wave op• Data kinds: cimage × cimage → cimage
• Call: import fullseye as fs; fs.ledger.four_f_filter(field, transfer, invert=True) (to call the implementation directly, import optics; optics.four_f_filter(field, transfer, invert=True); from the registry, opsoptics.get("four_f_filter"))
4f optical processor: the output after passing through 2 lenses and the filter in the Fourier plane.
Input plane → lens 1 → Fourier plane (multiply by transfer) → lens 2 → output plane. In the ideal 4f system of scalar diffraction the output is
out(x, y) = (u * h)(-x, -y)
—— the convolution comes with a coordinate inversion. Each of the 2 lenses performs a *forward* Fourier transform, and F{F{u}}(x) = u(-x). This is where a computation that "just multiplies in the Fourier plane" differs from a real 4f system, so the inversion is on by default.
The inversion is exact in the discrete case too: it is the index map n -> (-n) mod N (np.roll(a[::-1], 1)). The inversion itself is an index permutation and carries no error —— the only remaining error is the 1e-16 level added by the FFT round trip (measured with transfer set to 1:
max absolute difference 3.4e-16).
Setting *invert* to False skips the inversion (when you want to see the filtered field in input coordinates. It is not a physical 4f system, so it is not the default).
Ground truth (what this op reproduces):
• identity filter → matches the input rotated by 180 degrees (measured 3.4e-16).
• block filter → exactly zero.
• derivative_x order=n → spatial derivative of order *n*. For a Gaussian (w=8, 64x64), relative error 5.8e-08 for the 1st order / 5.4e-08 for the 2nd order —— it does not reach 1e-16 because the Gaussian is not strictly band-limited (the truncated part).
• Linearity 4f(a*u1 + b*u2) = a*4f(u1) + b*4f(u2).
• Composition 4f(4f(u, H1), H2) = 4f(u, H1*H2) (it is offset by the inversion applied 2 times, so set *invert* to False when testing composition).
• A filter with |H| = 1 preserves the total power (Parseval).
Raises ValueError: *field* / *transfer* not 2-D, different shapes, smaller than 2x2, above MAX_FIELD_ELEMENTS, non-finite, masked.
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_four_f_processor — py -3.11 examples/optics_four_f_processor.py
cimage as input)angular_spectrum_propagate · jones_apply
wave)airy_pattern · angular_spectrum_propagate · fraunhofer_pattern · fourier_plane_filter · gaussian_beam · defocus_from_shift · pupil_psf · pupil_blur
*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.