Analysis and Post-Processing

Convergence Check

for res in [32, 64, 128]
    solver = Solver(TMWave(), geo, (res, res); cutoff=7)
    bands = compute_bands(solver, kpath; bands=1:4)
    gap = find_bandgap(bands, 1, 2)
    println("Resolution $res: gap = $(gap.gap)")
end

Group Velocity

k = [0.5, 0.0]  # Point in BZ
vg = group_velocity(solver, k; bands=1:4)
# vg[i] = [∂ω/∂kx, ∂ω/∂ky] for band i

Band Tracking

When bands cross or come close together, eigenvalue sorting by frequency can cause apparent discontinuities in the band structure. The track_bands option uses eigenvector overlap to maintain band continuity.

# Without tracking (default): bands sorted by frequency at each k-point
bands = compute_bands(solver, kpath; bands=1:6)

# With tracking: bands tracked using eigenvector overlap
bands = compute_bands(solver, kpath; bands=1:6, track_bands=true)

How it works:

1. At each k-point, compute overlap matrix between eigenvectors of adjacent k-points
2. Find optimal permutation that maximizes diagonal overlap
3. Reorder bands to maintain continuity

When to use:

• 3D calculations where band crossings are common
• Near degeneracy points
• When smooth band dispersion is needed for further analysis (e.g., group velocity)

Example: 3D FCC phononic crystal

lat = fcc_lattice(1.0)
geo = Geometry(lat, Epoxy, [(Sphere([0,0,0], 0.2), WC)])
solver = Solver(FullElastic(), geo, (12,12,12), KrylovKitMethod(); cutoff=3)

kpath = simple_kpath_fcc()
bands = compute_bands(solver, kpath; bands=1:12, track_bands=true)

K-path Options

Simple k-paths

kpath = simple_kpath_square(a=1.0, npoints=30)    # Γ-X-M-Γ
kpath = simple_kpath_hexagonal(a=1.0, npoints=30) # Γ-M-K-Γ

Brillouin.jl integration

using Brillouin

# Automatic high-symmetry path
kpi = irrfbz_path(2, lat.vectors)
kpath = kpath_from_brillouin(kpi, npoints=100)

Plotting

Requires Plots.jl:

using Plots
plot_bands(bands)

Saving and Loading Results

PhoXonic.jl uses JLD2.jl for saving and loading results.

Bands

# Save
bands = compute_bands(solver, kpath)
save_bands("bands.jld2", bands)

# Load
bands_loaded = load_bands("bands.jld2")

Eigenmodes

# Save with optional k-point and frequencies
ω, modes = solve(solver, k; bands=1:5)
save_modes("modes.jld2", modes; k=k, frequencies=ω)

# Load
data = load_modes("modes.jld2")
# data.modes, data.k, data.frequencies

Material Arrays

# Photonic: saves ε, ε⁻¹, μ
save_epsilon("epsilon.jld2", solver)

# Phononic: saves C11, C12, C44, ρ
save_epsilon("elastic.jld2", solver_phononic)

# Load
eps = load_epsilon("epsilon.jld2")
# eps.ε, eps.resolution, eps.wave_type

File Contents

API Reference

• Solver API - BandStructure, findbandgap, groupvelocity
• Plotting API - plot_bands (requires Plots.jl)
• I/O API - save/load functions (requires JLD2.jl)