Illustrated Physics
physics, drawn
Optics

The delay a flat lens has no room for

A glass lens is thick in the middle for a reason that is easy to miss. To focus light of every colour at one point, a lens must do more than bend each colour by the right amount: it must make every part of a light wave arrive at the focus at the same moment, and the light passing the edge has further to go. The glass in the middle is what holds the central light back to wait for it. A lens made flat — a kinoform, a metasurface a micrometre thick — has nowhere to keep that delay. The geometry of the focus sets how much delay is needed, the thickness sets how much is available, and between them they cap how large an achromatic flat lens can be: tens of micrometres, not centimetres.
16 min read 5 figures Where the model stopsThe shape decides

Assumes: The lens with its glass taken out · Two glasses that cancel a derivative

The lens with its glass taken out followed Fresnel’s insight that the glass inside a lens is only scaffolding for its surfaces. A lens refracts at its surfaces, so the bulk behind them can be cut away in rings, and a lighthouse lens loses most of its weight. Carried to the limit, the rings become a kinoform, a surface whose profile is the lens’s phase reduced modulo one wavelength, a wavelength or so thick everywhere. The newest flat lenses, metasurfaces of nanoscale posts a fraction of a micrometre tall, impose the same wrapped phase with structures that need no curved surface at all. That essay ended by noting that such lenses inherit the kinoform’s backward colours, and that the tricks which broaden their band narrow their aperture, with a physical limit somewhere inside the trade.

The limit can be found, and it is the reason a glass lens is thick in the middle. The argument is about time.

How long a lens must hold back its central light. The time by which a lens must delay light passing through its centre relative to light passing its edge, so that every part of a short pulse reaches the focus together, against the lens's radius, for focal lengths of 0.1 mm, 1.0 mm, 10.0 mm (solid), in femtoseconds on logarithmic axes: the extra path from the edge, √(R² + f²) − f, over c. The dashed line is what a flat structure 1 μm thick can supply if its building blocks span a group index of 1: 3.3 fs. A 1 mm focal-length lens reaches it at a radius of 45 μm; a 10 mm one at 141 μm. A glass lens of 10 mm focal length and 5 mm radius holds its centre back by 3937 fs, which it does by being 2.27 mm thicker in the middle. A flat lens has no room for that.
Fig. 1 The time by which a lens must delay light passing its centre relative to light passing its edge, so that every part of a short pulse reaches the focus together, against the lens’s radius, for focal lengths of 0.1, 1 and 10 mm, in femtoseconds on logarithmic axes. The dashed line is what a 1 μm flat structure spanning a group index of 1 can supply, 3.3 fs: a 1 mm focal-length lens reaches it at a radius of 45 μm, a 10 mm one at 141 μm. A glass lens of 10 mm focal length and 5 mm radius holds its centre back by 3,937 fs, by being 2.27 mm thicker in the middle.

Every part of the wave arriving together

The path that takes the longest time and the surface that images one point exactly found what a lens does in Fermat’s terms: it makes every path from a point of the object to the point of its image take the same time. At a single wavelength that condition can be met by phase alone. A ray through the edge of a lens has further to go to the focus than a ray through the centre, by R2+f2−f\sqrt{R^2 + f^2} - f for a lens of radius RR and focal length ff, and the lens makes up the difference by retarding the central light’s phase, by an amount whose whole cycles do not matter. That is the freedom the kinoform uses: a phase that would require millimetres of glass to delay is wrapped back every wavelength, and only the fraction of a cycle left over is imposed.

The freedom vanishes when there is more than one wavelength. A phase delay that is right for green light is wrong for red unless the lens’s delay varies with frequency in exactly the right way, and the right way is that the delay — measured now not as a phase but as a time, the group delay, the derivative of phase with frequency — is the same for every ray. A lens that focuses every colour at one point is a lens that brings every part of a short pulse of light to the focus at the same moment. The distinction between the two kinds of delay is the one the packet that moves at another speed drew between a wave’s crests and its envelope: the phase records where the crests are, the group delay records where the pulse is, and a lens that gets the crests right at one colour can still deliver the pulse at the wrong time. The light through the edge travels further and arrives late unless the light through the centre is held back by just as long.

The first figure computes how long. For a lens of ten millimetres’ focal length and a five-millimetre radius, a modest camera lens, the edge ray’s extra path is 1.2 millimetres, four picoseconds of travel. A glass lens supplies that delay as a matter of course: light in glass travels at its group velocity, about two-thirds of its speed in air, so a lens 2.3 millimetres thicker at its centre than its edge holds the central light back by exactly the needed four picoseconds. The thickness that refraction requires and the thickness that the delay requires are nearly the same, and that near coincidence is why a simple glass lens is almost achromatic — almost, because the two differ by the glass’s dispersion, which is the chromatic aberration that two glasses that cancel a derivative corrected by pairing glasses.

What a micrometre can hold

A flat lens has no such thickness. A metasurface a micrometre tall, made of posts whose shapes give them different effective indices, can delay the light passing one post relative to another by at most the thickness times the spread of group index the posts can span, divided by the speed of light. For a micrometre of height and a group-index spread of one, generous for transparent materials at visible wavelengths, that is 3.3 femtoseconds. Why a spread of about one is generous needs a sentence. The posts are made of a transparent dielectric — titanium dioxide or silicon nitride at visible wavelengths, whose indices are about 2.4 and 2.0 — standing in air, so the effective index a post can present ranges from a little above 1, for a thin post that the light mostly misses, to a little below the material’s own, for a wide one. Dispersion near a resonance of the post can push the group index further for a narrow band, at the cost of absorption and of the band itself. In the infrared, silicon’s index of 3.5 widens the spread to about 2.5, and infrared flat lenses can be correspondingly larger. The dashed line of the first figure is that budget, and each lens’s requirement crosses it at a definite radius: forty-five micrometres for a focal length of a millimetre, about a hundred and forty for ten millimetres.

The largest flat lens that keeps every colour in focus. The largest diameter an achromatic flat lens can have, against its numerical aperture, on a logarithmic vertical axis, for structures 0.5 μm thick spanning a group index of 1; 1.0 μm thick spanning a group index of 1; 1.0 μm thick spanning a group index of 2, from R(1 − √(1 − NA²))/NA ≤ LΔn. At NA = 0.1, a micrometre-thick lens spanning a group index of 1 can be at most 40 μm across; at NA = 0.5, 7.5 μm. Doubling the thickness or the index range doubles every diameter. The achromatic metalenses demonstrated since 2018 sit near these curves, tens of micrometres across. A camera lens of 5 mm diameter at NA 0.2 would need a delay budget 253 times that of a micrometre of index-1 contrast.
Fig. 2 The largest diameter an achromatic flat lens can have, against its numerical aperture, on a logarithmic axis, for structures 0.5 μm thick spanning a group index of 1, 1 μm thick spanning 1, and 1 μm spanning 2, from R(1−1−NA2)/NA≤LΔnR(1 - \sqrt{1 - \mathrm{NA}^2})/\mathrm{NA} \le L\Delta n. At NA 0.1 a micrometre-thick lens spanning a group index of 1 can be at most 40 μm across; at NA 0.5, 7.5 μm. The achromatic metalenses demonstrated since 2018 sit near these curves. A 5 mm camera lens at NA 0.2 would need 253 times the delay budget.

Expressed through the numerical aperture, the lens’s ability to collect light at steep angles, the condition becomes a cap on diameter, and the second figure draws it. A flat lens of modest numerical aperture, 0.1, can be achromatic only up to forty micrometres across if it is a micrometre thick; at a numerical aperture of 0.5, the kind a microscope objective needs, under eight micrometres. Doubling the thickness or the index range doubles every diameter, and nothing else helps. The bound does not depend on how the nanostructures are designed, whether as posts, fins or resonators; it is a statement about how much time a layer of a given thickness can hold light for, and every design must pay it.

The achromatic metalenses built since 2018 sit close to these curves. The first visible-light achromatic metalens, made at Harvard in 2018, was twenty-five micrometres across at a numerical aperture of 0.2, focusing light from 470 to 670 nanometres to one point, and later designs have reached larger sizes only by narrowing the band, lowering the numerical aperture or growing taller. A camera’s main lens, millimetres across, would need hundreds of times the delay any such layer provides.

Colours that run backwards

Where three lenses put the colours. The focal length against wavelength across the visible spectrum for three lenses designed for 1 mm at 550 nm: a glass singlet of BK7, its index from a two-term fit to its Abbe number of 64.2 (solid); a kinoform, whose wrapped phase makes its focal length inversely proportional to wavelength (dashed); and an achromat that holds one focus (dotted). From 450 to 650 nm the glass lens's focus moves by 2.1 per cent of its length, blue focusing short; the kinoform's by 38 per cent the other way, red focusing short, 18 times as far. A flat lens built on the kinoform's principle inherits its backward colours, and keeping them together needs exactly the group delay of the earlier figures.
Fig. 3 Focal length against wavelength across the visible spectrum for three lenses designed for 1 mm at 550 nm: a BK7 glass singlet, its index from a two-term fit to its Abbe number of 64.2 (solid); a kinoform, whose focal length is inversely proportional to wavelength (dashed); and an achromat (dotted). From 450 to 650 nm the glass lens’s focus moves 2.1 per cent, blue focusing short; the kinoform’s 38 per cent the other way, red focusing short, 18 times as far.

The third figure shows what a lens without the delay does instead. A kinoform’s phase is imposed modulo one wavelength, so the profile is fixed in units of the design wavelength, and a longer wavelength sees the same rings as a stronger lens: its focal length is inversely proportional to the wavelength. Across the visible spectrum its focus moves by more than a third of its length, eighteen times as far as a glass singlet’s, and in the opposite direction, red focusing short.

That reversal is useful. It is why the lens with its glass taken out noted that a kinoform’s colours can undo a glass lens’s: a weak diffractive surface on a refracting lens, its backward dispersion cancelling the glass’s forward one, makes a hybrid achromat lighter than any pair of glasses. Such hybrids are used in camera lenses and in the eyepieces of headsets. They work because the glass supplies the delay and the diffractive surface only adjusts the dispersion, so the bound of the previous figure is respected by the glass.

A flat lens alone has no glass to lean on. Its dispersion can be engineered — posts can be designed whose phase varies with wavelength in the way needed — but only within its delay budget, and the backward colours return wherever the budget runs out.

A pulse smeared at the focus

A short pulse focused by a lens that has no thickness. The light arriving at the focus against time, for a 10 fs pulse focused by a kinoform of 1 mm radius and 10 mm focal length (solid), each ring of the lens contributing its share of the pulse at the moment its extra path allows, beside the same pulse through an ideal achromat, arriving all at once (dashed, scaled to the same peak). The kinoform's edge light is 166 fs behind its centre's, so the pulse at the focus is smeared into a 166 fs plateau — more than ten times its own length — with its energy spread evenly, because equal intervals of lateness come from rings of equal area. A glass lens of the same focal length delays its centre by its thickness and brings the rings into step, apart from the much smaller effect of its own dispersion.
Fig. 4 The light arriving at the focus against time for a 10 fs pulse focused by a kinoform of 1 mm radius and 10 mm focal length (solid), each ring contributing at the moment its extra path allows, beside the same pulse through an ideal achromat, arriving all at once (dashed, scaled to the same peak). The edge light is 166 fs behind the centre’s, so the pulse is smeared into a 166 fs plateau, its energy spread evenly because equal intervals of lateness come from rings of equal area.

The time-domain version of the same failure is the most direct, and the fourth figure draws it. A pulse of light ten femtoseconds long, a few cycles of red light, is focused by a kinoform a millimetre in radius. The kinoform gives every ring the right phase at the centre of the pulse’s spectrum and no delay at all, being only a wavelength thick, so each ring’s light arrives at the focus when its own path allows: the centre’s first, the edge’s 166 femtoseconds later. The focus receives a smear, a flat-topped pulse sixteen times longer than the one that went in, its peak intensity reduced by the same factor. The smear is flat because a lens’s rings of equal area produce equal intervals of lateness, a coincidence of the parabola of delays with the area of annuli.

For ultrafast optics that is fatal, and it is why the field uses mirrors and thick lenses, or compensates. The dispersion made of angles found that a pair of gratings, which has the same backward dispersion as a kinoform, delays different colours by different amounts and can stretch or compress a pulse; the kinoform’s smear is that effect in space rather than in colour, the pulse front tilting and bowing until it arrives over a spread of times. The lens that would focus a femtosecond pulse without smearing it is the lens that has the delay, and a flat one cannot.

How much band a given size can keep

The band of colours a flat lens can keep together, against its size. The range of wavelengths around 550 nm over which a 1 mm focal-length flat lens can keep its edge rays within a quarter of a cycle of the focus, against its radius, for structures with no group delay of their own (a plain kinoform) and with 1 μm and 3 μm of group-index contrast 1; capped at 1000 nm. A kinoform 50 μm in radius keeps about 61 nm together; at 1 mm, 0.2 nm. A micrometre of delay buys full achromatism up to the radius where its budget runs out and then gives way to the same falling curve. The trade the curves draw — bandwidth for aperture, at fixed thickness — is a statement about how much time a thin structure can hold light for, and it applies to any flat optic, however its nanostructures are designed.
Fig. 5 The range of wavelengths around 550 nm over which a 1 mm focal-length flat lens keeps its edge rays within a quarter cycle of the focus, against its radius, for a plain kinoform and for structures with 1 and 3 μm of group-index contrast 1, capped at 1000 nm. A kinoform 50 μm in radius keeps about 61 nm together, at 1 mm 0.2 nm; a micrometre of delay buys full achromatism up to the radius where its budget runs out, then gives way to the same falling curve.

The last figure turns the bound into the trade that designers face. A flat lens that cannot supply all the delay can still keep colours together over a narrower band: the edge rays’ timing error is the shortfall in delay, and a timing error translates into a phase error that grows across the band, so the lens stays within a quarter cycle of focus over a band inversely proportional to the shortfall. A plain kinoform fifty micrometres in radius keeps sixty nanometres together, much of the visible spectrum; at a millimetre, a fifth of a nanometre, a single laser line. A micrometre of delay pushes the start of the fall out to the radius at which its budget is used up; after that the shortfall grows and the band falls on the same curve.

This is the content of the phrase that bandwidth is traded for aperture. At fixed thickness, a flat lens can be large and monochromatic, or broadband and small, or something between, and the product of the two is set by the time it can hold light. Applications choose accordingly, and the choice is visible in what flat lenses have been used for first: the infrared sensors and laser projectors of phones, the narrow-band light of a single laser, the depth sensors that work at one wavelength. A laser-scanning system or a fibre coupler at one wavelength can use a large flat lens freely; a camera that needs the whole visible spectrum through a five-millimetre aperture cannot, unless the flat lens is combined with refracting elements that supply the delay, which is how flat optics have entered cameras so far.

Several colours instead of a band

The bound is about a continuous band: every wavelength between two limits brought to one focus at once. It says much less about a handful of separate wavelengths, and that loophole is old. Cut a kinoform’s zones not one wavelength deep but several — so that the profile resets after a delay of, say, ten cycles at the design wavelength instead of one — and the lens focuses perfectly at every wavelength for which ten cycles is a whole number of that wavelength’s cycles: the design wavelength, and those for which the same depth holds nine, eleven, twelve. Such a multi-order, or harmonic, diffractive lens is a kinoform with ten times the delay budget, and the figures’ arithmetic applies to it unchanged: between its sharp wavelengths it is as blurred as the bound requires, and the sharp ones crowd closer together as the zones are cut deeper. It is the right lens for a system that uses a few laser lines, or the three narrow colours of a display, and the wrong one for daylight.

A second escape gives up on optics entirely. A flat lens can be left chromatic, each colour focused at its own distance, and the blur corrected afterwards by computation, using the fact that the blur at each wavelength is known exactly from the design. The bound is not broken: the information that colour carried is spread across the sensor rather than lost, and recovering it costs noise, since deconvolving a blur amplifies whatever the sensor added. Designs that shape the lens and the reconstruction together, so that the blur at every colour is the kind a computer undoes most cheaply, have produced flat cameras with recognisable colour images. What they trade is not delay but signal-to-noise, which is the same budget appearing once more under a different name.

The same bound in an antenna array

Radio engineers met this limit decades before optical engineers did, under the name beam squint. A phased-array antenna steers its beam by giving each element a phase offset, and when the source is not heard all at once found the sum over elements forming a beam in a chosen direction. Phase offsets are cheap to set and wrap round every cycle, exactly as a kinoform’s profile does. But a phase offset that steers the beam to one angle at one frequency steers it to a different angle at another, because the path difference between elements is a delay, not a phase, and a phase only imitates a delay at one frequency. A radar with a wide band of frequencies, needed for fine range resolution, has a beam that fans out across angles as the frequency changes. The cure is true time delay: lines of cable or optical fibre that delay each element’s signal by the real path difference, which keeps the beam pointing in one direction at every frequency.

A flat lens is a phased array of nanostructures, and a kinoform is an array steered by phase alone. An achromatic lens is an array with true time delay, and the delay budget of the previous figures is the length of delay line a micrometre of material can provide. The radar engineer’s solution, longer delay lines, is the glass lens’s solution, more thickness; neither has a version that takes up no room.

Where the bound is looser or tighter than drawn

The bound assumes the lens is a single thin layer through which light passes once, so that the delay available is the thickness times the index contrast. Resonant structures can store light for longer than a single pass, delaying it by more than their thickness suggests, but only over a band inversely proportional to the delay, and the product of delay and bandwidth for a resonator is bounded in a similar way; the trade reappears in a different form. Stacked layers add their budgets, at the cost of thickness and of the losses each layer brings. And the bound treats every ray as needing to arrive together; a lens that needs to be achromatic only over a limited field, or that can tolerate a known amount of focal shift, has a looser requirement. None of these escapes the conclusion that a lens much larger than its delay budget allows cannot be broadband.

The figures also treat the glass singlet with a simple two-term dispersion fit and the kinoform with the ideal one-over-wavelength law. A real kinoform’s efficiency falls away from its design wavelength, sending light into other diffraction orders, which a real lens must also contend with and the figures leave out.

What the delay picture leaves out

The figures draw delays and foci, and they leave out the wave optics of the focus itself: the size of the focal spot, the light lost to unwanted diffraction orders, the aberrations of rays off the axis, which the colour fringe no aperture can close found to be a separate problem from the colour of the axial focus. A lens can be achromatic on its axis and still colour the edges of its field. Nor do the figures show the nanostructures, whose design — which shape of post gives which phase and which delay at which wavelength — is the practical substance of the field. The bound does not care about any of that, which is its strength: it holds whatever the structures are.

Still open: how large a broadband flat optic can be made

The physical bound is understood; how close real devices can approach it, and how much the combinations that sidestep it — flat lenses on curved substrates, stacks of metasurfaces, hybrids with refracting elements — can deliver, is still being worked out. Multi-layer metasurfaces and three-dimensional nanostructures grown by lithography push the effective thickness up while keeping a flat outline, and computational imaging, which corrects the colour blur in software after the picture is taken, relaxes the requirement on the optics. Whether a millimetres-wide flat lens can ever form a sharp colour image unaided is not settled, and the delay arithmetic says it would have to be thick in some sense to do so.

The habit worth carrying away is to ask what a component must do in time as well as in space. Bringing every colour to one focus means bringing every part of a pulse there together, so a lens must hold its central light back by (R2+f2−f)/c(\sqrt{R^2 + f^2} - f)/c — four picoseconds for a 5 mm, 10 mm-focal-length lens, supplied by 2.3 mm of extra glass — and a flat structure LL thick can hold it back by at most LΔn/cL\Delta n/c, which caps an achromatic flat lens a micrometre thick at 40 μm across at NA 0.1. Removing material from an optical element removes the time it could store, and some jobs need the time.

Part 11 of 11

This essay is one argument about Refraction. The others:

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. The lens with its glass taken out Part 10 — 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.

Where this goes next

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. 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. 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. Colours that leave a grating on different roads. A pair of parallel diffraction gratings with 1,200 lines per millimetre and light arriving at 30°, drawn to scale for three wavelengths, 760, 800 and 840 nm. The first grating sends each colour off at its own angle — 24.3° for 760 nm, 27.4° for 800 nm, 30.5° for 840 nm — and the second turns them all back parallel to the input, side by side. Between the gratings the red travels the longest road: for gratings 10 mm apart the slant paths are 10.97 mm, 11.26 mm, 11.61 mm. Measured to a common wavefront after the second grating, the longer road still wins, so red arrives last: the geometry has given the pulse the opposite ordering to the one glass gives it. The dispersion made of angles Glass delays blue light more than red, so a short pulse that passes through glass comes out stretched with its red end first. At the wavelengths where most short-pulse lasers work, no transparent glass does the opposite. Two diffraction gratings do it with no material at all. The first spreads the colours in angle, the second turns them back parallel, and in between the red has travelled the longer road. A spread in angle, however it is made, always adds this kind of dispersion, of the sign glass cannot give. It is how every high-power ultrafast laser squeezes its pulse back after amplifying it.

The objects named here

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

Achromatic lensChromatic aberrationDispersionFermat's principleGroup delayKinoformMetalensNumerical aperture