Other solvers (Eg FDFD)

[sp94]

While plane-wave expansion method is efficient for modelling dielectric materials, it is inefficient at modelling materials such as metals or modelling finite (non-periodic) systems

It would therefore be useful to add other methods of solution to Peacock, such as the finite-difference frequency-domain (FDFD) method. We could begin with an implementation of 2D FDFD to solve for eigenmodes of a periodic structure - these results could be compared directly against the existing plane-wave expansion method in Peacock.

A priority would be to keep the interface simple and consistent between the different solvers.

[kabume]

Hi @sp94 , I have written the slover-function of 2D FDFD, and it support full anisotropy, which can be seen at https://github.com/kabume/Peacock.jl/tree/master/src. Besides, the demo of anisotropic square lattice (time-broken case) can be seen at https://github.com/kabume/Peacock.jl/blob/master/test/test_FDFD_aniso.jl, the
relative permittivity and permeability are expressed as eps=[14 -12.4im 0; 12.4im 14 0; 0 0 15] and mu=[14 12.4im 0; -12.4im 14 0; 0 0 15]. The result is the same as Zhao's paper. The result of getting started of Peacock is also listed .

Some existing functions of Peacock can be used directly, like Peacock.sample_path and geometry, which are very useful.

Some problems still need to be solved:

1. the geometry function is so complicated for fully anisotropic case;
2. The FDFD solver that I have written is very simple, we still need to make some improvements so that existing functions of Peacock (eg plot_band_diagram) can be used on the new solver;
3. The FDFD solver only supports the eigenvalue problem now, and the source function and PML (UPML) need to be added.
4. FDFD seems a better fit for working on the GPU because of the sparse matrix.

[sp94]

Thank you @kabume, this looks really good :)

Here are my first thoughts on how we can merge these together in an easy to use way.

---

FDFD.__ and PWEM.__ submodules

I think we should create internal submodules Peacock.FDFD and Peacock.PWEM, so that you can do

1. Using both solvers in the same script (probably less common):

using Peacock

solver1 = FDFD.Solver(...)
solver2 = PWEM.Solver(...)

modes1 = solve(solver1, ...)
modes2 = solve(solver2, ...)

2. Or just PWEM:

using Peacock.PWEM
solver = Solver(...)
modes = solve(solver, k, polarisation)
plot(modes[1])

3. Using just FDFD:

using Peacock.FDFD
solver = Solver(...)
modes = solve(solver, k, polarisation)
plot(modes[1])

Then solve(solver, k, polarisation) can use Julia's automatic dispatch to behave differently depending whether solver is a FDFD.Solver or PWEM.Solver, and can return either FDFD.Eigenmode or PWEM.Eigenmode.

Then we will make sure that the solvers and eigenmodes have a consistent interface that can be used by eg plot(mode), plot_band_diagram(solver,...).

---

Anisotropy

I think we can use the same Geometry for both FDFD and PWEM, but allow epf and muf to return either a scalar, 3-vector, or 3x3-matrix.

Then, we could make multiple types of solvers, eg IsotropicSolver, AnisotropicSolverTE, AnisotropicSolverTM (and maybe later also AnisotropicSolver for when there are anisotropic terms that mix the TE/TM modes together). And similar for PWEM.

When we construct solver = Solver(geometry, ...) we can try and automatically pick the simplest option based on the structure of epf and muf. i.e. if all of epf and muf are scalars, return IsotropicSolver.

---

These are just my first thoughts, I will probably come back and edit or add more soon :)

Thanks again!

[kabume]

The method of submodules looks so clear! I'll try it soon.

Besides, the geometry function may add a dielectric averaging function to get more accurate results, like this slide.

[sp94]

Good idea, I've opened an issue for dielectric smoothing
#22