wave op• Data kinds: image2d → image2d
• Call: import fullseye as fs; fs.ledger.pupil_psf(pupil, defocus_waves=0.0, wavelength_um=0.55, f_number=5.6, oversample=4, pixel_pitch_um=None, opd_waves=None) (to call the implementation directly, import optics; optics.pupil_psf(pupil, defocus_waves=0.0, wavelength_um=0.55, f_number=5.6, oversample=4, pixel_pitch_um=None, opd_waves=None); from the registry, opsoptics.get("pupil_psf"))
Diffraction PSF of an arbitrary pupil shape with defocus (sums to 1).
*pupil* is a square `(n, n)` amplitude transmittance (0 = opaque, 1 =
clear; a binary mask is the usual case) drawn on a grid whose **full width
is the pupil's clear diameter** `D` — so a circle filling the grid is a
conventional round stop, a W-shaped band or an off-axis hole inside the
grid is just a different mask, and `f_number = f / D` refers to that
full width in every case. The wavefront over the grid is
`W(rho) = defocus_waves * rho^2 (+ opd_waves) with rho` the radius
from the grid centre normalised to `1` at the grid half-width (the Seidel
defocus `W20; :func:defocus_from_shift` converts an axial shift to it),
and the PSF is the Fraunhofer intensity of the pupil function
`PSF = | FFT{ pupil * exp(i 2 pi W) } |^2`
on a zero-padded `M x M grid, M = n * oversample` (rounded up to
even), centred on sample `M//2` and normalised to unit sum. The image
plane sample spacing is
`dx = lambda * N * n / M ~= lambda * N / oversample` [um]
— with *pixel_pitch_um* the fine PSF is area-integrated onto detector
pixels of that pitch (odd `(K, K)`, centred on a pixel, unit sum), which
is what an image convolution needs; the pitch must not be finer than
`dx. Without it the fine PSF is returned and dx` is yours to compute
from the formula (an image cannot carry it).
Returns a float64 `image2d`.
Ground truth it reproduces (measured, `tests/test_optics.py`):
• a circle filling a 64-sample grid, `oversample = 16`: the first dark
ring at `1.2197 lambda N` within 0.5 % of the Airy value (three
wavelength / f-number pairs), and the same ring at the same
*micrometre* radius within 5 % after binning to a pixel pitch of
`lambda N / 8` (2.1 % measured — the parabolic minimum on a 9.8-pixel
ring, not the binning) — so the pitch bookkeeping is right in physical
units, not only in samples; the binned spot is centro-symmetric to
1e-17 and correlates with :func:airy_pattern sampled at the same
pitch at 0.99999 (0.9999 at `lambda N / 4, 0.9997 at lambda N / 3`);
• pure defocus of a circular pupil: the on-axis intensity relative to the
unaberrated peak is the closed-form `[sin(pi W20)/(pi W20)]^2`
(`0.405 at half a wave, 0` at one wave — the dark centre of the
one-wave defocused Airy spot), within 1 %;
• `defocus_waves = 0` and a clear circular pupil is the Airy pattern of
:func:airy_pattern to the sampling of the disc edge;
• the sign identity: for a real pupil, `-W` is the complex
conjugate of `+W, so PSF(-W)(x) = PSF(+W)(-x)` exactly. The test
pins it on a W-shaped band: the two PSFs are mirror images through the
centre to 1e-12, and they are *not* equal to each other (the W pupil
is asymmetric, so the direction of defocus is visible in the blur),
while for the circle they are equal (a symmetric pupil cannot tell
the sign). Rotating the W pupil by 90/180/270 degrees rotates the PSF
the same way (checked, so the asymmetry is the pupil's, not the grid's).
Raises `ValueError`: *pupil* is not 2-D, not square, smaller than 2x2,
over the size cap, complex, masked or non-finite; negative transmittance;
an all-opaque pupil (nothing to diffract, the normalisation would be
0/0); *opd_waves* not the same shape as *pupil*; non-finite
*defocus_waves*; non-positive or non-finite *wavelength_um* /
*f_number* / *pixel_pitch_um*; *oversample* outside `[1, 64]`; an FFT
side over :data:MAX_PUPIL_FFT; an aliased phase — more than
:data:MAX_WAVES_PER_SAMPLE waves between neighbouring pupil samples (the
message says how many samples the grid needs); a pixel pitch finer than
the fine sample spacing (raise *oversample*).
Scalar Fraunhofer optics: no polarisation, no high-NA obliquity, no
pupil apodisation by the lens itself. The defocus term is the paraxial
`rho^2 (see :func:defocus_from_shift`). A pupil that reaches the grid
edge is fine (the zero padding is the field stop); a pupil *larger* than
the grid cannot be expressed — widen the grid and lower `f_number`.
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
image2d as input)fraunhofer_pattern · pupil_blur · psf_to_mtf · polarization_demosaic · polarization_demosaic_color · illumination_uniformity · render_through_lens · surface_defect
wave)airy_pattern · angular_spectrum_propagate · fraunhofer_pattern · gaussian_beam · defocus_from_shift · 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.