Optical Aberrations: The Five Seidel Terms on a Single Lens

What spherical aberration, coma, astigmatism, field curvature and distortion actually look like, plus the two color terms, all seven traced on one piece of glass you can reshape.

First of three pages on aberrations. This one is what the five are, on the simplest lens that has them. Then chromatic aberration, and the two-glass lens that cancels it. Then correcting the five, which takes three elements.

A point of light, and five ways to miss it

Point a simple lens at a star and the star does not come out as a point. Everyone expects that. What is less obvious is that it does not come out as a formless blob either. It comes out as one of a small number of specific shapes: a disk with a hard bright rim, a comet with its tail pointing outward, a short line that becomes a different short line if you move the sensor a few millimeters, an image that is sharp only on a curved surface, a picture whose straight lines bow.

There are exactly five of these shapes in one color. Let the color change and two more appear. This page puts all seven on one lens, and shows what each one answers to.

What an aberration is

An aberration is the gap between where a real ray lands and where first-order optics said it would land. Every section below is a name for one pattern in that gap.

Here is what that looks like on one piece of glass, and it is the lens for the rest of the page: a biconvex singlet of N-BK7, the most common optical glass there is, with radii of 60 mm and −60 mm, 8 mm thick, taking in a beam 20 mm wide. The sign on a radius says which way that surface bows, so this one is fat in the middle. First-order optics, the arithmetic you get by assuming every ray stays close to the axis, says its focal length is 59.399 mm and its focus sits 56.700 mm behind the back face.

Now trace real rays instead of assuming. The ray entering at the very edge of the opening crosses the axis 2.649 mm short of that focus. Put a sensor at the first-order focus and that ray lands 465.5 µm off axis, against the 2.1 µm radius diffraction alone would allow at this opening. A factor of 220, from a lens with nothing wrong with it.

Why exactly five, and then two more

A lens that is a solid of revolution cannot tell up from down or left from right, and that symmetry is a strong constraint. At the lowest order that produces a blur it allows exactly five failures, no more and no fewer. Their classical names are spherical aberration, coma, astigmatism, field curvature and distortion, and the set is called the Seidel aberrations, after Ludwig von Seidel, who first wrote them down.

Every surface contributes to each of the five and the lens’s total is the sum, and with only two surfaces one entry is wildly lopsided: at a 5 degree field the back surface makes 93 % of the spherical aberration on its own. The ray arrives at it already converging steeply, so it meets that surface at a much larger angle, and these terms grow fast with angle. That is the whole reason the correction in the next section works.

The five answer to the field angle differently, and the exponents are exact: spherical aberration not at all, coma with the first power, astigmatism and field curvature with the second, distortion with the third.

Then dispersion adds two more terms, because glass has a different refractive index at every wavelength: a single lens has a different focal length in blue than in red, and a different magnification too. Those two close the page.

Spherical aberration

One sentence: rays through the edge of the opening focus at a different distance from rays through the middle, so a point on the axis becomes a disk, and it is the only one of the five that is still there when the object sits dead on the axis.

The edge ray crosses 2.649 mm short of the first-order focus. Divide each zone’s shortfall by the square of that zone and the answer barely moves across the whole opening:

A drift of under 4 % from the middle of the opening to its rim. So the shortfall is close to proportional to the square of the zone, which makes the sideways miss close to proportional to its cube.

The cube is testable in one move. Halve the opening and the outermost ray falls from 465.5 µm to 54.7 µm, a factor of 8.5. Halve it again and the factor is 8.1. Both sit a little above eight, not on it.

The best sensor position is not the first-order focus either. Slide it toward the edge rays and the blur passes through a minimum 1.995 mm short of that focus, where the outermost ray is 115.0 µm instead of 465.5. Four times smaller for nothing. That is 0.753 of the way to where the edge rays cross, against the classical rule of thumb of three quarters and a quarter.

Reshaping one lens: bending, and an aspheric surface

Spherical aberration is the one term you can attack without adding a second piece of glass, and there are two ways to do it. Both work for the reason the last section gave: the error grows fast with the angle at which a ray strikes a surface.

Hold the focal length and the thickness fixed and slide the curvature from one face to the other. Opticians call that family a bending and label it with a shape factor q: +1 when the back face is flat, 0 when the two faces curve equally, −1 when the front face is flat, and past ±1 when both faces bow the same way. Same focal length all the way along, different spherical aberration at every step:

The minimum sits where the two surfaces share the work, and the two bars below are where you watch it happen. Everywhere else one face does most of it, and one face doing all of it is always worse than two doing half each.

A plano-convex lens of this focal length gives 330 µm at q = +1, the flat face at the back, and 1429 µm with the flat face toward the star: same glass, same power, same thickness, 4.3 times the blur from mounting it backward.

The other move is to give one surface a shape that is not part of a sphere. Put a conic constant K on the front face, which flattens or steepens it away from the axis while leaving its curvature at the axis alone, and the focal length does not move: the tool shows 59.399 mm at every value of K. Turning it collapses the disk, from 465.5 µm at K = 0 to 7.30 µm at K = −6.766, the value that makes the disk smallest. That is the largest single improvement on the page, and it is one number typed into one field: Try it is that experiment.

Coma

One sentence: off the axis, each ring of the opening images to a circle of a different size and a different center, so a point becomes a comet with its tail pointing away from the axis.

Coma is the first term that needs the object to be off the axis, and it grows with the first power of the field angle. The next four figures work with the opening stopped down to 8 mm of the 20: at the full opening the spherical disk hides everything the field does.

The sensor in the picture below is not at the image plane. Off the axis this lens focuses in front of a flat sensor, on the curved image surface two sections on, and leaving the sensor back there buries the comet in a much larger blur.

Mark four rays: the two entering at the edges of the opening along the field direction, and the two across it. The pair across lands at one common height, 24.8 µm out; the pair along straddles 76.6 µm, three times as far. Three to one is the classical signature of coma, and here it is a measurement, read at the sensor position the figure uses, in micrometers:

The ratio drifts from 3.0 to 3.3 as the field grows, so three to one is a small-angle statement.

Moving the opening along the axis changes coma too, and far enough in front of the lens it kills the third-order term outright. That knob gets its own section below.

Astigmatism

One sentence: off the axis, the rays in the plane containing the axis and the rays in the plane across it come to a focus at two different distances, each one a short line, so there is no plane anywhere in which the point is a point.

Those two directions have names: tangential for the first, sagittal for the second.

Take the field out to 20 degrees and the two foci land 9.985 mm and 5.057 mm in front of the image plane, five millimeters apart on a lens 8 mm thick. In between, the spot turns inside out:

Both lines are more than ten times the width of the diffraction spot at this opening, so they are honestly geometric and not artifacts of the drawing.

Field curvature

One sentence: even after you have dealt with astigmatism, the surface on which a lens forms sharp images is curved, and a sensor is flat.

This is the term the knobs in the previous sections barely touch.

The sagittal focus stands 0.33 mm in front of the sensor at 5 degrees, 1.30 mm at 10 and 5.06 mm at 20, and the tangential focus is always close to twice as far out: 9.98 against 5.06 at 20 degrees.

Underneath both lies a third surface, the one that would be left if astigmatism were removed altogether. It has a name, the Petzval surface, and here it is a sphere of radius 88.05 mm curving toward the lens, about one and a half times the focal length. Nothing you can reach from outside the glass moves it: put the opening anywhere from 30 mm in front of the lens to 12 mm behind it and the tool returns the same field-curvature term. Why it is the term that forces a designer to add a third element is the correction story.

Distortion, and where you put the opening

One sentence: the magnification changes with field angle, so straight lines bow, but every point is still a sharp point; distortion moves the image rather than blurring it.

Every lens has an aperture stop, the opening that decides which rays get through. On this page it has been sitting on the front face. Slide it along the axis and distortion changes size, and then sign.

Put the opening 30 mm in front of the lens and the corner of a 20 degree field comes in 8.57 % short: barrel. On the front face it is 1.09 % short. Push it 12 mm behind the back face and the corner goes 12.36 % long: pincushion.

Somewhere between, distortion changes sign. The crossing sits 4.59 mm from the front face, which is inside the glass, and so not anywhere you can put a diaphragm.

Take the same lens toward zero thickness with the opening against it and distortion goes with it, from 1.09 % to 0.0016 %. A thin lens with the opening against it has no distortion. Distortion is not a property of the glass alone. It is a property of which part of the lens each field angle is allowed to use, and the opening decides that.

Color does the same thing off the axis. Blue and red land at different image heights, which is lateral color, and the difference tracks the opening as distortion does: at a 20 degree field it is 200 µm with the opening well in front, 21 µm on the front face, 175 µm with it behind, and zero 5.02 mm inside the glass, near the distortion crossing but not on it. It vanishes with thickness too.

Color: the term no shape can fix

One sentence: the refractive index of glass changes with wavelength, so a single lens has a different focal length in every color, and no sensor position is in focus for all of them.

This is the term no amount of reshaping will touch, which is the point of putting it last.

The focus sits 56.700 mm behind the back face in yellow light, 56.085 in blue and 56.978 in red. Put the sensor at the yellow focus and blue arrives as a disk 105 µm in radius.

Here is the part that matters, and the reason one piece of glass cannot fix it. The gap between the blue focus and the red focus is 0.89 mm and it does not depend on the opening at all: open the lens to its full 20 mm or stop it to a quarter and the tool returns the same number. What the opening changes is the blur, which shrinks in proportion, from 105 µm to 26 µm. So stopping down hides color without removing it.

Bending does not help. An aspheric surface does not help. Moving the opening does not help. The only thing that helps is a second piece of glass that disagrees with the first about color, which is the achromatic doublet.

Three demands, one knob

The opening’s position turned out to be a real control: it changed distortion’s sign, it changed lateral color’s sign, and it changes coma too. So put the question the way a designer would.

Coma wants it 20.24 mm in front of the front face. Distortion wants it 4.59 mm behind that face, which is 4.59 mm inside the glass. Lateral color wants it 5.02 mm in, also inside the glass. Field curvature does not care where the opening goes at all and cannot be moved by it.

One knob, three demands, and a fourth term that ignores the knob entirely. Two of the three want the diaphragm in a place you cannot physically put one, and no setting satisfies even two of them at once. That is the honest end of what one lens can do, and it is where correcting aberrations begins.

Try it

Open the Ray-Optics Designer, and under Starting design choose Biconvex Singlet (teaching); the manual lists them all. Then do three things this page did.

Halve the entrance pupil diameter, which is the tool’s name for the opening, and watch the on-axis spot fall about eightfold; the spot diagram needs a free account. Put a conic constant of −6.8 on the first surface and watch the focal length not move. Insert a plane surface after the first one, carrying the same glass, and move the stop to it.

Those last two, and the whole layout, need no account at all; the spot diagrams, the ray fans and the table the tool calls the third-order sums come with a free one. A two or three surface lens is well inside the anonymous limit of 6.

References

• W. T. Welford, Aberrations of Optical Systems (Adam Hilger, 1986), chapters 7 and 8: the five sums in the form the tool computes them.
• R. Kingslake and R. B. Johnson, Lens Design Fundamentals, 2nd ed. (Academic Press, 2010), chapters 6 to 11: aberration by aberration, bending, and the Petzval theorem.
• W. J. Smith, Modern Optical Engineering, 4th ed. (McGraw-Hill, 2008), chapter 3: the transverse and longitudinal pictures of each aberration.
• M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge, 1999), chapters 4 and 5: the symmetry argument for the five primary aberrations.

Traced at 587.6 nm unless a figure says otherwise; the two color sections also use 486.1 nm and 656.3 nm. The stopped-down figures run at 8 mm, where diffraction alone allows 5.3 µm, and the through-focus strip is drawn in a window of ±659 µm, on 8 mm of glass.