appearance op• Data kinds: signal → signal
• Call: import fullseye as fs; fs.ledger.thin_film_reflectance(nm, thickness_nm=350.0, n_film=1.33, n_sub=1.0, cos_theta=1.0, n_out=1.0, polarization='unpolarized', under_layers=()) -> 'np.ndarray' (to call the implementation directly, import matappear; matappear.thin_film_reflectance(nm, thickness_nm=350.0, n_film=1.33, n_sub=1.0, cos_theta=1.0, n_out=1.0, polarization='unpolarized', under_layers=()) -> 'np.ndarray'; from the registry, opsoptics.get("thin_film_reflectance"))
The spectral reflectance of a thin film (thickness d, index n_film). The exact characteristic-matrix solution (all multiple reflections included).
nm: wavelength [nm] (array allowed).
thickness_nm: film thickness [nm]. 200–800 for a soap bubble, 50–300 for an anodized coating. An array broadcastable with nm
is allowed too (a thickness field; e.g. (H, W, 1) thicknesses and (K,) wavelengths → (H, W, K)).
n_film/n_sub: refractive index of the film / substrate. Any of: a real number, a complex n + ik (k ≥ 0; metal substrates
and absorbing films), a material name (the shared material table materials, e.g. "TI" / "AL" / "SI" —— its n,k
at each wavelength), a material record (dict), or a per-wavelength array broadcastable with nm (film dispersion).
cos_theta: cos of the angle of incidence on the incident side (array allowed, in a shape broadcastable with nm).
n_out: refractive index of the incident side (a real number or a material name, no absorption).
polarization: "unpolarized" (default, mean of s and p) / "s" / "p".
under_layers: layers between the film and the substrate ([(index, thickness nm), ...] from the incident side; the index is written
like n_film). Empty (default) means a single film. For multilayer AR coatings, dielectric mirrors, a nitride under an oxide, etc.
Returns: reflectance R (0–1), in the broadcast shape of nm, thickness_nm and cos_theta.
★Sign convention: the complex index is N = n + ik with k ≥ 0. For a source that writes N = n − ik, pass the same k as a positive number (an input with a negative imaginary part raises ValueError —— the convention is not guessed).
Physics: the characteristic matrix of each layer [[cos δ, −i sin δ / η], [−i η sin δ, cos δ]], δ = 2π N d cosθ̃ / λ (the complex Snell law gives cosθ̃ inside the layer, taking the decaying branch), with the tilted optical admittance η = N cosθ̃ (s) / N / cosθ̃ (p). For one layer this gives the same value as the Airy formula R = |r1 + r2 e^{2iβ}|² / |1 + r1 r2 e^{2iβ}|² (compared in tests with a separately written formula). Reflection peaks at odd multiples of λ/4 (when n_film > n_sub), and the colour shifts as the thickness changes —— that is the colour of a soap bubble. At film thickness 0, R matches the Fresnel reflection of the bare substrate. The calculation is done in one place, the characteristic matrix of filmoptics (the same function as filmoptics.multilayer_reflectance).
★2026-10-14: the three indices used to be real only (a complex number raised TypeError). Approximating a metal substrate by a real index (heat-tinted titanium, 90 nm, with n_sub = 2.6) gave reflectances of 0.15–0.19, against 0.35–0.61 (380–780 nm) with the n,k of Ti —— a silent error that turns the colour into a different one. Now writing "TI" for the substrate brings in the n,k of the material table.
>>> r = thin_film_reflectance([450.0, 550.0, 650.0], 90.0, "TIO2_AMORPHOUS", "TI")
>>> bool(r.min() > 0.3)
True
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.
• colorimetry — Colorimetry and spectra — colour is fixed only by spectrum × illuminant × observer
• mv_illumination_practice — Lighting in practice — wavelength, polarisation, drive modes, ambient light, safety
• 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.
• appearance_structural_colour — py -3.11 examples/appearance_structural_colour.py
• simcal_tour — py -3.11 examples/simcal_tour.py
signal as input)emva_photon_transfer · emva_quantum_efficiency · emva_linearity_error · emva_snr_curve · emva_dark_current · emva_total_snr · emva_exposure_for_snr · emva_photon_exposure
appearance)cie_xyz_from_wavelength · spectrum_to_srgb · grating_wavelengths · grating_rgb · thin_film_rgb · ward_anisotropic
*Provenance: matappear.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.