Jones Calculus

Mueller matrices work at the intensity level: they handle depolarization but discard absolute phase. Jones is the amplitude-level counterpart — 2×2 complex matrices and 2-component vectors that preserve phase, which is what you want for fully-polarized coherent light.

A structure's reflection coefficients already are a Jones matrix:

J = [[r_pp, r_ps],
     [r_sp, r_ss]]     rows = output p/s, columns = input p/s

Basic usage

Every snippet on this page follows on from:

payload = {
    "ScenarioData": {"type": "Incident"},
    "Layers": [
        {"type": "Ambient Incident Layer", "permittivity": 50.0},
        {
            "type": "Semi Infinite Anisotropic Layer",
            "material": "Calcite",
            "rotationY": 90,
        },
    ],
}

from hyperbolic_optics.structure import Structure
from hyperbolic_optics.jones import Jones

structure = Structure()
structure.execute(payload)

jones = Jones(structure)
jones.set_incident_polarization("linear", angle=0)    # p-polarized
jones.add_optical_component("sample")
jones.add_optical_component("linear_polarizer", 90)   # crossed analyzer
extinction = jones.get_intensity()

Ideal components (linear_polarizer, quarter_wave_plate, half_wave_plate, rotator) are angle-independent and broadcast over any scenario sweep.

Eigenpolarizations and exceptional points

The eigenvectors of J are the states that reflect without polarization conversion — J·v = λ·v, so the state comes back scaled by a complex λ but otherwise unchanged.

data = jones.eigenpolarizations()
data["eigenvalues"]          # [..., 2]
data["eigenpolarizations"]   # [..., 2, 2], columns are the eigenvectors
data["discriminant"]         # -> 0 at an exceptional point
data["eigenvector_overlap"]  # -> 1 at an exceptional point

Index 0 is the more p-like eigenvector, index 1 the more s-like. That labelling matters: np.linalg.eig returns them in no particular order, so without it the two swap arbitrarily across a sweep and any per-channel map inherits seams that are not physical. Ordering by |λ| — the obvious alternative — is much worse, seaming along every |λ₀| = |λ₁| contour.

A reflection Jones matrix from a lossy anisotropic sample is generally non-normal: its eigenvectors are not orthogonal and can coalesce at an exceptional point, where the matrix stops being diagonalizable.

found = jones.find_exceptional_points()
found["ep_index"]       # strongest candidate on the grid
found["near_ep"]        # boolean mask on eigenvector overlap
found["defectiveness"]  # scale-free, 0 where the eigenvectors truly coalesce

No labelling is globally continuous near an EP

Encircling an exceptional point exchanges the two eigenvalue sheets, so every labelling scheme owns at least one branch cut per EP. Ordering by polarization character puts its cuts on the EP chains — where a discontinuity is physically correct — and nowhere else.

Ellipsometry

jones.ellipsometric_parameters()    # psi and delta

Composing elements

from hyperbolic_optics.jones import compose_jones

analyzer = jones.linear_polarizer(90)
total = compose_jones(structure, analyzer)   # beam order

Two Structure elements must share the same kx and frequency grids — the in-plane wavevector is conserved, so composing samples evaluated at different angles is not a meaningful product, and the guard rejects it.

Bridging to Mueller

jones.to_mueller()

Reuses the same Jones-to-Mueller transform Mueller does, so the two formalisms stay consistent — including the handedness convention for circular polarization.