Transfer Matrix Method (1D)

The Transfer Matrix Method (TMM) provides exact solutions for wave propagation in 1D multilayer structures. PhoXonic.jl supports TMM for both photonic and phononic crystals.

Overview

TMM uses an API similar to PWE, making it easy to compare results between the two methods. Both use the same wave types (Photonic1D, Longitudinal1D) and return compatible BandStructure objects, enabling direct validation of 1D calculations.

TMM is particularly useful for:

• Computing transmission and reflection spectra of multilayer structures
• Analyzing Bragg mirrors and Fabry-Pérot cavities
• Band structure calculation for 1D periodic systems
• Handling oblique incidence and polarization effects
• Modeling lossy materials (complex permittivity)

Quick Start

Photonic Bragg Mirror

See also: examples/601_tmm_bragg_mirror.jl

using PhoXonic

# Define materials
mat_hi = Dielectric(2.5^2)  # n = 2.5
mat_lo = Dielectric(1.45^2) # n = 1.45

# Design wavelength
λ0 = 1.55

# Quarter-wave layers
d_hi = λ0 / (4 * 2.5)
d_lo = λ0 / (4 * 1.45)

# Create multilayer
unit_cell = [Layer(mat_hi, d_hi), Layer(mat_lo, d_lo)]
ml = periodic_multilayer(unit_cell, 10)

# Create solver
solver = TMMSolver(Photonic1D(), ml)

# Compute transmission spectrum
λ_values = range(0.8*λ0, 1.2*λ0, length=201)
R, T = tmm_spectrum(solver, collect(λ_values))

Phononic Superlattice

See also: examples/603_tmm_phononic.jl

using PhoXonic

# Define elastic materials
steel = IsotropicElastic(ρ=7800.0, λ=115e9, μ=82e9)
epoxy = IsotropicElastic(ρ=1180.0, λ=4.43e9, μ=1.59e9)

# Create multilayer
unit_cell = [Layer(steel, 0.005), Layer(epoxy, 0.005)]
ml = periodic_multilayer(unit_cell, 10)

# Create solver for longitudinal waves
solver = TMMSolver(Longitudinal1D(), ml)

# Compute band structure
bands = tmm_bandstructure(solver; k_points=51, bands=1:5)

Layer and Multilayer Types

Layer

A single layer is defined by its material and thickness:

Layer(material, thickness)

Supported materials:

• Dielectric(ε) - lossless dielectric
• LossyDielectric(ε) - complex permittivity for absorption
• IsotropicElastic(ρ, λ, μ) - elastic material for phononic waves

Multilayer

A complete structure including surrounding media:

# Manual construction
Multilayer(layers::Vector{Layer}, left_material, right_material)

# Periodic structure
periodic_multilayer(unit_cell::Vector{Layer}, n_periods)

TMMSolver

The solver is created by specifying the wave type and structure:

# Photonic (electromagnetic) waves
solver = TMMSolver(Photonic1D(), multilayer)

# Phononic (longitudinal elastic) waves
solver = TMMSolver(Longitudinal1D(), multilayer)

Transmission Spectrum

Normal Incidence

# Single wavelength
result = tmm_spectrum(solver, λ)
# Returns: result.R (reflectivity), result.T (transmissivity)

# Multiple wavelengths
R, T = tmm_spectrum(solver, λ_values)
# Returns: vectors of R and T values

Oblique Incidence (Photonic only)

# TE polarization (s-polarized)
result = tmm_spectrum(solver, λ; angle=π/6, polarization=:TE)

# TM polarization (p-polarized)
result = tmm_spectrum(solver, λ; angle=π/6, polarization=:TM)

Features:

• Handles total internal reflection automatically
• Brewster angle effects for TM polarization
• Energy conservation: R + T = 1 (for lossless materials)

Band Structure

Compute the dispersion relation for periodic structures:

bands = tmm_bandstructure(solver; k_points=51, bands=1:5)

The result is a BandStructure object compatible with PWE results:

• bands.k_points - wave vectors along the path
• bands.frequencies - eigenfrequencies (kpoints × nbands matrix)
• bands.k_labels - high-symmetry point labels

Bloch's Theorem

For a periodic structure with period a, the eigenfrequencies satisfy:

cos(ka) = Tr(M) / 2

where M is the transfer matrix for one period. Bandgaps occur where |Tr(M)/2| > 1.

Lossy Materials

TMM supports complex permittivity for modeling absorption:

# Complex permittivity: ε = ε' + iε''
mat_lossy = LossyDielectric(complex(4.0, 0.1))

# Use in multilayer structure
ml = Multilayer([Layer(mat_lossy, d)], mat_air, mat_air)
solver = TMMSolver(Photonic1D(), ml)

# For lossy materials: R + T + A = 1
result = tmm_spectrum(solver, λ)
A = 1 - result.R - result.T  # Absorption

Note: LossyDielectric is only supported for TMM, not PWE.

Examples

Bragg Mirror

See also: examples/601_tmm_bragg_mirror.jl

Transmission spectrum of a dielectric Bragg mirror with varying number of layer pairs.

unit_cell = [Layer(mat_hi, d_hi), Layer(mat_lo, d_lo)]
ml = periodic_multilayer(unit_cell, 20)
solver = TMMSolver(Photonic1D(), ml)

λ_values = range(0.8*λ0, 1.2*λ0, length=201)
R, T = tmm_spectrum(solver, collect(λ_values))

Fabry-Pérot Cavity

See also: examples/602_tmm_fabry_perot.jl

Resonance behavior of a Fabry-Pérot cavity with Bragg mirrors.

# Symmetric structure: (HL)^N HH (LH)^N
left_layers = vcat([l for _ in 1:n_pairs for l in left_mirror_cell]...)
defect_layer = Layer(mat_hi, 2*d_hi)
right_layers = vcat([l for _ in 1:n_pairs for l in right_mirror_cell]...)

all_layers = vcat(left_layers, [defect_layer], right_layers)
ml = Multilayer(all_layers, mat_lo, mat_lo)

Phononic Superlattice

See also: examples/603_tmm_phononic.jl

Steel/Epoxy multilayer for acoustic wave bandgaps.

steel = IsotropicElastic(ρ=7800.0, λ=115e9, μ=82e9)
epoxy = IsotropicElastic(ρ=1180.0, λ=4.43e9, μ=1.59e9)

unit_cell = [Layer(steel, d_steel), Layer(epoxy, d_epoxy)]
ml = periodic_multilayer(unit_cell, 10)
solver = TMMSolver(Longitudinal1D(), ml)

bands = tmm_bandstructure(solver; k_points=51, bands=1:4)

PWE vs TMM Comparison

See also: examples/604_tmm_vs_pwe.jl

Verifies consistency between PWE and TMM for 1D structures.

# TMM
solver_tmm = TMMSolver(Photonic1D(), ml)
bands_tmm = tmm_bandstructure(solver_tmm; k_points=21, bands=1:5)

# PWE
solver_pwe = Solver(Photonic1D(), geo, 256; cutoff=20)

Comparison with PWE

For 1D periodic structures, TMM and PWE should give identical band structures within numerical precision.

API Reference

Types

• Layer{M} - single layer with material type M
• Multilayer{M} - complete multilayer structure
• TMMSolver{W,M} - solver for wave type W and material type M

Functions

• tmm_spectrum(solver, λ) - compute R, T at wavelength λ
• tmm_spectrum(solver, λ_values) - compute R, T spectrum
• tmm_bandstructure(solver; ...) - compute band structure
• periodic_multilayer(unit_cell, n) - create periodic structure
• refractive_index(material) - get refractive index
• acoustic_impedance(material) - get acoustic impedance (phononic)

See Also

• Core API - Dielectric, IsotropicElastic materials