Illustrated Physics
physics, drawn
Optics

The colour at which glass disappears

Lower a glass rod into an oil of the same refractive index and it vanishes, except for a faint coloured outline. The outline is the clue. Glass and oil disperse light differently, so their indices can agree at only one wavelength; at every other the glass is still there, bending and scattering a little. Grind the glass to powder and pack it in the oil, and the mixture is clear at that one wavelength and opaque at the rest — a colour filter made of nothing but two transparent materials, whose colour moves by a whole hue for a few degrees of warming.
15 min read 5 figures The same equation againMade into an instrument

Assumes: Two glasses that cancel a derivative · Everything a scatterer removes, from one direction

Two glasses that cancel a derivative found that no two glasses can be made to agree in their dispersion everywhere: an achromatic lens cancels the slope of focal length against wavelength at one point and leaves a residue, because the curves of index against wavelength have different shapes. That was a defect to be minimised. Turned round, the same fact is a resource. Two materials whose indices are made to agree at one wavelength will disagree at every other, and the disagreement grows on either side of the match. Anything that depends on the difference of index — and the scattering of light at a boundary depends on nothing else — then becomes a function of colour with a sharp zero.

The result is one of the stranger optical objects there is: a colour filter made of a heap of crushed glass in a jar of oil, neither of which absorbs any light at all.

Two curves that meet

The refractive index of every transparent material falls slowly with wavelength across the visible spectrum, rising steeply towards the ultraviolet where its electrons resonate. How steeply is summarised by the Abbe number, the ratio of the index’s excess over one to its spread between blue and red: a high Abbe number, like crown glass’s 64, means little dispersion; a low one, like a dense flint’s 30 or many oils’ 35, means a lot.

Two dispersion curves that cross once. The refractive index against wavelength of N-BK7 glass, from its Sellmeier fit, and of an immersion liquid chosen to match it at the sodium d line, 589.3 nm, where both are 1.5167; the liquid disperses more, with an Abbe number of 35 against the glass's 64.2. At 450 nm the liquid's index exceeds the glass's by 73.2 × 10⁻⁴; at 650 nm it falls short by 17.5 × 10⁻⁴. The difference changes by −34.1 × 10⁻⁶ per nanometre near the crossing. Only at the crossing are the two materials optically the same; everywhere else light passing from one to the other is bent and reflected, a little more the further from the crossing it is.
Fig. 1 The refractive index of N-BK7 glass, from its Sellmeier fit, and of an immersion liquid of Abbe number 35 matched to it at 589.3 nm, where both are 1.5167. At 450 nm the liquid exceeds the glass by 73 × 10⁻⁴; at 650 nm it falls short by 18 × 10⁻⁴; the difference changes by 34 × 10⁻⁶ per nanometre at the crossing.

The figure takes the commonest optical glass and a liquid adjusted to have exactly the same index at the yellow sodium line. Liquids are easy to adjust: blending two oils of different index gives any value in between. What cannot be adjusted together with the index is the dispersion, and the liquid’s is larger. At the sodium line the two are identical. Towards the blue the liquid’s index climbs faster and exceeds the glass’s by seven parts in a thousand by 450 nanometres; towards the red it falls below by two parts in a thousand.

Seven parts in a thousand is small. The bend at the boundary at such a mismatch is a fraction of a degree for any ordinary angle of incidence. But for scattering, which is what makes a clear material visible in a different clear material, the question is not how big the mismatch is but how it compares with zero.

Where glass vanishes

A glass object in air is seen because its surfaces reflect a few per cent of the light and refract the rest at visible angles, distorting what lies behind. Both effects scale with the difference of index across the surface. In a liquid that matches it, both vanish — but only at the matching wavelength.

The coloured outline of a vanished rod. The fraction of light reflected at normal incidence by the surface of N-BK7 glass immersed in the matching liquid, against wavelength, on a logarithmic scale, from Fresnel's formula, the square of the difference of the two indices over their sum. In air the same glass reflects 4.2 per cent. In the liquid it reflects 5.7·10⁻⁶ of blue light at 450 nm and 3.4·10⁻⁷ of red at 650 nm, and nothing at all at 589 nm. A glass rod lowered into such a liquid disappears, except that its outline glows faintly: at the edges, where light grazes the surface and even a small index difference bends and reflects it, the wavelengths on either side of the match are turned aside and the match is not, so the outline is tinged blue on one side of the colour of the match and orange on the other.
Fig. 2 The fraction of light reflected at normal incidence by N-BK7 in the matched liquid, on a logarithmic scale: 5.7 × 10⁻⁶ of blue light at 450 nm, 3.4 × 10⁻⁷ of red at 650 nm, and none at 589 nm. In air the same glass reflects 4.2 per cent.

The reflectance at a boundary between indices n1n_1 and n2n_2 is ((n1−n2)/(n1+n2))2((n_1-n_2)/(n_1+n_2))^2, so it goes as the square of the mismatch and falls by four orders of magnitude between air and a near-match. At the match itself it is exactly zero. A glass rod lowered into such a liquid disappears, a demonstration that schools perform with Pyrex and vegetable oil, whose indices happen to be close. What betrays the rod is its outline. Light grazing the rod’s edge meets the boundary at so steep an angle that even a small index difference reflects and bends it strongly, and it does so only at the wavelengths where the indices differ: blue on one side of the match, orange on the other. The vanished rod is drawn in faint coloured lines, and the colours are the dispersion curves made visible.

Mineralogists have used this for a century and a half. A grain of an unknown mineral, placed in a drop of liquid of known index under a microscope, shows a bright halo — the Becke line — that moves into whichever material has the higher index as the focus is raised, and disappears when the two match. A set of calibrated liquids identifies the mineral by the one in which it vanishes, and the coloured fringes that remain at the match tell how its dispersion compares with the liquid’s.

Why the liquid disperses more

The two curves have different slopes for a reason, and it is the same reason every transparent material has dispersion at all. A material’s index comes from its electrons responding to the light’s field, and the constant that depends on how fast it is asked found that the response depends on how close the light’s frequency is to the frequencies at which those electrons resonate. For glass the strong resonances are deep in the ultraviolet, around a tenth of a micrometre, and visible light is far below them: the index is nearly flat and creeps upward towards the blue. The oils used for matching are built of molecules with rings of carbon atoms, whose loosely held electrons resonate at much longer ultraviolet wavelengths, a quarter of a micrometre or so. Visible light is closer to those resonances, so the oil’s index rises more steeply towards the blue.

A liquid can therefore be blended to match glass’s index at one wavelength, but its curvature is set by where its resonances lie, and blending cannot move them. The crossing is a meeting of two curves with different shapes, not two parallel lines that happen to coincide. Choosing the liquid’s resonances further into the ultraviolet — a liquid more like the glass in its chemistry — flattens the difference and broadens the filter; choosing them nearer — a more aromatic, more dispersive oil — sharpens it.

A heap of glass that is clear

Now grind the glass into grains and pack them into the liquid. Every grain is a small lens, refracting light passing through it slightly and scattering a little of it aside, and a layer of thousands of grains in a row scatters a beam completely: crushed glass in air is white and opaque, like snow or sugar. In the matched liquid it is clear — at one wavelength.

A filter made of glass powder and oil. The fraction of light transmitted straight through a 5 mm layer of N-BK7 grains packed to half its volume in the matching liquid, against wavelength, for grains of radius 5, 10 and 20 μm, from van de Hulst's extinction for large, weakly refracting particles. At the crossing the grains vanish and the layer is clear. A few tens of nanometres away each grain shifts the light passing through it in phase by enough to scatter a little of it out of the beam, and a thousand grains in a row scatter all of it: the passband is 51.7 nm wide at half height for 5 μm grains, 36.3 nm for 10 μm, 25.5 nm for 20 μm. Larger grains make a narrower filter, because the phase a grain imposes grows with its size; the same mismatch of index that a fine powder lets through, a coarse one scatters.
Fig. 3 The light transmitted straight through a 5 mm layer of N-BK7 grains, half the volume, in the matched liquid, for grains of radius 5, 10 and 20 μm. The layer is clear at 589 nm and opaque far from it; the passband is 52, 36 and 26 nm wide at half height.

Christian Christiansen described the effect in 1884, using powdered glass in a mixture of benzene and carbon disulphide, and saw the mixture glow in a single brilliant colour when held up to the light, changing colour as it warmed in his hand. Such Christiansen filters were used for decades to isolate bands of light, particularly in the infrared, where good filters were scarce and suitable crystals and liquids could be found.

The width of the passband depends on the grains as well as on the dispersions. A grain whose index differs from the surrounding liquid’s by Δn\Delta n delays the light passing through its middle relative to light passing beside it, by a phase of about 2ka Δn/n2ka\,\Delta n/n for a grain of radius aa. The larger the grain, the larger the phase for the same mismatch, so coarse grains begin scattering closer to the match and make the narrower filter: 26 nanometres wide for grains of 20 micrometres, twice that for grains of five. It is the one place in optics where a coarser powder gives the sharper result.

The width can be estimated without the figure. The layer goes half opaque when the index difference reaches a certain size, set by the grains and the thickness — six parts in ten thousand for ten-micrometre grains in five millimetres — and the dispersion curves part at thirty-four parts in a million per nanometre. The ratio is about eighteen nanometres on each side of the match, thirty-six in all, which is the computed width. Every design choice enters through one of those two numbers: the grains and the depth through the first, the choice of liquid through the second. Halving the thickness of the layer, or the size of its grains, raises the tolerable mismatch by the square root of two and widens the filter by the same factor; choosing a liquid twice as dispersive halves the width at a stroke.

How much one grain takes

What a single grain removes from a beam follows from everything a scatterer removes: the light lost from the forward direction is fixed by what the grain does to the wave passing straight through it.

How much of the light a grain removes. The extinction efficiency of a large, weakly refracting sphere — the light it removes from a beam, as a multiple of its cross-section — against the phase difference ρ = 2ka|m − 1| between a ray through its middle and one past it, on logarithmic scales, from van de Hulst's anomalous-diffraction formula. For small ρ the efficiency goes as half the square of ρ: halve the index mismatch and the grain removes a quarter as much. Near ρ = 4 it peaks at 3.17 and at large ρ it settles at 2, twice the grain's shadow, because it diffracts as much light as it blocks. A single 10 μm grain at 550 nm reaches ρ = 1 only when the indices differ by 0.0066, far from the crossing. But a packed layer holds so many grains that it goes opaque while each is still deep in the weak regime where extinction goes as the square of ρ: the 5 mm layer drawn beside this halves its light at ρ = 0.09, an index difference of about 0.0006.
Fig. 4 The extinction efficiency of a large, weakly refracting sphere against the phase shift through it, from van de Hulst’s anomalous diffraction: ρ2/2\rho^2/2 for small shifts, a peak of 3.17 near ρ = 4, and 2 for large shifts. A 5 mm packed layer of 10 μm grains halves its light at ρ = 0.09, an index difference of about 0.0006.

For a grain much larger than the wavelength that refracts only slightly, Hendrik van de Hulst showed in the 1950s that the extinction — the light removed, as a multiple of the grain’s cross-section — depends only on the phase shift ρ\rho through its middle. For a small shift it is ρ2/2\rho^2/2: halve the mismatch and the grain removes a quarter as much light. For a large one it is two, twice the grain’s geometric shadow, because a large obstacle diffracts as much light out of the beam as it intercepts. The figure shows the transition, and the overshoot to 3.17 where the light through the grain arrives half a cycle late.

The filter works far down on the left of that curve. A single ten-micrometre grain needs a mismatch of nearly seven parts in a thousand to reach a phase shift of one radian. But a five-millimetre layer of grains packed to half its volume puts several hundred grains in the path of every ray, and the layer is already half opaque when each grain shifts the phase by only a tenth of a radian — a mismatch of six parts in ten thousand, which the dispersion curves reach about twenty nanometres from the match. Many weak scatterers in series make a strong one, exactly as the cloud light has to walk through is opaque though each droplet in it is nearly transparent.

A filter tuned by a thermometer

The index of a liquid falls as it warms, by about four parts in ten thousand per kelvin for oils, because the liquid expands and fewer molecules occupy each cubic centimetre. Glass’s index barely changes, by a few parts in a million. So warming the mixture lowers the liquid’s curve relative to the glass’s, and the crossing slides along the spectrum.

A filter tuned by a thermometer. The same layer of 10 μm grains at 15, 20 and 25 °C. The liquid's index falls by 4.0 × 10⁻⁴ for each kelvin of warming while the glass's barely changes, so the crossing slides along the spectrum: 661 nm at 15 °C, 589 nm at 20 °C, 538 nm at 25 °C — −11.5 nm for every kelvin. A Christiansen filter is therefore a tunable filter whose knob is temperature, and an untuned one is useless unless held still to a fraction of a degree; the passband changes colour in the hand as the hand warms it.
Fig. 5 The same layer of 10 μm grains at 15, 20 and 25 °C: the passband is centred at 661, 589 and 538 nm, moving 11.5 nm towards the blue for every kelvin of warming.

The crossing moves by the ratio of the two slopes: the change in index difference per kelvin divided by its change per nanometre. With four parts in ten thousand per kelvin and thirty-four parts in a million per nanometre, the passband moves eleven and a half nanometres for every degree — from red through yellow to green between 15 and 25 °C. Christiansen’s filter changed colour in his hand for this reason, and every practical Christiansen filter since has been a thermostat with a jar of glass in it, or else a deliberately tuned instrument, its colour set by its temperature.

That sensitivity has been turned into thermometers and into ways of measuring the dispersion of powders that cannot be made into prisms. A powder of unknown dispersion in a liquid of known dispersion, heated slowly while the transmitted colour is watched, gives the wavelength of the match at each temperature, and so the powder’s index at each wavelength, without ever cutting a surface.

The colour seen from the side

What the layer removes from the straight-through beam is not destroyed. Neither material absorbs, so every photon scattered out of the beam leaves the jar sideways. Looked at against a bright window, a Christiansen mixture shows the colour of its passband; looked at from the side, against a dark background, it glows with everything else — white light with the passband removed, the complementary colour. A yellow filter in transmission is violet-blue in scattered light. Christiansen’s own observations described exactly this pair of colours, and it is a quick test that the colour comes from scattering rather than absorption: a dyed liquid looks the same colour from every direction.

A match with empty space

The match need not be with a liquid. A powder of mineral grains in air has a match wherever the mineral’s own index passes through one, and every silicate mineral’s index does so in the infrared, near the strong vibrations of its silicon–oxygen bonds, where the index swings rapidly through values below one, as the index that falls below one found happens near any resonance. At that wavelength, about eight micrometres for common rocks, the grains neither reflect nor scatter, and a dusty surface emits heat radiation as freely as a perfect black body would — an emission peak at a wavelength fixed by the mineral’s composition.

That peak is called the Christiansen feature, and it is how the composition of the Moon’s surface is mapped from orbit. An infrared radiometer measuring the regolith’s thermal emission finds the wavelength of the peak, which moves with the silica content of the rock: shorter for silica-rich rocks, longer for iron- and magnesium-rich ones. The same physics that makes a jar of glass and oil glow yellow identifies the minerals of the lunar highlands, with the vacuum of space as the matching medium.

The same trick elsewhere

The principle — two materials matched at one value of some parameter and scattering everywhere else — turns up wherever a heterogeneous material is meant to be clear. Glass fibres reinforcing a transparent plastic must be matched to it, and the composite is clear only over the band where the two dispersions keep the mismatch small; outside it the fibres show as a haze. Biologists clearing tissue for microscopy soak it in liquids that match the index of its proteins and lipids, making whole mouse brains transparent, and the remaining haze is largest in the blue, where the dispersions part fastest. And the sand that darkens when it is wet does so partly by the same physics: water, at index 1.33, is far closer to quartz’s 1.54 than air is, so each wet grain scatters more weakly and more forwards, and light travels further into the bed before turning back.

The rainbow is the opposite case. The angle the rainbow has to be is set by water’s index, and the colours separate because water’s dispersion moves that angle by about two degrees across the spectrum; there the dispersion is the whole effect and nothing is matched to anything. A Christiansen filter uses dispersion in the subtler way: not to spread the colours apart, but to choose one colour at which a boundary stops existing.

What the smooth spheres leave out

The figures treat the grains as smooth spheres of one size, randomly packed, each scattering independently, with light removed from the beam by the anomalous-diffraction formula. Real crushed glass is angular and comes in a range of sizes, which broadens the passband; grains packed to half the volume are close enough together that their scattering is not independent, and the layer behaves partly as a continuous medium with a fluctuating index rather than as separate particles. The figures count only light transmitted straight through; light scattered forward by a small angle still passes a filter of finite acceptance and fills in the wings. The liquid is a model with a stated Abbe number and temperature coefficient, not a particular product, and any real liquid absorbs somewhere, which cuts the passband’s edges. And a real glass is not perfectly homogeneous: strain, small bubbles and variations of composition between grains each shift the match slightly from grain to grain, which in practice is the largest broadening of all.

The domain of the argument is a mixture of non-absorbing materials whose indices cross at one wavelength, with grains large compared with the wavelength, and light measured in a narrow beam. Within it, the mixture is clear only where the boundary has disappeared, and the colour is the dispersion’s. Outside it — grains comparable in size with the wavelength, or liquids that absorb — the same crossing still exists, but the passband is shaped by resonances of the grains or by the liquid’s own colour, and the simple estimate of its width no longer holds.

Still open: how sharp a matched medium can be made

The narrowest passbands come from large, uniform grains and liquids whose dispersion differs most from the glass’s, and both run into limits: large grains settle and are hard to pack evenly, and the most dispersive liquids absorb in the blue. Engineered versions of the effect use particles made deliberately uniform — monodisperse polymer spheres, or glass beads sorted by size — in matrices whose index and dispersion can be designed, and they reach passbands of a few nanometres in principle. How narrow a stable, solid Christiansen filter can be made, and whether such matched composites can be designed to be clear across the whole visible spectrum by matching dispersion as well as index — the problem that the tissue-clearing liquids and transparent composites both face — is a question about materials chemistry as much as optics, and it has not been settled.

The physics underneath is a crossing of two curves. Two transparent materials disperse differently, so their indices agree at one wavelength and differ elsewhere — by 7 × 10⁻³ in the violet for N-BK7 in a liquid of Abbe number 35 — and every boundary between them vanishes at that wavelength only: a glass rod disappears leaving a coloured outline, and a packed layer of glass grains transmits a band 26 to 52 nm wide that moves 11.5 nm for each kelvin of warming. A heap of crushed glass in a jar of oil is a filter, and its colour is wherever two dispersion curves meet.

Part 7 of 7

This essay is one argument about Dispersion. The others:

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 Part 6 — 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.

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. Why only a sideways scattered wave takes anything away. The transmitted amplitude behind a thin scatterer, drawn as a phasor: the incident wave of unit length along the axis, plus a forward-scattered wave of length 0.12 at 0°, 60°, 90°, 150°. What a detector reads is the square of the total length. A scattered wave along the incident one lengthens or shortens the sum in proportion to itself; one at right angles changes the length only in second order, because a small perpendicular addition to a long vector barely alters its length. So a scatterer that removes energy from the beam at first order must scatter forward with a component perpendicular to the incident wave, and the size of that component is the whole extinction — which is the optical theorem. Everything a scatterer removes, from one direction How much light a particle takes out of a beam — by scattering it anywhere at all, and by absorbing it — is fixed entirely by what it does in the forward direction, where its scattered wave cannot be told apart from the incident one. The mechanism is interference, and it also gives the refractive index. How much darker a surface looks under a film of water. The diffuse reflectance of a surface covered by a thin film of clear water (solid), against its reflectance dry, with the dry value for comparison (dashed): (1 − Rₑₓₜ) a (1 − Rᵢₙₜ)/(1 − a Rᵢₙₜ), where the film traps and returns light that falls outside its escape cone. A bright surface of reflectance 0.9 falls to 0.77; a mid-grey one of 0.5 to 0.32, 64 per cent of its dry value; a dark one of 0.2 to 0.11, 54 per cent. Dark surfaces darken proportionally most, because each return trip costs them most; and a coloured surface grows more saturated, because the colours it absorbs are absorbed again on every return while the colour it reflects survives. The sand that darkens when it is wet Wet sand is darker than dry sand, a wet stone darker than a dry one, a splashed pavement darker in patches, and a wet white shirt turns grey and then see-through. Water is clear and absorbs almost nothing, so it cannot be adding darkness. What it does is change where the light goes. A film of water on a surface traps light that has been scattered back up, returning most of it for another chance to be absorbed. Water in the pores between grains replaces air with something closer to the grains' own refractive index, so each grain turns light aside less, and light wanders deeper before it gets out. Both lengthen the path, and on a longer path more light is absorbed by the material that was there all along.

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 numberDispersionExtinctionFresnel reflectionIndex matchingOptical filterRefractive indexScattering