Abstract
Considering diffraction alone, the larger the pupil, the more contrast it passes for fine gratings. This paper adds to the eye values of Paper 919 a spherical aberration that grows toward the rim of the pupil, and numerically computes, for each pupil diameter, the contrast of the finest grating the cone mosaic can pick up (the Nyquist fineness) with defocus chosen for best contrast. No new theorem or law is claimed.
First, without aberration, the larger the pupil the better: the numerical computation reproduced the diffraction-only formula ($0.266568$ against $0.266680$ at a $3$ mm pupil, $0.612981$ against $0.612983$ at $6$ mm), and there was no interior maximum up to the $8$ mm examined ($0.707238$ at $8$ mm) (Section 2).
Second, with aberration, the best pupil settles at a middle size: setting the rim aberration at a $6$ mm pupil to $0.5$, $1$, $2$ and $4$ wavelengths, the best pupils were $6.149$, $5.207$, $4.523$ and $\mathbf{3.970}$ mm, with contrasts $0.573231$, $0.511423$, $0.445752$ and $0.378165$ (Section 3).
Third, the best pupil barely responds to the strength of the aberration: making the aberration $8$ times stronger shrank the best pupil only to $\mathbf{0.645625}$ times its size. The power relating diameter to aberration strength was $-0.210410$ (Section 3).
Fourth, for small pupils aberration hardly matters: at a $2.5$ mm pupil the contrast changed only from $0.151378$ to $0.151299$ as aberration went from $0$ to $4$ wavelengths (Section 3).
Fifth, the separator is whether the gain from widening the pupil or the loss from rim aberration wins: widening the pupil reduces diffraction blur, but rim aberration grows as the fourth power of the diameter. The gain wins for small pupils, the loss for large ones, and the boundary between them is the best pupil (Section 4).
Keywords: spherical aberration, pupil diameter, modulation transfer function, diffraction, Nyquist fineness, optics of the eye.
Scope (what is not claimed)
No new theorem or law is claimed: obtaining the optical transfer function as the autocorrelation of the pupil function, the modulation transfer function formula for a diffraction-only circular pupil, that spherical aberration grows as the fourth power of position measured to the rim, and that balancing aberration with defocus improves imaging are all known. These are set values: wavelength, cone spacing and nodal distance are as in Paper 919, and the strength of spherical aberration is a set value for comparison. It corresponds to no one’s eye. Other aberrations and the retina are not included: chromatic aberration, astigmatism, the tear film, scatter, and the retina’s different response to light entering through different parts of the pupil are not included. No visual acuity is predicted: contrast is a value for showing the comparison; anyone’s acuity and changes of pupil with brightness are not treated. Relation to earlier papers: Paper 919, in its scope note, did not include aberrations, leaving out the eye’s spherical aberration, chromatic aberration, tear film and scatter, and set up diffraction-only optics; this paper sits in the seat of spherical aberration (no earlier occupant). Paper 358, in its scope note, also did not build optics and did not enter the theory of diffraction limits, aberrations or point spread functions; this paper also sits in that seat of aberrations. Paper 919 showed that with the pupil wider than $1.85$ mm, the eye’s limit of fineness is set not by diffraction but by the spacing of cones. Here it is counted how aberration erodes the contrast at that limit as the pupil opens further. What is added: reproducing the diffraction-only formula numerically, finding the best pupil and its contrast for four aberration strengths, showing by a power that the best pupil barely responds to aberration strength, setting out that aberration hardly matters for small pupils, and placing the separator on “whether the gain from widening the pupil or the loss from rim aberration wins”.
1. Notation and the Question
Setting: as in Paper 919, the wavelength is $555$ nm, cone spacing $2.5$ μm and the nodal point to retina distance $16.7$ mm. The finest grating the cone mosaic can pick up (the Nyquist fineness) is $58.2940$ cycles per degree. With pupil diameter $D$ and position $\rho$ normalised to $1$ at the rim, the wavefront error is set as follows.
$$W(\rho)=W_{020}\,\rho^2+W_{040}\,\rho^4,\qquad W_{040}=A\left(\frac{D}{6\ \mathrm{mm}}\right)^4\ \text{(wavelengths)}$$$A$ is the rim spherical aberration at a $6$ mm pupil and $W_{020}$ is defocus, chosen for each pupil to give the highest contrast. The modulation transfer function (contrast) at the Nyquist fineness is computed numerically as the magnitude of the autocorrelation of the laterally shifted pupil function.
Question: with spherical aberration, which pupil gives the highest contrast at the Nyquist fineness, and how does it move with the strength of the aberration?
2. Without Aberration
Computation 2.1: numerical values without aberration and the diffraction-only formula.
| pupil | numerical | formula (values of Paper 919) |
|---|---|---|
| $3$ mm | $0.266568$ | $0.266680$ |
| $6$ mm | $0.612981$ | $0.612983$ |
Contrasts for pupils of $1.5$, $2$, $2.5$, $3$, $3.5$, $4$, $5$, $6$, $7$ and $8$ mm were $0.0000$, $0.0235$, $0.1514$, $0.2666$, $0.3587$, $0.4318$, $0.5385$, $0.6130$, $0.6669$ and $0.7072$; the highest was at the edge of the range examined, $8$ mm ($0.707238$).
Observation 2.2 without aberration, the larger the pupil the better, with no interior maximum. The numerical computation reproduced the diffraction-only formula within a difference of $5\times10^{-3}$ at $3$ mm and $6$ mm. This is the starting point for the computation with aberration.
3. With Aberration
Computation 3.1: contrast by pupil (with defocus chosen for best contrast).
| rim aberration $A$ | $3$ mm | $4$ mm | $5$ mm | $6$ mm | $7$ mm | $8$ mm |
|---|---|---|---|---|---|---|
| $0.5$ | $0.2665$ | $0.4309$ | $0.5305$ | $0.5729$ | $0.5299$ | $0.3660$ |
| $1$ | $0.2665$ | $0.4283$ | $0.5071$ | $0.4646$ | $0.2505$ | $0.2116$ |
| $2$ | $0.2661$ | $0.4178$ | $0.4215$ | $0.2257$ | $0.1840$ | $0.1372$ |
| $4$ | $0.2648$ | $0.3781$ | $0.1941$ | $0.1628$ | $0.1112$ | $0.0859$ |
Computation 3.2: best pupil (narrowed by golden-section search).
| rim aberration $A$ | best pupil | contrast | defocus $W_{020}$ (wavelengths) |
|---|---|---|---|
| $0.5$ | $6.149$ mm | $0.573231$ | $-0.5068$ |
| $1$ | $5.207$ mm | $0.511423$ | $-0.5110$ |
| $2$ | $4.523$ mm | $0.445752$ | $-0.5809$ |
| $4$ | $\mathbf{3.970}$ mm | $0.378165$ | $-0.7025$ |
Increasing the aberration eightfold from $0.5$ to $4$ made the best pupil $\mathbf{0.645625}$ times as large, a power of $-0.210410$ ($-0.239947$ between $0.5$ and $1$, $-0.203297$ between $1$ and $2$). At a $2.5$ mm pupil the contrast for aberrations $0$, $0.5$, $1$, $2$ and $4$ was $0.151378$, $0.151377$, $0.151373$, $0.151358$ and $0.151299$.
Observation 3.3 the best pupil settles at a middle size and barely responds to aberration strength. Even with eight times the rim aberration, the best pupil shrank only from $6.149$ mm to $3.970$ mm: for small pupils the aberration strength hardly matters, and at $2.5$ mm the contrast was $0.151299$ even at $4$ wavelengths. The best defocus was opposite in sign to the aberration, from $-0.5068$ to $-0.7025$ wavelengths.
4. The Separator
Observation 4.1 the separator is whether the gain from widening the pupil or the loss from rim aberration wins. Widening the pupil reduces diffraction blur and raises contrast at the Nyquist fineness: meanwhile rim aberration grows as the fourth power of the diameter. Without aberration only the gain remains and the larger pupil is better; with aberration the loss wins for large pupils, and the boundary became a middle-sized best pupil.
Observation 4.2 what is not said. The aberration strength of real eyes, chromatic aberration, astigmatism, tear film and scatter, retinal response, changes of pupil with brightness, and anyone’s acuity are not treated. What is shown is that under a set spherical aberration the pupil maximising contrast at the Nyquist fineness settles at a middle size, and that size barely responds to aberration strength.
5. Roll Call and Closing
- wavefront error and numerical transfer function (Section 1): rim aberration as the fourth power of diameter.
- formula reproduced without aberration (Computation 2.1): the larger the pupil the better.
- best pupil from $6.149$ to $\mathbf{3.970}$ mm (Computation 3.2): settles at a middle size.
- $\mathbf{0.645625}$ times for eightfold aberration (Computation 3.2): power $-0.210410$.
- $0.151378$ to $0.151299$ at $2.5$ mm (Computation 3.2): aberration does not matter for small pupils.
- gain from widening, or loss at the rim (Observation 4.1): the separator; the scope notes of Papers 919 and 358.
Closing: Spherical aberration gives the eye a middle-sized best pupil that barely moves with the aberration’s strength. With diffraction alone the larger pupil was better, with contrast $0.707238$ at $8$ mm: setting the rim aberration to $0.5$, $1$, $2$ and $4$ wavelengths, the best pupils were $6.149$, $5.207$, $4.523$ and $3.970$ mm, and eight times the aberration made the pupil only $0.645625$ times as large. At a $2.5$ mm pupil aberration hardly mattered. The separator is whether the gain from widening the pupil or the loss from rim aberration wins. Placed among the earlier papers: to the limit of the eye’s fineness that Paper 919 set with diffraction alone, the aberration that Papers 919 and 358 left out was added, and how the contrast at that limit is eroded as the pupil opens was counted. To be honest: a real eye also has chromatic aberration and scatter. Under a set spherical aberration, where the best pupil settles and how far it moves, and no further.
References
- Born, M., & Wolf, E. (1999). Principles of Optics (7th ed.). Cambridge University Press.
- Goodman, J. W. (2005). Introduction to Fourier Optics (3rd ed.). Roberts & Company.
- Yamagishi, Y. (2026). The “Depth” of Depth of Field Is a Convention, Not a Measurement [Paper 358]. YOI: https://www.yamagishiyuuki.com/yoi/358; DOI: 10.5281/zenodo.22697216
- Yamagishi, Y. (2026). With the Pupil Wider Than $1.85$ mm, the Eye’s Limit of Fineness Is Set Not by Diffraction but by the Spacing of Cones [Paper 919]. YOI: https://www.yamagishiyuuki.com/yoi/919
Reference
Yamagishi, Y. (2026). Spherical Aberration Gives the Eye a Middle-Sized Best Pupil That Barely Moves with the Aberration’s Strength [Paper 949]. YOI: https://www.yamagishiyuuki.com/yoi/949
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 949. This paper is a preprint and has not been peer reviewed.