Optics

The rough surface that is a mirror at a glance

A sheet of office paper is matt from above and shines when sighted along its surface. A choppy sea scatters a radar beam from overhead and returns it like a mirror near the horizon. An X-ray mirror polished to a third of a nanometre is still rough to X-rays except at angles below a degree. All three are one rule: two rays reflected from points at different heights differ in path by twice the height times the sine of the grazing angle, so roughness is measured in wavelengths times that sine — and every surface, however rough, is a mirror for light that skims it closely enough.

15 min read 5 figures The shape decides What happens at the edge

Assumes: The surface that cancels whatever skims it · How accurate a mirror has to be

The surface that cancels whatever skims it found that every flat surface, whatever it is made of, reflects a wave arriving at grazing incidence completely and turns it upside down, so that the reflection cancels the direct wave along the surface itself. It assumed the surface was flat. It ended by asking what happens over a real sea, covered with waves of every size, where the reflection near the horizon matters most to radars and radio links.

The answer belongs to a rule that is simpler than the sea and reaches much further. Roughness is not a property a surface has. It is a property of the surface as seen by a particular wave arriving at a particular angle, and the angle matters as much as the wavelength. Seen at a steep enough angle every surface is rough; seen at a shallow enough one every surface is a mirror.

Two rays and a bump

Take two parallel rays of the same wave, arriving at a grazing angle $\psi$ — the angle between the ray and the surface, not the normal — and let one reflect from a facet at the top of a bump and the other from a facet a height $h$ lower. Both leave in the mirror direction. The one reflected lower down travels further, by exactly $2h\sin\psi$ .

The extra path a bump adds, steep and glancing. Two parallel rays reflecting from a facet at the top of a rough surface and from one a height h lower, drawn as two short mirrors, at a grazing angle of 40° on the left and 6° on the right; the reflected rays leave parallel. The ray reflected from the trough travels further, by twice the height times the sine of the grazing angle: 1.29 h at 40°, only 0.21 h at 6°. Whether the two reflections add in step — whether the surface acts as a mirror — depends on that extra path compared with the wavelength. A bump that spoils the reflection at a steep angle is invisible to light that skims it. — Fig. 1 Two parallel rays reflecting from facets a height h apart, at grazing angles of 40° and 6°. The lower reflection travels further by 2h sin ψ: 1.29h at 40°, only 0.21h at 6°.

If that extra path is small compared with the wavelength, the two reflections arrive in step and add like the reflections from a single flat mirror. If it is comparable with a wavelength, some pairs of reflections arrive in step and some out, and the mirror direction receives a jumble that averages to much less. The height of the bumps enters only through the extra path, and the extra path carries a factor $\sin\psi$ that goes to zero as the ray skims the surface. A bump that spoils the reflection at 40° spoils it six times less at 6°.

Lord Rayleigh stated the criterion in the nineteenth century: a surface reflects like a mirror if its bumps add less than a quarter of a wavelength of path, $h\sin\psi < \lambda/8$ . The criterion has an unusual consequence for a physicist used to thinking of smoothness as absolute. There is no such thing as a surface rough enough that no wave can use it as a mirror. Bring the angle down far enough and $\sin\psi$ will make the bumps irrelevant.

How much stays a mirror

A real surface has bumps of every height, distributed randomly. If the heights vary about their mean with a standard deviation $\sigma$ and follow the bell-shaped distribution, the average over all of them of the phase factor each one gives can be computed exactly, and the part of the reflection that stays in the mirror direction — the coherent part — is reduced in power by

\exp(-g^2), \qquad g = \frac{4\pi\sigma\sin\psi}{\lambda}.

This is the factor that how accurate a mirror has to be found at normal incidence, where it says that a telescope mirror with surface errors of a fourteenth of a wavelength keeps most of its peak; here it carries the extra $\sin\psi$ , and at grazing incidence it is much kinder.

The factor follows in two lines from the picture of two rays. A ray reflected from a point at height $h$ comes back with a phase $2kh\sin\psi$ relative to one reflected from the mean plane, $k$ being $2\pi/\lambda$ . The coherent reflection is the average of $e^{i2kh\sin\psi}$ over all the heights, and for heights spread with the bell-shaped distribution that average is $e^{-2k^2\sigma^2\sin^2\psi}$ — the bell curve’s own transform. Power is the square of amplitude, which doubles the exponent and gives the formula. Nothing in it depends on the material, which sets the overall reflectance, or on how the bumps are arranged, which decides where the lost light goes.

$The share of the reflection that stays a mirror. The fraction of the reflected power that goes into the mirror direction rather than being scattered, which falls exponentially with the square of the phase roughness g = 4πσ sin ψ/λ, against the grazing angle on a logarithmic scale, for surfaces whose height varies with a standard deviation σ of 0.05, 0.5 and 5 wavelengths. A surface rough by a twentieth of a wavelength is a good mirror at every angle: 67 per cent at normal incidence. One rough by half a wavelength keeps only 7·10⁻¹⁸ at normal incidence but 74 per cent at 5°. One rough by five wavelengths reflects nothing specularly until the angle is below about 0.9°, where g = 1, and is a mirror below a few tenths of a degree. The angle at which a surface turns from matt to glossy scales as the wavelength over the roughness; no surface is too rough to be a mirror for light that grazes it closely enough.$

Fig. 2 The share of the reflected power that stays in the mirror direction,

e^{-g^2}

, against grazing angle, for roughness of 0.05, 0.5 and 5 wavelengths. Rough by half a wavelength: 7 × 10⁻¹⁸ at normal incidence, 74 per cent at 5°. Rough by five wavelengths: specular only below about a degree.

A surface rough by a twentieth of a wavelength is a respectable mirror at any angle. One rough by half a wavelength reflects nothing specularly when looked at from above — seven parts in $10^{18}$ — and three-quarters of its reflection at five degrees. One rough by five wavelengths turns into a mirror below about a degree. The transition in each case is quick, because the exponent goes as the square of the angle: a factor of three in angle changes $g^2$ by a factor of nine.

What leaves the mirror direction is not lost. For a surface that absorbs nothing, the total reflected power is set by the material, and roughness only decides where it goes: what the coherent reflection loses, the diffuse scattering gains, spread over a cone of directions whose width is set by the slopes of the bumps. That diffuse light is what makes a matt surface visible from every direction, and when it is lit coherently it is the grain that is in the light, the speckle pattern a laser makes on a wall.

Every surface shines when skimmed

The grazing angle at which $g$ reaches one — above it the reflection is mostly diffuse, below it mostly specular — depends only on the roughness in wavelengths: $\sin\psi = \lambda/4\pi\sigma$ .

Fig. 3 The grazing angle at which the roughness parameter reaches one against roughness in wavelengths: 26° for glossy paper in light (0.1 μm), 1.3° for office paper (2 μm), 0.003° for asphalt (1 mm); a sea with 0.5 m of height variation, 0.3° for 3 cm radar and 9.2° for 1 m radio.

The cases on the figure can be checked at a desk. A sheet of office paper, rough by a couple of micrometres, is matt seen from above: its reflection of a lamp is a diffuse glow. Sighted along its surface towards a window, within a degree or two, it shows a clear reflection — a sheen that grows into a recognisable image as the angle falls. Glossy photographic paper, smoother by a factor of twenty, is a mirror from much steeper angles, which is what glossy means. A road surface, rough by a millimetre, needs an absurdly shallow angle for light, three thousandths of a degree, and is never a mirror to the eye; the shimmering “water” seen on a hot road is a different thing, the ray that bends without a surface in the hot air above it. A wet road at night, its texture filled with water, becomes smooth and reflects headlights like a pond.

The rule also explains why sanding a surface finer makes it glossier gradually rather than suddenly, and why gloss meters, which measure the specular reflection of a lamp at fixed angles, use a steep angle of 20° from the normal for very glossy surfaces and a grazing 85° for nearly matt ones: each angle is chosen to put the surfaces it sorts on the steep part of the exponential.

A sea for a radar

A sea rough enough to look grey and choppy to the eye has waves a metre or two high, a standard deviation of surface height of perhaps half a metre. To light it is impossibly rough. To radio waves of a metre or more it is close to smooth.

How much of a radio wave a choppy sea returns as a mirror. The amplitude of the coherent, mirror-like reflection from a sea whose surface height varies with a standard deviation of 0.5 m — waves about 2 m high — as a fraction of a calm sea's, against the grazing angle, for radio wavelengths of 3 cm, 10 cm and 1 m. At 1 m wavelength the sea keeps more than half its mirror reflection at every angle drawn — 55 per cent of a calm sea's at 10°. At 10 cm it keeps half at 1° and almost nothing by 4°. At 3 cm only the lowest angles survive: 19 per cent at half a degree and 3·10⁻¹² at 2°. The interference lobes a calm sea makes, between the direct wave and its mirror image, fade as the coherent reflection does; a radar low over the sea still sees the lobes near the horizon, where they matter most for detecting low-flying targets and for the fades of a microwave link. — Fig. 4 The amplitude of the coherent reflection from a sea with 0.5 m of height variation, as a fraction of a calm sea’s, against grazing angle: for 1 m radio waves more than half survives at every angle drawn, 55 per cent at 10°; for 10 cm, half at 1° and almost none by 4°; for 3 cm, 19 per cent at half a degree.

For a radar looking out over the sea the consequence is the interference pattern that Lloyd’s mirror produces between the direct wave and its reflection from a calm surface, lobes of strong and weak signal stacked up in elevation. Over a rough sea the lobes fade as the coherent reflection does — except close to the horizon, at grazing angles of a fraction of a degree, which is exactly where a radar hunting a low-flying target or a sea-skimming missile needs to look. There the mirror image survives at every practical radar wavelength, and the target and its reflection in the sea interfere, producing fades and false elevations that make low-angle tracking one of the hardest problems in radar.

The rule has also been turned into an instrument. Navigation satellites broadcast signals with wavelengths of about nineteen centimetres, and those signals reflect off the sea towards satellites in low orbit placed to catch them. How much of the reflection stays coherent, and how widely the rest is scattered around the mirror point, depends on the sea’s roughness, which depends on the wind; since 2016 a constellation of small satellites has mapped wind speeds over the tropical oceans, including inside hurricanes, by reading the roughness off reflected navigation signals.

A glitter path that measures slopes

The sun low over a rippled lake does not make one reflection. It makes a long glittering streak reaching towards the observer, made of thousands of brief flashes, each from a facet of water tilted at just the angle that sends sunlight to the eye. This is the diffuse part of the reflection made visible: the ripples are far rougher than a wavelength of light, there is no coherent mirror image at all, and every flash comes from a slope rather than a height. The length of the streak measures how widely the slopes are spread, and it grows with the wind.

In 1954 Charles Cox and Walter Munk photographed the sun’s glitter on the Pacific from an aircraft and turned its extent into the first measured distribution of sea-surface slopes, finding that its spread grows in proportion to the wind speed. Satellites use the same glitter to spot oil slicks, which damp the smallest ripples and show as dark patches in it. And as the sun sinks towards the horizon the streak narrows and sharpens: at grazing angles the reflection from even a rippled surface begins to keep a coherent part, and a lake that glittered at noon gives a recognisable, if trembling, image of the setting sun.

The full Moon, by contrast, shows no glitter and no sheen at any angle, though sunlight reaches its surface at every grazing angle somewhere along the terminator. Its soil is rough on every scale from mountains to grains, its grains are dark and shadow one another, and as the full moon that is too bright found, what it does instead is return light preferentially straight back towards the Sun. A surface rough on every scale at once never reaches the regime where the factor $\sin\psi$ rescues it, because there is always a scale of bump comparable with the wavelength at whatever angle is chosen.

Mirrors for X-rays

At the other end of the spectrum the wavelengths are so short that no surface is smooth. An X-ray of 8 keV has a wavelength of 0.15 nanometres, about the size of an atom, and the best-polished mirrors in existence are rough by a few tenths of a nanometre. Worse, the index that falls below one found that every material’s refractive index for X-rays is a few parts in a million less than one, so a surface reflects X-rays strongly only below a critical grazing angle of a fraction of a degree.

An X-ray mirror read for its roughness. The reflectivity of a silicon mirror for 8 keV X-rays against the grazing angle, on a logarithmic scale, smooth and with surface roughness of 0.3, 1 and 3 nm, from Fresnel's formula with silicon's index 1 − δ − iβ (δ = 7.53·10⁻⁶ from its electron density) and the roughness factor exp(−(4πσ sin θ/λ)²). Below the critical angle, 0.222°, every surface reflects almost totally. Above it the smooth mirror's reflectivity falls as the fourth power of the angle, to 0.0014 at 0.6°; with 1 nm of roughness, to 6.6·10⁻⁴; with 3 nm, 2.1·10⁻⁶. An X-ray's wavelength is 0.15 nm, so even an atomically smooth surface is rough to it at steep angles, and every X-ray mirror works at grazing incidence. Measuring how fast the reflectivity falls is a standard way to measure a surface's roughness to a tenth of a nanometre. — Fig. 5 The reflectivity of a silicon mirror for 8 keV X-rays against grazing angle, smooth and with 0.3, 1 and 3 nm of roughness. Below the critical angle, 0.222°, all reflect almost totally; above it the smooth surface falls as the fourth power of the angle, to 1.4 × 10⁻³ at 0.6°, and 3 nm of roughness takes that to 2.1 × 10⁻⁶.

Both facts push X-ray mirrors to grazing incidence, and the roughness rule is what makes the shallowest angles usable at all. Below the critical angle the reflection is total and roughness of a nanometre barely touches it, because $\sin\psi$ is a few thousandths. Above it, where the reflectivity of even a perfect surface falls steeply, roughness multiplies the fall: a surface rough by three nanometres loses three orders of magnitude at 0.6° that a perfect surface keeps.

X-ray telescopes are therefore nested shells of polished mirror struck at grazing angles of well under a degree, the design Hans Wolter worked out in 1952 and that NASA’s Chandra observatory flies with shells polished to a few tenths of a nanometre. And the same curves, run backwards, are a measuring instrument: shining X-rays on a surface and recording how quickly its reflectivity falls with angle above the critical angle is the standard way to measure a surface’s roughness, and the thickness of films on it, to a fraction of a nanometre.

The smoothest surfaces ever made

The X-ray case has a counterpart that shows what happens when grazing incidence is not available. The machines that print the finest features on computer chips use extreme-ultraviolet light of 13.5 nanometres, and they must bounce it off a dozen mirrors in sequence at nearly normal incidence, where $\sin\psi$ is close to one and gives no help at all. No single surface reflects such light well at steep angles, so each mirror is a stack of some forty pairs of alternating molybdenum and silicon layers, each a few nanometres thick, whose many weak reflections add in step — the kind of stack of alternating layers that reflects one colour almost completely, here tuned to 13.5 nanometres. Every interface in the stack must obey the roughness rule at normal incidence with a wavelength of 13.5 nanometres: to keep most of the reflection, the roughness must be a small fraction of $\lambda/4\pi$ , about a nanometre, and in practice it is held to about a tenth of that, a few atoms’ worth over areas of hundreds of square centimetres.

These are the smoothest large surfaces ever made, and they are smooth because the angle was not allowed to help. An X-ray telescope, working at grazing incidence with wavelengths ten times shorter, can tolerate mirrors of similar roughness only because its angle divides the bumps by a hundred. The exponent in the specular factor is the same for both; the factor $\sin\psi$ is the whole of the difference.

Where the scattered light goes

The exponential describes the coherent reflection only, and it is exact only for a surface whose heights are randomly distributed with the bell-shaped distribution and whose bumps are broad compared with the wavelength. Where the light that leaves the mirror direction goes depends on something the exponential does not contain: how far apart the bumps are, or equivalently how steep their sides are. A surface with high, gentle undulations scatters into a narrow cone close to the mirror direction, which the eye reads as a blurred reflection; one with the same heights but short, steep bumps scatters widely, which the eye reads as matt. Two surfaces with the same $\sigma$ can therefore have the same specular share and look quite different, and a full description of how a surface looks needs the spectrum of its roughness, not just its height.

The exponential also assumes each part of the surface is lit and seen. At the very lowest angles that fails: the crests hide the troughs from the incoming ray, and the troughs cannot reflect what never reaches them. Shadowing removes the deep parts of the surface from the reflection altogether, so the effective roughness falls further than the factor $\sin\psi$ alone would say, and for a sea the crests themselves — steeper and sharper than the troughs, unlike the symmetric bell-shaped surface — dominate what is seen. Waves breaking into foam add scatterers that are not a surface at all.

What the bell-shaped surface leaves out

Every figure here uses a surface with Gaussian heights, a single roughness, one material’s index, and the Kirchhoff approximation that treats each patch of surface as a locally flat mirror. Real surfaces are rough on many scales at once, from the waves of a sea down to its capillary ripples, and a wavelength sees the scales comparable with itself most strongly; a single $\sigma$ is a summary that depends on what range of scales is included. The approximation fails for bumps sharper than a wavelength, and for surfaces so rough that light reflects more than once before leaving. The figures do not separate the two polarisations, which reflect differently at a rough sea — vertically polarised radio waves have a Brewster-like dip at a few degrees over sea water that horizontally polarised ones do not. And the X-ray figure uses the simplest roughness factor; more careful treatments, which account for the different wavevector inside the material, change the numbers near the critical angle by tens of per cent.

The domain of the argument is a surface whose bumps are broad compared with the wavelength and whose heights vary randomly about a mean plane, lit at angles where the troughs are not hidden by the crests. Within it, roughness is the height in wavelengths times the sine of the grazing angle, and every surface is a mirror for light that skims it closely enough.

Still open: what the sea does at the lowest angles

The case the grazing-reflection argument ended on remains the hard one. At grazing angles below about a degree, over a sea with waves of every size, theory has to combine the coherent reflection, the shadowing of troughs by crests, the asymmetric shape of real sea waves, the bending of the radio path by the air’s layering, and the diffraction of waves around crests that the geometric picture treats as opaque. Different treatments of these predict coherent reflection coefficients that differ by large factors, and measurements at sea, which are difficult to make with the sea state and the air both known, have not settled between them. Whether a single description works across sea states and wavelengths, or whether low-angle propagation over the ocean must be modelled case by case, is still argued in the radar literature.

The core of the rule is simple and certain. Two reflections from heights h apart differ in path by 2h sin ψ, so the share of reflection that stays a mirror is exp(−(4πσ sin ψ/λ)²): paper rough by 2 μm is a mirror below about 1.3°, a sea rough by half a metre is a mirror to 3 cm radar below 0.3° and to metre waves below 9°, and an X-ray mirror rough by a third of a nanometre works only below a few tenths of a degree. Gloss is not a finish; it is an angle.