In this article I will summarize the advantages of using our eigenmode expansion solver EME. Next I will discuss a few of the subtle challenges in EME comparing and contrasting it with FDTD. Finally I will highlight some diagnostic features that I find useful to overcome these complications.
EME is a powerful and rigorous method; ideal for analysis of guided electromagnetic modes. For a number of devices EME is more efficient than FDTD. In addition it handles propagation analytically, effectively eliminating grid dispersion phenomenon that are prohibitive in large FDTD volumes. For these reasons it is the preferred tool for devices like polarization converters , edge couplers or MMI’s . It also naturally handles structures that are periodic in the propagation direction like fiber bragg gratings .
For an introduction on the solver physics this page provides a concise overview MODE - EigenMode Expansion (EME) solver introduction – Lumerical Support . Essentially EME subdivides the device into a number of cells on which it finds the eigenmodes and then calculates the expansion coefficients for the sets of modes on corresponding cells overlap. These make it possible to calculate all reflection and transmission coefficients at each cell interface. Finally, it propagates these modes through the cells using their effective indices. For more information on the mechanics of using EME please see EME solver analysis window .
FDTD and EME are both rigorous formulations of Maxwell’s equations, meaning they are exact in the limit of an infinitesimal mesh. To phrase it another way we approximate the physics of continuous space on a discrete mesh; the smaller the mesh the closer the approximation. With numerical methods there will always be a trade-off between accuracy and efficiency so the idea is to refine it as much as you need and no more. For this reason convergence testing in EME and FDTD share some commonalities. In my experience FDTD is very general, and simple to implement; EME , however, is less intuitive to most users and convergence testing presents additional challenges which I will highlight below.
To take advantage of EME two additional closely related sources of error have to be considered.
- In contrast to FDTD, EME users have to determine how to manually discretize the structure in the propagation direction. This can be straightforward for devices that have natural interfaces like the MMI , and their are techniques to increase the accuracy for smoothly varying structures like the polarization converter . Methods of forcing field continuity or energy conservation may also be used depending on the application.
- EME requires a basis of eigenmodes calculated in each cell. This basis is actually infinite dimensional; therefore, any set of basis vectors we choose will be incomplete. It has the additional stipulation that to be exact one we need an infinite number of modes. That being said it is usually sufficient to take a small subset of this basis, particularly when analyzing guided modes. If your system has substrate leakage or radiative loss then you need to include those modes as well, and the analysis will necessarily become more complicated.
These phenomenon are described on the EME convergence testing page. Additionally the page understanding EME error diagnostics provides a wealth of detail on the various error diagnostics available. I’ve found it is easy to get lost in the details the diagnostics, and in my experience understanding where and why there is a problem with your EME simulation provides useful insight on how to fix it.
To minimize EME errors, you can proceed by asking yourself.
- have the cells been suitably defined?
- have I set the solver settings correctly on all cells?
- do I have a sufficient eigenmode basis everywhere?
The identify the cells where you may be experiencing problematic behavior, it is important to understand what is happening internally. The most direct visualization for this are the expansion coefficients. The forward and backwards transmission coefficients are returned as MxN complex valued matrix where N = number of modes and M = number of cells. Let’s use the edge couplers as an example.
If we reverse the row and columns we can look at the results in an orientation that aligns with the GUI.
It is often helpful to plot this as a function of the propagation distance x, especially when the cells are not linearly spaced.
Next we take the absolute value squared of the coefficients to get the power propagating in each mode for all cells$^1$. Here we have also used a log plot for the color bar with the same scale for backwards and forwards coefficients.
We immediately notice a couple of things. First the reflection is insignificant, and in the input/output cell most of the power is in the first mode. The inner cells help this transition, and we would expect some higher order modes to contribute. Remembering that the goal of this device is to squeeze a large fiber mode, into a tiny integrated single mode waveguide. Also that the modes in each cell represent a different eigenmode basis.
If it was a purely adiabatic transition then we might expect all the power to be confined to the first order mode throughout. Looking at the power in the coefficients it is usually possible to identify cells with unexpected loss. If their seems to be quite a bit of power in the higher modes it may mean that you need more modes or that you should look at redesigning this segment. Possibly using a propagating length sweep to make the device longer.
Returning to our example one interesting feature is how the higher order modes come into play around cell 2-4. This could indicate a problem, but here it can be easily be explained by looking at the structure.
Notice that in cell 3 the upper nitride layer is terminated, resulting in modes that have greater overlap, and the ordering of the modes has switched in cell 4.
Other features in the coefficients plot may be inspected in a similar way. In cell 9-11, for example, the middle nitride layer terminates causing the modes to be reordered. Once this is understood the transition seems much smoother, since the most significant change is associated with a reordering of the modes. Finally you can see mode 1 in cell 23 couples nicely onto the output TM port mode, while the power in mode 2, 3 and 5 is lost.
You can usually identify problematics cells from the coefficients plot and then investigate physical reasons for this. Unlike plotting the local or global diagnostics this approach can immediately suggest ways of fixing the problem.
For more information on how to use the diagnostics to identify discontinuities, advanced smoothing functionality, and convergence criteria I recommend this series of videos from our EDU course.
Convergence Testing - Error Diagnostics - Diagnostics Results
Convergence Testing - Error Diagnostics - Method for Using Diagnostics
- For real guided modes in low loss devices, we can associate this quantity with real power. For lossy materials, and evanescent modes where the fields are complex one loses power orthogonality. In this case, the coefficients are no longer directly related to the actual power; however, they can still be used to provide insightful diagnostics.