Optical Aberrations

Where aberrations come from

Ideal optics is built on the paraxial (Gaussian) approximation: rays stay close to the axis and make small angles, so we replace sin θ with just θ. In that world a thin lens of focal length f maps an object distance s to an image distance s' by the famous

1/s' − 1/s = 1/f

and every point goes to a perfect point. But the real Snell's law uses the full series

sin θ = θ − θ³/6 + θ⁵/120 − …

The instant you keep the θ³ term, perfect imaging breaks. Those leftover terms are the aberrations. Third-order theory (keeping up to θ³) gives the five monochromatic Seidel aberrations; each is a separate way the real wavefront departs from the ideal sphere converging on the image point. Chromatic aberration is a sixth, separate effect caused by dispersion rather than geometry.

Spherical aberration

A spherical lens surface is cheap to grind because a loose abrasive between two surfaces naturally wears them into matching spheres. But a sphere is not the shape that focuses parallel rays to a point. Rays hitting the lens far from the axis (marginal rays) are refracted too strongly and cross the axis nearer the lens than paraxial rays do. There is no single focus — instead there is a range of crossing points and a smallest blur, the circle of least confusion, sitting between the marginal and paraxial foci.

The wavefront error scales with the fourth power of the ray height h in the pupil (the transverse ray error scales with h³), so spherical aberration is dominated by the rim of the aperture. This is why stopping down a camera lens by one or two stops sharpens it dramatically: halving the aperture diameter cuts the spherical blur by roughly a factor of eight. The cures are an aspheric surface (deliberately departing from a sphere), splitting the power across multiple elements, or bending the lens (changing the ratio of front to back curvature for fixed power).

The most expensive lesson in spherical aberration is the Hubble Space Telescope. Its 2.4 m primary mirror was ground to the wrong conic constant — the edge was too flat by about 2.2 micrometres, a tiny error that produced gross spherical aberration. Light that should have landed in a 0.1-arcsecond core was smeared into a halo. The 1993 servicing mission installed COSTAR, a set of small corrective mirrors that exactly reversed the error.

Chromatic aberration

Glass is dispersive: its refractive index n falls as wavelength rises. For a common borosilicate (BK7), n ≈ 1.522 at the blue F line (486 nm) but ≈ 1.514 at the red C line (656 nm). Through the lensmaker's equation the focal length depends on n, so each color has its own focal length — blue focuses closest, red farthest. This longitudinal chromatic aberration blurs the whole field; lateral chromatic aberration changes image height with color, throwing colored fringes onto off-axis edges.

The standard measure of dispersion is the Abbe number:

V_d = (n_d − 1) / (n_F − n_C)

where d, F, C are the yellow, blue, and red Fraunhofer lines. High V means low dispersion. Crown glasses have V ≈ 55–65; dense flint glasses V ≈ 30–40. Isaac Newton wrongly believed chromatic aberration was incurable and turned to mirrors; in 1733 Chester Moore Hall and later John Dollond proved him wrong with the achromatic doublet — a converging crown element cemented to a diverging flint element. The crown's power and dispersion partly cancel the flint's, bringing red and blue to a common focus. An apochromat goes further, crossing three wavelengths, often using special low-dispersion (ED/fluorite) glass.

Coma, astigmatism, field curvature, distortion

Spherical and chromatic aberration appear even on-axis. The remaining Seidel aberrations only matter for off-axis object points:

• Coma. Magnification varies across the aperture for an off-axis point, so successive ray zones form displaced, growing circles that pile into a comet-shaped flare. Coma is the killer aberration in fast parabolic telescope mirrors.
• Astigmatism. Rays in the tangential plane and the sagittal plane focus at different distances. A point becomes a short radial line at one focus and a tangential line at another, with a blurred ellipse between.
• Field curvature (Petzval). The sharpest image lies on a curved surface, not a flat sensor, so corners go soft when the center is sharp. Flat-field designs add a field flattener near the focal plane.
• Distortion. Straight lines bow — barrel distortion (edges bulge out) or pincushion (edges pinch in). Distortion misplaces points but does not blur them, so software can undistort it after capture.

How each aberration scales

Third-order theory gives each aberration a clean dependence on aperture height h and field angle H. This table is the working designer's cheat sheet — it tells you whether to stop down, shrink the field, or redesign.

Aberration	Aperture dependence	Field dependence	Symptom
Spherical	h³	—	Soft, low-contrast everywhere; halo around bright points
Coma	h²	H	Comet tails on off-axis stars
Astigmatism	h	H²	Point → line; rotates 90° through focus
Field curvature	h	H²	Center and corners can't both be sharp
Distortion	—	H³	Barrel / pincushion bowing of straight lines
Chromatic (longitudinal)	h (focus shift ∝ 1/V)	—	Color fringes, soft focus that shifts with color

Spherical vs chromatic aberration

Property	Spherical aberration	Chromatic aberration
Root cause	Geometry of refraction (θ³ term)	Dispersion — n depends on wavelength
Present in mirrors?	Yes (unless aspheric/parabolic)	No — reflection is wavelength-independent
Color dependent?	No	Yes (defines it)
On-axis?	Yes	Yes
Main cure	Aspheric surface, stop down, bend lens	Achromatic doublet, ED glass, mirror optics
Famous example	Hubble primary mirror (1990)	Newton abandoning refractors for reflectors

Numerical examples

Scenario	Result
BK7 singlet, f = 100 mm: blue (486 nm) vs red (656 nm) focus	Δf ≈ f / V_d ≈ 100/64 ≈ 1.6 mm of longitudinal color
Stop a lens from f/2 to f/4 (halve aperture)	Spherical blur ∝ D³ falls ~8×; coma ∝ D² falls ~4×
Diffraction-limited spot, λ=550 nm, f/8	Airy radius ≈ 1.22 λ (f/D) ≈ 5.4 µm — the floor aberrations must beat
Hubble mirror conic error	2.2 µm edge displacement → ~0.5 wave of spherical at 633 nm
Achromatic doublet, crown V=60 + flint V=36	Powers chosen so φ_crown/V_crown + φ_flint/V_flint = 0

JavaScript — modeling aberrations

// Where a single ray crosses the axis after a thin spherical lens,
// keeping the third-order term that the paraxial model throws away.
// h = ray height in the pupil, f = paraxial focal length,
// k = spherical-aberration coefficient (depends on glass + bending).
function focusCrossing(h, f, k) {
  // Marginal rays focus shorter than paraxial: Δ ∝ h²
  const dz = -k * h * h;          // longitudinal shift, negative = closer to lens
  return f + dz;
}

const f = 100; // mm
const k = 4e-4;
for (const h of [2, 5, 10, 20]) {
  console.log(`h=${h}mm focuses at z=${focusCrossing(h, f, k).toFixed(2)} mm`);
}
// Edge rays (h=20) cross well before the paraxial f=100 -> spherical aberration

// Chromatic focal shift from the Abbe number.
// V = (n_d - 1)/(n_F - n_C); longitudinal color ~ f / V
function chromaticFocalSpread(f, V) {
  return f / V; // mm between blue (F) and red (C) foci
}
console.log(`Crown V=60: ${chromaticFocalSpread(100, 60).toFixed(2)} mm color spread`);
console.log(`Flint V=36: ${chromaticFocalSpread(100, 36).toFixed(2)} mm color spread`);

// Lensmaker's equation showing focal length depends on n (hence on color).
function focalLength(n, R1, R2) {
  return 1 / ((n - 1) * (1 / R1 - 1 / R2));
}
const R1 = 50, R2 = -50; // symmetric biconvex, mm
console.log(`Blue n=1.522: f=${focalLength(1.522, R1, R2).toFixed(2)} mm`);
console.log(`Red  n=1.514: f=${focalLength(1.514, R1, R2).toFixed(2)} mm`);
// Blue focuses shorter -> longitudinal chromatic aberration

// Achromatic doublet condition: cancel total chromatic power.
// phi1/V1 + phi2/V2 = 0  with phi1 + phi2 = phi_total
function achromatPowers(phiTotal, V1, V2) {
  const phi1 = phiTotal * V1 / (V1 - V2);   // crown, converging
  const phi2 = phiTotal - phi1;             // flint, diverging
  return { phi1, phi2 };
}
console.log(achromatPowers(0.01, 60, 36)); // 1/(100mm) total power, in 1/mm

// Diffraction floor: the smallest spot aberrations must beat (Airy radius).
function airyRadius(lambda_nm, fNumber) {
  return 1.22 * (lambda_nm * 1e-6) * fNumber; // mm
}
console.log(`f/8, 550nm Airy radius = ${(airyRadius(550, 8) * 1000).toFixed(2)} µm`);

Where aberrations matter

• Camera and phone lenses. Every modern lens is a balancing act of aspheres and exotic glass to suppress all six aberrations across the frame and zoom range.
• Telescopes. Ritchey–Chrétien designs cancel spherical aberration and coma with two hyperbolic mirrors; refractors use apochromatic triplets to kill color.
• The human eye. Eyes carry real spherical aberration and astigmatism; wavefront-guided LASIK and aspheric IOLs are aberration correction applied to people.
• Microscopy. High-NA objectives correct spherical aberration for a specific coverslip thickness; correction collars retune it.
• Astronomy at scale. Adaptive optics measures and cancels atmospheric and instrument aberrations hundreds of times per second with deformable mirrors.
• Lithography. Chip-making lenses are held to wavefront errors of a few nanometres — thousandths of a wave — to print the smallest features.

Common mistakes

• Thinking aberrations are defects. A perfectly made spherical lens still has spherical aberration. It is geometry, not a flaw.
• Confusing chromatic aberration with spherical. Color fringes are chromatic (dispersion); a soft, colorless halo is spherical. Mirrors get the second but never the first.
• Assuming stopping down fixes everything. A smaller aperture crushes spherical aberration and coma but does nothing for distortion or lateral color, and eventually diffraction takes over.
• Ignoring the diffraction floor. Even a perfectly corrected lens cannot focus below the Airy disk (~1.22 λ f/D). Chasing aberration past that point is wasted effort.
• Treating coma and astigmatism as the same. Coma scales linearly with field and quadratically with aperture; astigmatism is the reverse. They look different and need different cures.
• Forgetting that distortion isn't blur. Distortion misplaces points but keeps them sharp, so it is the one aberration you can fully fix in software after capture.