## 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!

## Why am I running out of memory?¶

GME requrest the diagonalization of dense matrices, and you might start running out of memory for simulations in which computational time is not that much of an issue. This is also because the scaling with gmax is pretty bad: the linear dimension of the matrix for diagonalization scales as gmax**2, and so the total memory needed to store it scales as gmax**4. So, unfortunately, if you’re running out of memory in a simulation there’s not much you can do but decrease gmax.

That said, if you’re running out of memory in a gradient computation, there could be something you can try. Reverse-mode autodiff is generally the best approach for optimization problems in terms of computational time, but this can sometimes come at a memory cost. This is because all of the intermediate values of the forward simulation have to be stored for the backward pass. So, if you are for example doing a loop through different k-points, the dense matrices and their eigenvectors at every k will be stored, which can add up to a lot. There is no easy way to fix this (and no direct way within autograd), but we’ve included a function that can provide a workaround. For details, have a look at this example.

Finally, it’s worth mentioning that there are probably improvements that can be made to the memory usage. If anybody wants to dive deep in the code and try to do that, it will be appreciated!

## What should I know about the guided-mode basis?¶

The expansion basis in the GME consists of the guided modes of an effective homogeneous structure (panels (a)-(b)) in the Figure. By default, the effective permittivities in (b) are taken as the average value in every layer. This is controlled by the gmode_eps keyword option in the run options. Setting gmode_eps = 'background' will take the background permittivity instead, while there’s also the option to have custom values by setting gmode_eps = 'custom'. In that case, every layer (including the claddings) in the PhotCryst object should have a pre-defined effective permittivity eps_eff, which will be used in the guided-mode computation. This is simply set as an attribute of the layer, e.g.

phc.layers[0].eps_eff = 10   # Slab custom effective epsilon
phc.calddings[0].eps_eff = 1 # Lower cladding


The guided modes can be classified as TE/TM, where in our notation the reference plane is the slab plane (xy). 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 (panel (c)). However, this classification is often broken by the photonic crystal structure (we discuss symmetries further below).

We only include the fully-guided modes in the computation (the ones that lie below both light lines in (c)). This is what makes the computation approximate, as the basis set is not complete.

## How do I incorporate symmetry?¶

The TE/TM classification of the guided modes of the homogeneous structure is often broekn by the photonic crystal permittivity. Here is how you can still incorporate some structural symmetries.

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).

## What if I only need the Q of some of the modes?¶

In some simulations, the computation of the radiative losses could be the time bottleneck. In some cases, e.g. when optimizing a cavity, you only need to compute the quality factor of a single mode. If you run the GME by default, the Q-s of all modes will be computed instead, but you can set the option compute_im = False to avoid this. Running the GME with this option will compute all modes, but not the imaginary part of their frequencies (which is done perturbatively after the first stage of the computation). Then, you can use the legume.GuidedModeExp.compute_rad() method to only compute the loss rates of selected modes.

## What’s the gauge?¶

Something to be aware of is the fact that the eigenmodes come with an arbitrary k-dependent gauge, as is usually the case for eigenvalue simulations. That is to say, each eigenvector is defined only up to a global phase, and this phase might change discontinously even for nearby k-points. If you re looking into something that depends on the gauge choice, you will have to figure out how to set your preferred gauge yourself.

Of course, apart from this global phase, all the relative phases should be well-defined (as they correspond to physically observable quantities). So for example if you compute radiative couplings to S and P polarization, the relative phase between the two should be physical.

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},
}