MPB Triangular Lattice Band Structure

This example computes the photonic band structure for a triangular lattice of air holes in a dielectric, demonstrating a complete photonic band gap (gap in both TE and TM polarizations).

Overview

Triangular hole lattices are important because:

Complete band gap: Gap for all polarizations and directions
Practical fabrication: Easier to etch holes than rods
Photonic crystal fibers: Hollow-core guidance
2D slab waveguides: Silicon photonics

A complete band gap occurs when the TE and TM gaps overlap. This requires careful optimization of the hole radius and dielectric contrast.

Parameters

Parameter	Value	Description
`r`	0.45	Hole radius (optimal for complete gap)
`eps`	12	Dielectric constant (Si-like)
`resolution`	32	Grid resolution
`num_bands`	8	Number of bands

Triangular Lattice

The lattice vectors for a triangular lattice:

basis1: (√3/2, 1/2)
basis2: (√3/2, -1/2)

K-Point Path

High-symmetry path through the hexagonal Brillouin zone:

Γ → M → K → Γ

Code Structure

from meep import mpb
import math

# Triangular lattice
geometry_lattice = mp.Lattice(
    size=mp.Vector3(1, 1),
    basis1=mp.Vector3(math.sqrt(3) / 2, 0.5),
    basis2=mp.Vector3(math.sqrt(3) / 2, -0.5),
)

# Air hole in dielectric
geometry = [mp.Cylinder(r=0.45, material=mp.air)]
default_material = mp.Medium(epsilon=12)

ms = mpb.ModeSolver(
    geometry_lattice=geometry_lattice,
    geometry=geometry,
    default_material=default_material,
    k_points=k_points,
    resolution=32,
    num_bands=8,
)

ms.run_te()
ms.run_tm()

Band Gap Analysis

For r=0.45 and ε=12:

TE gap: Between bands 1-2
TM gap: Overlaps with TE gap
Complete gap: Non-zero gap-midgap ratio

MPB Square Rods - Square lattice
Holey Waveguide Bands - Waveguide in photonic crystal

MPB Triangular Holes