Illustrated Physics
physics, drawn
Optics

The colour fringe no aperture can close

A lens bends blue light more than red, and that one fact produces two different faults. One puts the blue focus nearer the lens than the red, and closing the aperture shrinks it. The other puts the blue image of an off-axis point nearer the centre of the frame than the red one, and closing the aperture does nothing at all, because it is carried by the one ray every aperture keeps. It is why photographs have coloured fringes in their corners and not in their middles, and why a lens designer who cannot remove it with glass can sometimes remove it with symmetry, or with a distance.
15 min read 5 figures The shape decidesWhat happens at the edge

Assumes: The flat scene that comes back curved · Two glasses that cancel a derivative

The flat scene that comes back curved ends by pointing at the place where the field of view and colour interact: a surface of best focus that sits at a different place for each wavelength. That is one member of a family of faults that appear only away from the centre of the picture, and it is the least dramatic of them. The dramatic one is lateral colour, and it has a property that sets it apart from almost everything else that can go wrong in an image. It cannot be improved by closing the aperture.

Every photographer learns that stopping a lens down sharpens it. Spherical aberration, coma, most of the blur from field curvature and the coloured halo of two glasses that fail to cancel a derivative all shrink as the aperture closes, because each of them is a disagreement between rays passing through different parts of the lens, and a smaller aperture keeps fewer of the disagreeing rays. Lateral colour is a disagreement between colours along the same ray. Nothing an aperture does can remove it, and the drawings follow it from its cause to the tricks that do.

The ray through the centre of the stop

The ray through the centre of the stop lands in three places. A single N-BK7 lens of 50 mm focal length with its stop 20 mm in front of it, and the ray through the centre of the stop from a point 10° off axis — the chief ray — traced in blue, yellow and red light to the plane where yellow light focuses. Because the stop is away from the lens, the chief ray meets the lens 3.53 mm off centre and is bent there, and blue is bent more than red. At full scale the three colours are indistinguishable; magnified, they arrive at heights −38.1, 0 and 16.9 µm relative to yellow — a spread of 55.0 µm between blue and red, which is the height at which the ray met the lens divided by the glass's Abbe number of 64.2. Every other ray in the beam passes through the stop elsewhere, but all of them are centred on the chief ray, so the whole image of the point is displaced by colour, and closing the stop does not change where its centre goes.
Fig. 1 A single N-BK7 lens of 50 mm focal length with its stop 20 mm in front, and the chief ray from a point 10° off axis traced in blue, yellow and red light to the plane where yellow focuses. The ray meets the lens 3.53 mm off centre and is bent there, blue more than red. Magnified, the three colours arrive at −38.1, 0 and 16.9 µm relative to yellow — a spread of 55.0 µm, the height at the lens divided by the glass’s Abbe number of 64.2.

The central character is the chief ray — the ray from an off-axis point that passes through the centre of the aperture stop. Every other ray from that point passes through the stop somewhere else, and the bundle of them is centred on the chief ray, so wherever the chief ray lands is where the image of the point is centred. The chief ray is the one ray that belongs to every aperture: close the stop to a pinhole, and it is all that remains.

If the stop is at the lens, the chief ray passes through the lens’s centre, where a thin lens is a flat plate and bends nothing. Every colour goes straight through, and they all land at the same height. There is no lateral colour at all, which the generator checks before drawing anything. Most lenses cannot have their stop there: a camera lens has glass on both sides of its diaphragm, an eyepiece’s stop is the pupil of the eye in front of it, and a telescope’s stop is its objective, far from the eyepiece. Put the stop 20 millimetres in front of the lens, as in the drawing, and the chief ray meets the lens 3.5 millimetres off centre, where the lens is a weak prism — the shape of lens the three-ray construction treats as a stack of prisms. A prism disperses: it bends blue by more than red, for the reason that puts the rainbow at its angle. So the three colours leave the lens at slightly different angles and land at slightly different heights.

The spread is tiny at the scale of the drawing and large at the scale of a photograph: 55 micrometres, about fourteen pixels of a modern sensor. Its size follows from one line. The prism bends the ray by the height at which it meets the lens divided by the focal length, and the fractional difference between blue and red focal lengths is one over the Abbe number — 64 for this glass — so the spread on the image plane is the height at the lens divided by 64.

Stopping down removes one colour error and not the other

Stopping down removes one colour error and not the other. The two chromatic errors of a single 50 mm N-BK7 lens with its stop 20 mm in front, at an image point 10° off axis, against the f-number. Axial colour — blue and red focusing at different distances, seen as a coloured blur at the yellow focus — is about 775/N µm across: 387 µm at f/2, 97 at f/8, 35 at f/22. Lateral colour — blue and red images of the point centred at different heights — is 55.0 µm at every f-number, because it is carried by the ray through the centre of the stop. The two are equal near f/14, and past that stopping down leaves only the error it cannot touch.
Fig. 2 The two chromatic errors of the same lens at an image point 10° off axis, against f-number. Axial colour, a coloured blur at the yellow focus, is about 775/N µm: 387 at f/2, 97 at f/8, 35 at f/22. Lateral colour, a displacement between the blue and red images, is 55.0 µm at every f-number. The two are equal near f/14.

The same lens has both faults at once, and the drawing sets them side by side. Axial colour arises because the blue focus is nearer the lens than the red, by the focal length divided by the Abbe number — 0.78 millimetres here. At the yellow focus, the blue and red cones of light from a point are both slightly out of focus, and each spreads into a disc whose size is the focus error divided by the f-number. Close the aperture and the cones narrow, the discs shrink, and the coloured blur falls in proportion.

Lateral colour does not blur anything. The blue image of the point is exactly as sharp as the red one; it is simply centred somewhere else. Closing the aperture narrows both cones around their chief rays and leaves the chief rays where they were. So at wide apertures axial colour dominates and stopping down helps; past about f/14 in this lens, lateral colour dominates and stopping down does nothing. The focus that is a slab describes stopping down as buying depth of field at the cost of diffraction; for lateral colour it buys nothing at any cost.

This distinction also shows up in how lenses are tested. A lens’s axial colour appears as coloured halos round a point source at the centre of the field, which change size as the aperture changes. Lateral colour appears as a coloured fringe along edges that run tangentially across the corners of the frame — edges at right angles to the line from the centre — with blue on one side and red on the other, and a lens tester can tell the two apart by stopping down and watching which one stays.

Lateral colour grows with the field and with the stop’s distance

Lateral colour grows with the field and with the stop's distance. The spread between the blue and red images of a point, against how far from the centre of the frame the point is imaged, for a 50 mm N-BK7 singlet with its stop at 0, 10, 20, 40 mm in front of the lens. With the stop at the lens there is none anywhere in the field. Moving the stop away makes it grow in proportion to both the image height and the stop distance: at 20.2 mm off centre it is 63 µm with the stop at 10 mm, 126 µm with the stop at 20 mm, 252 µm with the stop at 40 mm. The centre of a photograph is free of it and the corners are where it lives, which is why a colour fringe appears on high-contrast edges at the edge of a frame and never in the middle.
Fig. 3 The spread between blue and red images, against image height, for the singlet with its stop 0, 10, 20 and 40 mm in front of the lens. With the stop at the lens there is none anywhere. Moving it away makes the spread grow in proportion to both the image height and the stop distance: at 20.2 mm off centre it is 63 µm with the stop at 10 mm, 126 µm at 20 mm and 252 µm at 40 mm.

The one-line formula has two lengths in it, and the drawing varies both. The height at which the chief ray meets the lens is the stop distance times the tangent of the field angle, so lateral colour is proportional to both. At the centre of the frame the chief ray is the axis, meets the lens at its centre and is not dispersed; at the corners the chief ray meets the lens far off centre and is dispersed most. Doubling the stop distance doubles the effect everywhere.

That is why lateral colour is a problem of wide-angle lenses and of long telescopes’ eyepieces, where the chief ray from the edge of the field meets the glass far from the axis, and why it hardly matters for a narrow-field lens like a telephoto’s rear element or a microscope objective used near its centre. It is also why lateral colour is a difference of magnification rather than a blur. The image in blue is a slightly smaller copy of the image in red, scaled about the centre of the frame, and the fringe at any point is the difference between the two copies there.

This makes lateral colour, alone among the aberrations, easy to remove after the picture is taken. Since each colour channel is sharp and differs from the others only in scale, resampling the blue and red channels by their own magnifications lines them up again. Every camera and every raw-photo converter now does this from a table of the lens’s measured lateral colour, and the fringes that were once the signature of cheap wide-angle lenses have largely vanished from photographs — not because the lenses changed but because a difference of magnification is information, and a blur is not.

Two lenses of one glass with a colour-free focal length

Two lenses of one glass with a colour-free focal length. The focal length of a pair of N-BK7 lenses of 60 and 20 mm, relative to its value in yellow light, across the visible spectrum, for three separations. Between blue and red the pair's focal length changes by −1.03 per cent with the lenses 20 mm apart, 0.01 per cent with the lenses 40 mm apart, 3.19 per cent with the lenses 60 mm apart. At 40 mm, half the sum of the focal lengths, the curve is flat at the centre of the spectrum: the pair's power no longer depends on wavelength to first order, although each lens's does. An eyepiece's magnification is set by its focal length, so this spacing removes its lateral colour with a single kind of glass; it is the spacing of the two-lens eyepiece Christiaan Huygens introduced in the 1660s.
Fig. 4 The focal length of a pair of N-BK7 lenses of 60 and 20 mm, relative to its value in yellow light, across the visible spectrum. Between blue and red it changes by −1.03 per cent with the lenses 20 mm apart and 3.19 per cent at 60 mm. At 40 mm, half the sum of their focal lengths, it changes by 0.01 per cent: the pair’s power is stationary with wavelength, although each lens’s is not.

The standard cure for colour is the achromatic doublet, two lenses of different glasses cemented together so that their dispersions cancel. It removes axial colour, and for a thin doublet with the stop at it, it removes lateral colour as well. But there is an older cure, needing only one kind of glass, and it works on lateral colour specifically.

Two thin lenses separated by a distance dd have a combined power 1/F=1/f1+1/f2−d/(f1f2)1/F = 1/f_1 + 1/f_2 - d/(f_1 f_2). Each lens’s power varies with wavelength in proportion to n−1n - 1, and if both are made of the same glass the variations are proportional. Differentiating the combined power with respect to the index and setting the result to zero gives a condition on the spacing alone:

d=f1+f22.d = \frac{f_1 + f_2}{2}.

At that spacing the combined focal length is stationary with wavelength, although neither lens’s is, and the drawing confirms it: the curve for 40 millimetres is flat across the centre of the spectrum. The pair still has axial colour — its focus moves with wavelength, because its principal planes move — but its focal length, and therefore the magnification it gives to an eye looking through it, does not. For an eyepiece, whose job is to magnify an image already formed by a telescope’s objective, that is the fault that matters: an eyepiece with lateral colour shows every star near the edge of the field as a short spectrum.

This is the spacing of the two-lens eyepiece Christiaan Huygens introduced in the 1660s, before Newton had shown that white light is a mixture of colours, and it remained the commonest eyepiece for two centuries. A design that removes one colour fault with one glass is also the clearest demonstration that “the colour correction of a lens” is not one condition but several, each on a different derivative of a different quantity, and each removable by a different trick.

The coloured edge at the corner of a photograph

The coloured edge at the corner of a photograph. The image of a sharp dark-to-light edge 15 mm from the centre of the frame, through a 50 mm N-BK7 singlet with its stop 5 mm in front, in red, green and blue light, each blurred by 5 µm and centred where lateral colour puts it: the blue edge 25.8 µm nearer the centre of the frame than the red. Across the 26 µm between them the three colours are not in balance: on one side blue has risen and red has not, and on the other red has and blue has not, so a neutral edge acquires a blue fringe on one side and a red one on the other. On a sensor with four-micrometre pixels that is a band several pixels wide, running tangentially round the frame and growing towards the corners — the fringe that photographers call chromatic aberration and that software now removes by rescaling each colour channel, which works precisely because lateral colour is a difference of magnification and not a blur.
Fig. 5 The image of a sharp dark-to-light edge 15 mm from the centre of the frame, through a 50 mm N-BK7 singlet with its stop 5 mm in front, in red, green and blue light, each blurred by 5 µm and centred where lateral colour puts it: the blue edge 25.8 µm nearer the centre than the red. Across the shaded band the colours are out of balance — blue risen and red not on one side — so a neutral edge acquires a blue fringe on one side and a red one on the other.

The drawing shows what lateral colour looks like when it reaches a sensor. Each colour’s image of the edge is equally sharp, a smooth step a few micrometres wide. The three steps are simply in different places, and across the band between them the light is out of balance: on the inner side blue has already risen to full brightness while red is still dark, and on the outer side the reverse. A grey edge becomes a blue-violet line on one side and a red one on the other, a band several pixels wide running along every tangential edge near the corners.

The band’s width is set by the stop position and the field height, and nothing else about the picture: not the subject, not the focus, not the exposure. It is the same in every photograph taken with the same lens at the same zoom setting. That repeatability is what lets software correct it from a table, and it distinguishes it from the violet fringes that appear on overexposed highlights in any part of the frame, which come from a mixture of axial colour, the sensor’s own response and the spreading of charge between saturated pixels, and which a table cannot remove.

Why a mirror has no colour

The most complete cure for colour of either kind is not to use glass. A mirror reflects every wavelength at the same angle, because the law of reflection contains no refractive index, so a mirror has neither axial nor lateral colour however far off axis the light arrives and wherever its stop is. Isaac Newton built his reflecting telescope in 1668 for exactly this reason. He had concluded, wrongly as it turned out, that the colour of a lens could never be corrected, because he believed every glass dispersed in proportion to its refraction; a mirror side-stepped the problem.

Mirrors have faults of their own — a sphere cannot focus parallel light to a point, and a paraboloid that can is ruined by coma a fraction of a degree off axis — but colour is not among them. Every large astronomical telescope is therefore a reflector, and the lateral colour that remains in a telescopic image comes from whatever glass the light meets after the mirrors: a corrector plate, a field flattener, or an eyepiece. The Huygens eyepiece’s single-glass correction was, for two centuries, the finishing step of a system whose main optics had no colour to begin with.

The same division appears in instruments that must work across wide bands of wavelength. Infrared and ultraviolet instruments, and spectrographs that must bring many wavelengths to one focus, are built from mirrors wherever possible, and a designer adds glass only where a mirror cannot do the job — accepting, with each lens, a lateral colour that has to be corrected by the tricks above.

Symmetry, the other cure

There is one more way to remove lateral colour, and it removes several other faults at once. Arrange the elements of a lens symmetrically about its stop — the same lenses, in mirror image, on either side — and use it at unit magnification, as a copying lens does. Every ray’s path through the first half is then reversed through the second, and any fault that depends on an odd power of the chief ray’s height at the stop is introduced by one half and cancelled by the other. Lateral colour is such a fault; so are distortion and coma. It is the reason that so many classic camera lenses — the Rapid Rectilinear, the Double Gauss that nearly every standard lens since the 1950s descends from — are built as two nearly symmetric halves around a central diaphragm, and why their lateral colour is small even though no glass was chosen to correct it.

The symmetry is only exact at unit magnification, and a camera lens photographing a distant scene is far from that. But a design that starts symmetric starts with those faults nearly cancelled, and the designer’s work is to recover the small remainder rather than to fight the whole of it. The condition a lens must meet describes coma’s cure as the sine condition; symmetry is the same cure reached by geometry rather than by calculation.

Where the thin-lens picture stops

Thin lenses and paraxial rays. Every number here is from tracing rays close to the axis through lenses of zero thickness. A real lens’s lateral colour also has higher-order terms that grow faster than the field height, and they are why many modern lenses have lateral colour that rises, falls and changes sign across the frame.

Three wavelengths. The drawings trace the F, d and C lines, or red, green and blue. Real light is a continuous spectrum, and glass’s index curves across it in a way no pair of glasses can straighten entirely — the curvature a fibre exploits to hold a pulse together — and a lens corrected so that red and blue land together usually leaves green slightly elsewhere — the secondary spectrum, which for lateral colour shows as a green–magenta fringe rather than a red–blue one.

A stop that stays put. In a zoom lens the stop moves relative to the groups of glass as the focal length changes, so the lateral colour changes with zoom, and the correction tables have to be indexed by focal length as well as by lens.

What the pictures cannot show

The drawings show heights and spreads on a line. A photograph shows lateral colour in two dimensions, and it combines with the finite sharpness every lens owes to diffraction, as a pattern of fringes that point radially: every point’s blue image is displaced towards the centre of the frame and its red image away from it, so the fringes run along tangential edges and vanish on radial ones. A spoke pattern photographed with such a lens shows clean spokes and coloured rings. None of that geometry fits on a graph of one variable, and it is the geometry that tells an experienced eye which aberration it is looking at.

Still open: whether glass will still correct colour

Lens design is moving away from choosing glasses. Diffractive surfaces — fine gratings on a lens surface — disperse light in the opposite sense to glass, with an effective Abbe number of about −3.5, and a weak diffractive surface can correct the colour of strong glass lenses with far less material; several camera and telescope lenses use them. Flat metalenses, built from arrays of nanostructures, can in principle shape the phase of each wavelength separately. Both carry new problems — stray light diffracted into unwanted orders, efficiency that falls away from the design wavelength — and whether a flat lens can be made achromatic over the whole visible spectrum at an aperture and field of view useful to a camera is being argued about with theorems on both sides, bounding the bandwidth such a surface can correct against its size and numerical aperture.

The habit worth carrying away is to ask of any fault which ray carries it. An aberration carried by rays that pass through different parts of the aperture shrinks when the aperture closes; one carried by the chief ray does not, because the chief ray is the ray every aperture keeps — and a fault that is a difference of magnification rather than a blur can be removed after the fact by anyone who knows how large it is.

Part 7 of 7

This essay is one argument about Imaging. The others:

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. The flat scene that comes back curved Part 6 — A lens does not image a plane onto a plane. It images it onto a bowl, and the curvature of that bowl is fixed by the powers and the glasses alone — not by the shapes of the surfaces, not by where the stop is, and not by stopping down. Everything a designer usually plays with leaves it exactly where it was.

Where this goes next

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. The flat scene that comes back curved A lens does not image a plane onto a plane. It images it onto a bowl, and the curvature of that bowl is fixed by the powers and the glasses alone — not by the shapes of the surfaces, not by where the stop is, and not by stopping down. Everything a designer usually plays with leaves it exactly where it was. Cancelling a derivative, and what is left over. How far the focus of a 500 mm lens moves with wavelength, for a single crown element and for a cemented pair of crown and flint whose powers satisfy the achromatic condition. The singlet's focus runs over 18.1 mm across the visible — a fifth of a per cent of its focal length, and utterly ruinous at any useful aperture. The doublet's runs over 2.268 mm, some 8× less, and — this is the whole content of the figure — it is not flat. The condition sets the rate of change of power with wavelength to zero, so the curve is stationary rather than constant: it returns to the corrected focus at exactly 2 wavelengths — 486 nm and 656 nm, which are the two Fraunhofer lines the condition was written at — and departs from it everywhere else, most at 400 nm. That residual is the secondary spectrum, it has the same sign at both ends of the visible, and no pair of ordinary glasses removes it, because two conditions cannot be met with one free ratio. Two glasses that cancel a derivative A single lens focuses blue light closer than red, and the difference ruins the image. Cementing a second lens of another glass behind it fixes the fault at two wavelengths and at no others, because the condition sets a slope to zero rather than a value. Image distance against object distance. Image distance in focal lengths against object distance in focal lengths. At exactly one focal length the image runs off to infinity; inside it the image distance goes negative, which means virtual. The focus that is a slab, not a plane A lens images one plane and no other, which would make every photograph and every micrograph almost entirely out of focus. What rescues them is a tolerance — and there are two of them, one from rays and one from waves, which give different answers and stop being interchangeable exactly where microscopes work.

The objects named here

The third axis, after the field and the reading path: the things themselves, and every essay that touches each one.

Abbe numberAperture stopChief rayChromatic aberrationDispersionF numberLateral colourLens design