1D Calculations

1D photonic and phononic crystal calculations for multilayer structures (Bragg reflectors, superlattices).

1D Wave Types

Wave Type	Field	Description
`Photonic1D`	E	Electromagnetic wave in dielectric multilayer
`Longitudinal1D`	u	Longitudinal elastic wave (P-wave) in elastic multilayer

PWE vs TMM

PhoXonic provides two methods for 1D band structure:

Method	Description	Best For
PWE (Plane Wave Expansion)	Fourier-based eigenvalue solver	Band structure, eigenmodes
TMM (Transfer Matrix)	Exact matrix propagation	Transmission spectra, finite structures

Both methods give identical band structures for periodic systems. TMM additionally provides transmission/reflection spectra.

1D Photonic Crystal

Bragg Reflector

A quarter-wave stack of two dielectric materials.

using PhoXonic

lat = lattice_1d(1.0)
mat1 = Dielectric(1.0)
mat2 = Dielectric(12.0)

# Quarter-wave stack: d1/d2 = n2/n1
d1 = 0.5 * sqrt(12) / (1 + sqrt(12))
geo = Geometry(lat, mat1, [(Segment(0.0, d1), mat2)])

solver = Solver(Photonic1D(), geo, 128; cutoff=20)

# Compute bands
kpath = simple_kpath_1d(; a=1.0, npoints=50)
bands = compute_bands(solver, kpath; bands=1:5)

Joannopoulos Ch4 Benchmark

Reproduces Figure 2 from "Photonic Crystals: Molding the Flow of Light" Chapter 4.

1D Phononic Crystal

Elastic Superlattice

Alternating layers of two elastic materials (e.g., steel/epoxy).

using PhoXonic

lat = lattice_1d(0.01)  # 1 cm period
steel = IsotropicElastic(ρ=7800.0, λ=115e9, μ=82e9)
epoxy = IsotropicElastic(ρ=1180.0, λ=4.43e9, μ=1.59e9)
geo = Geometry(lat, epoxy, [(Segment(0.0, 0.5), steel)])

solver = Solver(Longitudinal1D(), geo, 128; cutoff=20)

# Compute bands
kpath = simple_kpath_1d(; a=0.01, npoints=50)
bands = compute_bands(solver, kpath; bands=1:5)

Iterative Solvers

1D problems support all solver methods:

# Dense (default)
solver = Solver(Photonic1D(), geo, 128; cutoff=20)

# KrylovKit
solver = Solver(Photonic1D(), geo, 128, KrylovKitMethod(); cutoff=20)

# LOBPCG
solver = Solver(Photonic1D(), geo, 128, LOBPCGMethod(); cutoff=20)

For phononic problems with large eigenvalues, KrylovKitMethod applies automatic scaling.