Illustrated Physics
physics, drawn
Optics

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.
15 min read 5 figures The shape decidesMade into an instrument

Assumes: Two glasses that cancel a derivative · The wavelength a fibre does not smear

Every dispersion met so far on this subject has come from a material. Two glasses that cancel a derivative balanced the index slopes of a crown and a flint. The wavelength a fibre does not smear found fused silica’s second derivative passing through zero at 1,273 nanometres, set by where its ultraviolet and infrared absorptions balance. The delay that is a random variable followed a random walk of polarisation delays through a slightly elliptical core. In each case the physics was in the glass.

There is a second source, and it needs no glass at all. When the colours in a beam are sent off at different angles and then brought back parallel, they have travelled paths of different lengths, and a difference in path is a difference in delay. It is dispersion made of geometry. It has a fixed sign — the opposite of what every ordinary glass gives in the visible and near infrared out to about 1.3 micrometres — and its size is set by angles and distances that an experimenter can choose. That makes it the one form of dispersion that can be dialled, and the instrument built on it is in every laser that makes pulses a few femtoseconds long.

Two roads between two gratings

The simplest arrangement is a pair of identical diffraction gratings facing each other, parallel. The first grating diffracts each wavelength at its own angle, given by the grating equation. The second, identical but reversed, undoes the angular spread, so every colour leaves travelling parallel to the input. The colours come out side by side, shifted sideways by different amounts, and they have not travelled the same distance.

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.
Fig. 1 A pair of parallel gratings with 1,200 lines per millimetre and light arriving at 30°, drawn to scale for 760, 800 and 840 nm. The first grating sends the colours off at 24.3°, 27.4° and 30.5°; the second turns them back parallel. For gratings 10 mm apart the slant paths between them are 10.97, 11.26 and 11.61 mm: the red takes the longest road.

The figure draws the arrangement to scale for three wavelengths spanning the spectrum of a short pulse at 800 nanometres. The grating has 1,200 lines per millimetre, the kind a thousand slits buy for spectroscopy. The first grating sends 760 nanometres off at 24.3 degrees, 800 at 27.4 and 840 at 30.5, because a longer wavelength is diffracted through a larger angle. The redder light therefore crosses the gap more obliquely, and its slant path between gratings 10 millimetres apart is 11.61 millimetres against 10.97 for the blue.

That is not yet the delay, because the colours also leave the second grating at different places, and a fair comparison measures each to the same wavefront after the pair. Doing that carefully, as Treacy did in 1969, shows that the longer slant path wins: the red is delayed relative to the blue. The group delay equals the length of the ray’s path divided by the speed of light, and the grating adds nothing of its own to it.

The ordering is the opposite of the one glass gives. In ordinary glass in the visible and near infrared, blue travels more slowly than red — the index rises towards the ultraviolet absorptions — and a pulse emerges red first. After a grating pair it emerges blue first. The pair provides what is called negative, or anomalous, group-delay dispersion, and it does so in air.

Why an angle always delays the same way

The sign is not a feature of gratings. It follows from any spread of colours in angle, and a short argument shows why. A beam whose colours travel at slightly different angles, measured from a common direction, has for each colour a phase that advances along the colour’s own direction. Projected onto the common direction, each tilted colour advances less per unit of forward distance, by a factor cos⁡θ\cos\theta. Because cos⁡θ\cos\theta falls on either side of zero, a colour tilted either way lags behind. The lag grows as the square of the tilt, and the tilt is proportional to how far the colour is from the centre, so the delay is quadratic in frequency with a fixed negative sign. Fork, Martinez and Gordon wrote it in 1984 in a form that applies to any angularly dispersive system:

d2ϕdω2=−λ3L2πc2(dθdλ)2,\frac{d^2\phi}{d\omega^2} = -\frac{\lambda^3 L}{2\pi c^2}\left(\frac{d\theta}{d\lambda}\right)^2,

with LL the distance over which the angles act. The square of the angular dispersion guarantees the sign. Nothing about the device appears except how strongly it spreads the colours and for how long they are allowed to diverge.

That makes the result unusually general. A prism pair obeys it. So does a grating pair, a combination of a grating and a lens, and any optical system in which colours are separated in angle and later recombined. The dispersion of a glass, by contrast, has whatever sign the positions of its absorptions give it at the wavelength in question. Geometry supplies negative dispersion at any wavelength, across the whole band, with essentially no loss.

Why glass cannot usually give the other sign

A material’s dispersion is tied to its absorption by causality. A medium cannot respond before it is driven, and the answer that cannot come first turns that fact into the Kramers–Kronig relations, which fix the refractive index at every wavelength once the absorption at every wavelength is known. A transparent glass in the visible sits between strong absorptions in the ultraviolet, from its electrons, and in the infrared, from the vibrations of its atoms. The ultraviolet resonances are the stronger and the nearer, and they make the index rise towards the blue, with an upward curvature: blue light travels more slowly, and the group-delay dispersion is positive. That is normal dispersion, and every transparent glass has it throughout the visible.

The sign can turn over only where the infrared absorption begins to dominate the curvature. For fused silica that happens at 1,273 nanometres, the zero-dispersion point of the wavelength a fibre does not smear. Beyond it, silica’s dispersion is anomalous, and telecommunications fibre at 1,550 nanometres carries pulses whose red end travels slower. That is why optical solitons, which need anomalous dispersion to balance a nonlinearity, live in fibres at those wavelengths. At 800 nanometres, where titanium–sapphire lasers make the shortest pulses, and in the visible, no transparent glass offers the anomalous sign, and near an absorption line, where a material does, it absorbs the light it is meant to compress.

So the geometric term is not merely convenient. At the wavelengths where most ultrafast optics is done it is the only source of negative dispersion that is both broadband and lossless, and the group velocity it acts on is, as the speed that carries no signal makes clear, only a description of how the envelope moves. Neither glass nor gratings do anything to the speed at which information can travel. They only rearrange, in time, the colours that make up a pulse.

A delay that undoes a delay

The practical use is to cancel the dispersion of glass.

A delay from glass, and one from geometry that undoes it. Group delay against wavelength, relative to 800 nm, for 100 mm of fused silica, for a 300-line-per-millimetre grating pair passed twice at the 19.9 mm spacing that cancels the silica's group-delay dispersion, and for the two together. Through glass, blue is slower: the delay falls with wavelength, by 857 fs across 760–840 nm. Through the gratings, red is slower, by nearly the same amount. The sum is flat at 800 nm and curves away at the edges, to 52 fs at 760 nm and 46 fs at 840 nm, because the two delays have different curvatures: the grating pair's third-order dispersion adds to the glass's instead of cancelling it.
Fig. 2 Group delay against wavelength, relative to 800 nm: through 100 mm of fused silica, blue is slower by 857 fs across 760–840 nm; through a 300-line-per-millimetre grating pair passed twice at 19.9 mm spacing, red is slower by nearly as much. Together (dashed) the delay is flat at 800 nm and curves away to 52 fs at 760 nm and 46 fs at 840 nm.

The figure plots group delay against wavelength for three things. The first is 100 millimetres of fused silica, through which blue at 760 nanometres arrives 857 femtoseconds later than red at 840. Its group-delay dispersion at 800 nanometres is 36 femtoseconds squared per millimetre, 3,620 for the whole length. The second is a grating pair with 300 lines per millimetre, used twice — the light is sent back through the pair by a mirror, which also undoes the sideways shift between the colours. Its spacing, 19.9 millimetres, is chosen so that its group-delay dispersion is exactly minus the silica’s. The third curve is the sum.

At 800 nanometres the sum is flat: its slope, the second-order dispersion, has been cancelled. But it is not flat everywhere. It curves up at both edges, reaching 52 femtoseconds at 760 nanometres and 46 at 840. The glass’s delay and the gratings’ delay have different curvatures, so cancelling their slopes at one wavelength leaves a difference in how fast those slopes change. That third-order dispersion does not cancel. For glass and for gratings of this kind its signs are the same, so the two add. The situation is the one two glasses that cancel a derivative met with lenses, where an achromat corrects the focus at two colours and leaves a residual, the secondary spectrum, at all others. There it was the first derivative that cancelled; here it is the second.

What a pulse gets back

Whether the residue matters depends on the pulse. A pulse lasting 20 femtoseconds at 800 nanometres has a spectrum about 47 nanometres wide, spanning most of the range drawn, and it is sensitive to every curvature there.

The spacing that gives a pulse its shortness back. The duration (full width at half maximum) of a 20 fs pulse at 800 nm after 100 mm of fused silica and a 300-line-per-millimetre grating pair passed twice, against the grating spacing, computed from the whole spectral phase rather than from its second derivative alone. The glass alone stretches the pulse to 501 fs. The gratings bring it back to 28.8 fs at a spacing of 19.9 mm, where the second-order dispersions of glass and gratings cancel (19.9 mm). It does not return to 20 fs, because the third-order terms of glass and gratings add rather than cancel, and near the best spacing that residue sets the length: two millimetres either side the pulse is still only 30.3 fs long, whichever way the spacing is wrong.
Fig. 3 The duration of a 20 fs pulse at 800 nm after 100 mm of fused silica and a 300-line grating pair passed twice, against the grating spacing, from the whole spectral phase. The glass alone stretches it to 501 fs. At the best spacing, 19.9 mm, it comes back to 28.8 fs; two millimetres either way it is 30.3 fs.

The figure computes the pulse directly, adding up its spectrum with the full phase that glass and gratings give each frequency and measuring the width of the result. The silica alone stretches the 20-femtosecond pulse to 501 femtoseconds, twenty-five times longer, because the blue and red ends of the spectrum have been pulled apart in time. Adding the gratings and varying their spacing brings the pulse back, to a minimum of 28.8 femtoseconds at 19.9 millimetres, where the second-order dispersions cancel.

It does not come back to 20. The third-order residue of the previous figure leaves the spectrum’s edges out of step with its centre, and the pulse acquires a tail of weaker pulses on one side. Near the best spacing, that residue rather than the spacing sets the length: two millimetres either way, the pulse is still only 30.3 femtoseconds long. The minimum is shallow because the third-order error dominates it. That is why real compressors are designed with the third-order term in mind as well, by choosing grating densities and angles, adding prisms whose third order has the other sign, or using mirrors with layered coatings that supply a specified dispersion curve.

This is the arithmetic behind chirped-pulse amplification, which the packet that will not keep its shape described in outline. A short pulse is too intense to amplify directly: its peak power would damage the amplifier’s glass. So it is deliberately stretched a thousandfold or more, by a device of the same kind with its dispersion reversed, amplified while long and harmless, and then compressed by a grating pair. The compressor’s gratings sit in vacuum at the end of the chain because no glass could survive the compressed pulse. The same scheme, stretching a signal in time to reduce its peak and compressing it afterwards, was already used in radar, where a chirped pulse is compressed by correlation, as the shift a mirror gives twice notes. In optics the compressor is geometry, and the idea won the 2018 Nobel Prize in physics.

A front that is not a wavefront

Angular dispersion does something else to a pulse that a material cannot: it tilts it.

A pulse whose front is not its wavefront. A short pulse just after a diffraction grating, drawn for gratings of 600, 1200, 1800 lines per millimetre at 800 nm with 30° incidence. The wavefronts, the surfaces of constant phase, are perpendicular to the direction the light travels, as always. The pulse front — the surface where the intensity peaks — is not: angular dispersion tilts it, by an angle whose tangent is the wavelength times the rate the angle changes with wavelength. 600 lines per millimetre: 25.6°; 1200 lines per millimetre: 47.2°; 1800 lines per millimetre: 76.7°. A tilted front sweeps across a crystal at a speed that can be chosen by choosing the tilt, which is how short pulses in lithium niobate are made to keep pace with the terahertz waves they generate.
Fig. 4 A short pulse just after a grating at 800 nm with 30° incidence. Wavefronts (grey) are perpendicular to the propagation direction; the pulse front, where the intensity peaks (shaded), is tilted from them by 25.6° for 600 lines per millimetre, 47.2° for 1,200 and 76.7° for 1,800.

In ordinary light the pulse front — the surface at which the intensity peaks at a given moment — coincides with the wavefront, the surface of constant phase, and both are perpendicular to the direction of travel. Just after a grating they are not the same. The different colours leave at different angles, and the envelope they build between them, where they add in phase, is inclined. The tilt γ\gamma satisfies tan⁡γ=λ dθ/dλ\tan\gamma = \lambda\, d\theta/d\lambda: one wavelength times the angular dispersion. The figure draws it for three gratings at 800 nanometres with 30 degrees of incidence. With 600 lines per millimetre the front is tilted by 25.6 degrees; with 1,200, by 47.2; with 1,800, by 76.7, so steeply that the pulse sweeps almost sideways across whatever it meets.

A tilted front is a tool. When it crosses a crystal, the point where the pulse is brightest slides along the crystal’s face at a speed that depends on the tilt, and it can be made to match the speed of a wave the pulse generates inside the crystal. The standard way of producing strong terahertz radiation uses exactly that. In lithium niobate a terahertz wave travels at roughly half the speed of the optical pulse, so the two drift apart and the generation stops. Tilting the pump’s front by about 63 degrees makes the bright region move across the crystal at the terahertz wave’s speed, and the generated wave keeps building as the pulse crosses. The same tilt is a nuisance in other places: a compressor that is not perfectly aligned leaves a residual tilt that lengthens the pulse at a focus, and it is diagnosed by the colours’ positions.

A prism pair: glass and angle in one device

Prisms offer the same geometric term, with a complication that turns out to be useful. Light passing through a prism pair gains negative dispersion from the angular spread between the prisms and positive dispersion from the glass it traverses. Both depend on the index, the first through its slope and the second through its curvature.

Two prisms, one term from glass and one from angle. The group-delay dispersion of a prism pair at 800 nm against the distance between the prisms' apexes, with 10 mm of glass in the beam, for fused silica and for the dense flint SF10, from the formula of Fork, Martinez and Gordon: a positive term from the glass the light passes through and a negative term from the angular spread, growing with the separation. The two balance at 334 mm for silica and 173 mm for SF10. A strongly dispersive glass needs a shorter separation, because the angular term goes as the square of the index's slope while the glass term goes as its curvature. Sliding a prism along its own face changes the glass path without moving the beam, which is how the dispersion of a laser cavity is tuned by hand.
Fig. 5 The group-delay dispersion of a prism pair at 800 nm against apex separation, with 10 mm of glass in the beam, for fused silica and SF10 flint. The glass term is positive; the angular term, negative, grows with separation. They balance at 334 mm for silica and 173 mm for SF10.

The figure plots the total against the separation of the prisms, with ten millimetres of glass in the beam, for fused silica and for the dense flint SF10. At zero separation only the glass counts and the dispersion is positive. As the prisms are moved apart, the angular term grows in proportion to the distance and the total falls through zero: at 334 millimetres for silica and 173 for SF10. The flint needs a shorter separation because its index changes faster with wavelength, and the angular term goes as the square of that slope while the glass term goes only as the curvature.

Prism pairs have one practical advantage: the glass path can be changed without moving the beam. Sliding a prism along its own face inserts more or less glass, changing the positive term smoothly while the geometry stays fixed. That is how the dispersion inside a titanium–sapphire laser cavity was tuned by hand in the lasers that first made pulses shorter than 20 femtoseconds. Prisms lose less light than gratings, which matters inside a laser cavity where the light passes the dispersive element many times. Gratings give far more dispersion per unit length and are used where large stretches must be undone, which is why the two devices divide the work: prisms inside oscillators, gratings in amplifier compressors.

What the geometry cannot correct

The geometric picture explains the sign and the size of the effect; three limits come with it.

The orders beyond the second. A grating pair’s dispersion is fixed by its line density and its angles. Its second order can be set to anything by choosing the spacing, but its third order then follows, and it adds to glass’s rather than cancelling it. Correcting the third order needs a second degree of freedom — another material, another angle, or a mirror designed for the purpose — and correcting the fourth needs a third. Pulses of a few optical cycles demand phase control over a spectrum an octave wide, and the dispersion there is shaped by programmable devices rather than by a pair of gratings.

Spatial side effects. A single pass through a grating pair leaves the colours side by side, a spatial chirp, and tilts the pulse front. The double pass cancels both in principle, and imperfect alignment reintroduces them. Every compressor therefore has an alignment problem as well as a dispersion problem, and the two are measured together.

Efficiency and damage. Gratings diffract only part of the light into the useful order, and each pass loses a few per cent. The compressor’s last grating also receives the full compressed pulse, and its coating’s damage threshold, not the optics of dispersion, sets how much energy the whole laser can deliver.

Still open: how short a pulse geometry can make

The shortest pulses at visible and near-infrared wavelengths are now a few femtoseconds long, one or two cycles of the light’s oscillation, and they are compressed by combinations of chirped mirrors, thin wedges of glass and occasionally gratings, over spectra so broad that no two-element device can match their dispersion. Below a cycle, the definition of the pulse’s duration itself becomes delicate, because the phase of the carrier under the envelope matters. Pulses of attoseconds, generated by driving atoms with such short pulses and then emitted in the extreme ultraviolet, need dispersion control in a range where no transparent glass exists and gratings are inefficient. Metal foils and multilayer mirrors are used there, and whether an attosecond pulse can be compressed further than its generation allows is limited by the lack of any geometric or material dispersion of the right sign that does not also absorb the light.

The habit worth carrying away is to look for dispersion wherever colours are separated, not only where there is matter. A spread of colours in angle turns into a spread of delays, always of the same sign, because a tilted path advances less along the common direction whichever way it is tilted. Glass gives dispersion through its resonances and cannot choose its sign across a band; geometry gives it through angles and distances, with its sign fixed and its size free, and that difference is why the shortest and most powerful pulses of light are made by a pair of gratings in a vacuum.

Part 6 of 6

This essay is one argument about Dispersion. The others:

A delay that grows as the square root of the length. The differential group delay between the fastest and slowest polarisation states of a fibre built from sections 100 m long, each a slightly birefringent waveplate with its axis at a random angle, against length up to 400 km. Three individual fibres are drawn faint, and the root-mean-square delay over 300 of them heavy. The ensemble grows as a power 0.50 of the length — the square root, because each section rotates the polarisation it receives before adding its own delay, so the delays add like the steps of a random walk in three dimensions rather than like lengths laid end to end. At 100 km the mean delay is 5.18 ps against 5.00 ps for a coefficient of 0.5 ps/√km. Had the axes all been aligned, the same sections would have added to 172 ps at that length, growing in proportion, and the drawn dashed line leaves the frame within a few kilometres. The randomness is what keeps the delay small, and it is also what makes it impossible to compensate with a fixed device. The delay that is a random variable Part 5 — A fibre's core is very slightly elliptical, so its two polarisations travel at very slightly different speeds — and the ellipse turns, at random, every hundred metres or so. The delays of the pieces do not add. They random-walk, so the total grows as the square root of the length, follows the same distribution as the speeds of gas molecules, differs from one wavelength to the next, and on any particular day may be three times its average.

Where this goes next

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. The narrow packet is the one that spreads. Three packets on deep water, ω = √(gk), all built on the same 4 m carrier and differing only in bandwidth — 10%, 18%, 28% of the carrier wavenumber. Each curve is the width of the emitted envelope, measured as the second moment of its intensity about its own centroid in a frame moving at the group velocity, divided by that width at the start. The starting widths are 4.50 m, 2.50 m, 1.61 m, and the order of the curves is the reverse of the order of the widths: the shortest packet, which is the one with the widest spectrum, is the one that comes apart first. The dashed curves are √(1 + (t/τ)²) with τ = σ₀²/|d²ω/dk²| — computed from the dispersion relation, not fitted. They agree with the measured widths to 0.3% at 10% bandwidth, 3.3% at 18% bandwidth, 8.5% at 28% bandwidth, and that ordering is the second thing the figure says: the closed form keeps only the curvature of ω(k), so it is exact for a narrow spectrum and starts to fail for a wide one, by about as much as the cubic term is worth. A packet with no bandwidth would never spread at all, and would also never begin or end. The packet that will not keep its shape A group velocity is only the first thing a dispersion relation says. The second is that the packet spreads — at a rate fixed by its own bandwidth and by the curvature of ω(k) — so a short pulse comes apart quickly and a long one hardly at all. It is a trade quantum mechanics is usually given credit for, and classical waves make it too. The wavelength at which a fibre stops smearing a pulse. Group-delay dispersion against wavelength for fused silica, computed by differencing Malitson's Sellmeier fit twice rather than read off a table, in picoseconds of spread per nanometre of source width per kilometre of fibre. The material curve crosses zero at 1273 nm — a property of the glass, fixed by where the ultraviolet and infrared absorptions balance, and not adjustable. Below it a fibre is normally dispersive and above it anomalously so, and at 1550 nm, where silica is most transparent, it is 21.9. The second curve adds a waveguide term, which is negative because a mode confined by a core spreads into the cladding differently at different wavelengths, and which depends on the fibre's geometry rather than on the glass; it is drawn here as a constant offset chosen to put the total zero at 1310 nm, which is what standard single-mode fibre is built to do. The consequence in a system is the thing worth carrying away: a source one nanometre wide sent 100 km through fibre at 1550 nm arrives 1833 picoseconds broader than it left, against 0.0 at the zero. That is the whole reason a network runs at one wavelength rather than another, and the reason the two quantities an engineer wants — lowest loss and zero dispersion — sit at different wavelengths and have to be reconciled by design rather than chosen. The wavelength a fibre does not smear Fused silica's index has a second derivative that passes through zero at 1,273 nanometres, and that is not a design choice — it is where the ultraviolet and infrared absorptions balance. A pulse sent at that wavelength arrives the shape it left. A pulse at the wavelength of least loss arrives 1,800 picoseconds wider after a hundred kilometres.

The objects named here

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

Angular dispersionChirped pulse amplificationDiffraction gratingDispersionGroup delay dispersionGroup velocityPrismPulse front tilt