Correcting Aberrations: The Cooke Triplet, Stop Shift and the Petzval Sum

How a designer trades five errors against eight free numbers, on the 1896 lens that started it.

Third of three, and the only one that leans on the other two. If you want what an aberration is, start with the five terms on a single lens; for color, the achromatic doublet. This page is about what a designer does about them, on a lens with enough freedom to do it.

The lens, and the eight numbers that shape it

A lens is corrected when large errors cancel, not when small ones are absent. On the Cooke triplet of H. D. Taylor’s 1896 patent, every one of the six glass surfaces bends light badly on its own, and the design’s whole art is the arithmetic that makes those six contributions add to nearly nothing. Here is the lens, and then the ledger that shows it happening.

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, which is what lets a negative flint cancel two positive crowns’ color error. 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.

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 ledger, surface by surface, and what cancels

The five things a lens can do wrong to a point of light, in one color and to lowest order, are spherical aberration, coma, astigmatism, field curvature and distortion, each defined, and each traced on a single lens, on the page before this one. Here they are as five numbers you can add up.

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, and that symmetry is what allows exactly five. Expanded to the order where a blur first appears, the wavefront error reads:

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.

A ray runs perpendicular to its wavefront, so a tilt in W is a miss at the image: each column below is a wavefront coefficient and every fan on this page is its derivative. Everything below, the ledger, the fans and the spots, is that one polynomial and its derivative.

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. Every one of the six surfaces contributes more astigmatism than the finished lens has, the strongest of them 200 times more, the weakest still 5 times more, and they sum to 0.50 % of the largest. 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.

Those five percentages are the same at every field angle. Each per-surface contribution carries exactly the same power of tan θ as its column sum does, so the field factor cancels out of the ratio: the cancellation is a property of the design, not of where you look.

Six curvatures and two airspaces are 8 numbers you can choose. The focal length, the five sums above and the two color errors are 8 things you must satisfy. Eight against eight is why the triplet is the simplest lens that can satisfy all of them at once and why a doublet cannot; the color half of that count is not computed here, because this lens is specified at one wavelength, and Taylor’s crown-flint pairing is what answers it. The numbers for that half are on the achromatic-doublet page.

What is left after correction

The ledger’s five columns are five different-looking failures. Here they are drawn at the corner of the plate, one at a time, in one window and at one scale.

After correction: rays through the rim of the pupil still focus nearer the lens than rays through the center, which is spherical aberration, and it is the one column this design leaves largest: the six surfaces cancel to 11.95 %, against half a percent for astigmatism. Alone among the five, spherical aberration is the same size everywhere in the field, so this panel does not change when you move the ledger’s field slider.

After correction: a point off axis still images as a small comet, but the third-order coma the ledger reports is down to 1.79 % of one surface’s share, so what you are looking at is mostly the next order, which the ledger cannot see.

After correction: the tangential and sagittal fans still focus at two different distances, which is astigmatism, and this is the column the triplet cancels best: 0.50 % of its largest single surface. The rays separate the two foci further than third order says, with the same sign, and where third order stops is where that is read.

After correction: the sharp image still lies on a curved surface, because field curvature depends only on the curvatures and the indices, not on bending and not on where the stop sits, and that is the one error you cannot balance away with the other seven numbers. It is also the largest of the five residuals in the picture above, which is the next section.

After correction: the image is still not quite where a perfect lens would put it, which is distortion, and nobody optimized it: the per-surface entries were the largest numbers in the table and the sum is thirty times smaller, cancelled by accident of the stop position at 3.26 %.

Why a flat field needs a third element

After correction: field curvature is the one term the other seven numbers cannot balance away. 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.

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.

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.

Neither of the two knobs the rest of this page uses reaches it. Bending an element does not move S_IV, and neither does the stop: S_IV* = S_IV is one of the five lines in Moving the stop, and it holds to the last printed digit at every stop position the widget offers. So the only lever left is a negative element, which is the third piece of glass. On this lens the Petzval radius is -327 mm against a focal length of 98.3 mm, a ratio of -3.325.

The slider’s readout is the 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.

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.

Color, already in hand

The two color terms are the seventh and eighth of the eight requirements above, and this lens cannot show them: the patent prints one index per glass, so everything on this page is at one wavelength. Real glass has an index that falls with wavelength, which is why Taylor’s two outer elements are crown glass and the middle one flint: that pairing is the color correction, and it is the reason the middle element is the one it is. The numbers are on 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.
• Optical aberrations: the five Seidel terms on a single lens: what each of the five does to a point image, traced on a biconvex singlet where every one of them is large enough to photograph.
• Chromatic aberration and the achromatic doublet: the two color terms this lens is specified at one wavelength to show, and the two-glass lens that cancels the first of them.