polarization op• Data kinds: matrix × stokes → stokes
• Call: import optics; optics.mueller_apply(mueller, stokes) (or opsoptics.get("mueller_apply"))
Push a Stokes vector through a Mueller matrix: `S' = M @ S`.
> The detailed description below is the original text — the summary and the headings are translated.
*mueller* is a `(4, 4)` real matrix (build one with
:func:mueller_element, or multiply several together) and *stokes* a
4-component Stokes vector `[S0, S1, S2, S3]`.
The input is checked for physical realisability: `S0 >= 0` and
`sqrt(S1^2+S2^2+S3^2) <= S0` (degree of polarisation at most 1). Handing
an impossible state to a Mueller matrix returns a plausible-looking result
that means nothing, so it is refused instead. The output is not
re-checked, deliberately: an unphysical output is real evidence that
*mueller* is not a physical Mueller matrix, and swallowing it would hide
the bug — inspect it with :func:stokes_analyze, which will say so.
Returns a `(4,)` float64 Stokes vector.
`mueller を (..., 4, 4)、stokes を (..., 4)` にすると
画素ごとに違う行列を通せる。両者は numpy の規則で broadcast するので、
`(H, W, 4, 4) と (4,)` の組(素子だけが場所で変わる)や、
`(4, 4) と (H, W, 4)` の組(光だけが場所で変わる)も書ける。
足した理由は実測。`examples/poc_photoelasticity.py` は画素ごとに
位相差の変わる位相子を通す必要があり、**この口が無いために (H,W,4,4) を
自分で組む羽目になっていた**。行列そのものは正しい(暗視野円偏光系を
組んで教科書の `I = sin²(δ/2)` と 125 通りで最大差 2.2e-16)ので、
足りなかったのは形だけだった。
実現可能性の検査は画素ごとに行い、破っている画素があれば
その数と最悪値を挙げて拒否する(1 点でも通さない = fail-closed)。
Ground truth it reproduces exactly: unpolarised `[1,0,0,0]` through an
ideal polariser gives `S0 = 0.5` with degree of polarisation 1; through
two polarisers at relative angle theta, `0.5*cos^2(theta)` (Malus, to
1e-16 over a full sweep); the identity matrix returns the input unchanged.
Raises `ValueError: *mueller* is not (4, 4)`, *stokes* is not a
1-D 4-vector, either is complex / masked / non-finite, the input Stokes
vector is unphysical, or the product overflows float64.
Every optics op validates its input before computing (nothing slips through silently):
• Units are baked into the argument name — _mm / _um / _deg / _mrad. Confusing mm with µm does not crash; it yields a plausible-looking wrong answer, so the name prevents it. Nothing here guesses the unit from the magnitude.
• **Strings raise ValueError** — float('50') succeeds, so an unparsed configuration value would slip through as a length (measured: thin_lens('50', '200') returned a plausible 66.667 mm). bool is refused too, as the implicit promotion True == 1.
• **complex / masked arrays raise ValueError (real-valued slots only; silently dropping the imaginary part or peeling off the mask is refused). NaN/Inf raises ValueError on every input.**
• Division by zero and its relatives are refused by name: focal length 0, radius of curvature 0, refractive index <= 0, a fully opaque aperture (all zeros, so the normalisation is 0/0), a PSF whose sum is <= 0, a Stokes vector with S0 = 0, and an object sitting at the front focal point (the image is at infinity).
• Only two ops return a non-finite value, and both state it as a contract: depth_of_field returns far_mm = inf beyond the hyperfocal distance (that is what the hyperfocal distance means), and gaussian_beam returns wavefront_radius_mm = inf at the waist (the radius of curvature of a plane wavefront). Both also return a finite companion (far_is_infinite / curvature_per_mm). **Any other silent NaN/Inf is detected internally and raises ValueError** — "float64 overflowed" and "the answer is infinite" are different claims, so the first is never returned wearing the face of the second.
• Size caps: generated grids are capped by optics.MAX_GRID (4096); supplied fields/PSFs/apertures by optics.MAX_FIELD_ELEMENTS (2^24); ABCD element chains by optics.MAX_SYSTEM_ELEMENTS (1024); Zernike by MAX_ZERNIKE_TERMS (512) / MAX_ZERNIKE_ORDER (40) / MAX_ZERNIKE_BASIS (2^25). This closes, fail-closed, the paths where a small argument triggers a huge internal allocation (measured: n_max=40 × 4096² needs 108 GB).
• Physically impossible states are refused too: a Stokes vector with degree of polarisation > 1, negative transmittance, negative intensity, and invalid Zernike indices such as n-|m| odd.
• Sample-data catalog (download URLs / licences) — 2-D uses skimage.data (BSD/public domain) plus synthetic images; 3-D lists download URLs for real data sources (Stanford, PDS, …).
• Operator provenance and references — the sources of the research/methods this op family came from.
• The canonical algorithm (author, year) and its uses are named in the family usage guide above.
• optics_imaging — py -3.11 examples/optics_imaging.py
• poc_photoelasticity — py -3.11 examples/poc_photoelasticity.py
stokes as input)polarization)jones_element · jones_apply · stokes_from_jones · mueller_element · stokes_analyze
*Provenance: optics.py — OPTICS operator registry. This per-op note is generated by tools/opdocs.py md (do not hand-edit).*
© 2026 Kazufumi Furuse — Fullseye operator documentation. Licensed under Apache-2.0.