Mueller Matrices¶

Mueller matrices describe how light's polarization state changes upon reflection from a surface.

Basic Usage¶

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

# Create structure
structure = Structure()
structure.execute(payload)

# Create Mueller analyzer
mueller = Mueller(structure)

# Set incident polarization
mueller.set_incident_polarization('linear', angle=45)

# Add the sample
mueller.add_optical_component('anisotropic_sample')

# Calculate results
params = mueller.get_all_parameters()

Incident Polarization States¶

Linear Polarization¶

# p-polarized (0°)
mueller.set_incident_polarization('linear', angle=0)

# s-polarized (90°)
mueller.set_incident_polarization('linear', angle=90)

# 45° linear
mueller.set_incident_polarization('linear', angle=45)

Circular Polarization¶

# Right-handed circular
mueller.set_incident_polarization('circular', handedness='right')

# Left-handed circular
mueller.set_incident_polarization('circular', handedness='left')

Elliptical Polarization¶

# Elliptical with azimuth 30° and ellipticity 20°
mueller.set_incident_polarization('elliptical', alpha=30, ellipticity=20)

Optical Components¶

Anisotropic Sample¶

The main sample being analyzed:

mueller.add_optical_component('anisotropic_sample')

Linear Polarizer¶

# Horizontal polarizer (0°)
mueller.add_optical_component('linear_polarizer', 0)

# Vertical polarizer (90°)
mueller.add_optical_component('linear_polarizer', 90)

# 45° polarizer
mueller.add_optical_component('linear_polarizer', 45)

Quarter-Wave Plate¶

# QWP with fast axis at 45°
mueller.add_optical_component('quarter_wave_plate', 45)

Half-Wave Plate¶

# HWP with fast axis at 22.5°
mueller.add_optical_component('half_wave_plate', 22.5)

Multiple Components¶

Place components in series:

# Polarizer → Sample → Analyzer
mueller.set_incident_polarization('linear', angle=0)
mueller.add_optical_component('linear_polarizer', 0)  # Input polarizer
mueller.add_optical_component('anisotropic_sample')    # Sample
mueller.add_optical_component('linear_polarizer', 90)  # Analyzer (crossed)

reflectivity = mueller.get_reflectivity()

Stokes Parameters¶

The Stokes parameters describe the polarization state:

S0: Total intensity (reflectance)
S1: Horizontal vs vertical linear polarization
S2: +45° vs -45° linear polarization
S3: Right vs left circular polarization

stokes = mueller.get_stokes_parameters()

print(f"Total intensity: {stokes['S0']}")
print(f"Linear H/V: {stokes['S1']}")
print(f"Linear ±45°: {stokes['S2']}")
print(f"Circular R/L: {stokes['S3']}")

Polarization Properties¶

Degree of Polarization (DOP)¶

dop = mueller.get_degree_of_polarisation()
# DOP = 0: unpolarized
# DOP = 1: fully polarized

Ellipticity¶

ellipticity = mueller.get_ellipticity()
# Ellipticity angle in radians
# 0: linear
# ±π/4: circular

Azimuth¶

azimuth = mueller.get_azimuth()
# Orientation of polarization ellipse major axis

All Parameters at Once¶

params = mueller.get_all_parameters()

# Contains: S0, S1, S2, S3, DOP, Ellipticity, Azimuth
for key, value in params.items():
    print(f"{key}: {value}")

Complete Example¶

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

# Define structure
payload = {
    "ScenarioData": {
        "type": "Incident"
    },
    "Layers": [
        {"type": "Ambient Incident Layer", "permittivity": 50.0},
        {"type": "Isotropic Middle-Stack Layer", "thickness": 0.1},
        {
            "type": "Semi Infinite Anisotropic Layer",
            "material": "Quartz",
            "rotationX": 0,
            "rotationY": 70,
            "rotationZ": 0
        }
    ]
}

# Run simulation
structure = Structure()
structure.execute(payload)

# Mueller analysis
mueller = Mueller(structure)

# Try different incident polarizations
for angle in [0, 45, 90]:
    mueller.set_incident_polarization('linear', angle=angle)
    mueller.add_optical_component('anisotropic_sample')

    params = mueller.get_all_parameters()
    print(f"\nIncident angle: {angle}°")
    print(f"Average reflectance: {params['S0'].mean():.4f}")
    print(f"Average DOP: {params['DOP'].mean():.4f}")

    mueller.reset()  # Reset for next calculation

Resetting the Mueller Object¶

# Clear all settings and start fresh
mueller.reset()

This clears: - Mueller matrix - Stokes parameters - Optical components - Resets incident polarization to unpolarized

Physical Interpretation¶

Reflectance (S0)¶

Total reflected power, sum of all polarization components.

Degree of Polarization¶

DOP = 1: Fully polarized (pure state)
0 < DOP < 1: Partially polarized
DOP = 0: Unpolarized (random)

Understanding S1, S2, S3¶

S1 > 0: More horizontal than vertical
S1 < 0: More vertical than horizontal
S2 > 0: More +45° than -45°
S2 < 0: More -45° than +45°
S3 > 0: More right-circular
S3 < 0: More left-circular