moire_align op• Data kinds: none → table (an op determined by its arguments alone — it takes no image or data input)
• Call: import fullseye as fs; fs.ledger.moire_phase_crlb(noise_sigma=0.01, amplitude=0.2, n_rows=256, n_cols=1024, p_mask_um=2.0, p_wafer_um=2.04, moving='wafer', window='hann', noisy_reference=True) (to call the implementation directly, import moirealign; moirealign.moire_phase_crlb(noise_sigma=0.01, amplitude=0.2, n_rows=256, n_cols=1024, p_mask_um=2.0, p_wafer_um=2.04, moving='wafer', window='hann', noisy_reference=True); from the registry, opsoptics.get("moire_phase_crlb"))
How finely the fringe phase and the displacement can be read under white Gaussian noise (closed-form lower bound).
> The detailed description below is the original text — the summary and the headings are translated.
周波数が既知の正弦 A cos(2πx/P + φ) を N = n_rows × n_cols 画素の独立な雑音 σ の中で読むとき、var(φ) ≥ 2σ² / (N A²)(Rife & Boorstyn 1974 の位相の下限)。窓 w で重みを付けたロックインは係数 k_w = n Σw² / (Σw)² だけ悪い(Hann で約 1.5)。基準像にも同じ雑音があれば位相差の分散は 2 倍。ずれの標準偏差 = σ_φ / |dφ/dδ| = σ_φ p / (2π)。差動の 2 対(各対が同じ画素数)は σ_δ,diff = √(σ_A² + σ_B²) / |g_A − g_B|。★これは雑音だけの下限。実際の位置合わせはマークの非対称・ギャップ・傾き・レジストなどの系統誤差で決まり、公表値は 10〜30 nm(Li ら 2006、Zhu ら 2013)、空間位相の干渉法で 1 nm 未満の検出能(Moon & Smith 2005)。
返り値: dict(phase_std_rad, shift_std_nm(対 A だけ)、shift_std_nm_differential、window_factor, n_pixels)。Raises ValueError: σ < 0、振幅・画素数が正でない、綴り違い、同じピッチ。
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.
• poc_moire_alignment — py -3.11 examples/poc_moire_alignment.py
• poc_moire_screen — py -3.11 examples/poc_moire_screen.py
table as input)abcd_matrix · wavefront_stats · sfr_from_edge · paraxial_trace · seidel_coefficients · spot_stats · tolerance_analysis · wavefront_from_opd
moire_align)moire_magnification · moire_talbot_gap · moire_overlay_image · moire_phase · moire_phase_to_shift · moire_differential_shift · moire_mark_read
*Provenance: moirealign.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.