The cloud light has to walk through

Everything a scatterer removes found the total light a single particle takes out of a beam in one direction, straight ahead, and it ended by naming the thing that theorem cannot do. It is a statement about one particle in a plane wave. In a medium many scattering lengths thick, the light arriving at each particle has already been scattered by others, and nothing about a single particle’s forward amplitude says what comes out of the far side.

The question is not exotic. Why a litre of water is not blue observed that a cloud is white because its optical depth is tens or hundreds, so that every colour has certainly been scattered, and when the particle is the size of the wave noted that so many scatterings wash out a droplet’s own angular pattern. Both are correct statements about colour and pattern. Neither says how much light gets through a cloud, or why the answer is so much larger than an exponential would allow. That needs a different kind of calculation, and the kind of calculation it needs is the most useful fact in the subject.

A walk instead of a beam #

The direct approach is to follow light through a slab one photon at a time. A photon travels a random distance, drawn from the exponential distribution whose mean is the scattering mean free path. It then scatters into a new direction drawn from the particle’s angular pattern, travels another random distance, scatters again, and so on until it leaves through the top or the bottom.

The paths are tangles. A photon entering the top goes a mean free path or so, turns, turns again, and most of the time finds its way back out of the surface it came in through. The ones that reach the bottom do so after wandering sideways as far as they have travelled down. Of twenty thousand photons sent into a slab eight mean free paths thick, 17.5 per cent emerge from the bottom.

The number to compare that with is the fraction a straight beam would keep, $e^{-8}$, which is 0.034 per cent, and the photons that crossed without scattering at all are indeed about that fraction — ten of them, against 6.7 expected, which is inside the counting error of so few. Five hundred times more light comes out of the bottom than crossed without scattering. Nearly everything transmitted has walked, and has forgotten the direction it arrived from.

Ohm’s law for light #

A photon’s walk through a thick slab is a random walk, and a crowd of random walkers obeys the equation that only runs forwards: the density of photons diffuses, with a diffusion coefficient set by how far a photon goes before forgetting its direction. Light entering the top of a slab is a source just inside the top surface; light leaving through either face is a sink. The steady state is a density that falls linearly from one face to the other.

A linear density profile has a definite consequence. The flux of photons through the slab is proportional to the slope of the profile, and the slope is inversely proportional to the thickness. So the transmitted fraction falls as one over the thickness, not exponentially — the relation between current and length in a wire, with the mean free path in the role of the conductivity. The correct form, from the diffusion equation with the surfaces handled properly, is

$$T = \frac{1 + z_e}{L/\ell^* + 2z_e},$$

where $\ell^*$ is the transport mean free path, introduced below, and $z_e = 0.7104$ is the extrapolation length, measured in transport paths: the distance past each face at which the diffusing density would fall to zero if the medium continued. That 0.7104 comes from solving the exact transport problem at a surface — Milne’s problem — and it is the one number in the formula that a diffusion argument alone cannot supply.

The formula has a second reading that needs no flux at all. A walker diffusing between two absorbing walls leaves through the far one with a probability equal to its distance from the near one divided by the distance between them — the one-dimensional case of walkers stopping on a boundary to solve Laplace’s equation, where the answer is always the harmonic function and in one dimension the harmonic function is a straight line. A photon entering the top makes its first effective scattering about one transport length in. The walls it diffuses between are the two extrapolated faces, $z_e$ transport lengths outside the slab on either side. So it starts $1 + z_e$ from the top wall, the walls are $L/\ell^* + 2z_e$ apart, and the chance of leaving through the bottom is the ratio: the transmission formula, term for term.

On logarithmic axes the difference between the two laws is the difference between a straight line of slope minus one and a cliff. For particles that scatter evenly, eight mean free paths transmit 0.180 and thirty transmit 0.060: nearly four times the thickness, a third of the light. Diffusion theory, which adds the extrapolation lengths to both thicknesses, gives 0.182 and 0.054, a factor of 3.3, and the photons at thirty paths sit within their counting error of it. The unscattered beam over the same range falls by nearly ten orders of magnitude. Even a slab one mean free path thick is not described by the beam: it transmits 0.662 where $e^{-1}$ is 0.368, so nearly half of what crosses it has already scattered at least once.

The correspondence with Ohm’s law is closer than an analogy. Electrons in a disordered metal also scatter elastically off impurities, forget their direction after a transport mean free path, and diffuse, and the conductance of a wire is proportional to that mean free path divided by the wire’s length for exactly the reason the transmission of a cloud is. A cloud is a resistor for sunlight, and the diffusion equation does not care whether what walks through it is a photon or an electron.

Forward scattering only stretches the step #

Cloud droplets are not even scatterers. When the particle is the size of the wave computed their angular pattern and found it tipped strongly forward, with a mean scattering cosine — the asymmetry parameter $g$ — near 0.8 for droplets many wavelengths across. A photon scattered by such a droplet is usually deflected by only a few degrees, and needs several scatterings before its direction has been randomised.

That suggests a single correction, and it is the right one. If each scattering keeps a fraction $g$ of the photon’s forward momentum on average, the direction is forgotten after about $1/(1 - g)$ scatterings, so the distance over which it is forgotten — the transport mean free path — is

$$\ell^* = \frac{\ell}{1 - g}.$$

For $g = 0.85$ that is 6.7 scattering lengths. The claim is then that a slab of forward scatterers behaves, once it is thick, exactly like a slab of even scatterers whose mean free path is $\ell^*$.

The photons agree. Measured in transport lengths, slabs of four very different kinds of scatterer — from particles that scatter evenly to droplets whose mean cosine is 0.95, which need about twenty scatterings to forget a direction — transmit within nine per cent of one another from four transport lengths onward. The collapse is the similarity principle of radiative transfer, and it is why an atmospheric scientist summarises a cloud’s droplets by two numbers, an optical depth and an asymmetry parameter, and gets the brightness of the cloud right without knowing anything else about how the droplets scatter.

The collapse fails for thin slabs, and the way it fails is instructive. A slab half a transport length thick lets through more light when its scatterers are forward-peaked, because the few scatterings a photon undergoes all deflect it only slightly and it emerges still heading roughly downward. The walk has not had room to start, and the angular pattern still matters. Far from the surfaces only the distance over which a direction is forgotten counts; near them, everything about the pattern does.

White on top, grey underneath #

The same calculation, applied to a cloud of water droplets with $g = 0.85$ and no absorption, answers a question anyone who has flown through weather has asked.

A thin cloud, optical depth five, lets three quarters of the sunlight through and reflects a quarter, which is why a veil of cloud dims the Sun without darkening the day much. At an optical depth of thirty — an ordinary overcast deck a few hundred metres thick — the numbers have reversed, 71 per cent reflected and 29 transmitted. At a hundred, a deep storm cloud, the base receives a tenth of the light falling on the top.

Nothing in that calculation absorbs, so every per cent missing from the base has gone upward. The dark underside of a thick cloud is the same light as the dazzling white of its top seen from a satellite, and a cloud’s darkness from below is a measure of how bright it is from above. That arithmetic is one of the largest terms in the Earth’s energy balance: clouds cover about two thirds of the planet, and the fraction of sunlight they send back to space before it reaches the surface is set by the transmission law in the figure. Clouds supply roughly half of the Earth’s albedo of about 0.3, the number that sets the temperature a planet radiates at, and they send back something like 47 watts on every square metre of the planet averaged over day and night; a change of under ten per cent in that is a push on the climate as large as doubling carbon dioxide.

The base is grey rather than simply dim for a second reason the walk explains. The light emerging from the bottom of a thick slab has forgotten where the Sun is. It comes out nearly equally in all downward directions, so the underside of an overcast sky is evenly lit with no hint of the Sun’s position, and shadows on the ground below are soft or absent. The same forgetting is why the light’s polarisation is washed out after passing through fog: a photon that has lost its direction has lost its plane of polarisation along with it.

Why a trace of absorption matters #

The last number in the cloud figure is the most surprising, and it follows from counting scatterings.

A random walker needs a number of steps proportional to the square of the distance it has to cover — the law that makes light take so long to leave the Sun, whose interior is a ball of plasma some tens of billions of free paths in radius — and the photons confirm it: the scatterings per transmitted photon grow as the square of the slab’s effective thickness, with a measured exponent of 2.08 and 2.06. Measured against the bare thickness the same counts give 1.76 and 1.74, because each face adds a fixed extrapolation length that matters less the thicker the slab — a reminder that the thickness diffusion sees is not quite the thickness a ruler measures. Forward-scattering droplets take $1/(1 - g)$ times as many scatterings for each transport step, so a photon crossing sixteen transport lengths of cloud scatters about a thousand times.

A thousand scatterings multiply any absorption a thousandfold. Water absorbs almost nothing in the visible, but in the near infrared, at wavelengths where a droplet absorbs a tenth of a per cent of the light it scatters, a cloud with an optical depth of a hundred absorbs eighteen per cent of what falls on it. Weak absorption in a strongly scattering medium is amplified by the length of the walk, and satellite instruments exploit exactly this: comparing a cloud’s brightness at a wavelength where water absorbs slightly with one where it does not measures the droplets’ size, because larger droplets absorb more per scattering.

The amplification runs through every milky medium. A white paint contains pigment particles that scatter strongly and absorb weakly, and a small trace of absorbing contaminant greys it far more than its concentration suggests. Near-infrared light can penetrate several centimetres of living tissue because tissue scatters strongly and, between the absorption bands of water and haemoglobin, absorbs weakly; the light that emerges has walked, and clinical instruments that measure blood oxygen through the skin read how much of that walk was absorbed.

A cone that interference adds back #

The diffusion picture treats light as particles and discards phase, and there is one place where that is measurably wrong — and wrong in a way that connects this cloud to the electrons in a wire once more.

Consider light scattered back out of the top of a thick slab along a closed-looking path: in at one point, a sequence of scatterings, out at another. The same sequence of scatterings traversed in the opposite order is a second path with exactly the same length. In exactly the backward direction the two paths leave in phase and interfere constructively, doubling the intensity of that pair. Away from exact backscattering they drift out of phase. So a white slab lit by a laser sends back a narrow cone around the exact backward direction, up to twice as bright at its peak as the diffuse background, with an angular width of about the wavelength divided by the transport mean free path.

That enhanced backscattering was observed in suspensions of particles in 1984 and 1985, and it is the optical version of weak localisation in metals, where the same pair of time-reversed paths makes a disordered conductor’s resistance slightly higher than the classical Drude value. The cone is interference surviving a random walk, and it is the first correction diffusion misses. Its width measures the transport mean free path directly, which is how $\ell^*$ is measured in a material too dense to follow photons through.

Where the walk stops being a diffusion #

The slab must be several transport lengths thick. Diffusion describes the walk, and a walk needs room. Below a few transport lengths the transmission depends on the angular pattern and the diffusion curve overestimates it, which the photons show for every phase function at half a transport length.

The scatterers are independent. The mean free path is computed as though each particle scattered alone. In a densely packed medium — a powder, a white pigment at high concentration — the particles’ positions are correlated, their scattered waves interfere, and the transport length departs from what one particle would give. That is why a paint does not simply get whiter with more pigment.

The geometry is a plane-parallel slab. A real cloud has sides, gaps and towers, and light leaks out of its edges and finds its way down through its thin places. Averaging a broken cloud field as a uniform slab overestimates its reflection, a correction that matters in climate models and that the one-dimensional calculation cannot supply.

The phase function is Henyey and Greenstein’s. The photons here scatter according to a one-parameter pattern that has the right asymmetry and the wrong details; a real droplet’s pattern has a diffraction spike, a rainbow and a glory. In a thick cloud the similarity collapse makes the details irrelevant to the brightness, which is the point of the figure; for anything that depends on the angle of the emerging light, such as the halo of a thin cloud around the Sun, they are everything.

And phase has been discarded. Every photon here is a particle. The coherent backscattering cone, the speckle a laser makes on a white wall, and the whole question of whether disorder can trap light are wave effects the walk contains none of.

What the photon paths hide #

The paths figure draws seven walks projected onto the page, and a projection flattens three dimensions of wandering into two; the sideways excursions a photon makes perpendicular to the page are invisible, which makes every path look more tangled across and less tangled in depth than it is. And seven photons say nothing statistical: the reflected and transmitted fractions come from twenty thousand others that are not drawn.

The transmission and cloud figures report fractions and not where the light goes once it leaves. A cloud sends its reflected light back in all upward directions, brighter near the edges of its top for some geometries and near the centre for others, and it is that angular distribution — not the total — that a satellite looking straight down actually measures. The total is the number the energy balance needs; the angular distribution is the number an instrument sees, and the figures give only the first.

Still open: whether disorder can stop light #

For electrons, enough disorder does more than raise the resistance. Past a critical strength the interference of multiply scattered paths traps them entirely, and a disordered metal becomes an insulator — Anderson localisation. The backscattering cone is the first hint of that interference; localisation is the same effect grown until diffusion stops.

Whether light can be localised the same way in three dimensions is an open question with an uncomfortable history. Claims to have observed it in strongly scattering powders were later traced, at least in part, to absorption, which also makes transmission fall faster than one over the thickness and is hard to distinguish from localisation. Calculations for dense collections of point-like scatterers suggest that the vector nature of light and the near-field coupling between closely packed scatterers may prevent three-dimensional localisation of light altogether, while localisation of sound and of matter waves in three dimensions has been reported. What a disordered material would have to be, if anything, to stop light by interference alone — and how an experiment could tell that from absorption — has not been settled.

The habit worth carrying away is to ask whether a quantity is crossing a medium or walking through it. An exponential describes what has not yet been scattered; an inverse law describes what has forgotten where it was going, and in a thick enough medium the second is almost all of what arrives.