The cone light has to find to get out

The first light-emitting diodes were made from gallium arsenide and emitted infrared light, and for years their efficiency was baffling. The semiconductor was good: nearly every electron that crossed the junction recombined by giving off a photon. The devices gave off a few per cent of that light. Nothing was absorbing the rest, or not at first. It was simply not getting out.

Around the same time, people making silicon solar cells were finding that etching the surface of a wafer into a field of tiny pyramids made the cell better, by more than the reduced reflection could explain. The light the pyramids let in was absorbed as if the wafer had been many times thicker.

Those are the same effect seen from the two sides of one boundary. The brightness no lens can increase introduced the quantity that decides both — étendue, the product of area and solid angle — and noted that it carries a factor of the refractive index squared. The cone a fibre will accept used it to fix how much light a guide can take in. This essay is about that factor of $n^2$, and about what it does to light on its way out of a dense material rather than on its way in.

The n² in the invariant #

Snell’s law bends a ray leaving a dense material away from the normal, so a narrow cone of directions inside fans out into a wide one outside. The widest possible fan is the whole hemisphere, and the cone inside that fills it has the half-angle at which the outgoing ray is grazing the surface: the critical angle, $\arcsin(1/n)$. The angle past which light cannot leave is that same angle met from the point of view of a ray that fails.

The figure checks something stronger than Snell’s law ray by ray. It measures the cone inside by its projected solid angle — the solid angle weighted by the cosine of the angle to the normal, which is the measure of how much light a cone of directions carries through a surface — and finds $\pi\sin^2\theta_c = \pi/n^2$. The hemisphere outside has projected solid angle $\pi$. So $n^2$ times the inside cone equals the outside hemisphere, with no remainder.

That is the optical invariant written across a boundary: area times projected solid angle times $n^2$ is the same on both sides. Its meaning is that light in a dense medium is concentrated. A given flux crossing the face inside occupies fewer directions, and so it has a radiance $n^2$ times higher; outside it spreads over more directions at a lower radiance. Nothing has been gained or lost at the face — the same light has been rearranged between how bright it is and how many directions it occupies.

Two per cent of the light #

Now make the light inside, in every direction equally, as a recombining electron in a semiconductor does. Only the part travelling within the escape cone of a given face can leave through it, and the fraction of all directions inside a cone of half-angle $\theta_c$ is $(1 - \cos\theta_c)/2$. For a large index the critical angle is small, the cosine is close to $1 - 1/2n^2$, and the fraction becomes $1/4n^2$.

For gallium arsenide, with an index near 3.6, that is two per cent through the top face, and the reflection of the light that does reach the face inside the cone takes it down to 1.3. A cube of the material has six faces and so six cones, which is still only about eight per cent. The early infrared diodes were emitting at almost exactly the level this arithmetic allows.

The light that does not escape is not destroyed. It is totally reflected and goes on travelling inside the chip, and what happens to it next decides everything about the device. If it is absorbed by the substrate, the contacts or the material itself, it is gone. If something changes its direction before that happens, it has another chance to find a cone. In a very pure semiconductor the absorbed light can even be re-emitted — an absorbed photon lifts an electron across the gap between the bands, and the pair recombines and gives back a single photon in a new random direction — so trapped light is recycled rather than lost, which is part of why the best gallium arsenide devices do so well.

Moving the cone into a dome #

The first thing that helps is changing what the chip’s face looks out into. The critical angle is set by the ratio of the two indices, not by the chip’s index alone, so covering the chip with a transparent material of higher index than air widens the cone.

A gallium nitride chip under epoxy sends two and a half times as much light across its face as the same chip in air. That looks like a way round the invariant, and it is not. The light that crossed into the epoxy is now inside the epoxy, at the radiance appropriate to an index of 1.5, and it still has to get from the epoxy into the air. A flat layer of epoxy would simply move the problem: the light would meet the epoxy’s top surface outside its own escape cone and be reflected.

What makes the covering work is its shape. Make it a dome much larger than the chip, and every ray leaving the chip meets the dome’s surface close to the normal, whatever direction it started in, so none of it is outside the dome’s escape cone. The chip is a small bright source near the centre of a large sphere, and the dome’s surface is a large area seen from it through a small solid angle. The product of area and solid angle crossing each surface in turn is the same, which is the invariant being obeyed at both faces rather than evaded at one.

The same idea, pushed further, is behind most of what has happened to light-emitting diodes since. Chips are shaped so that light bouncing inside meets some face near normal incidence; surfaces are roughened so that trapped light is scattered into new directions on every bounce; mirrors are put under the chip so that light travelling downward is sent back up for another try. Every one of those is a way of giving trapped light more chances to find a cone, and none of them changes the size of the cone.

The comparison with a chip of higher index shows which part of the problem a covering solves. For a large chip index the escape fraction is close to $n_\text{out}^2/4n_\text{chip}^2$, and a covering multiplies it by the square of its own index — 2.25 for epoxy — whatever the chip is made of: the figure finds a factor of 2.3 for the index-3.5 chip and 2.4 for gallium nitride, the difference being the approximation’s error at the lower index. What the covering cannot do is make up the difference between the chips. Under the same epoxy the higher-index chip still sends less than half as much light across its face, because its own cone is narrower by the square of the ratio of the indices.

That is why red and infrared diodes made from high-index arsenides and phosphides had a harder extraction problem than the gallium nitride diodes that came after them, and why so much of their engineering went into the shaping described above. Red diodes cut into a truncated inverted pyramid, so that light trapped by the top face meets the sloping sides close to normal incidence, roughly doubled their external efficiency at the end of the 1990s without changing either index. The invariant did not move. The geometry gave the trapped light more cones to find.

The same cone keeps light in #

Now turn the boundary round. Light from outside enters a slab of index $n$ and is refracted into a narrow cone, so it crosses the slab almost straight and, if the slab is weakly absorbing, most of it goes straight through. Put a mirror behind the slab and it crosses twice. Either way, a thin slab of a weak absorber absorbs little.

Roughen the front surface so that it scatters the light entering it into every direction inside, and the situation changes completely. The light is now travelling at all angles, most of them oblique, and when it returns to the front surface it meets the same problem the light inside a diode meets: it can leave only if it is within the escape cone. For light spread evenly over directions inside, the fraction in the cone at each arrival is not $(1-\cos\theta_c)/2$ but $\sin^2\theta_c = 1/n^2$, because the light arriving at a surface is weighted by the cosine of its angle. The rest is reflected back for another round trip.

Counting round trips gives the result. Each one crosses the slab twice at an average obliquity that makes the path twice the thickness per crossing, so a round trip absorbs a fraction $4\alpha d$ for weak absorption; and each one ends with a chance $1/n^2$ of escape. The light makes about $n^2$ round trips before leaving, and the absorption is $4n^2$ times the single-pass value. For silicon, with an index of 3.57, that is a factor of 51. The figure does not assume it: it computes the absorptance of the randomised slab in full and recovers $4n^2$ from its slope at weak absorption.

The practical consequence is large. Silicon absorbs weakly near its band edge — at 1100 nanometres a 100-micrometre wafer takes out less than seven per cent of the light in a double pass — and a randomising surface takes that to two-thirds. That is why silicon solar cells are textured, and why a cell a fraction of a millimetre thick can collect infrared light that would need a centimetre of flat silicon. The result was worked out by Eli Yablonovitch in 1982, and the $4n^2$ is often called the Yablonovitch limit.

The sky from under water #

The factor of $n^2$ can be checked without a laboratory. A swimmer looking up from under a calm surface sees the whole sky compressed into a circle about 97° across — Snell’s window, the escape cone of water seen from the inside — and the angle past which light cannot leave explains its size. It says nothing about how bright the circle is.

The invariant does. Radiance divided by $n^2$ is conserved across the surface, apart from the few per cent reflected there, so the sky seen from under water has 1.78 times the radiance of the same sky seen from above: water’s index, 1.33, squared. The light that was spread over a hemisphere of directions has been squeezed into a cone of 48.6° half-angle, and the same flux carried in fewer directions is brighter per direction by exactly the ratio of the projected solid angles. Outside the window the surface is a mirror showing the dim bottom, which makes the contrast larger still.

An overhead Sun makes the same point for a single source. Seen from below, its disc is shrunk by refraction by a factor of 1.33 in diameter, so it covers 1.78 times less sky, and its radiance is 1.78 times greater. The two cancel exactly in the heat it delivers to a surface under the water, which is the invariant stated for one star rather than for a whole sky: nothing has been amplified, and the light has only been rearranged between how bright it is and how many directions it fills.

Why it is a limit and not a trick #

The same factor appearing on both sides of the boundary is not a coincidence, and the cleanest way to see it is the argument that makes it a limit.

Put the textured slab in a closed box at a uniform temperature, full of thermal radiation. In equilibrium the slab must absorb exactly as much radiation as it emits — the balance of a surface that glows at the rate it absorbs — and the radiation inside it must be in equilibrium with the radiation outside. Thermal radiation inside a medium of index $n$ has $n^2$ times the energy per unit volume of the radiation outside it, which is the invariant again. For the inside to be $n^2$ times as intense while the flow out balances the flow in, the light inside must be escaping at $1/n^2$ of the rate at which it arrives at the surface — exactly the escape-cone fraction for randomised light.

So the trapping factor is not a property of pyramids or of any particular roughness. It is what equilibrium requires of any slab that randomises the light in it, and a surface that randomised perfectly would reach it and one that randomised imperfectly would fall short. Doing better than $4n^2$ for light arriving from all directions over a broad range of wavelengths would mean light inside the slab reaching an intensity above what equilibrium with the outside allows, which is a brightness no lens can increase run in the other direction.

The limit is on a randomising slab over all directions and all wavelengths, and that qualifier is where it can be beaten. A structure that confines light to fewer directions — accepting it only from a narrow cone, or trapping it in guided modes of a periodic stack over a narrow band of wavelengths — can exceed $4n^2$ inside that restricted range. Structures of that kind have been designed and demonstrated for thin films, and they trade breadth of acceptance for depth of trapping in exactly the proportion the invariant permits.

A concentrator that changes the colour #

The escape cone has one more use, and it is the case that looks most like breaking the invariant. A sheet of plastic doped with a dye absorbs sunlight falling on its face and re-emits it, in every direction, at a longer wavelength. The re-emitted light is trapped in the sheet by total internal reflection — for an index of 1.5 the fraction outside both escape cones is $\sqrt{1 - 1/n^2}$, three-quarters of it — and is guided to the sheet’s edges, where solar cells collect it.

That is a luminescent solar concentrator, and it concentrates diffuse light, which no arrangement of lenses and mirrors can do, because a lens cannot compress the étendue of light arriving from the whole sky. The concentrator escapes the bound because the dye changes the light’s colour. The emitted light is at a lower photon energy than the absorbed light, the difference goes into heat, and that heat carries away the entropy that concentration would otherwise have had to violate. The larger the shift in wavelength, the more concentration thermodynamics permits — which ties the geometric invariant of the previous figures directly to the second law, through a number that a chemist chooses when picking the dye.

Where this stops being right #

Everything here is ray optics. A slab a few wavelengths thick does not have well-defined rays inside it; the light is in a finite number of modes, as the cone a fibre will accept found for a guide, and the counting of directions becomes a counting of modes. The $4n^2$ limit survives for thick slabs and is replaced by a mode count for thin films, where it can be exceeded in narrow bands.

The randomisation is taken as perfect. A real textured surface scatters light into a distribution that is not exactly uniform, and some light makes it out through the escape cone on its first return because it was scattered into it. Real textured cells reach a large fraction of the $4n^2$ enhancement, not all of it.

The emission is taken as isotropic. Light from a thin quantum well in a modern light-emitting diode is not emitted equally in all directions, and a well designed to emit more steeply towards the face than along the plane puts more of its light inside the cone to begin with.

And the mirrors and faces are perfect. Every real mirror absorbs a few per cent, every real surface carries an absorbing contact or a coating, and a light ray that must bounce fifty times is exposed to every one of those losses fifty times. The trapping factor is a ceiling that a cell reaches only as closely as its other losses allow.

What the pictures cannot show #

The escape-cone figure draws rays, and every ray it draws is travelling in the plane of the page. Real light inside a chip travels in three dimensions, and the cone is a cone — which is why the fraction that escapes involves $1 - \cos\theta_c$ rather than an angle ratio, and why a picture of a fan of rays understates how small the escaping share is.

Nor do the absorption curves show a path. The trapped light in a textured slab has no single path length; it has a distribution of them, close to exponential, in which most light leaves after a few passes and a small fraction stays for hundreds. The factor of 51 is the mean of that distribution, and what a thin cell actually absorbs depends on how the long tail of it meets the absorption.

Still open: how much of the light’s energy the invariant lets anything keep #

Every result here counted light, not what can be done with it. A diode’s light that escapes is useful light; a solar cell’s light that is trapped is absorbed, but absorption is not yet work. How much work can be taken from a beam of light is a question about its entropy as well as its energy, and the étendue is, within constants, that entropy.

Sunlight arriving at the Earth has the spectrum of a body at nearly six thousand kelvin spread over a solid angle a hundred thousand times smaller than the sky, and that dilution is part of its entropy. How much of its energy a converter could keep in the limit — a heat engine, a solar cell, anything — depends on how the light’s étendue compares with the converter’s, and the answer is not the Carnot efficiency between the Sun’s surface and the Earth’s. Working out what it is, and how much concentration changes it, is the question the invariant leaves when it is asked about energy rather than brightness.

The habit worth carrying away is the one both halves of this essay turned on. When light seems to be lost at a boundary, ask what fraction of its directions the boundary can pass. The early diodes and the flat solar cells were not failing to generate or absorb light; they were failing to move it between a dense material and a thin one, and a factor of $n^2$ — which the invariant had written down all along — was the whole of the difference.