Optical Aberrations: The Five Seidel Terms, on a Real Lens

What each one does to an image, traced live on a real camera lens from an 1896 patent.

The gap: four and a half micrometers, or a hundred and eighty-eight

Take a lens with a focal length of 98.27 mm and an aperture of 15.38 mm — the patent calls it f/6.5, and its printed radii give f/6.4 — and light it with a single color, the sodium D line at 589.3 nm. If the lens were perfect, a star on its axis would image as a bright disk with a first dark ring at a radius of 4.59 µm, the size diffraction alone would make it. That is the floor physics sets.

Now trace real rays through a real lens of exactly those numbers: the Cooke triplet from H. D. Taylor’s 1896 patent, US 568,052, read straight from the printed table. The 127 rays the Ray-Optics Designer sends through the lens land on the image plane in a blob whose outermost ray is 187.7 µm from the axis. Forty-one times the diffraction floor, on axis, in one color.

What you are looking at. A Cooke triplet is three separate pieces of glass on one axis, a positive element, a negative one, and a second positive one, with the aperture stop, the ring that decides how wide a bundle of light the lens accepts, on the back surface of the middle element. It is the simplest lens with enough adjustable quantities, six curvatures and two air gaps, to hold the focal length and bring all seven primary aberrations under control at once: the five on this page and two color errors. That is why it is still the first lens a design textbook works through, and the ancestor of a large family of camera lenses.

Three pieces of glass mean six curved surfaces, numbered 1 to 6 from the object side in the diagram above, and every table on this page is a sum of one contribution per surface: 1 and 2 are the front crown, 3 and 4 the flint in the middle, 5 and 6 the rear crown. Crown and flint are the two broad families of optical glass, told apart by how strongly they spread colors, not by density or index: a flint fans white light into a wider spectrum than a crown does. That stronger color spread is what lets a negative flint cancel the color error of two positive crowns without canceling their focusing power. On this page, in one color, the middle element’s visible job is a different one: flattening the field, which the ledger below shows surface by surface.

The radii, the thicknesses and the two glasses’ refractive indices, the lens’s prescription, are the numbers printed in H. D. Taylor’s 1896 patent, US 568,052. Nothing on this page is fitted, tuned or rounded to look better.

Where did the factor of forty-one go? Into five named places. Not because a lens has five kinds of flaw, but because, to lowest order, rotational symmetry allows exactly five ways for a wavefront to depart from a sphere. They are spherical aberration, the blur that is there even on axis; coma, the comet tail off axis; astigmatism, two focus distances and no sharp plane; field curvature, a sharp surface that is not flat; and distortion, the image moves but does not blur. The rest of this page takes them one at a time: what each does to a point image, how each grows with aperture and with field angle, and how a designer reads them off a real prescription. The lens stays on the page throughout: every widget below re-traces those same six surfaces in your browser, with the same code the Ray-Optics Designer runs, and you move the sliders.

What an aberration is

A lens that images a point perfectly turns the light from that point into a spherical wavefront converging on one image point. Any real lens produces a wavefront that is not quite spherical. The wavefront aberration W is the gap between the two: pick a reference sphere centered on the ideal image point and passing through the center of the exit pupil (the lens aperture as seen from the image side), and measure, ray by ray, how far the actual wavefront sits from that sphere along the ray. W is a function of where the ray crosses the pupil.

A few words carry the rest of the page. A ray is labeled by where it crosses the pupil, from 0 at the center to 1 at the rim.

Field point

: a direction in object space, its angle θ from the axis, always set in the vertical plane here, so “up” in the image is the field direction; this lens covers fields out to 26°.

Chief ray

: the ray through the center of the pupil.

Marginal ray

: the ray at the rim, in the vertical plane.

Tangential

: the vertical section of the pupil. Sagittal is the horizontal one, and most aberrations treat the two differently.

Paraxial

: computed for rays so close to the axis that the lens is ideal.

The wavefront picture and the ray picture are the same information. A ray runs perpendicular to its wavefront, so a tilt in W is a bend in the ray, which is a miss at the image: the transverse ray aberration ε, where a ray meets the image plane minus where the chief ray of the same field meets it. With R the radius of the reference sphere, n′ the refractive index in image space, and (x_p, y_p) the ray’s position in the exit pupil,

Everything else on this page, the sums, the fans and the spots, is that one derivative applied to one polynomial.

Why exactly five

A rotationally symmetric system cannot tell x from y, or a field point on one side of the axis from its mirror image on the other. So W can depend on pupil and field only through three combinations that survive a rotation: ρ², how far out in the pupil; h², how far out in the field; and h·ρ·cos φ, their product, with φ the angle round the pupil from the field direction. Expand W in those three. The first-order terms are a constant, a focus shift and a change of magnification, absorbed into the choice of image plane and scale. The next order, fourth-order in the coordinates, is where the aberrations begin, and there are exactly five:

This is the third-order, or Seidel, wavefront aberration, and S_I to S_V are the Seidel sums, in millimeters on this page: spherical aberration, coma, astigmatism, field curvature (the Petzval term) and distortion, in that order. Third-order, because the ray misses that follow from this W are cubic in the coordinates. Sums, because each surface contributes to each of the five separately and the contributions simply add, which is what makes the ledger below readable.

Look at how the pupil and field enter each term, because that is the whole art of reading a Seidel table:

Every row sums to three powers in ε. Spherical aberration is all aperture and no field; distortion is all field and no aperture; the other three sit between. The exponents are exact in the engine, and the widget prints them.

Here is the ledger at the full 26° field and the patent’s aperture. One row per surface, numbered as in the diagram at the top, each with the radius it is ground to and the glass it sends light into; all values in mm:

Read down any column and you see what a triplet is for. The astigmatism column has individual surfaces contributing ±0.5 mm and a sum of −0.0025 mm: a 200-to-1 cancellation, arranged by hand in 1896. The field-curvature column shows the two crowns (surfaces 1–2 and 5–6) putting in +0.37 and +0.08 mm and the flint (surfaces 3–4) taking out −0.41 mm. That is the job of the negative element in the middle, and it is why a triplet has three pieces of glass and not two. Surface 5 contributes almost nothing to S_I and S_II because the marginal ray meets that nearly flat surface (R = 362 mm) close to its normal, so the refraction invariant A — n times the angle at which the ray meets the surface — is near zero there.

Spherical aberration

One sentence: rays through the edge of the pupil focus at a different distance from rays through its center, so a point on the axis becomes a disk, and it is the only one of the five that does not vanish on axis.

What it looks like: rays from the rim of the pupil cross the axis short of where the central rays do, and the further out in the pupil a ray enters, the sooner it crosses; the figure below is that crossing, drawn from real rays. On any plane you choose the star is a disk with a bright core and a halo. In the ray fan — a plot of how far each ray in the pupil misses, which the last section of this page draws — it is a cubic through the origin, ε ∝ ρ³; in the spot diagram, where every ray is a dot, it is round, dense at the center and thin at the rim.

How it scales: pupil power 3, field power 0: the two exponents, one for the aperture and one for the field, that tell the five apart. The blur grows as the cube of the aperture and does not care where in the field you look. On this lens, S_I converts to a 220 µm miss for the marginal ray, third order; the real marginal ray misses by 188 µm, the number the page opened with, forty-one times the 4.6 µm floor.

Stop the lens down and the blur collapses far faster than the diffraction floor grows. Near 6 mm the geometric blur and the Airy radius are about equal, 13.0 and 11.8 µm: below that the lens is diffraction-limited on axis, above it geometry wins. Halve the patent’s aperture and the outermost ray comes in from 188 to 27 µm: the cube law says eightfold, the rays say sevenfold. The slider above stops at the patent’s aperture; our camera-triplet guide opens it further.

Coma

One sentence: off axis, each ring of the pupil images to a circle of a different size and center, so a point becomes a comet, its tail stretched along the line between the image point and the axis.

What it looks like: take a ring of rays around the pupil and follow them to the image. Off axis they do not land in a ring around the chief ray’s image point; they land on a circle pushed to one side of it, toward the axis on this lens.

Smaller rings give smaller circles, pushed less far, all tangent to a pair of lines 60° apart: that wedge is the comet. The rays from the top and bottom of the pupil land three times as far from the chief ray as the rays from its left and right, so tangential coma is three times sagittal coma, and because the whole pattern lies to one side of the chief ray, its average position is displaced from it.

How it scales: pupil power 2, field power 1. Double the field and coma doubles, which the field slider shows; double the aperture and it quadruples. On this lens at 10° the rays’ average position sits 46.3 µm inside the chief ray, toward the axis, and the RMS spot radius, the root-mean-square distance of the same 127 rays from their average position, has grown from 119 µm on axis to 142 µm. Taylor got the third-order coma small (it moves the rim of the pupil 46 µm at 10°, against 220 µm of spherical), so the tail you see is mostly the next order up, whose own top-to-side ratio is five rather than three, and which the ledger cannot see.

Astigmatism

One sentence: off axis, the tangential fan of rays and the sagittal fan focus at two different distances, each to a short line, so there is no plane where the point is a point.

What it looks like: in the wavefront it is a cylinder added to the sphere, so the wavefront curves more in the tangential section than in the sagittal one, and the two sections come to a focus at two different distances from the lens.

At the tangential focus the point is a short line perpendicular to the field direction (the tangential rays have come together in y but are still spread in x); at the sagittal focus it is a short line the other way round; halfway between sits a round blur, the circle of least confusion, and nowhere is it a point.

How it scales: pupil power 1, field power 2, so doubling the field separates the two foci four times as far and makes each line twice as long; the field slider shows both. At 26° the rays put the foci somewhere else entirely, one of them on the far side of the sensor, and that is the story of Where third order stops.

Field curvature and the Petzval surface

One sentence: even with astigmatism removed, the surface on which a lens forms sharp images is curved, and how curved depends only on the curvatures and refractive indices of its surfaces, never on where the stop is.

What it looks like: S_IV is the one sum with no ray heights and no ray slopes in it, only each surface’s curvature and the index change across it. Take the astigmatism away and both line foci coincide on one curved surface, the Petzval surface, which on this lens bows toward the glass. A flat sensor behind a curved image surface is sharp at the center and soft at the corners, or the reverse if you refocus; no focus setting fixes both.

For thin lenses in air this is the Petzval theorem: every positive element curves the field inward, toward the lens, in proportion to its power divided by its index, and the only way to flatten it with spherical surfaces is a negative element.

How it scales: pupil power 1, field power 2, exactly like astigmatism. On this lens the Petzval surface has a radius of -327 mm, curving toward the lens, a little over three focal lengths; the slider’s readout is its sag at the field you set. The traced-ray surfaces the picture also draws tell a different story at 26°, told in Where third order stops, because it is not a story about S_IV.

Distortion

One sentence: the magnification changes with field angle, so straight lines bow, but each point is still a point; distortion moves the image, it does not blur it.

What it looks like: the distortion term of the wavefront is a pure tilt across the pupil, and a tilt does not spread a bundle, it steers it. So every point is still a point; it is just not where a perfect lens would have put it, which for a lens focused at infinity is the focal length times the tangent of the field angle, f·tan θ (the photographic convention, and the Ray-Optics Designer’s).

When the outer points land too far out, the corners of a square grid are pulled outward and its sides curve inward: pincushion. When they land too far in, the corners are pulled in and the sides bulge outward: barrel.

How it scales: pupil power 0, field power 3. Stopping down does nothing to it, and as a fraction of the image height it grows as the square of the field. One part in eight hundred at the corner of the plate is invisible to the eye, which is why the picture above has a magnifier, and it was one of the triplet’s selling points.

Notice that the per-surface S_V entries in the ledger were the largest numbers in the table (±0.8 mm) and the sum is 30 times smaller. Distortion is not corrected surface by surface; it is corrected by balancing large contributions of opposite sign, which is what the stop position controls, and that is the next section.

Moving the stop

Here is the single most useful theorem in the whole subject. Move the aperture stop along the axis and the marginal ray does not change at all, so S_I is unchanged. S_IV knows only curvatures and indices, so it is unchanged too. Only the chief ray moves, and it moves by a fixed multiple of the marginal ray; the multiple is ε, the stop-shift parameter, the change in chief-ray height at the first surface divided by the marginal height there (the substitution is in How the sums are computed, above). Write S* for a sum after the move; the stop-shift equations are one line per sum:

• S_I* = S_I
• S_II* = S_II + ε S_I
• S_III* = S_III + 2ε S_II + ε² S_I
• S_IV* = S_IV
• S_V* = S_V + ε (3 S_III + S_IV) + 3ε² S_II + ε³ S_I

On the widget, look at the bars first, S_I and S_IV pinned while the other three slide along their curves; then the field surfaces; then the distortion.

Coma moves linearly with the stop, astigmatism quadratically, distortion cubically, and the two aperture-only terms not at all. So if a lens has spherical aberration, there is always a stop position that removes its third-order coma: ε = −S_II/S_I. This is exactly what the position of the diaphragm is for in a triplet, and why Taylor’s patent specifies it.

On the Cooke, the patent puts the stop “as closely as possible behind” the middle element, which the saved lens this page uses puts at the rear surface of the flint. Slide it backward through the 11.2 mm airspace toward the third element, and the engine reports (26° field, patent aperture, sums in mm):

S_I and S_IV do not move, and neither does the focal length (98.268 mm at every position). Third-order coma passes through zero at 2.27 mm behind the flint. Astigmatism dips and then climbs; distortion grows monotonically, because the ε(3S_III + S_IV) term is dominated by S_IV and never changes sign.

The real rays follow, roughly. At 13°, the spot RMS goes from 147 µm at the patent’s stop to 121 µm at the third-order coma zero, keeps improving to 106 µm at 5 mm, then degrades to 277 µm at the far end of the airspace. The ledger tells you the direction and roughly where; the rays tell you where.

Where third order stops

The Seidel sums are the first term of a series, and this lens is a good place to watch the series stop converging. Here are the real-ray field surfaces beside the third-order ones, in mm from the paraxial image plane, negative meaning inside, toward the lens:

At 5° they agree to within 6.4 %. At 10° third order overshoots the tangential focus by a quarter and the sagittal by an eighth, and every “third order says, the rays say” clause in the sections above is this row. By 26° the real tangential surface has swung to the far side of the paraxial plane while third order says it should be 2.9 mm inside: the two have opposite signs. That is fifth-order astigmatism and oblique spherical aberration, and it is not a defect of the lens: it is how the lens works. Taylor did not have the Seidel sums at zero and the field flat; he had them small and the higher orders bending the field surfaces back so that the whole 26° plate was tolerably sharp. The real tangential surface recrosses the image plane near 20°, and the spot RMS across the field falls with it, from 142 µm at 10° to 114 at 20°.

Here are the fans and the spot of the whole lens at once, at 26° with the third-order prediction switched on.

Two things to read off the fans: the two leave the origin at visibly different slopes, the tangential climbing and the sagittal dipping, and that difference is astigmatism, 5.7 times what third order drew, though with the same sign; that the tangential fan climbs at all, where its dashed twin dips, is the field-curvature sign flip of the table above. And the tangential fan is no longer odd about the origin, the top and bottom rays missing the chief ray on the same side, which is coma.

The Seidel view in the Ray-Optics Designer is the third-order ledger and says so; the spot diagrams and ray fans are exact real rays through the exact surfaces, and where they disagree with the ledger, the rays are right. The fan labels under each plot are fitted to the real fan and name what dominates: at 26° on this lens “Coma-dominant, with fifth-order” on the tangential side, which is why the comet in the coma figure, dragged to the full field, points the way it does and the drawn circles do not. Learn to read the sums for what to change and the rays for whether it worked.

What this page left out: color

Everything above was at one wavelength, because the patent prints only one index per glass. Real glass has an index that falls with wavelength, so a single lens has a different focal length in blue than in red (axial color) and a different magnification (lateral color), and both are as large as anything in the Seidel table. Correcting them is the reason the Cooke’s two outer elements are crown glass and the middle one is flint, and the reason the Abbe number is the first thing a designer reads off a glass map. That is the next tutorial: chromatic aberration and the achromatic doublet.

Try it on the tool

Open the Ray-Optics Designer, load the Cooke Triplet f/6.5 preset from the preset list, and look at the Seidel aberrations view, then Ray fans and Spot diagrams at 0°, 10° and 26°. Then do the three things this page did: halve the entrance pupil diameter and watch the on-axis spot fall about sevenfold; insert a plane surface after surface 4 (the back of the middle element — see the diagram at the top) and move the stop to it; and nudge the first radius, which is where the camera-triplet guide begins. Loading this lens, editing it and seeing it drawn need no account. The Seidel table, spot diagrams and ray fans need a free one, and so does the seventh surface the stop shift adds, because without an account the editor holds six; nothing here needs more than that.

References

• W. T. Welford, Aberrations of Optical Systems (Adam Hilger, 1986), chapters 7 and 8: the Seidel sums in the A, Ā, δ(u/n) form used here, the stop-shift equations, and the wavefront expansion in this normalization.
• R. Kingslake and R. B. Johnson, Lens Design Fundamentals, 2nd ed. (Academic Press, 2010), chapters 6 to 11 and 14: the aberration-by-aberration treatment, the Petzval theorem and the triplet.
• W. J. Smith, Modern Optical Engineering, 4th ed. (McGraw-Hill, 2008), chapter 3: the transverse and longitudinal pictures of each aberration and the ray-fan reading.
• M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge, 1999), chapter 5: the symmetry argument for the five primary aberrations and the wavefront-to-ray relation.
• H. H. Hopkins, Wave Theory of Aberrations (Oxford, 1950): the wavefront formulation and the insensitivity of W to the reference-sphere radius.
• H. D. Taylor, “Lens”, US Patent 568,052 (1896), Series III prescription, the lens on this page. The preset’s provenance, including the convexity-to-directional radius-sign conversion, is recorded in the Ray-Optics Designer manual.