Optics

The lens with its glass taken out

Snell's law turns a ray at a surface by an angle set by the slope of the surface where the ray crosses, and the glass behind the surface does not enter it. So a lens can be sliced into rings, the glass between the slopes thrown away, and the rays will not notice. Fresnel did it for lighthouses in 1822. The waves notice — every step cuts a few hundred wavelengths out of them — unless the steps are made exactly one wavelength deep, and then the lens becomes something else: a diffractive element whose colours run backwards, and which can cancel a glass lens's colour with a twentieth of its power.

16 min read 5 figures The shape decides Made into an instrument

Assumes: The law that only asks about one component · The lens that is a set of rings

In 1819 the French lighthouse commission asked a young engineer named Augustin Fresnel to improve the lights along the coast of France. The lights of the time were lamps backed by polished metal reflectors, which lost about half their light to the tarnished metal, and the obvious improvement — a large glass lens in front of the lamp, focusing its light into a beam — had been tried and had failed. A lens big enough to gather a useful share of a lamp’s light, with the short focal length a lighthouse needs, would have had to be tens of centimetres thick at the centre. Nobody could cast a block of glass that size free of bubbles and strain, it would have absorbed much of the light it was meant to focus, and it would have cracked under the heat of the lamp.

Fresnel’s solution, first lit at the Cordouan lighthouse at the mouth of the Gironde in 1823, was to make the lens without most of its glass. He built it from concentric rings of glass, each ring a thin slice of the surface of a much thicker lens, mounted in a frame. The rings focused the lamp’s light into a beam visible tens of kilometres out to sea, and within decades every major lighthouse in the world had one. It works because of something about refraction that is easy to state and easy to overlook.

Only the slope does any work

The law that only asks about one component found that Snell’s law is a statement about the component of a wave’s direction along the surface: that component is the same on both sides, because the wave’s crests have to match up along the boundary. The angle a ray turns through is therefore set entirely by the direction of the surface where the ray meets it — its slope — and by the two refractive indices. Nothing about how thick the glass is behind the surface, or what shape the surface has anywhere else, enters.

A lens is two surfaces. In a plano-convex lens with its flat face towards a parallel beam, the rays cross the flat face square-on and do not turn; they travel straight through the glass and turn only at the curved face, by an angle set by its slope at the height they arrive at. The glass between the two faces is crossed without turning anything. It is there only to hold the curved face at the right slope.

A lens with its glass taken out, focusing the same. Two lenses of focal length 200 mm and aperture 100 mm in acrylic (n = 1.49), each with its flat face to a parallel beam and the exact aspheric curve that sends every ray to one focus: solid (top), 12.2 mm thick at the centre, and the same curve cut into 13 rings 1 mm deep with the glass between them removed (bottom), on a backing 0.4 mm thick. Thickness is drawn six times larger than height, and the distance to the focus is not to scale. Every ray leaves each lens at the same angle from the same height, because the angle a ray turns through at a surface depends only on the slope of the surface where it crosses, and every ring keeps the slope it had. What was removed was glass the rays travelled through without turning. — Fig. 1 A solid acrylic lens of focal length 200 mm and aperture 100 mm, with the exact aspheric surface that sends every ray to one focus (top), 12.2 mm thick at the centre; and the same surface cut into 13 rings 1 mm deep with the glass between them removed (bottom). Thickness is magnified sixfold; the distance to the focus is not to scale. Each ray leaves both lenses from the same height at the same angle.

So the glass can go. Cut the curved face into narrow rings, keep each ring’s slope, and drop each ring back towards the flat face until it is only a millimetre or so thick. The profile becomes a sawtooth of rings with the same slopes as the original surface at the same heights, and every ray crosses a ring of the right slope and turns by the right angle. The figure traces it: the solid lens and its Fresnel version send the same rays to the same focus.

The surface drawn is not a sphere. A spherical surface would send rays from different heights to slightly different foci, the spherical aberration the mirror that cannot focus found in a spherical mirror. Each ring’s slope here is chosen so that the ray through it goes exactly to the focus: for a ray at height $r$ that must turn through $\delta = \arctan(r/f)$ , Snell’s law at a facet tilted by $\alpha$ requires $n\sin\alpha = \sin(\alpha + \delta)$ , which gives

\tan\alpha = \frac{\sin\delta}{n - \cos\delta}.

Integrated over the aperture, that slope makes the aspheric surface of the surface that images one point exactly. A solid aspheric lens is hard to make, because the surface has to be ground and polished to a shape that is not a sphere. A Fresnel lens moulded in plastic can have any facet slopes at all, which is one reason the cheap lenses are often better corrected than the solid lenses they replace.

The glass a lens would need

The saving grows quickly with size. A lens’s thickness is set by how far its surface rises over its aperture, and for a given focal ratio — focal length divided by aperture — that rise is proportional to the aperture.

Fig. 2 The centre thickness a solid acrylic lens needs, beyond its edge, against its aperture, for focal ratios f/2, f/1 and f/0.7, each with the exact aspheric profile; the Fresnel version of any of them is 1 mm deep (black). A 100 mm f/1 lens needs 22 mm of glass at its centre, a 500 mm one 109 mm.

A lighthouse needs a fast lens — a short focal length for its aperture — because it has to gather light from a lamp spread over a wide range of angles. At f/1, a lens 100 mm across needs 22 mm of glass at its centre, and one 500 mm across needs 109 mm. The thickness grows in proportion to the size and the volume as its cube, so the weight of a lens two metres high, the size of Fresnel’s largest, would have been measured in tonnes. A Fresnel lens of any size is as deep as its grooves, and its weight grows only as its area.

Fresnel’s own lighthouse lenses went one step further. Rays from the lamp that met the outermost rings at a steep angle would have needed facets so steeply tilted that much of the light reflected off them instead of passing. For those rings he replaced refraction by total internal reflection: prisms that took the light in through one face, turned it by reflecting it internally off a second, and let it out through a third, parallel to the beam. The finished lens was a beehive of glass rings, refracting in its middle band and reflecting above and below, gathering more than half of the lamp’s light into a beam — against a few per cent for a lamp and mirror.

The idea had been proposed before. The naturalist Buffon suggested in 1748 that a large burning lens could be made in steps, carved from a single block, and the Marquis de Condorcet later proposed building one from separate rings. Fresnel did not know of either, and he was the first to work out the facet angles from the theory of refraction and to build one that worked. The lenses still bear his name in every theatre spotlight, overhead projector, car headlamp, solar concentrator and virtual-reality headset — where the thin plastic Fresnel lenses in front of each eye cost the faint radial streaks around bright objects that users call god rays.

The engineer who also wrote the wave theory

There is an irony in the lens carrying Fresnel’s name. In the same years he was designing lighthouse optics, he was also writing the wave theory of light that would eventually explain why his lenses could never form sharp images. In 1818 he submitted a memoir on diffraction to the French Academy’s prize competition, treating light as a wave whose every point acts as a source, and computing the fringes at the edge of a shadow — the rings that belong to the edge. One of the judges, Siméon Denis Poisson, objected that the theory predicted something absurd: a bright spot at the centre of the shadow of a disc. Arago looked, and found it, the result where rays stop being enough begins from.

Fresnel’s lighthouse lens was designed entirely with rays, and it did not need his wave theory, because a lighthouse beam does not need to be sharp. The phase drift between his rings was there in every lens he built, and it cost nothing that a sailor could see. A century and a half passed before anyone needed a stepped lens to image, and when they did, the theory that said what had to be done to its steps was Fresnel’s own.

What the waves notice

The rays cannot tell the Fresnel lens from the solid one. Light is not rays, and the waves can.

A lens forms an image because the waves leaving every point of its aperture arrive at the focus in step: the extra glass in the middle delays the central part of the wave by exactly as much as the longer path from the edge delays the edge, so all the contributions add at the focus. That was the argument of what a lens is doing, and it rests on the glass being there. Removing a step of glass of depth $d$ advances the wave crossing it by $(n-1)d$ of optical path. For acrylic and a 1 mm step that is 0.49 mm, or 890.9 wavelengths of green light.

What the steps do to the focus, as waves. The intensity across the focal plane at 550 nm for a cylindrical version of the lens, computed from the waves leaving every point of the aperture, relative to the solid lens's peak: the solid lens (grey), the Fresnel lens with 1 mm grooves (blue), and with grooves exactly one wavelength of optical path deep, λ/(n − 1) = 1.12 μm (red). Rays cannot tell the three apart. Waves can: each 1 mm step cuts 890.9 wavelengths of path out of the wave, a fractional part that differs from one ring to the next, so the rings no longer arrive in step and the peak falls to 16 per cent. With steps of whole wavelengths the cut is invisible to the wave at the design wavelength and the focus is the solid lens's exactly — a kinoform. — Fig. 3 The intensity across the focal plane at 550 nm for a cylindrical version of the lens, computed from the waves leaving every point of the aperture: solid (grey), Fresnel with 1 mm steps (blue), and with steps exactly one wavelength of optical path deep, $\lambda/(n-1) = 1.12$ μm (red). The 1 mm steps lower the peak to 16 per cent of the solid lens’s; the one-wavelength steps restore it exactly.

The whole wavelengths do not matter. The fraction does: each step advances the wave by 0.9 of a cycle more than a whole number, and the rings beyond it by 0.9 of a cycle times the number of steps crossed. So the rings arrive at the focus with phases that drift by nine-tenths of a cycle from one ring to the next, no longer in step, and they add only partly. In the figure the central peak of a cylindrical Fresnel lens falls to 16 per cent of the solid lens’s, and the light that should have been in it is spread across a band as wide as the image a single ring would make by itself. In effect each ring focuses its own light independently, and the sharpness is set by the width of a ring rather than by the whole aperture.

For a lighthouse this does not matter at all. A lighthouse does not form an image; it collimates the light of a lamp whose flame is far larger than any diffraction blur, and its beam needs to be bright, not sharp. The same goes for a spotlight, a solar concentrator, a projector condenser, or a headlamp. For a camera lens it would be fatal, which is why Fresnel lenses were for a century and a half a tool for moving light rather than for forming pictures.

Steps one wavelength deep

The repair is in the figure’s red curve. Make every step exactly one wavelength of optical path deep — $d = \lambda/(n-1)$ , 1.12 micrometres in acrylic at 550 nanometres — and each step advances the wave by exactly one cycle, which the wave cannot distinguish from no advance at all. The rings arrive in step again, and at the design wavelength the focus is identical to the solid lens’s. A lens of this kind was named a kinoform by Lesem, Hirsch and Jordan in 1969.

A kinoform is no longer, in any useful sense, a refracting lens with the glass taken out. It is a grating. Its rings are so shallow that the slope of each facet is irrelevant to the rays — they are a micrometre deep and a fraction of a millimetre wide — and what the structure does is impose a phase on the wave that rises across each ring and drops by one cycle at its edge. It is the lens that is a set of rings — a zone plate — with each zone’s phase made to ramp smoothly rather than switch between two values, so that all of the light goes into the zone plate’s first focus instead of a tenth of it. The zone plate threw most of its light away; the kinoform blazes it all into one order, like a blazed grating doing what a thousand slits buy with every groove tilted towards the order it serves.

Colours that run the other way

A grating bends long wavelengths more than short ones. A kinoform is a circular grating whose rings narrow towards its edge so that every part of it diffracts its light towards one point, and the angle it diffracts by is proportional to the wavelength. Its focal length is therefore proportional to one over the wavelength: red light focuses closer than blue.

Glass does the opposite. Its refractive index is higher for blue than for red, so a glass lens bends blue more and focuses it closer, the colour error the colour fringe no aperture can close traced across an image. Opticians measure a material’s dispersion by its Abbe number, $V = (n_d - 1)/(n_F - n_C)$ , which for a common crown glass is about 64. The same definition applied to a kinoform, whose power is proportional to wavelength, gives

V_{\text{diff}} = \frac{\lambda_d}{\lambda_F - \lambda_C} = \frac{587.6}{486.1 - 656.3} = -3.45,

negative and nearly twenty times smaller in size than any glass.

The lens whose colours run the other way. How far the focus of a 200 mm lens moves with wavelength, relative to its position at 587.6 nm: a refractive glass lens with an Abbe number of 64.2 (blue), focusing blue light slightly short; a kinoform (red), whose focal length goes as one over the wavelength, focusing blue light long by much more — its Abbe number is −3.45, negative and nineteen times smaller in size; and a hybrid (black) that takes 5.1 per cent of its power from a kinoform surface on the glass lens. Between 486 and 656 nm the glass lens's focus moves 3.10 mm and the kinoform's 62.7 mm the other way; the hybrid's returns to the same place at both ends. — Fig. 4 How far a 200 mm lens’s focus moves with wavelength from its position at 587.6 nm: a glass lens with Abbe number 64.2 (blue), a kinoform with −3.45 (red), and a hybrid taking 5.1 per cent of its power from a kinoform surface (black). Between 486 and 656 nm the glass lens’s focus moves 3.10 mm, the kinoform’s 62.7 mm the other way, and the hybrid’s returns to the same place.

That sign is a gift. The classical way of cancelling a glass lens’s colour, two glasses that cancel a derivative, pairs a converging lens of low-dispersion crown glass with a weaker diverging lens of high-dispersion flint, so that their colour errors cancel while their powers do not. The price is that the diverging element takes away a large part of the power, so both elements must be strong and steeply curved. A kinoform’s dispersion has the opposite sign to glass, so a converging kinoform cancels a converging glass lens’s colour. The condition is the same — the powers divided by their Abbe numbers must sum to zero — and because the kinoform’s Abbe number is so small, it needs only a sliver of the total power:

\frac{\phi_g}{V_g} + \frac{\phi_d}{V_d} = 0 \quad\Rightarrow\quad \frac{\phi_d}{\phi_g + \phi_d} = \frac{-V_d}{V_g - V_d} = \frac{3.45}{64.2 + 3.45} \approx 5.1\ \text{per cent}.

A glass lens with a kinoform pattern a micrometre deep moulded onto one surface, supplying a twentieth of the power, brings blue and red light to the same focus. Since 2001 camera makers have sold telephoto lenses built this way, shorter and lighter than their all-glass equivalents because the diffractive surface does the colour correction that would otherwise need extra heavy elements.

The light a kinoform does not focus

The kinoform’s steps are exactly one wavelength deep at one wavelength only. At any other wavelength they are a little more or less than a cycle, and the rings drift out of step again, though far less than the 1 mm steps did.

$Where a kinoform sends the light it does not focus. The share of light a kinoform designed for 550 nm sends to its focus (red), straight on undeviated (blue) and to a second focus at half the focal length (green), against wavelength, for steps one design wavelength of path deep in a glass of Abbe number 64.2. At the design wavelength all of it reaches the focus. At 450 nm 83 per cent does, at 650 nm 92 per cent; the rest is spread as faint defocused light by the other two orders, which is why a camera lens with a diffractive element can show coloured flare around a bright point that a purely refractive lens would not.$

Fig. 5 The share of light a kinoform designed for 550 nm sends to its focus (red), straight through undeviated (blue) and to a second focus at half the focal length (green), against wavelength. All of it reaches the focus at 550 nm; 83 per cent at 450 nm and 92 per cent at 650 nm.

The light that misses the focus is not lost; it goes into the grating’s other orders. The zeroth order passes straight on as if the kinoform were not there, and the second order comes to a focus at half the focal length. For a step depth of $\alpha$ cycles, the share in order $m$ is $\mathrm{sinc}^2(m - \alpha)$ , and across the visible spectrum that leaves a few per cent to over a tenth of the light defocused. In a photograph of an ordinary scene that light is spread so thinly that it is invisible. Around a bright point source — a street lamp at night, the sun glinting off chrome — it appears as a faint coloured halo, which is the characteristic flaw of diffractive camera lenses and the reason makers stacked two kinoforms of different materials to flatten the efficiency across the spectrum.

The ray figures assume the facets are perfectly sharp and their risers — the vertical steps between rings — are exactly parallel to the rays. Real moulded facets have rounded tips and drafted risers, and light hitting the risers is scattered or sent the wrong way, a loss that grows with the number of rings and is the source of the god rays in a virtual-reality headset’s Fresnel lens. The wave figures are computed for a cylindrical lens, focusing in one direction only, because the calculation is a sum along a line rather than over an area; a circular lens gives the same qualitative result, with the drift between rings weighted by the rings’ areas.

The efficiency figure uses the scalar theory of a thin grating, which is good when the rings are many wavelengths wide and poor near a fast kinoform’s edge, where the rings narrow to a few wavelengths and the steps’ finite depth and the light’s polarisation start to matter. The colour figure models the glass’s index with a two-term Cauchy formula matched to its Abbe number, which is adequate over the visible spectrum and wrong outside it. And nothing here includes the absorption in the glass that a Fresnel lens saves, which for the thick lighthouse lenses that were never built would have been one of their worst failings.

Still open: how flat a lens can become

The kinoform removes the glass and keeps the phase; the next step removes the steps and keeps the phase too. A metalens imposes the same phase profile with a carpet of nanoscale posts or fins, each a fraction of a wavelength across, whose shape sets the delay the light suffers in passing it. Such lenses are a few hundred nanometres thick, can be made with the same lithography as computer chips, and have been demonstrated at visible wavelengths since 2016. Whether they can match a refracting lens over a broad band of colours and a wide field of view at the same time is not settled: they inherit the kinoform’s backward dispersion, and the tricks that broaden their bandwidth narrow their aperture. Small metalenses are already in phones; whether one will ever replace a camera’s main lens is an open engineering question with a physical limit somewhere inside it.

What Fresnel saw is the first step of that sequence. Refraction at a surface is decided by the slope of the surface where the ray crosses, so the glass behind the slope is only scaffolding — and the waves care about the scaffolding only modulo one wavelength. Taking the glass out gave lighthouses their beams. Making the steps exactly a wavelength deep turned a lens into a grating, and a grating’s colours run the other way, which a glass lens can use to undo its own.