavelengths. This is because shorter wavelengths experience less attenuation for a given bend radius and have lower coupling for a given gap distance, resulting in higher Q factors and consequently longer simulation times. When the simulation time is too short, the time signal is truncated which leads to errors in the frequency domain results after Fourier transform and incorrect results such as negative group delay may be seen (see more information Simulation time and Frequency domain monitors). If the simulation time is too long in MODE, the simulation can diverge due to numerical instabilities created by the PML boundaries.
Improvements
The above results can be generated quickly and are useful for demonstration purposes. If the settings below are used instead, the accuracy is significantly improved. However, this increases the required simulation times by 10-100 times, which is particularly noticeable in the 3D FDTD simulations.
3D FDTD Simulation:
Gap: 0.1 um
Simulation time: 30 ps
wavelength range: 1.5 - 1.6 um
Mesh accuracy: 3
2.5D MODE Propagator:
Gap: 0.1 um
Simulation time: 50 ps
wavelength range: 1.4 - 1.7 um
Mesh accuracy: 3
See also
Ring resonator MODE (design and initial simulation)