## What do I do to test convergence?¶

The best way to make sure that your GME computation is converged is to increase the parameters controlling the precision of the simulation until you no longer see change in the eigenmodes of interest. We recommend doing this in the following order:

• First, make sure you have set a high enough gmax, which is defined upon initialization of GuidedModeExp.
• Then, increase the number of guided bands included in the simulation by adding more indexes to the gmode_inds list supplied to GuidedModeExp.run(). Note that after including more modes in gmode_inds, you should test again the convergence w.r.t. gmax.
• If your bands look particularly weird and discontinuous, there might be an issue in the computation of the guided modes of the effective homogeneous structure (the expansion basis). Try decreasing gmode_step supplied in GuidedModeExp.run() to 1e-3 or 1e-4 and see if things look better.

Finally, note that GME is only an approximate method. So, even if the simulation is converged with respect to all of the above parameters but still produces strange results, it might just be that the method is not that well-suited for the structure you are simulating. We’re hoping to improve that in future version of legume!

## How do I incorporate symmetry?¶

The expansion basis in the GME consists of the guided modes of an effective homogeneous structure. These can be classified as TE/TM, where in our notation the reference plane is the slab plane. The guided modes alternate between TE and TM, such that gmode_inds = [0, 2, 4, ...] are TE and gmode_inds = [1, 3, 5, ...] are TM. However, this classification is often broken by the photonic crystal permittivity.

For gratings (permittivity is periodic in one direction and homogeneous in the other), the TE/TM classification holds. You can selectively compute the modes by supplying gmode_inds with either only even or only odd numbers.

For photonic crystals with a mirror plane, like a single slab with symmetric claddings, the correct classification of modes is with respect to reflection in that plane. The positive-symmetry guided modes are gmode_inds = [0, 3, 4, 7, 8, ...], while the negative-symmetry modes are gmode_inds = [1, 2, 5, 6, 9, 10, ...]. Low-frequency positive-symmetry modes that are mostly fromed by the gmode_inds = 0 guided band are sometimes referred to as quasi-TE, and low-frequency negative-symmetry modes that are mostly formed by the gmode_inds = 1 guided band are sometimes referred to as quasi-TM.

Without any mirror planes, all the guided modes are generally mixed. There can still be symmetry if the k-vector points in a high-symmetry direction, but there is currently no way to take advantage of that in legume.

## When should I use approximate gradients?¶

When running GME with the autograd backend, one of the run() options you can specify is 'gradients' = {'exact' (default), 'approx'}. The approximate option could be faster in some cases, and could actually still be exact in some cases. This is the high-level computational graph of the guided-mode expansion:

The 'approx' option discards the gradient due to the top path in this graph, i.e. the gradient due to the changing basis. Only the gradient from the diagonalization path is included. Here are some rules of thumb on what to use:

• If you’re optimizing hole positions, or more generally parameters that don’t change the average permittivity, you’re in luck! In this case, the 'approx' gradients should actually be exact!
• If you’re optimizing dispersion (real part of eigenfrequencies), you could try using 'approx' gradients, as they might be within just a few percent of the exact ones.
• If you’re optimizing loss rates or field profiles and/or if your parameters include the layer thicknesses, then the 'approx' gradients could be significantly off, 'exact' is recommended (and is the default).

Our paper gives a lot of detail both on the guided-mode expansion method and on our differentiable implementation.

## How should I cite legume?¶

If you find legume useful for your research, we would apprecite you citing our paper. For your convenience, you can use the following BibTex entry:

@article{Minkov2020,
title = {Inverse design of photonic crystals through automatic differentiation},
author = {Minkov, Momchil and Williamson, Ian A. D. and Gerace, Dario and Andreani, Lucio C. and Lou, Beicheng and Song, Alex Y. and Hughes, Tyler W. and Fan, Shanhui},
year = {2020},
journal = {arXiv:2003.00379},
}