3D Calculations

3D photonic and phononic crystal calculations have special considerations due to the vector nature of the fields and the presence of spurious modes.

3D Wave Types

3D Wave Polarizations

Wave Type	Field Components	Description
`TransverseEM`	H_⊥ (2N basis)	Recommended for 3D photonic crystals
`FullVectorEM`	Hx, Hy, H_z	Full vector H-field (3N basis, includes spurious modes)
`FullElastic`	ux, uy, u_z	Full elastic (1 P + 2 S modes per k)

3D Photonic Crystals

TransverseEM vs FullVectorEM

For 3D photonic crystal calculations, TransverseEM is strongly recommended over FullVectorEM:

Feature	TransverseEM	FullVectorEM
Matrix size	2N × 2N	3N × 3N
Spurious modes	None (∇·H = 0 enforced)	N longitudinal modes at ω ≈ 0
Shift required	No	Yes (to filter spurious modes)
Memory usage	~44% less	Full

TransverseEM expands the H-field in a transverse polarization basis where each plane wave has two orthonormal polarization vectors e₁, e₂ perpendicular to (k+G). This automatically satisfies the transversality constraint ∇·H = 0, eliminating spurious longitudinal modes.

H-field Formulation

The 3D implementation uses the H-field formulation:

∇ × (ε⁻¹ ∇ × H) = (ω/c)² μ H

TransverseEM (Recommended)

TransverseEM expands the H-field in a transverse polarization basis:

H_G = h₁(G) e₁(k+G) + h₂(G) e₂(k+G)

where e₁ and e₂ are orthonormal vectors perpendicular to (k+G). This construction automatically satisfies ∇·H = 0, eliminating all spurious longitudinal modes.

# TransverseEM: no spurious modes, no shift needed
solver = Solver(TransverseEM(), geo, (16, 16, 16), DenseMethod(); cutoff=5)

FullVectorEM (Legacy)

FullVectorEM uses Cartesian components (Hx, Hy, Hz), producing:

N longitudinal modes with ω ≈ 0 (unphysical, violate ∇·H = 0)
2N transverse modes with physical frequencies

When using FullVectorEM, use shift > 0 to filter out spurious modes:

# DenseMethod: post-hoc filtering
solver = Solver(FullVectorEM(), geo, (12, 12, 12), DenseMethod(shift=0.01); cutoff=7)

# KrylovKitMethod: shift-and-invert
solver = Solver(FullVectorEM(), geo, (16, 16, 16), KrylovKitMethod(shift=0.01); cutoff=7)

See Solver Methods for details on shift-and-invert.

Examples

FCC Lattice with Spheres

MPB Benchmark

PhoXonic 3D calculations have been validated against MIT Photonic Bands (MPB).

cutoff	Plane waves	Error vs MPB
3	123	24%
5	515	7%
7	1419	< 1%

Cutoff Convergence

For 3D photonic crystals with high dielectric contrast (ε > 10), sufficient plane waves are required for accurate band structure calculation.

Recommended cutoff values:

Dielectric contrast	Minimum cutoff
ε < 4	3–5
4 ≤ ε < 10	5–7
ε ≥ 10	7+

Example convergence test:

for cutoff in [3, 5, 7]
    solver = Solver(TransverseEM(), geo, (16,16,16), DenseMethod(); cutoff=cutoff)
    bands = compute_bands(solver, kpath; bands=1:6)
    println("cutoff=$cutoff: ω₁ = $(bands.frequencies[1,1])")
end

Note: Higher cutoff increases computation time significantly.

3D Phononic Crystals

FullElastic (Experimental)

FullElastic is an experimental implementation for 3D phononic crystal calculations. It computes the full 3D elastic wave equation with three displacement components (ux, uy, u_z), producing 3 modes per k-point (1 longitudinal P-wave + 2 transverse S-waves).

lat = cubic_lattice(0.01)  # 1 cm period
epoxy = IsotropicElastic(ρ=1180.0, λ=4.43e9, μ=1.59e9)
steel = IsotropicElastic(ρ=7800.0, λ=1.15e11, μ=8.28e10)
geo = Geometry(lat, epoxy, [(Sphere([0.0, 0.0, 0.0], 0.004), steel)])

solver = Solver(FullElastic(), geo, (16, 16, 16), KrylovKitMethod(shift=0.01); cutoff=5)

Why is FullElastic Challenging?

3D phononic calculations are more difficult than 3D photonic calculations for several reasons:

No transverse basis: Unlike TransverseEM for photonic crystals, there is no established transverse basis for elastic waves that eliminates spurious modes. Electromagnetic waves satisfy ∇·D = 0 (transverse condition), allowing a 2-component basis that automatically excludes spurious longitudinal modes. Elastic waves have both longitudinal (P) and transverse (S) components with no such simplification.
Large contrast in elastic moduli: Real materials have high elastic moduli contrast (e.g., Csteel / Cepoxy ~ 50), which causes numerical challenges regardless of the unit system.
Shift parameter sensitivity: Too small a shift fails to filter spurious modes; too large a shift may miss physical low-frequency modes. The optimal value depends on material properties and k-point.
High memory and computation time: 3N × 3N matrices with N > 1000 plane waves require significant resources.
Limited benchmarks: Unlike 3D photonic crystals (validated against MPB), 3D phononic benchmarks are less standardized in the literature.

Common Topics

3D K-paths

# FCC lattice
kpath = simple_kpath_fcc(a=1.0, npoints=20)

# Simple cubic
kpath = simple_kpath_cubic(a=1.0, npoints=20)

# BCC lattice
kpath = simple_kpath_bcc(a=1.0, npoints=20)

Memory Considerations

3D calculations require significantly more memory than 2D:

Resolution	Plane waves (cutoff=7)	Dense matrix memory
12×12×12	~500	~6 GB
16×16×16	~1400	~47 GB
20×20×20	~2700	~175 GB

For large 3D systems, use matrix-free methods. See Matrix-Free Methods.

API Reference

Solver API - TransverseEM, FullVectorEM, FullElastic, Solver methods
Advanced API - Matrix-free operators