Bragg filter

Reproducing an example of [1] (and soon [2])

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from collections import OrderedDict

import matplotlib.pyplot as plt
import numpy as np
import shapely
from skfem import Basis, ElementTriP0
from skfem.io import from_meshio

from femwell.mesh import mesh_from_OrderedDict
from femwell.mode_solver_2d_periodic import plot_periodic, solve_periodic

height = 1
a = 0.330
b = 0.7
c = 0.2

k0 = 0.7 / a  # 1.05/a

left = shapely.LineString([(0, y) for y in np.linspace(-height, height, 20)])
right = shapely.LineString([(a, y) for y in np.linspace(-height, height, 20)])
top = shapely.LineString([(x, height) for x in np.linspace(0, a, 2)])
bottom = shapely.LineString([(x, -height) for x in np.linspace(0, a, 2)])

box = shapely.box(0, -height, a, height)
structure = shapely.box(0, -b / 2, a, b / 2)
hole = shapely.box(a / 4, -c / 2, a / 4 * 3, c / 2)

resolutions = {"hole": {"resolution": 0.1, "distance": 1}}

mesh = from_meshio(
    mesh_from_OrderedDict(
        OrderedDict(
            left=left,
            right=right,
            top=top,
            bottom=bottom,
            hole=hole,
            structure=structure,
            box=box,
        ),
        resolutions=resolutions,
        filename="mesh.msh",
        default_resolution_max=0.05,
        periodic_lines=[("left", "right")],
    )
)

basis_epsilon_r = Basis(mesh, ElementTriP0(), intorder=4)

epsilon_r = basis_epsilon_r.zeros(dtype=np.complex64) + 1.45
epsilon_r[basis_epsilon_r.get_dofs(elements="structure")] = 3.5
epsilon_r **= 2
basis_epsilon_r.plot(np.real(epsilon_r), ax=mesh.draw(), colorbar=True).show()

ks, basis_phi, phis = solve_periodic(basis_epsilon_r, epsilon_r, k0)


idx = np.abs(np.imag(ks * a)) < 0.5
ks = ks[idx]
phis = phis[:, idx]

# print(ks)

# plt.plot(np.real(ks))
# plt.plot(np.imag(ks))
# plt.show()

for i, k in enumerate(ks):
    fig, axs = plt.subplots(1, 2, figsize=(13, 5), gridspec_kw={"width_ratios": [1, 10]})
    mesh.draw(ax=axs[0], boundaries=True, boundaries_only=True)
    basis_phi.plot(np.real(phis[..., i]), shading="gouraud", colorbar=True, ax=axs[0])
    axs[0].set_aspect(1)
    plt.title(f"{k*a}")
    # axs[0].set_aspect(1)
    plot_periodic(k, a, basis_phi, phis[..., i], 10, axs[1])
    plt.show()

Bibliography

[1]

Jelena Notaros and Miloš A. Popović. Finite-difference complex-wavevector band structure solver for analysis and design of periodic radiative microphotonic structures. Optics Letters, 40(6):1053, March 2015. URL: https://doi.org/10.1364/ol.40.001053, doi:10.1364/ol.40.001053.

[2]

Chris Fietz, Yaroslav Urzhumov, and Gennady Shvets. Complex k band diagrams of 3d metamaterial/photonic crystals. Optics Express, 19(20):19027, September 2011. URL: http://dx.doi.org/10.1364/OE.19.019027, doi:10.1364/oe.19.019027.