Correct realizations of hexagonal lattice using MPB?

[knifelees3]

I want to realize the real hexagonal lattice using MPB, there will be six rods in the unit cell，

import math
import meep as mp
from meep import mpb
import numpy as np
import matplotlib.pyplot as plt

num_bands = 4

# 定义好扫描的k波矢
k_points = [mp.Vector3(),          # Gamma
            mp.Vector3(1/3,2/3),       # K
            mp.Vector3(0.5,0.5),  # M
            mp.Vector3()]          # Gamma

# 对k波矢进行插值
k_points = mp.interpolate(40, k_points)

#  units of nm
a=1



# 基本结构和材料性质
# 折射率设置
nslab = 2.48
slab = mp.Medium(index=nslab)

nair = 1
air = mp.Medium(index=nair)



s=0.25
geometry = [mp.Cylinder(0.1,center=mp.Vector3(-1*s,0), material=air),
            mp.Cylinder(0.1,center=mp.Vector3(-0.5*s,math.sqrt(3)/2*s), material=air),
            mp.Cylinder(0.1,center=mp.Vector3(0.5*s,math.sqrt(3)/2*s), material=air),
            mp.Cylinder(0.1,center=mp.Vector3(1*s,0), material=air),
           mp.Cylinder(0.1,center=mp.Vector3(0.5*s,-math.sqrt(3)/2*s), material=air),
            mp.Cylinder(0.1,center=mp.Vector3(-0.5*s1,-math.sqrt(3)/2*s), material=air)]



geometry_lattice = mp.Lattice(
    size=mp.Vector3(1, 1),
    basis1=mp.Vector3(0.5, np.sqrt(3) / 2),
    basis2=mp.Vector3(1,0),
)


resolution = 32

# mode solver设置
ms = mpb.ModeSolver(num_bands=num_bands,
                    k_points=k_points,
                    geometry=geometry,
                    geometry_lattice=geometry_lattice,
                    default_material=slab,
                    resolution=resolution)

ms.run_te()
te_freqs = ms.all_freqs
te_gaps = ms.gap_list
md = mpb.MPBData(rectify=True, periods=1, resolution=64)
eps = ms.get_epsilon()
converted_eps = md.convert(eps)

plt.imshow(converted_eps.T, interpolation='spline36', cmap='binary')
plt.axis('off')
plt.show()

However, the generated material distribution is not correct (I have used rectify=True to correct the distributions ), How can I resolve this?

[stevengj]

Technically, this is a honeycomb structure in a hexagonal lattice. The MPB primitive unit cell is a rhombus (not a hexagon — that is a Wigner–Seitz cell, whereas in MPB you need to specify the primitive lattice vectors, which give a parallelogram/parallelopiped), and has two holes per primitive cell.

(You also need to understand MPB's coordinate system: the geometry centers are not Cartesian coordinates, they are in the basis of the lattice vectors.*)

You should start with the triangular lattice (i.e. hexagonal lattice) tutorial and add a second hole to the unit cell.

Actually, I just remembered that there is an MPB example file for a honeycomb lattice, albeit in Scheme — should be easy for you to copy from.

[stevengj]

PS. Even if you use a hexagonal Wigner–Seitz cell, there are still only two cylinders per primitive cell. In your diagram, you drew your hexagons too big — the primitive Wigner–Seitz cell is a hexagon whose corners are the centers of the cylinders. So, the cell contains only 1/3 of each cylinder, and 6/3 = 2 whole cylinders per Wigner–Seitz cell.

[Ron-007-git]

Hello, did you find the solution of the above problem? Why do these hexagons look distorted?

[stevengj]

By default, all vectors in MPB are in the basis of the lattice vectors. The default edges of a block are e1 = (1,0) and e2 = (0,1) — but again, these are in the lattice basis, which is not Cartesian in your case.

All you need to do is to call the cartesian_to_lattice function to convert your desired edge orientations to lattice coordinates, and set these as the e1 and e2 directions of the block.

[Ron-007-git]

Thank you for your reply. Now the blocks look fine but they seem too be rotated. It would be really helpful if you could point out where I went wrong.

geometry_lattice = mp.Lattice(
    size=mp.Vector3(1, 1),
    basis1=mp.Vector3(0.5, np.sqrt(3) / 2),
    basis2=mp.Vector3(1,0),
)



e1 = mp.cartesian_to_lattice(mp.Vector3(1, 0, 0), geometry_lattice)
e2 = mp.cartesian_to_lattice(mp.Vector3(0, 1, 0), geometry_lattice)
size=mp.Vector3(0.2, 0.2)

geometry = [mp.Block(size, e1, e2, center=mp.Vector3(-0.25/2, -np.sqrt(3)/4/2), material=air),
            mp.Block(size, e1, e2, center=mp.Vector3(0.25/2,np.sqrt(3)/4/2), material=air)
                     ]

[knifelees3]

Hello, did you find the solution of the above problem? Why do these hexagons look distorted?

I didn' t find a solution

[stevengj]

Looks like you also want to change the center to use cartesian_to_lattice?