geometric op• Data kinds: none → pairs (an op determined by its arguments alone — it takes no image or data input)
• Call: import optics; optics.relative_illumination(half_angle_deg=20.0, samples=64, exponent=4.0) (or opsoptics.get("relative_illumination"))
Natural vignetting: relative image-plane illuminance versus field angle.
The cosine-fourth law `E(theta)/E(0) = cos(theta)^4` — one cosine from the
inverse-square increase in distance to the off-axis point (twice), one from
the tilt of the exit pupil as seen from there, one from the tilt of the
image plane. Sampled uniformly in angle from 0 to *half_angle_deg*.
Returns an `(samples, 2) float64 pairs` array: column 0 the field
angle in degrees, column 1 the relative illuminance in [0, 1].
*exponent* exists because the fourth power is the *ideal symmetric* case:
a lens with pupil aberration, or a telecentric design, or one with a tilted
entrance pupil, falls off closer to `cos^3` (or is deliberately corrected
flatter still). Setting the exponent is how you say which lens you have —
it is not a fudge factor to be tuned after the fact.
Ground truth it reproduces exactly: `cos^4(45 deg) = 1/4` and
`cos^4(60 deg) = 1/16`, both to machine precision; the curve is 1.0 on
axis and monotonically decreasing.
Raises `ValueError: *half_angle_deg* outside (0, 90)` — at 90
degrees the illuminance is exactly 0 and the "relative" curve carries no
information; *samples* outside `[2, MAX_GRID]`; a negative or non-finite
*exponent*.
This is the *natural* falloff only. Mechanical vignetting (a stop clipping
the oblique beam) is a separate, lens-specific effect that no closed form
covers — measure it with a flat field.
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
pairs as input)—
geometric)thin_lens · abcd_matrix · abcd_trace · depth_of_field
*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.