Seidel Aberrations

Paraxial optics is a beautiful lie. It keeps only the first term of $\sin\theta=\theta-\theta^3/6+\dots$ , and in that first-order world every lens is perfect: a point maps to a point, straight lines stay straight, and the image sits on a plane. Real lenses collect rays that are not infinitesimally close to the axis, and for those the discarded $\theta^3$ term reasserts itself. Its consequences are the five Seidel aberrations — the primary monochromatic defects that survive even in perfectly figured spherical surfaces made of a single, dispersion-free glass. The name to hold onto is the first of the five, spherical aberration: a spherical surface does not bring a wide parallel beam to a single point. Rays through the rim of the lens cross the axis closer than rays near the centre, smearing a star into a bright cusped surface — a caustic — instead of a dot. The other four appear once the object moves off-axis: coma flares a point into a comet, astigmatism splits it into two perpendicular line foci, field curvature bends the sharp-image surface into a bowl, and distortion leaves points sharp but misplaces them, bowing straight lines into barrels or pincushions. Why exactly five, and not four or six? Because a rotationally symmetric system can only build its fourth-order wavefront error from three rotational invariants — the pupil radius, the field height, and their dot product — and there are exactly five independent quartic combinations of these that blur or displace the image. That counting argument, and the $\theta^3$ scaling that makes marginal rays misbehave as the cube of aperture, are the load-bearing results of this article. Everything a lens designer does — bending elements, splitting powers, adding stops, going aspheric — is a negotiation with these five numbers.

Key takeaways

Paraxial imaging keeps only $\sin\theta\approx\theta$ ; the next term $-\theta^3/6$ generates the five primary (third-order) monochromatic aberrations, present even for perfect spheres of a single glass.
Spherical aberration: marginal rays focus short of paraxial rays; the longitudinal focus shift scales as the square of ray height, $\mathrm{LSA}\propto h^2$ , and the transverse blur as the cube, $\propto h^3$ .
The five primary Seidel aberrations are spherical aberration, coma, astigmatism, field curvature, and distortion. For a rotationally symmetric optical system, these are the five third-order monochromatic aberration terms that arise in the wavefront expansion.
For an on-axis object point, spherical aberration is the primary Seidel aberration that produces image blur; the other four are field-dependent and vanish on axis in the ideal third-order description. Distortion misplaces without blurring; field curvature bends the focal surface.
Stopping down reduces aberrations that depend strongly on pupil height, especially spherical aberration and coma, but it does not remove distortion or the geometrical field curvature. At sufficiently small apertures, diffraction eventually becomes the dominant limit on resolution.
Chromatic aberration is a separate defect (dispersion, not geometry) and is not one of the five Seidel terms.
Key idea: keep one term past paraxial and a spherical, single-glass lens acquires exactly five monochromatic defects; the worst on-axis is spherical aberration, whose focus error grows as the square of the aperture height because the offending ray term is $\theta^3$ .

The core insight

Real rays disobey $\sin\theta\approx\theta$ . Reinstating the next term, $-\theta^3/6$ , and propagating it through a rotationally symmetric system produces five and only five primary monochromatic aberrations: spherical aberration, coma, astigmatism, field curvature, and distortion. Spherical aberration alone spoils the axial image, with a longitudinal focus shift $\propto h^2$ and a transverse blur $\propto h^3$ in the ray height $h$ .

Where this lives today

These five terms form a useful starting point for understanding optical design, and modern design tools extend the analysis to higher-order aberrations as well. Lens-design software can optimize aberration coefficients across many surfaces, while bending elements and distributing positive and negative optical power help control coma, spherical aberration, and field curvature. This is one reason compact phone-camera lenses often use several aspheric elements despite their small apertures. Aspheric and freeform surfaces, once nearly unmanufacturable, are now molded and diamond-turned by the million, erasing spherical aberration with a single element. At the largest scale, every giant telescope mirror is a conic figured to nanometre tolerance; the Hubble episode remains the canonical warning of what one part-in-a-million figuring error in a conic constant costs. Adaptive optics closes the loop in real time: a deformable mirror, guided by wavefront measurements, dynamically corrects wavefront distortions, including atmospheric and instrumental aberrations, in ground-based telescopes and increasingly in deep-tissue microscopy.

The physical picture

Take a fat converging lens and shine a wide parallel beam straight down its axis. Paraxial optics promises one focal point. What you get instead is a short, bright, trumpet-shaped surface of light: the rays near the rim bend too much and cross the axis first, the rays near the centre cross later, and no single plane catches them all sharply. This is spherical aberration, and it is entirely geometric — it would happen with monochromatic light and a flawless spherical surface.

Figure

Spherical aberration: marginal rays (blue) cross the axis short of the paraxial focus (amber), forming a caustic; the tightest spot — the circle of least confusion — sits about ¾ of the way toward the marginal focus

Now slide the star off-axis and three more defects wake up. A point off-axis images as a little comet (coma) because different annular zones of the lens map to displaced, differently sized rings. Push further and the point becomes two short perpendicular streaks at two different distances (astigmatism), and the surface on which the image is actually sharp bows away from the flat film into a shallow bowl (field curvature). Finally, even where the image is perfectly sharp, its position can be wrong: magnification that changes with field angle bends a square into a barrel or a pincushion (distortion).

Figure

Punchline — longitudinal spherical aberration LSA versus ray height h for a lens with f₀ = 100 mm: a clean parabola LSA = f₀ a h², marked at h = 5 mm (0.50 mm), 10 mm (2.0 mm), 15 mm (4.5 mm) — the focus error grows as the square of the aperture

The punchline is the scaling. Because the offending ray term is cubic in angle, the longitudinal focus shift climbs as the square of the ray height and the transverse blur at the paraxial focus as the cube. Double the aperture and the axial spot smears eight times worse. This one fact — third-order in the ray, quartic in the wavefront — is why fast (large-aperture) lenses are hard, why designers "bend" and split elements to cancel the coefficient, and why a single asphere can erase on-axis spherical aberration entirely.

The core equations

Everything starts from the paraxial approximation and its first failure. The refraction and imaging laws of geometrical optics come from expanding $\sin\theta$ and keeping the linear term; the aberrations are what the next term does:

\sin\theta=\theta-\frac{\theta^3}{6}+\frac{\theta^5}{120}-\dots \qquad (1)

where: $\theta$ — ray angle (radians) to the axis or normal; the retained linear term $\theta$ is paraxial optics; the first correction $-\theta^3/6$ is the seed of the five primary aberrations, so they are called third-order (in the ray angle) or Seidel aberrations. Keeping through $\theta^3$ is the third-order theory; $\theta^5$ gives the fifth-order aberrations. It is cleaner to book-keep the error in the wavefront — the optical-path departure of the real emerging wavefront from the ideal sphere converging on the paraxial image. For a rotationally symmetric system this wavefront aberration function depends only on the field height $h$ and the pupil coordinates $(\rho,\varphi)$ , and to fourth order it has exactly five terms:

W(h,\rho,\varphi)={}_0C_{40}\,\rho^4+{}_1C_{31}\,h\rho^3\cos\varphi+{}_2C_{22}\,h^2\rho^2\cos^2\!\varphi+{}_2C_{20}\,h^2\rho^2+{}_3C_{11}\,h^3\rho\cos\varphi \qquad (2)

where: $\rho$ — normalized radial pupil coordinate; $\varphi$ — pupil azimuth measured from the meridional plane; $h$ — field (object) height; the five coefficients name the five aberrations — $\rho^4$ spherical, $h\rho^3\cos\varphi$ coma, $h^2\rho^2\cos^2\varphi$ astigmatism, $h^2\rho^2$ field curvature, $h^3\rho\cos\varphi$ distortion. The transverse ray error is the pupil gradient of $W$ , $\varepsilon\propto\partial W/\partial\rho$ , so each wavefront term of degree $n$ in $\rho$ gives a ray error of degree $n-1$ . Reading off the on-axis term ( $h=0$ , so only $\rho^4$ survives) gives the spherical-aberration scalings directly:

\varepsilon_{\text{tr}}\propto\frac{\partial}{\partial\rho}\big({}_0C_{40}\rho^4\big)\propto\rho^3,\qquad \mathrm{LSA}\propto\rho^2 \qquad (3)

where: $\varepsilon_{\text{tr}}$ — transverse ray aberration at the paraxial focus (blur radius); $\mathrm{LSA}$ — longitudinal spherical aberration, the axial distance between the marginal and paraxial foci. Written in the physical ray height $h_{\text{ray}}$ in the pupil these are $\varepsilon_{\text{tr}}\propto h_{\text{ray}}^3$ and $\mathrm{LSA}\propto h_{\text{ray}}^2$ — the parabola of the punchline figure. Field curvature has its own closed invariant, the Petzval sum, which depends only on the powers and indices, not on how the elements are bent or where the stop sits:

\frac{1}{R_{\text P}}=-\sum_i\frac{1}{n_i f_i}\qquad\text{(thin lenses in air)} \qquad (4)

where: $R_{\text P}$ — radius of the Petzval surface on which the image is naturally sharp; $n_i,f_i$ — index and focal length of each element. Because every positive lens contributes inward curvature, flattening the field forces the designer to add negative elements or field flatteners — the reason a camera lens is a stack, not a singleton.

Derivation ASpherical aberration from the third-order term

This is the structural heart of the article: we show that keeping the single cubic term past the paraxial approximation forces the marginal focus to shift as the square of the ray height and the transverse blur as the cube, with no reference to any particular glass or shape. The strategy is to model the surface's power as acquiring a small height-dependent excess, read the focus shift from the change in focal length, and then propagate a marginal ray to the paraxial focal plane to get the blur — the two scalings that define spherical aberration and set the shape of the punchline parabola.

Step 1. Paraxial optics gives a single focal length $f_0$ . The $\theta^3$ term makes the effective power depend weakly on where the ray crosses the pupil; to leading order write $f(h)=f_0\,(1-a\,h^2)$ with $a>0$ , so a ray at height $h$ focuses shorter. Why $h^2$ and not $h$ : the surface is rotationally symmetric, so the first allowed correction to an even quantity like power is quadratic in $h$ — the cubic ray term becomes a quadratic power error.

Step 2. The longitudinal spherical aberration is the focus shift itself, $\mathrm{LSA}(h)=f_0-f(h)=f_0\,a\,h^2\propto h^2$ . For $f_0=100$ mm and $a=2\times10^{-4}\ \mathrm{mm}^{-2}$ this is $0.50$ mm at $h=5$ , $2.0$ mm at $h=10$ , $4.5$ mm at $h=15$ . Why it is called longitudinal: it is measured along the axis, and it is the gap the eye sees as a "soft" focus that never quite sharpens.

Step 3. Follow the marginal ray to the paraxial focal plane. Having crossed the axis a distance $\mathrm{LSA}$ early, it lands off-axis by $\varepsilon_{\text{tr}}=h\cdot\mathrm{LSA}/f(h)\approx h\,a\,h^2\cdot(f_0/f)\propto h^3$ . At $h=10$ mm, $\varepsilon_{\text{tr}}=10\times2.0/98=0.204$ mm. Why the cube: one power of $h$ is the ray's lever arm and two more come from $\mathrm{LSA}\propto h^2$ , so the transverse blur is one order steeper than the longitudinal.

Step 4. The tightest image is not at either focus. Modelling the transverse error at a defocus $c$ (in units of the marginal aberration) as $\varepsilon(\rho)=\rho^3-c\,\rho$ for normalized $\rho\in[0,1]$ , minimizing the maximum $|\varepsilon|$ gives $c=\tfrac34$ with blur radius $\tfrac14$ of the marginal value: the circle of least confusion sits three-quarters of the way from paraxial to marginal focus and is four times smaller than the paraxial-plane smear. Why this matters: it is where you actually set focus, and it quantifies the best a spherical singlet can do. *These results are self-consistent: the $\sin\theta$ series $\theta-\theta^3/6+\theta^5/120$ underpins the scaling; $\mathrm{LSA}(h)\propto h^2$ gives 0.50/2.0/4.5 mm at $h=5/10/15$ mm; $\varepsilon_{\text{tr}}\propto h^3$ means the transverse blur ratio between $h=10$ and $h=5$ is exactly 8; and the circle of least confusion sits at $c=3/4$ with blur radius $1/4$ of the marginal value.*

Go deeper · Derivation B · 1 minWhy there are exactly five — the symmetry count

Derivation BWhy there are exactly five — the symmetry count

Here we explain why third-order monochromatic aberration theory contains five primary Seidel aberrations. The counting follows from rotational symmetry and the lowest-order wavefront terms, rather than from detailed ray tracing. The strategy is to note that the wavefront error can depend on the field and pupil only through rotational invariants, enumerate the invariants of the right degree, and discard the one that does not affect the image — leaving exactly five, each of which is one of Seidel's terms.

Step 1. By rotational symmetry $W$ can depend on the field vector $\mathbf h$ and pupil vector $\boldsymbol\rho$ only through the three scalar invariants $\mathbf h\!\cdot\!\mathbf h=h^2$ , $\boldsymbol\rho\!\cdot\!\boldsymbol\rho=\rho^2$ , and $\mathbf h\!\cdot\!\boldsymbol\rho=h\rho\cos\varphi$ . Why only these: any rotation of the whole system must leave $W$ unchanged, and these three dot products are the complete set of rotational invariants built from two vectors.

Step 2. The primary aberrations are the fourth-order (in the combined coordinates) terms. The degree-4 monomials in the invariants are $(\rho^2)^2,\ (h\rho\cos\varphi)(\rho^2),\ (h\rho\cos\varphi)^2,\ (h^2)(\rho^2),\ (h^2)(h\rho\cos\varphi),\ (h^2)^2$ — six in all. Why fourth order: it is the lowest order beyond the paraxial (second-order) piston/defocus/tilt that actually varies across the pupil to blur an image.

Step 3. The last term $(h^2)^2=h^4$ has no pupil dependence — it is a constant phase (piston) over the pupil for a given field point and does not blur or displace the image, so it is dropped. Why drop it: an aberration must vary across the pupil (to blur) or scale the mapping (to distort); a pure field-dependent constant does neither.

Step 4. The surviving five are exactly $\rho^4$ (spherical), $h\rho^3\cos\varphi$ (coma), $h^2\rho^2\cos^2\varphi$ (astigmatism), $h^2\rho^2$ (field curvature), $h^3\rho\cos\varphi$ (distortion) — Seidel's five sums. Why this is powerful: it fixes the entire menu of primary defects before any lens is specified; design only sets their five coefficients, and a symmetry that kills one (e.g. a symmetric doublet about the stop kills coma, distortion, and lateral colour) does so for a structural reason. *The count is self-consistent: enumerating degree-4 monomials in the three invariants returns six; removing the pupil-independent $h^4$ leaves exactly five, each with the $(h,\rho,\varphi)$ powers tabulated above.*

Limiting cases

Case	Result	Meaning
On-axis ( $h=0$ )	only $\rho^4$ survives — pure spherical aberration	the axial star blurs into a caustic regardless of field
Small aperture $\rho\to0$	$W\to0$ faster than the mapping error	stopping down sharpens, but distortion ( $\propto\rho$ ) and field curvature persist longest
Symmetric system about the stop	coma, distortion, lateral colour cancel	odd-in- $h$ terms vanish by symmetry — the basis of the double-Gauss lens
Paraboloidal mirror, axial beam	spherical aberration $=0$	the conic exactly cancels the $\rho^4$ term on-axis (why telescopes use paraboloids)
Flat field demanded	Petzval sum $\to0$	needs negative elements or a field flattener; a lone positive lens always bows inward
Aperture $\to$ diffraction limit	geometric blur $\to$ Airy disk	past a point, stopping down trades aberration for diffraction — the two meet at the optimum $f$ -number

Worked examples

1 · The spherical-aberration budget of a fast singlet (confidence builder)

A converging singlet has paraxial focal length $f_0=100$ mm and a spherical-aberration coefficient such that a marginal ray at $h=10$ mm focuses at $f=98$ mm. Find the longitudinal spherical aberration at $h=5,10,15$ mm, the transverse blur at the paraxial focus for the $h=10$ mm ray, and the circle of least confusion.

Step 1. From $f(h)=f_0(1-a h^2)$ and $f(10)=98$ mm: $a=(1-98/100)/10^2=2\times10^{-4}\ \mathrm{mm}^{-2}$ . The longitudinal spherical aberration is $\mathrm{LSA}=f_0 a h^2$ : at $h=5$ , $\boxed{0.50\ \mathrm{mm}}$ ; at $h=10$ , $\boxed{2.0\ \mathrm{mm}}$ ; at $h=15$ , $\boxed{4.5\ \mathrm{mm}}$ — a clean $h^2$ parabola.

Step 2. The marginal ray crosses the axis $2.0$ mm early, so at the paraxial focal plane it has climbed back off-axis by $\varepsilon_{\text{tr}}=h\,\mathrm{LSA}/f(h)=10\times2.0/98=\boxed{0.204\ \mathrm{mm}}$ . Halving the aperture to $h=5$ gives $5\times0.5/99.5=0.025$ mm — smaller by a factor $8=2^3$ , confirming the cube law.

Step 3. The best focus is the circle of least confusion, at three-quarters of the way from paraxial toward marginal focus (a defocus of $0.75\times2.0=1.5$ mm) with a blur radius one-quarter of the paraxial smear: $\tfrac14\times0.204=\boxed{0.051\ \mathrm{mm}}$ . This is where you would set the film — four times sharper than either focus alone.

2 · Field curvature of a single lens (exam level)

A thin lens of focal length $f=100$ mm is made of glass $n=1.50$ . Estimate the radius of its Petzval field-curvature surface.

Step 1. For a single thin lens in air the Petzval sum is $1/R_{\text P}=-1/(n f)$ , so $R_{\text P}=-n f=-1.50\times100=\boxed{-150\ \mathrm{mm}}$ . The minus sign means the sharp-image surface bows toward the lens — a bowl, not a plane.

Step 2. A flat detector of half-width $y=10$ mm therefore sits out of focus at its edge by roughly the sagitta $y^2/(2|R_{\text P}|)=100/300=0.33$ mm — enough to soften the corners of an otherwise sharp frame. To flatten the field you must add a negative element (raising $R_{\text P}$ toward infinity) — which is exactly why fast camera lenses carry a diverging group they seem not to "need."

Physics of the portrait

Spherical aberration is the aberration that named a scandal and a rescue. It is the simplest of the five, the only one that ruins the on-axis image, and the one whose cure — the aspheric or paraboloidal surface — is oldest and cleanest.

Origins · 1 minspherical aberration and how this was discovered

spherical aberration

the archetypal monochromatic aberration · Seidel's five sums 1856 · why a sphere will not focus a wide beam

A spherical surface is easy to grind — rub two glass blanks together and they wear naturally toward spheres — but it is the wrong shape for perfect focusing, and everyone from Alhazen to Newton knew a wide beam came to a blurred cusp, not a point. The quantitative theory waited for Ludwig von Seidel, who in 1856 expanded the imaging error to third order and separated it into the five independent sums that carry his name, turning "the lens is soft" into five computable numbers a designer could attack one at a time. The paraboloid cures spherical aberration on-axis exactly, which is why Gregory, Newton, and Hadley built reflecting telescopes and why every giant telescope mirror is a conic, not a sphere. The most public lesson came in 1990, when the Hubble Space Telescope's 2.4 m primary was ground to a perfect but wrong conic — its edge too flat by about 2 μm — giving textbook spherical aberration that smeared starlight into a halo until the COSTAR corrective optics, installed in 1993, cancelled it and restored the diffraction limit. One misfigured conic, five aberrations' worth of theory, and a servicing mission: spherical aberration's whole story in one instrument.

History

300 BCE

— Euclid and later Greek opticians know curved mirrors do not focus a wide beam to a point

1637

— Descartes shows the aplanatic "Cartesian oval" surfaces that would focus perfectly, but they are impractical to grind

1668

— Newton, convinced refractors cannot be freed of colour, builds his reflecting telescope with a spherical mirror and accepts its residual spherical aberration

1840

— Petzval computes his portrait-lens design and the field-curvature sum that bears his name

1856

— Ludwig von Seidel publishes the five third-order aberration sums, the foundation of lens-design theory

1930

— Bernhard Schmidt builds his corrector-plate camera, removing a spherical mirror's aberration over a wide field

1990

— Hubble's primary mirror is found to suffer spherical aberration; COSTAR corrects it in 1993

Going further

Texts. Hecht, Optics, 5th ed., §6.3 (Aberrations) · Born & Wolf, Principles of Optics, §5.1–5.5 (theory of aberrations, the Seidel sums) · Mahajan, Optical Imaging and Aberrations · Welford, Aberrations of Optical Systems · Smith, Modern Optical Engineering, ch. on aberrations. The key numerical anchors are: $\sin\theta$ series $\theta-\theta^3/6+\theta^5/120$ ; spherical-aberration scalings $\mathrm{LSA}\propto h^2$ (0.50/2.0/4.5 mm at $h=5/10/15$ mm, $f_0=100$ mm, $a=2\times10^{-4}\,\mathrm{mm}^{-2}$ ) and $\varepsilon_{\text{tr}}\propto h^3$ (0.204 mm at $h=10$ , ratio 8 relative to $h=5$ ); circle of least confusion at defocus $3/4$ with blur radius $1/4$ ; five surviving quartic invariants after dropping $h^4$ ; single-lens Petzval radius $R_{\text P}=-nf=-150$ mm. Historical dates: Seidel 1856; Hubble/COSTAR 1990/1993.

Builds on: Refraction at a Spherical Surface · Thin Lens and Lensmaker's Equation · Fermat's Principle · Image Formation and Magnification · Snell's Law. Leads to: Thick Lenses and Principal Planes · Dispersion and Prisms · Plane and Spherical Mirrors.

Are Seidel aberrations the same as chromatic aberration?

No — they are entirely separate, and it is worth keeping them apart. The five Seidel aberrations are monochromatic: they occur in a single wavelength, from the geometry of spherical surfaces, and would persist even if glass had no dispersion at all. Chromatic aberration comes from the wavelength dependence of the refractive index, $n(\lambda)$ , so a lens has a different focal length for each colour — a defect of dispersion, not of third-order geometry. A design must correct both: the Seidel sums with shapes, stops, and aspheres; the chromatic error by pairing glasses of complementary dispersion (an achromat). The two are budgeted independently, which is why a lens can be sharp in green but colour-fringed, or colour-corrected but soft at the rim.

If a paraboloid removes spherical aberration, why don't we use paraboloidal lenses everywhere?

A paraboloid cancels spherical aberration only for an axial parallel beam — exactly the job of a telescope's primary mirror, which is why those are paraboloids. Off-axis, a paraboloid has strong coma, so its sharp field is tiny; and aspheric refracting surfaces are far harder to grind and test than spheres, historically to the point of impracticality. Modern molding and diamond-turning have changed the economics — aspheres now appear in phone cameras and projector lenses — but a single asphere still fixes only spherical aberration, leaving the four field aberrations to a multi-element design. The sphere survives because it is cheap, self-generating under grinding, and correctable in combination.

What actually is the "circle of least confusion"?

When a lens has spherical aberration there is no plane where the star is a true point, so you look for the plane where the blur is smallest. Rays from different pupil zones cross the axis at a spread of distances; as you move the screen from the paraxial focus toward the marginal focus, the outer-zone blur shrinks while the inner-zone blur grows, and somewhere between them the overall spot is tightest. For pure third-order spherical aberration that best plane sits about three-quarters of the way toward the marginal focus, and the spot there has about a quarter the radius of the smear at the paraxial focus. That tightest disk is the circle of least confusion — the practical "focus" of an aberrated lens.

Does a smaller f-number (faster lens) always mean worse aberrations?

For a given design, yes for the aperture-dependent aberrations. Spherical aberration's transverse blur grows as the cube of the ray height, so opening from $f/8$ to $f/2$ (doubling twice the aperture) worsens on-axis softness dramatically — which is why lenses are sharpest stopped down a stop or two and soft "wide open." But it is not a law of nature: a well-corrected fast lens spends its extra elements buying the aberrations back down, so a modern $f/1.4$ can outperform a cheap $f/4$ . The trade is real but designable; the only floor no design escapes is diffraction, which grows as you stop down.

Why does distortion not blur the image?

Distortion is a mapping error, not a focusing error. Every point still images to a sharp point — the wavefront is not smeared — but the transverse magnification changes with field height, so a point that should land at radius $y$ lands at $y+\text{(cubic term)}$ . Straight lines off-axis therefore bow: outward-growing magnification makes a pincushion, inward makes a barrel. Because it moves rather than blurs, distortion is uniquely correctable after the fact in software (remap the pixels), which is why phone-camera lenses are often left with visible barrel distortion that the image processor quietly undoes — a luxury none of the blurring aberrations allow.

How did the Hubble mirror end up with spherical aberration if it was "perfectly" made?

It was perfectly made to the wrong shape. The primary was polished to an extraordinarily smooth surface — but the tool that measured its conic figure, a "null corrector," was assembled with a spacing error of about 1.3 mm, so the mirror was ground to a conic constant that was slightly off, leaving its edge too flat by roughly 2 μm (about a fiftieth of a hair). That tiny conic error is textbook spherical aberration: marginal rays focused short of paraxial ones, wrapping every star in a faint halo. The optics themselves were flawless in smoothness; the specification they were figured to was wrong. COSTAR, installed in 1993, inserted small mirrors figured with the equal-and-opposite aberration, cancelling it — the five-aberration theory used as a rescue.