Berreman

F# implementation of the Berreman matrix method for analyzing electromagnetic wave propagation through stratified (layered) optical media.

Overview

The Berreman matrix is a 4x4 transfer matrix method used in computational electromagnetics to calculate reflection and transmission coefficients in multilayer thin films and anisotropic crystals. This library provides a comprehensive, type-safe implementation suitable for research and engineering applications in thin film optics, crystal optics, polarization analysis, and related fields.

Features

• Full Berreman 4x4 matrix implementation for anisotropic and magneto-optically active media
• Support for arbitrary optical properties: electric permittivity (epsilon), magnetic permeability (mu), and magneto-electric coupling (rho)
• Multilayer thin film calculations with arbitrary number of layers
• Thick plate and wedge substrate handling with multiple internal reflections
• Dispersive materials with wavelength-dependent optical properties
• Complete polarization analysis: Stokes parameters, Jones vectors, Mueller matrices, ellipticity, azimuth
• Active (magneto-optic) crystals: planar, cubic (point group 23), type 3/4/6, and type 32/42/62 crystals
• Gaussian beam support via FFT-based propagation
• Type-safe units of measure (nm, um, mm, meter)
• Interactive visualization with Plotly.NET charts
• Comprehensive test suite with property-based testing

Installation

Requirements

• .NET 9.0 SDK
• x64 platform

Building

dotnet build Berreman.sln -c Release

Project Structure

Berreman/
├── Berreman/                    # Core library (Softellect.Berreman.Core)
│   └── Berreman/
│       ├── MathNetNumericsMath.fs   # Math abstractions and wrappers
│       ├── FourierTransform.fs      # FFT for Gaussian beams
│       ├── MatrixExp.fs             # Matrix exponential (Pade approximation)
│       ├── Constants.fs             # Physical constants and units
│       ├── Geometry.fs              # Vector/matrix types, rotations
│       ├── MaterialProperties.fs    # Optical properties (eps, mu, rho)
│       ├── Fields.fs                # Electromagnetic field definitions
│       ├── Media.fs                 # Optical media and systems
│       ├── BerremanMatrix.fs        # Core 4x4 matrix implementation
│       ├── Solvers.fs               # Reflection/transmission solvers
│       ├── Dispersion.fs            # Wavelength-dependent properties
│       └── FieldFunctions.fs        # Field analysis functions
├── OpticalProperties/           # Material definitions (Softellect.Berreman.OpticalProperties)
│   ├── Standard.fs                  # Standard materials (glass, crystals)
│   ├── Active.fs                    # Magneto-optic active crystals
│   └── Dispersive.fs                # Dispersive materials (Langasite, Silicon)
├── Analytics/                   # Visualization and analysis (Softellect.Berreman.Analytics)
│   ├── Variables.fs                 # Ranged variables for parametric studies
│   ├── StandardLightVariables.fs    # Common light configurations
│   ├── StandardSystems.fs           # Pre-defined optical systems
│   ├── Charting.fs                  # Plotly.NET integration
│   └── Examples/                    # Example scripts (see below)
├── BerremanTests/               # Test suite (xUnit, FsCheck)
├── BerremanRunner/              # Console application
└── BerremanModelGenerator/      # Model generation utility

Core Concepts

Optical Properties

Materials are defined by three 3x3 complex tensors:

• Eps (epsilon): Electric permittivity tensor
• Mu: Magnetic permeability tensor
• Rho: Magneto-electric coupling tensor (for active crystals)

// Isotropic glass with refractive index 1.52
let glass = OpticalProperties.fromRefractionIndex (RefractionIndex 1.52)

// Uniaxial crystal
let uniaxial = OpticalProperties.uniaxialCrystal e11 e33

// Active planar crystal with magneto-electric effect
let active = OpticalProperties.planarCrystal e11 e33 g12

Optical Systems

An optical system consists of:

• Upper medium: Semi-infinite medium where light originates
• Films: List of thin film layers with thickness
• Substrate: Optional thick plate or wedge
• Lower medium: Semi-infinite exit medium

let system = {
    description = Some "Multilayer coating"
    upper = OpticalProperties.vacuum
    films = [
        { properties = glass; thickness = Thickness.nm 100.0 }
        { properties = vacuum; thickness = Thickness.nm 150.0 }
    ]
    lower = OpticalProperties.vacuum
}

Incident Light

Light is characterized by:

• Wavelength: With unit safety (nm, um, mm)
• Incidence angle: From normal (0 degrees) to grazing
• Polarization: s, p, or arbitrary angle
• Ellipticity: For elliptically polarized light

let light = {
    wavelength = WaveLength.nm 600.0
    incidenceAngle = IncidenceAngle.normal
    polarization = Polarization.s
    ellipticity = Ellipticity.zero
}

Output Functions

Available output quantities:

• R, T: Total reflectance and transmittance
• Rs, Rp, Ts, Tp: s- and p-polarized components
• Is, Ip: Incident intensity components
• AzimuthR, AzimuthT: Polarization azimuth angles
• EllipticityR, EllipticityT: Polarization ellipticity

Examples

The Analytics/Examples/ folder contains F# script files demonstrating various optical scenarios. To run an example:

1. Build the solution in Release mode
2. Open the .fsx file in an F# interactive environment (FSI)
3. Execute the script

Glass.fsx

Simple glass substrate and film analysis

Demonstrates the difference between treating glass as a thin film vs. a thick substrate. Compares reflectance and transmittance as functions of incidence angle for a glass layer (n=1.78) at 600nm wavelength.

Key features:

• Thin film interference effects vs. incoherent thick plate behavior
• Incidence angle dependence (0 to 90 degrees)
• Wavelength dependence (300-700nm)

Glass_Dispersive.fsx

Dispersive glass with wavelength-dependent refractive index

Uses a Sellmeier-type dispersion formula for the ordinary refractive index:

n^2 = 1 + 1.43*lambda^2/(lambda^2 - 0.073^2) + 0.65*lambda^2/(lambda^2 - 0.12^2)

Demonstrates how to define custom dispersive materials and plot refractive index vs. wavelength.

MultilayerThinFilm.fsx

Multilayer dielectric coating (anti-reflection)

A 40-layer alternating glass/vacuum quarter-wave stack designed for 600nm. Each layer is lambda/(4n) thick for its material:

• Glass layers (n=1.52): 98.7nm
• Vacuum layers (n=1.0): 150nm

Shows characteristic interference patterns in reflectance vs. wavelength and incidence angle.

MultilayerThinFilm_02.fsx

High-reflectance multilayer mirror

100 layer pairs of alternating high/low index materials (n=1.75/1.50) on glass substrate. Demonstrates:

• High reflectance at design wavelength
• Spectral bandwidth engineering
• Angular dependence of multilayer mirrors

MultilayerThinFilm_EUV.fsx

EUV (Extreme Ultraviolet) multilayer mirror

100 Mo/Si layer pairs for 10.6nm (EUV) wavelength using complex refractive indices:

• Molybdenum: absorption and phase shift at EUV
• Silicon: spacer layers

Used in EUV lithography mirrors and X-ray optics.

MultilayerThinFilm_EUV__002.fsx

Custom EUV mirror parameters

Similar to above but with user-defined delta and beta values for the complex refractive index:

n = (1 - delta) + i*beta

LangasiteOnSilicon.fsx

Dispersive active crystal (Langasite) on Silicon substrate

Langasite (La3Ga5SiO14) is an optically active piezoelectric crystal used in sensors. This example:

• Uses full dispersive optical properties from literature
• Shows thin film vs. thick substrate comparison
• Demonstrates wavelength-dependent behavior (250-600nm)

ActiveCrystal.fsx

Magneto-optically active planar crystal plate

Demonstrates optical activity (polarization rotation) in crystals with non-zero rho tensor:

• Uniaxial crystal (n11=2.315, n33=2.226)
• Variable gyration coefficient g12
• Plots s- and p-polarized transmission vs. g12

ActiveCrystalComparison.fsx

Comparison of different active crystal types

Compares three crystal symmetry classes:

• Planar crystal: rho has only g12 component
• Type 3,4,6 crystal: rho has g11 and g33 components
• Type 32,42,62 crystal: rho has g11, g12, and g33 components

Shows transmitted azimuth and ellipticity as functions of incidence angle.

TotalRefl_Glass.fsx

Total internal reflection at glass/vacuum interface

Light traveling from glass (n=1.50) to vacuum demonstrates:

• Critical angle calculation
• Brewster angle identification
• s- and p-polarized reflectance behavior

Wedge_Glass.fsx

Glass wedge prism

Analyzes a glass prism with 40-degree wedge angle:

• Reflectance and transmittance vs. wedge angle (0-85 degrees)
• Handles multiple internal reflections

Wedge_BiaxialCrystal.fsx

Biaxial crystal wedge

Similar to glass wedge but with anisotropic biaxial crystal (nx=1.5, ny=1.65, nz=1.75):

• Demonstrates birefringence effects in wedge geometry
• 45-degree polarized input to excite both eigenmodes

Wedge_ActiveCrystal.fsx

Active planar crystal wedge

Combines wedge geometry with optical activity:

• 23-degree wedge of planar active crystal
• Variable gyration coefficient analysis
• Useful for polarimetric device design

Wedge_ActiveCrystalComparison.fsx

Comparison of active crystal wedges

Compares wedge behavior for three crystal symmetry classes:

• Transmitted intensity and polarization state
• Azimuth and ellipticity of transmitted light
• 23-degree wedge geometry

API Reference

Key Types

Type	Description
WaveLength	Wavelength with units (nm, um, mm)
Thickness	Layer thickness with units
IncidenceAngle	Angle of incidence
Polarization	Linear polarization angle
Ellipticity	Elliptical polarization parameter
OpticalProperties	Material properties (eps, mu, rho)
Film	Thin film layer
OpticalSystem	Complete optical system
IncidentLightInfo	Light source specification

Key Functions

// Create optical system info
let info = { incidentLightInfo = light; opticalSystem = system.dispersive }

// Plot results
plot info [ R; T ] incidenceAngleRange
plot info [ Rs; Rp; Ts; Tp ] wavelengthRange
plotComparison [ system1; system2 ] [ R; T ] variable
plot3D info [ R ] xRange yRange

Dependencies

• MathNet.Numerics: Linear algebra and numerical methods (customized version included)
• Plotly.NET: Interactive charting
• FSharp.Collections.ParallelSeq: Parallel processing
• xUnit: Testing framework
• FsCheck: Property-based testing
• FluentAssertions: Test assertions

License

This project is licensed under the GNU General Public License v3.0. See the LICENSE file for details.

Individual components may have different licenses:

• MatrixExp.fs: LGPL (ported from C code)
• Core library packages: MIT license

Author

Konstantin Konstantinov Softellect Systems, Inc.

Repository

https://github.com/kkkmail/Berreman

References

The Berreman matrix method is described in:

• D. W. Berreman, "Optics in Stratified and Anisotropic Media: 4x4-Matrix Formulation," J. Opt. Soc. Am. 62, 502-510 (1972)

Version

Current version: 9.0.100.5 (targeting .NET 9.0)