Hyperbolic Optics Simulation Package

This package provides a comprehensive suite of tools to study the reflective properties of hyperbolic materials and anisotropic structures using the 4×4 transfer matrix method. It enables easy configuration of multilayer systems, calculation of reflection coefficients, and analysis using Mueller matrices.

Table of Contents

• Features
• Installation
• Quick Start
• Usage Examples
• Advanced Features
• Contributing
• Citation
• Known Issues / Limitations
• Papers & Further Reading
• License
• Getting Help

Features

• 4×4 transfer-matrix engine for reflection coefficients of arbitrary anisotropic multilayers, plus an opt-in numerically-stable scattering-matrix backend for thick / lossy / evanescent stacks
• Transmission, layer-resolved absorption, and field profiles (E, H, Sₙ vs depth) computed numerically from the Poynting flux — energy-conserving R + T + ΣA = 1
• Polarization toolkit: Mueller and Jones calculus, ellipsometry (Ψ/Δ), eigenpolarizations / exceptional points, co- vs cross-polarized power, and Poincaré-sphere trajectories
• Materials library: Quartz, Calcite, Sapphire, Ga₂O₃ (monoclinic), α-MoO₃ (biaxial), AlN, SiC, hBN, GaN — plus arbitrary permittivity/permeability tensors
• Scenarios: incident-angle, azimuthal, k-space dispersion (kₓ–k_y), full 3-D sweep, single-point — and a swept layer-thickness axis
• Fully vectorized over angle, azimuth, frequency, and thickness via a canonical batch convention
• Publication-quality plotting for spectra, k-space maps, and polarization

Installation

The package is now available on PyPI and can be installed using pip or uv:

Using pip

pip install hyperbolic-optics

Using uv (recommended for modern Python development)

uv add hyperbolic-optics

Development Installation

For development or to get the latest features:

git clone https://github.com/hyperbolic-optics/hyperbolic-optics.git
cd hyperbolic_optics
pip install -e .

Quick Start

Here's a simple example to get you started:

import json
from hyperbolic_optics.structure import Structure
from hyperbolic_optics.mueller import Mueller

# Define a simple multilayer structure
payload = {
    "ScenarioData": {
        "type": "Simple",
        "incidentAngle": 45.0,  # degrees
        "azimuthal_angle": 0.0,  # degrees
        "frequency": 1460.0      # cm^-1
    },
    "Layers": [
        {
            "type": "Ambient Incident Layer",
            "permittivity": 50.0
        },
        {
            "type": "Semi Infinite Anisotropic Layer",
            "material": "Calcite",
            "rotationX": 0,
            "rotationY": 90,
            "rotationZ": 0
        }
    ]
}

# Create and execute the simulation
structure = Structure()
structure.execute(payload)

# Calculate reflectivities
R_pp = abs(structure.r_pp)**2
R_ss = abs(structure.r_ss)**2
print(f"p-polarized reflectivity: {R_pp}")
print(f"s-polarized reflectivity: {R_ss}")

# Perform Mueller matrix analysis
mueller = Mueller(structure)
mueller.set_incident_polarization('linear', angle=45)
mueller.add_optical_component('anisotropic_sample')

# Get Stokes parameters and polarization properties
all_params = mueller.get_all_parameters()
print(f"Reflectance (S0): {all_params['S0']}")
print(f"Degree of polarization: {all_params['DOP']}")

Usage Examples

The examples/ folder contains simple scripts demonstrating various capabilities.

Run any example with:

python examples/calcite.py

Advanced Features

Custom Materials

Define materials with arbitrary permittivity and permeability tensors:

custom_material = {
    "eps_xx": {"real": 2.27, "imag": 0.001},
    "eps_yy": {"real": -4.84, "imag": 0.755}, 
    "eps_zz": {"real": -4.84, "imag": 0.755},
    "eps_xy": {"real": 0.0, "imag": 0.0},
    "eps_xz": {"real": 0.0, "imag": 0.0},
    "eps_yz": {"real": 0.0, "imag": 0.0},
}

Multiple Scenario Types

• Incident: Frequency vs incident angle analysis
• Azimuthal: Frequency vs azimuthal rotation analysis
• Dispersion: k-space dispersion at fixed frequency
• Simple: Single-point calculation for specific conditions

Built-in Visualization

from hyperbolic_optics.plots import plot_mueller_dispersion, plot_kx_frequency

# Generate publication-quality plots
plot_kx_frequency(structure, reflectivity, save_name="my_plot")

Contributing

We welcome contributions! Please see our contributing guidelines for details on:

• Reporting bugs and requesting features
• Setting up a development environment
• Code style and testing requirements
• Submitting pull requests

Citation

If you use this package in your research, please cite:

Software Citation

@software{cunningham2025hyperbolic,
  title={Hyperbolic Optics Simulation Package},
  author={Mark Cunningham},
  year={2026},
  version={0.3.0},
  doi={10.5281/zenodo.14946556},
  url={https://pypi.org/project/hyperbolic-optics/},
  howpublished={PyPI},
  note={Python package for 4×4 transfer matrix method simulations}
}

Related Publications

This package was used to generate results in:

M. Cunningham et al., "Optical footprint of ghost and leaky hyperbolic polaritons," Photonics Research, vol. 13, no. 8, pp. 2291-2305 (2025). DOI: 10.1364/PRJ.558334

Transmission, Absorption & Field Profiles

Reflection is computed automatically by Structure.execute. Power transmittance, layer-resolved absorption, and field profiles are computed numerically from the propagated fields (energy-conserving R + T + ΣA = 1) via FieldProfile:

from hyperbolic_optics.structure import Structure
from hyperbolic_optics.fields import FieldProfile

structure = Structure()
structure.execute(payload)

fp = FieldProfile(structure)
print(fp.summary("p"))            # R, T, per-layer absorption, conservation residual
T = fp.transmittance("p")          # power transmittance (same shape as r_pp)
A = fp.layer_absorption("p")       # per-interior-layer absorptance
prof = fp.field_profile("p")       # z, Ex..Hz, S_z(z), cumulative absorption

For a single semi-infinite anisotropic layer there are no interior layers to resolve, so T = 1 − R is the power delivered into the bulk, and field_profile shows it being absorbed with depth.

Polarization conversion

Cross-polarization (the r_ps/t_ps channels) can be analysed several ways:

• FieldProfile.polarization_resolved("p") (experimental) — splits R and T into co- and cross-polarized power and reports the conversion fractions. The transmitted s/p split is rigorous for an isotropic exit.
• FieldProfile.stokes_from_field_profile("p") — the Stokes vector and polarization ellipse (azimuth ψ, ellipticity χ) of the transverse field versus depth, so you can watch the state evolve through a birefringent layer.
• Mueller.calculate_transmission_mueller_matrix() — the transmission Mueller matrix from the t coefficients. Its |t|² intensity equals true power transmittance only for a symmetric prism = substrate system; into a lower-index medium it breaks in the evanescent regime (use FieldProfile.transmittance for power there).

examples/layer_resolved_absorption.py reproduces the azimuthal layer-resolved absorption of a MoO₃/AlN/SiC heterostructure in the Otto geometry (Passler, Jeannin & Paarmann, J. Opt. Soc. Am. B 37, 1060 (2020)). The amplitude transmission coefficients are also available via FieldProfile.transmission_coefficients() (and Structure.calculate_transmissivity()).

Numerical backends

execute defaults to the 4×4 transfer-matrix method, which is fast but numerically unstable for thick / lossy / strongly-evanescent layers (the propagation terms grow exponentially and the matrix product overflows to NaN). For those cases, opt into the numerically-stable scattering-matrix backend:

structure.execute(payload, backend="scattering")

It cascades per-layer scattering matrices with the Redheffer star product so only decaying exponentials ever appear. It returns the same reflection/transmission coefficients as the transfer method where that is well-conditioned, and correct ones where the transfer matrix is (near-)singular — e.g. a thick evanescent Otto gap, where backend="transfer" gives NaN but backend="scattering" correctly gives total reflection. Method/conventions follow PyLlama (Bay et al., J. Opt. Soc. Am. A 39, 1431 (2022)).

Sweeping Layer Thickness

A layer's thickness may be a list instead of a scalar, which sweeps it as a fourth canonical axis (T) alongside incident angle, azimuth, and frequency — computed in a single execute (the eigendecomposition is thickness-independent; only the propagation phase broadcasts over T). It composes with any scenario and is absent (size 1) by default:

{"type": "Crystal Layer", "material": "Calcite",
 "thickness": [0.5, 1.0, 1.5, 2.0], "rotationY": 90}   # -> appends a length-4 T axis

Outputs append the trailing T axis (e.g. Simple → r_pp.shape == (4,), Incident → (F, angle, 4)). At most one layer may carry a list thickness. For a 2-D thickness × thickness grid, combine a list thickness on one layer with the ThicknessSweep helper (from hyperbolic_optics import ThicknessSweep), which re-runs the stack and stacks results along a leading index.

Known Issues / Limitations

• Multiple Optical Components: While you can place multiple Mueller matrix components in series, matching incident angles between them isn't yet implemented
• Field profiles over full sweeps: field_profile is batched but intended for Simple / single-point use — a full angle/frequency sweep times the depth axis is memory-heavy.

Testing

This package includes a comprehensive test suite covering all major functionality. Run tests locally with:

pytest
pytest --cov=hyperbolic_optics --cov-report=html  # with coverage report

Please open an issue if you encounter any bugs or have suggestions for improvements.

Papers & Further Reading

For background on the physics and mathematical methods, see:

Key References:

• N. C. Passler and A. Paarmann, "Generalized 4 × 4 matrix formalism for light propagation in anisotropic stratified media," J. Opt. Soc. Am. B 34, 2128-2139 (2017)
• P. Yeh, "Electromagnetic propagation in birefringent layered media," J. Opt. Soc. Am. 69, 742-756 (1979)
• N. C. Passler et al., "Layer-resolved resonance intensity of evanescent polariton modes in anisotropic multilayers," Phys. Rev. B 107, 235426 (2023)
• M. Cunningham et al., "Optical footprint of ghost and leaky hyperbolic polaritons," Photonics Research 13, 2291-2305 (2025)

License

This project is licensed under the MIT License – see the LICENSE file for details.

Getting Help

• Documentation: Check the examples/ folder and docstrings
• Issues: Report bugs or request features via GitHub Issues
• Discussions: Start a discussion for usage questions or feature ideas

Thank you for your interest in the Hyperbolic Optics Simulation Package!

Happy simulating! 🔬✨