The flat scene that comes back curved

Everything about how a lens forms an image begins with the assumption that a flat object gives a flat image. What a lens is doing draws three rays from a point and finds where they cross, and repeating that for every point of a plane object gives a plane image at a definite distance. The construction never suggests otherwise, and neither does the wave picture that replaces it in where rays stop being enough.

It is not true, and the failure is not small. A single lens focuses a flat scene onto a bowl, and for an ordinary photographic lens the bowl’s radius is comparable with the focal length. A sensor placed in it is correctly focused in the middle and wrong everywhere else, by an amount that grows as the square of the distance from the centre.

The bowl a lens actually focuses onto. Where a lens of 50 mm focal length brings each part of a flat scene to a focus, against distance from the centre of the field, out to 21.6 mm — the corner of a 35 mm frame. The surface of best focus is a sphere of radius 76 mm curving towards the lens, and the corner of the frame sits 3.13 mm in front of the plane the centre is focused on. The shaded band is the depth of focus at f/8, which is 0.070 mm — 45 times smaller than the sag it has to cover. The curvature is not an error in the lens. It is what Σ1/nf comes to for this stack, and it depends on the powers and the glasses and on nothing else: bending the surfaces, moving the stop or stopping down changes every other aberration and leaves this one exactly where it was. — Fig. 1 Where a 50 mm lens brings each part of a flat scene to a focus, against distance from the centre of the field. The surface of best focus is a sphere curving towards the lens, and the corner of a 35 mm frame sits several millimetres in front of the plane the centre is focused on. The narrow band is the depth of focus at f/8 — smaller by a factor of tens.

The one aberration nothing removes

What makes field curvature different from every other defect of a lens is the shortness of the expression for it.

For a stack of thin lenses in air, the curvature of the image surface is minus the Petzval sum, $\sum_j 1/(n_j f_j)$ : for each element, its power divided by its refractive index. That is the whole of it. The shapes of the surfaces do not appear. The separations do not appear. Where the aperture stop sits does not appear, and neither does the f-number, the object distance, or how well the glass was polished.

This is worth dwelling on, because it is not how aberrations usually behave. Spherical aberration depends on how the power is split between the two surfaces of an element, so bending a lens changes it. Coma and astigmatism depend strongly on where the stop is, which is why moving a stop is one of a designer’s main tools. Chromatic aberration is fixed by pairing glasses, as in two glasses that cancel a derivative. Field curvature responds to none of these.

Two lenses of the same focal length made of the same glass have the same field curvature, however differently they are built. The only way to change it is to change the powers or the glasses, and since the powers are what make the lens a lens, that means adding elements.

Where the bowl comes from

The shortness of the expression is a clue that its origin is simple, and it is.

Take a single refracting surface of radius $R$ separating two media. A point object on the axis is imaged at some distance; a point object off the axis is imaged too, and the question is where the locus of those images lies. The answer, for a surface, is a sphere — and the geometry that produces it is the same geometry that makes the surface itself a sphere. An off-axis ray bundle strikes the surface at a place whose distance from the object differs from the axial one, and the difference is quadratic in the field height, which is exactly the shape of a sphere near its pole.

What makes the result useful is that the sphere’s curvature comes out proportional to the power of the surface divided by the indices on either side, and is completely independent of where the object was. So each surface contributes its own fixed amount of image-surface curvature, the contributions add, and the total does not care how the surfaces are arranged.

That independence is the whole reason a designer cannot reach it. An aberration whose size depends on the object distance, the ray height at a surface or the stop position can be traded against another aberration by changing one of those. Field curvature depends on none of them, so it can only be traded against itself — which means adding an element whose contribution has the opposite sign.

What it costs on the picture

A defocus of $\Delta$ spreads a point into a disc of diameter $\Delta/N$ , where $N$ is the f-number. So the blur at the edge of the frame follows the sag of the bowl divided by the cone angle.

Fig. 2 The blur diameter on a flat sensor against distance from the centre of the field, at three apertures. The three curves are the same shape scaled apart, because stopping down changes the cone and not the defocus. A sensor pixel is four or five micrometres.

Whether a blur of a given size is visible is a separate question with its own answer, and how far apart two things have to be supplies it: a lens at f/8 cannot resolve better than about ten micrometres however perfect it is, so a corner blurred over ten micrometres is at the diffraction limit and a corner blurred over two hundred is not a lens with a problem but a lens with a different problem.

The numbers settle the practical question. Stopping a simple lens from f/2.8 to f/16 improves the corner by a factor of about six, which is a large improvement and leaves the corner blurred over dozens of pixels. There is no aperture at which the corner is sharp, because the aperture is not what is wrong.

That is the reason a photograph taken with a simple lens has a characteristic look — sharp in the middle, softening outwards, and softening faster than any vignetting would explain. It is also the reason the softness cannot be repaired in software: the information is not merely attenuated at the corners, it is defocused, and a defocus of that size is not invertible in the presence of noise.

The condition that flattens it

If the sum is $\sum 1/(n_j f_j)$ , then it can be made zero, and the way to do so falls out of the expression: use negative elements in a glass of high index and positive elements in a glass of low index. A negative dioptre in dense flint removes more curvature than a positive dioptre in light crown puts in.

Three ways to make the same power, and what each does to the field. Three stacks of thin lenses, all of 50 mm focal length, with the Petzval sum Σ1/nf computed for each. One crown singlet: sum 1.32e-2 per mm, a field radius of 76 mm; One singlet in dense flint: sum 1.16e-2 per mm, a field radius of 86 mm; A crown and a flint, separated: sum 1.30e-9 per mm, which is a flat field to any sensor. The sum is a weighted count of powers, so a negative element of high index subtracts more field curvature per dioptre than a positive element of low index adds — and a pair with equal and opposite powers, in glasses of different index, can have net power and no field curvature at all. That is the Petzval condition, and it is the reason a flat-field lens is built from more elements than its power needs. The cost is that the elements have to be separated to keep the power, which lengthens the lens, and that the negative element undoes some of the positive one's work, so both have to be stronger than the net — which makes every other aberration worse. — Fig. 3 Three ways of making a 50 mm lens, with the Petzval sum computed for each. A singlet in a denser glass is a little better and nowhere near enough. A crown positive paired with a flint negative of the matching power, separated to recover the focal length, has a sum indistinguishable from zero.

The trick that makes this work is that the sum does not depend on the separations and the focal length does. A pair of strong opposed elements can be given a sum of exactly zero and then moved apart until the combination has whatever power is wanted — which is the design Petzval published in 1840 and the reason his portrait lens outperformed everything before it.

The cost is visible in the same figure. The two elements are individually much stronger than the pair, so each is doing far more bending than the net power requires, and every aberration that grows with the power of a surface grows with it. A flat-field lens therefore needs more elements than a curved-field one of the same speed — not to flatten the field, which two elements can do, but to clean up the damage the flattening caused. That trade-off runs through the whole subject: the condition a lens must meet is another constraint that costs elements, and a modern lens has a dozen or more surfaces because it is satisfying half a dozen such conditions at once.

The lens that started it

Petzval’s portrait lens is worth a paragraph because it shows what the condition was worth in practice.

Photography in 1840 meant exposures of many minutes, and the limiting factor was the lens: the standard landscape meniscus worked at about f/16, and nobody could sit still long enough. Petzval was a mathematician rather than an optician, and he computed a four-element design that worked at f/3.6 — roughly twenty times as much light, taking the exposure from minutes to under a minute. Its front pair was cemented and its rear pair separated, and it was the separated rear group with a strong negative flint element that did the work described here.

The lens had a curved field even so, because Petzval traded the sum against the other aberrations rather than zeroing it, and the corners of a portrait made with it fall away sharply into a characteristic swirl. That was accepted, and eventually valued: a portrait lens whose corners dissolve is drawing the eye to the face. The aberration became a style once it stopped being a limitation, which is not a rare fate for an optical defect.

Three surfaces, not one

The account so far pretends that a point off the axis has a single best focus. It does not.

Not one surface but three. A real lens does not have a single surface of best focus off axis. A point away from the axis is imaged as a short line in one orientation at one distance and as a line at right angles to it at another, and the two surfaces those foci lie on straddle a third — the Petzval surface, which is where they would both be if the astigmatism were removed. The geometry fixes the spacing exactly: the tangential surface departs from the Petzval surface three times as far as the sagittal one does, which the figure checks rather than quotes. Here the Petzval surface reaches 3.13 mm at the corner, the sagittal 3.69 mm and the tangential 4.81 mm. That factor of three is what lets a designer cheat. Astigmatism of the right sign pulls the sagittal surface flat while pushing the tangential one further away, so a lens can be made to look sharp on a flat sensor for detail running one way and not the other — which is what a cheap wide-angle lens is doing at its corners, and why the smear there has a direction. — Fig. 4 The three surfaces. Astigmatism images an off-axis point as a short line in one orientation at one distance and a line at right angles to it at another; the Petzval surface is where both would be if the astigmatism were removed. The geometry fixes the spacing: the tangential surface departs three times as far as the sagittal one.

The factor of three is exact and it is the designer’s opening. Astigmatism of the right sign and size pulls the sagittal surface onto a flat sensor while pushing the tangential one three times further away — so a lens can be made to appear sharp at the corners for detail running one way and not the other.

Which surface is worth flattening is a question with an answer, and the answer depends on the subject. Radial detail — a spoke, a mast, anything pointing away from the centre of the frame — is imaged by the tangential surface; circumferential detail by the sagittal one. A landscape has more of the second than the first near the corners, since horizons and skylines run across rather than out. A test chart has equal amounts of both by construction, which is one of the ways a lens optimised for charts and a lens optimised for photographs differ.

Anyone who has looked closely at the corner of a photograph taken with a cheap wide-angle lens has seen the result: the blur is not a disc but a smear, and it has a direction, either radial or circumferential depending on which surface was flattened. A corner that is soft in one direction only has been corrected by that trade, and one that is soft in both has not been corrected at all.

The same three surfaces are why the plane of best focus for a real lens is a matter of judgement rather than of measurement. There is no distance at which everything is in focus, so the setting chosen depends on what the picture is of, and a lens tested on a flat target gives a different answer from the same lens tested on a scene.

Bending the detector instead

There is another way to solve the problem, and it was the first one used.

How much a detector would have to bend. The depth a detector would have to be dished by, at its corner, to sit on the surface this lens focuses onto — a sphere of radius 76 mm. a phone sensor: 0.11 mm; micro four thirds: 0.77 mm; a 35 mm frame: 3.13 mm; medium format, 44 by 33: 5.15 mm. The sag goes as the square of the format, so the problem is four times worse on a sensor twice the size, which is one of several reasons a small camera is easier to make sharp than a large one. Bending the detector is not a joke: the Schmidt telescope held its photographic plate against a curved former and was the widest-field instrument of its era because of it, and the eye does the same thing with a retina laid over the inside of a sphere. Both are cases where the detector was free to be curved and the optics were not free to be complicated. A silicon sensor is neither, which is why the correction went into the glass instead. — Fig. 5 The depth a detector would have to be dished by, at its corner, to lie on the surface this lens focuses onto. The sag goes as the square of the format, so the problem is four times worse on a sensor twice the size.

If the image is a bowl, put the detector in the bowl. The Schmidt telescope did exactly that, holding its photographic plate against a curved former, and was the widest-field instrument of its generation partly because of it. The eye does the same: the retina is laid over the inside of a sphere, which removes at a stroke a problem every camera has to spend glass on.

Photographic film could be bent and silicon cannot, which is why the correction moved into the glass when photography went electronic. Curved sensors have been made — the sag needed for a small sensor is only tens of micrometres, and silicon that thin will take it — and the argument for them is exactly this figure: a curved detector removes the need for the elements whose only job is flattening, which makes the lens shorter, faster and simpler at once. The argument against is that a curved sensor is matched to one lens.

There is a third answer, which is to change what counts as the image. A lens whose field is curved still delivers a sharp image everywhere — just not on one plane at one time. A scanner that moves a narrow sensor across the field can refocus as it goes; a telescope that tiles its focal surface with separate detector packages can step each one to its own best distance, which is what several large survey cameras do; and a system that takes several exposures at different focus settings can assemble a sharp frame from the sharp parts of each. None of these removes the curvature. They arrange for the detector to be in the right place at the moment each part of the field is read, which is a curved detector spread out in time instead of in space.

The scaling in the figure is why phone cameras are sharper across the frame than their size suggests. The sag is proportional to the square of the format, so a sensor a fifth the width of a 35 mm frame has a twenty-fifth of the field curvature to deal with — before any correction at all.

How it is measured

A quantity that cannot be adjusted is still one that has to be verified, and the measurement is more interesting than it sounds.

The direct method is to focus on a point source at one field angle, record the position of the sensor at best focus, and repeat across the field. What comes back is not one curve but two, because best focus for detail running radially is at a different distance from best focus for detail running around the frame — the tangential and sagittal surfaces, measured rather than assumed. Plotting the two together is the standard field-curvature chart in a lens test, and the gap between the two curves is the astigmatism at that field height.

The indirect method, which is what a published lens test usually shows, measures contrast at a fixed spatial frequency with the sensor held flat, at points across the field, at two orientations of the target. That produces four curves rather than two — contrast against field height, for two frequencies and two orientations — and the field curvature appears in it as a fall-off with field height that closing the aperture reduces only in proportion.

Neither measurement separates field curvature from the rest by itself. What separates them is the aperture dependence: field curvature and astigmatism produce a blur proportional to the aperture diameter, spherical aberration to its cube, and diffraction to its reciprocal. Measure at three apertures and the terms come apart, which is why a careful test is always made at more than one.

Where the model stops

The elements here are thin and the treatment is third-order. A real design uses thick elements, and the Petzval sum is then a sum over surfaces rather than over lenses, in the form $\sum (n' - n)/(n n' R)$ . Nothing structural changes; the sum is still independent of the shapes taken at fixed power, and still independent of the stop.

Third order is the first term of a series. At large field angles the higher-order terms matter, and a lens can be designed so that the fifth-order field curvature partly cancels the third-order term over a chosen part of the field. That is how a well-corrected wide-angle lens is flat over most of its frame and turns up sharply at the very corner.

The astigmatism coefficient is put in by hand. The relation between the three surfaces is exact, but how much astigmatism a given lens has depends on its construction and is one of the things a designer controls; the figure shows the geometry, not a prediction for a particular lens.

And nothing here is about focus at other distances. The field curvature of a lens changes with object distance, which is why a lens that is flat at infinity can be curved at close range, and why a macro lens is a separate design rather than an ordinary lens focused closer.

What the pictures cannot show

Every figure here is a cross-section through the field, and the field is two-dimensional. What a photograph shows is a set of concentric zones of increasing blur, and the smear direction rotating with position around the frame, and neither is available on a plot with field height on one axis.

Nor do the figures show the interaction with the other aberrations. A real corner is soft because of field curvature, astigmatism, coma and lateral colour together, and separating them requires either a design prescription or a measurement made at several apertures — the aberrations scale differently with the f-number, which is how they are told apart in practice. The single clean curve drawn here is the contribution of one term, and it is the term that survives when the others have been dealt with.

Where the ladder goes next

The imaging ladder began with what a lens is doing and its three rays, went on to the shapes that focus and the shapes that do not in the mirror that cannot focus, through the double transform of the image that is a diffraction pattern twice, the sine condition, and then to the focus that is a slab, not a plane, where the focus turns out to have depth. This rung finds that it also has shape: the slab is bent, and the bending is set by a sum that ignores almost everything else about the lens.

The rung after it is where the field and the colour interact — the fact that the surface of best focus is at a different place for each wavelength, so a lens can be flat in green and bowl-shaped in blue. The habit worth carrying is the one this rung turns on: when a defect refuses to respond to every parameter that usually works, look for the expression it is controlled by, because a short expression means a small number of things can change it and they are worth knowing by name.

The flat scene that comes back curved