Cascaded MZIs

This cascaded MZI example shows the remarkable scaling efficiency of Multi Topology (MT)- FIMMPROP applied to large scale PICs in comparison to FDTD’s scaling. 

MT-FIMMPROP uses the EigenMode Expansion (EME) method, a 3D rigorous solution to Maxwell’s equations, to provide a scattering matrix of the full device in the frequency domain (single wavelength, steady state).

The run-times for 5 scales of cascaded MZI, varying from a single MZI (~70um x 80um) all the way to multistage cascaded MZI at ~580um x 380um are shown below. Despite the ~40 times increase in the device’s size, the simulation time only doubles between the single MZI and the four stage cascaded set.

MT-FIMMPROP splits large devices into many small EME simulations and natively combines these computational regions to describe propagation through a full device. When a computational region is repeated, MT-FIMMPROP can re-use simulation results thanks to MT-FIMMPROP’s Sub-Device architecture.

This means that, despite the increasing size of these MZI circuits, the number of unique simulations to perform does not vastly increase and remains relatively constant.

Another popular method for waveguide simulations would be Finite Difference Time Domain FDTD. An FDTD simulation scales with the number computations it must perform: Number of Grid Cells × Duration ∝ Simulation Volume × Optical Path Length. The graph below shows the number of calculations necessary for the same cascaded MZIs.

For FDTD, one could theoretically expect the largest simulation to be ~350× the duration of the single MZI compared to roughly doubled as with EME.

These results may be considered an under-estimate given a value of 14 Grid Cells/ Wavelength was chosen and the simulation duration was conservatively approximated by the device optical path length multiplied by group velocity of the fundamental mode.

As of 2025, popular alternative cloud computing FDTD simulations show a cascaded MZI filter, roughly 1 billion cells in size to require 57 minutes for simulation. At a more sensible 30 Grid Cells/ Wavelength, the three stage cascaded MZI used in this example is over a billion cells in size and is simulated in EME instead in ~36 seconds though this is a single wavelength simulation.

The FDTD simulation gives a spectral response for each simulation it performs compared to EME’s steady state simulation. To replicate the same resolution of a spectral response rigorously, thousands of EME simulations would need to be performed.

However:

• With EME designers have the option to simulate specific desired wavelengths quickly. This could be useful when investigating how the operation at a single wavelength varies due to fabrication errors (changes in waveguide width, side wall tilt, corner rounding, etc).
   
• The spectral range of an FDTD simulation depends on the simulation's run time. Finding a spectral response at higher frequencies (lower wavelengths) takes longer whereas in EME simulation time is relatively invariant.
   
• For MZI’s, the difference in operation at different wavelengths is largely due to a changing phase difference (rather than mode profiles changing). As such, the same effect of a spectral response can be emulated by varying the device’s path difference. This can be achieved rapidly with a length scan performed natively in MT-FIMMPROP.
   
• Development is planned for the increased speed of FIMMPROP’s spectral response simulations.