Optics

The work a diluted beam will not do

Sunlight at the top of the atmosphere has the spectrum of a body at 5,762 kelvin and the energy flux of one at 394. The mismatch is not an accident of units: the light has been spread over a hundred thousand times more modes than it left in, and that dilution is entropy. Run it through a heat engine unconcentrated and five per cent of it is available as work.

17 min read 6 figures What stays the same Where the model stops

Assumes: The invariant that is a count · The brightness no lens can increase

The concentration bound found that a concentrator cannot raise a receiver above the temperature of the source, and that the bound comes from thermodynamics rather than from optics. The mode count found why the two subjects were talking about the same thing: étendue divided by the square of the wavelength is a count of modes, and a count of modes is what an entropy is made of.

This essay asks the question that follows and that none of the others has answered. Light has energy and it has entropy. How much of the energy is available as work?

The answer is not what it is usually quoted as, and the reason is a single number that is easy to state and easy to leave out.

Two temperatures for the same sunlight. Sunlight described two ways, against how much it has been concentrated. The flat line is the temperature its spectrum belongs to — 5,762 K, the Sun's surface, which concentration does not change because a mirror does not alter a photon's energy. The rising curve is the temperature a blackbody would need in order to radiate the flux actually arriving: 394 K unconcentrated, 2213 K under a parabolic dish, and 5771 K at the geometric limit, where the two meet — computed here and checked against the Sun's own temperature, because a perfect concentrator reproduces the source's radiance and cannot exceed it. The gap between the two curves is the dilution, and dilution is entropy: the same energy spread over a hundred thousand times more directions occupies a hundred thousand times more modes. That is what a converter has to carry, and it is why the ceiling on solar conversion is not the Carnot efficiency between 5,762 K and 300 K. — Fig. 1 Sunlight described two ways, against how much it has been concentrated. The flat line is the temperature its spectrum belongs to — the Sun’s surface — which concentration does not change, because a mirror does not alter a photon’s energy. The rising curve is the temperature a blackbody would need to radiate the flux actually arriving: 394 K unconcentrated, and 5,771 K at the geometric limit, where the two meet.

Two temperatures, and the gap between them

Sunlight at the top of the atmosphere carries 1,361 watts per square metre. A blackbody radiating that much is at 394 kelvin — a warm oven.

Its spectrum belongs to a body at 5,762 kelvin. Every photon in it has the energy it had when it left the Sun, distributed as Planck’s law says a body at that temperature distributes them, and nothing along the way has changed a single one.

The two numbers disagree by a factor of fifteen, and what accounts for the disagreement is the solid angle. The Sun subtends about seven ten-thousandths of a steradian from here, against the $\pi$ steradians a surface radiating into a hemisphere fills, so the light arriving occupies about one part in four thousand of the directions it could — which is the same counting of modes that made the invariant an integer. In the language of the mode count, it occupies many more modes than its energy flux would fill, at a low occupancy in each, and spreading a fixed energy over more modes is exactly what raising its entropy means.

That is what “diluted” means and it is the whole of the difficulty. A beam of light whose flux corresponds to 394 K and whose spectrum corresponds to 5,762 K is not in equilibrium with anything, and a converter placed in it has to deal with both facts at once.

Concentration undoes the dilution and does nothing else. A mirror or a lens gathers light from a large area into a small one, raising the flux by that ratio, and the spectrum is unchanged throughout. So the rising curve in the figure is the concentration doing its work, and the two curves meet exactly at the geometric limit — where the concentrator has reproduced the source’s own radiance and can do no more, which is the bound concentration established and which the figure recovers by computing the crossing and finding the Sun’s temperature there. The same ceiling met from inside a material is the escape cone a photon has to find, where the invariant’s factor of $n^2$ decides what fraction of the light can leave at all.

The most any concentrator is allowed. The greatest concentration a receiver can be given, against the half-angle it accepts, both logarithmically, for a receiver in a medium of index 1. The upper curve is a three-dimensional concentrator, n²/sin²θ; the lower is a trough, which concentrates in one direction only, n/sin θ. The Sun's angular radius is 0.267°, so sunlight can be concentrated by at most 46165 times in a dish and 215 times in a trough — marked. Nothing about glass, mirrors, wavelength or aperture appears in either expression. The bound is thermodynamic: a receiver that accepts light out to θ also radiates out to θ, and the concentration at which what it emits balances what it absorbs is exactly where these curves are. — Fig. 2 The geometry that sets the ceiling: the concentration achievable against the rim angle of the collector, with the 46,000-fold limit at the top. The whole of the previous figure’s rising curve is this one raised to the quarter power, since a flux goes as the fourth power of a temperature — which is why a hundredfold concentration raises the flux temperature only threefold.

What an absorber is willing to do

The obvious converter is an absorber that gets hot and drives a heat engine, and following it through gives the number this whole essay is about.

An absorber under a concentration $C$ takes in $C \times 1361$ watts per square metre and radiates $\sigma T^4$ away. What is left is heat delivered at temperature $T$ , and a Carnot engine between $T$ and the surroundings converts a fraction $1 - T_0/T$ of it.

Those two requirements pull in opposite directions. A hot absorber runs a good engine and radiates away most of its input; a cool one keeps its input and runs a poor engine. Somewhere between is an optimum.

How much of sunlight's energy is available as work. The greatest fraction of incoming sunlight a converter can deliver as work, against how much the light has been concentrated first. The rising curve is an absorber that reaches whatever temperature balances what it takes in against what it radiates away, driving a Carnot engine between that temperature and the ambient — with the absorber temperature chosen by scanning: 5.4 per cent at 1 suns, with the absorber at 348 K; 29.6 per cent at 10 suns, with the absorber at 504 K; 52.1 per cent at 100 suns, with the absorber at 761 K; 68.6 per cent at 1.0e+3 suns, with the absorber at 1174 K; 79.7 per cent at 1.0e+4 suns, with the absorber at 1830 K; 84.9 per cent at 4.6e+4 suns, with the absorber at 2468 K. The two flat lines are the Carnot efficiency between the Sun's surface and the ambient, 94.8 per cent, and the Landsberg limit for an ideal converter of undiluted radiation, 93.1. The first of those is quoted as the solar limit and is not one: it assumes the light arrives undiluted. Unconcentrated sunlight through a heat engine is worth five per cent, and the eighty points between that and the ceiling are bought by reconcentrating the light — which is to say by undoing the entropy of its dilution, and which no amount of engine design substitutes for. — Fig. 3 The greatest fraction of incoming sunlight available as work, against how much the light has been concentrated first, with the absorber temperature chosen by scanning at each concentration. Unconcentrated it is 5.4 per cent with the absorber at 348 K; at a thousand suns it is 68.6 per cent at 1,174 K; at the geometric limit 84.9 per cent at 2,468 K. The two flat lines above are the Landsberg limit and the Carnot efficiency between the Sun’s surface and the ambient.

Five point four per cent. That is what unconcentrated sunlight is worth to a heat engine, and it is the number that makes the point. The engine is ideal, the absorber is perfect, nothing is lost to convection or to imperfect optics, and ninety-five per cent of the energy is unavailable.

It is unavailable because the absorber cannot get hot. At 348 kelvin it is already radiating away most of what it receives, and any attempt to run it hotter loses more to re-emission than the improved Carnot factor recovers. The absorber is in the position of a body trying to reach 5,762 kelvin by standing in light that a 394-kelvin body could supply.

Concentrate the light and the absorber can be hot without radiating its input away, and the efficiency climbs the curve. The eighty points between five per cent and eighty-five are bought entirely by reconcentrating the light — by undoing the entropy of the dilution — and no improvement to the engine substitutes for any of it.

Why a cell is allowed to beat the engine

The cell in the next section reaches thirty per cent in light an ideal heat engine can only get five per cent out of, and that comparison looks at first like a violation of something. It is not, and the reason is worth having before the calculation.

A heat engine works by thermalising: the energy arrives, it becomes heat in a body at a definite temperature, and from that point on nothing remembers that it came from photons of five thousand kelvin. The entropy has been increased at the moment of absorption — a 5,762 K photon absorbed by a 348 K absorber creates entropy equal to its energy divided by 348 — and a Carnot engine downstream cannot recover what was lost upstream.

A photovoltaic cell does not thermalise. A photon lifts one electron across a gap and the electron is taken out through a wire before it has given its energy to the lattice, so the conversion happens at the photon’s own energy rather than at the absorber’s temperature. That is why the Carnot factor between the absorber and the ambient never appears in the detailed-balance calculation, and why the arrival of light in individual lumps is not a curiosity here but the mechanism.

What the cell pays instead is different and is the whole of its ceiling: it has one threshold, so it throws away the below-gap light and the excess energy of everything above. Two converters, two ways of wasting the same beam, and neither is the ceiling on every engine in the form that is usually quoted.

The number that is quoted and is wrong

The Carnot efficiency between 5,762 K and 300 K is 94.8 per cent, and it is widely quoted as the thermodynamic ceiling on solar conversion. It is not one, and the reason is exactly the dilution.

Carnot’s result is about heat drawn from a reservoir at the hot temperature. A reservoir at 5,762 K is a body which, placed against the converter, would deliver radiation at that body’s own radiance — an undiluted 6.3 × 10⁷ watts per square metre, not 1,361. Sunlight at Earth is not that reservoir; it is that reservoir seen through a very small hole, and a small hole is exactly what a dilution is.

The correct ceiling for undiluted blackbody radiation is a little below Carnot’s: 93.1 per cent, the Landsberg limit, the difference being that radiation carries its own entropy and a converter that absorbs it must either re-radiate or account for that entropy somewhere. And the correct ceiling for diluted radiation is the curve, which is below Landsberg everywhere and equals it nowhere.

The habit this illustrates is worth stating. A Carnot efficiency quoted between two temperatures is a claim that a reservoir exists at each of them. Where the hot “reservoir” is a beam rather than a body, the temperature of its spectrum is not the temperature of a reservoir, and using it overstates what is available — here by a factor of eighteen.

The quarter of the limit a focusing dish gets. Concentration against rim angle for a paraboloidal mirror aimed at the Sun, against the thermodynamic ceiling of 46165. The curve is sin²φ·cos²(φ+θ)/sin²θ, which is what the Sun's image on the focal plane of a paraboloid is worth. It rises, peaks at 11434 at a rim angle of 44.87° — located by scanning the drawn curve, and displaced from a round 45° by the Sun's own angular radius — and falls back to nothing at 90°. The best of them is 25 per cent of what is allowed. The shortfall is not a manufacturing tolerance and no figuring of the mirror recovers it: a dish that makes an image wastes étendue, because the edge of the image is formed by rays arriving at the wrong angle to be useful. Concentrators that reach the ceiling give up imaging altogether, and the compound parabolic funnel is one. — Fig. 4 And the optics the curve assumes: the concentration a paraboloidal dish reaches against its rim angle, with the geometric limit at the top. Reaching the right-hand end of the efficiency curve requires a rim angle close to ninety degrees — a collector that nearly surrounds its receiver — and every practical dish sits well to the left of it, which is why the achievable efficiency is read from the middle of the previous figure rather than its end.

What a cell does instead

A photovoltaic cell is not a heat engine and does not work by getting hot, which is why it beats the unconcentrated engine by a factor of six. It has its own ceiling, and computing it shows exactly what kind of limit it is.

What one threshold costs. The greatest fraction of unconcentrated sunlight a solar cell with a single band gap can deliver, against where that gap is, computed by detailed balance from a 5,762 K Planck spectrum diluted to 1,361 W/m² and a 300 K ambient. Every photon below the gap is lost entirely and every photon above it loses its excess energy as heat, and the cell must also emit, because a body that absorbs above its gap radiates above its gap. The curve peaks at 30.5 per cent at a gap of 1.25 eV; germanium at 0.67 eV reaches 21.1, silicon at 1.12 eV reaches 30.0, gallium arsenide at 1.42 eV reaches 30.0, cadmium telluride at 1.5 eV reaches 29.4. Set that against the thermodynamic ceiling for the same light — five per cent for an unconcentrated heat engine, eighty-five for a fully concentrated one — and the comparison says something specific. A single-gap cell is far better than a heat engine at the same concentration and far worse than thermodynamics allows, and the whole of the difference is the single threshold: one gap cannot be right for photons of every energy at once. — Fig. 5 The greatest fraction of unconcentrated sunlight a cell with a single band gap can deliver, against where the gap is, computed by detailed balance from a 5,762 K Planck spectrum diluted to the solar constant. The curve peaks at 30.5 per cent at 1.25 eV. Silicon at 1.12 eV reaches 30.0 and gallium arsenide at 1.42 reaches 30.0, which is why the choice between them was never made on this ground.

The calculation has three ingredients and no fitted parameters. Every photon above the gap makes one electron–hole pair and every photon below it is transmitted. Every pair delivers the gap energy and no more, so a photon of twice the gap wastes half of itself as heat. And the cell, being a body that absorbs above its gap, must radiate above its gap at the ambient temperature — which is its saturation current, and which sets how much voltage can be extracted before the emission eats the current. That last requirement is Kirchhoff’s law applied to a device with a threshold, and it is what makes the calculation a balance rather than a tally of losses.

The result is thirty per cent, and the shape of the curve says where it comes from. A small gap collects nearly every photon and pays for each of them at a low voltage; a large gap pays well for the few photons it collects and misses most of the spectrum. The peak is the compromise, and it is broad — anything between one and 1.6 electronvolts is within two points of the best, which is why silicon was never a bad choice and why the record cells of quite different materials sit so close together.

Now set that against the thermodynamic numbers. The cell reaches thirty per cent where an ideal heat engine in the same light reaches five, and where thermodynamics permits eighty-five if the light is concentrated. A single-gap cell is six times better than a heat engine and three times worse than the second law, and the whole of the second gap is the single threshold.

That diagnosis is testable, and it has been tested by removing the restriction. A stack of cells with different gaps splits the spectrum, so each photon meets a threshold closer to its own energy — and the gaps themselves are a property of how close the atoms in the crystal sit, which is what makes the stack an engineering choice rather than a wish: two junctions reach 42 per cent in principle, three 49, and an infinite stack approaches the thermodynamic limit. Measured multi-junction cells under concentration exceed 47 per cent, which is above anything a single gap allows at any concentration and is the direct evidence that the thirty per cent was never a statement about thermodynamics.

Fig. 6 And the optics such a cell needs: a compound parabolic concentrator accepting light within four degrees, which reaches the geometric limit for that acceptance. A concentrator of this kind trades acceptance angle for concentration exactly as the invariant requires, which is why a high-concentration system must track the Sun to a fraction of a degree and why the cheap fixed collectors that do not track are limited to a few suns.

The same argument, run the other way

Dilution is a disadvantage when the light is arriving and an advantage when it is leaving, and the case where it leaves is worth a section because it is the one people meet without recognising it.

Point a surface at a clear night sky. It radiates into a set of modes that are very nearly empty — space at 2.7 kelvin, seen through the atmospheric window between eight and thirteen micrometres, where the air is largely transparent. What comes back from those directions is almost nothing, so the surface loses energy and cools, and it goes on cooling until conduction and convection from the surrounding air balance the loss. With good enough insulation from the air it reaches several degrees below ambient, which is how ice was made in shallow trays in warm climates for centuries before refrigeration.

The entropy accounting is the first figure read backwards. The surface is at 290 kelvin and the modes it is radiating into are at 2.7, so the emission is enormously undiluted relative to what it would be into an equilibrium field — it is pouring energy into modes with nothing in them. Exactly as an absorber cannot exceed its source’s radiance, an emitter cannot fall below its sink’s, and the sink here is the coldest thing there is.

What makes it hard in practice is the same thing that makes concentration hard: the window is narrow. Outside eight to thirteen micrometres the atmosphere radiates back at nearly its own temperature, so a surface that emits broadly gains as much as it loses. A surface that emits strongly only in the window can cool below ambient in full sunlight, which was demonstrated in 2014 with a multilayer coating that also reflects the solar spectrum, and which is the same spectrally selective trick the solar absorber uses with the two roles exchanged.

What a leaf gets

One converter has been running on this light for three billion years, and its number is instructive because it is small for reasons that are mostly on the list above.

Photosynthesis uses photons between about four hundred and seven hundred nanometres — roughly forty-five per cent of the solar energy, the rest being infrared that no pigment absorbs usefully. Each usable photon is processed at the energy of the reaction centre’s threshold, about 1.8 electronvolts, so a blue photon of 3 eV loses a third of itself immediately: the single-threshold loss, in a system with two thresholds rather than one. Fixing one molecule of carbon dioxide takes eight to ten photons against a thermodynamic minimum of about four, and the losses of respiration take roughly a third of what is fixed.

Multiplying those gives a theoretical ceiling near eleven per cent for C3 plants and a little higher for C4. Measured field crops over a growing season reach one to two, and the best short-term laboratory values around five. The gap between eleven and two is not thermodynamics either — it is light saturation at high intensity, incomplete canopy cover, water and nitrogen limitation, and the plant spending energy on things other than mass.

The comparison worth drawing is that the shape of the leaf’s loss budget is the cell’s: a spectral window, a threshold, and a quantum requirement, none of which is a Carnot factor. Living converters and semiconductor ones are in the same class and the heat engine is in another, which is the classification this essay has been building towards.

What a blackbody Sun and a perfect absorber assume

Every entropy here is a count of ways. The four-thirds in the Landsberg limit, the dilution, the mode counting behind the invariant — all of them rest on entropy being a logarithm of a number of arrangements, and on the arrangements of a radiation field being its modes and their occupations. A reader who wants the radiation entropy derived rather than used will find that the derivation is the same counting that produced the blackbody spectrum with one more step taken.

The Sun is treated as a blackbody at a single temperature. It is not: the emergent spectrum is shaped by absorption in its atmosphere, it departs from Planck’s law substantially in the ultraviolet, and the effective temperature depends on what is being fitted. Every number here would move by a few per cent using a measured spectrum, and the detailed-balance peak would move by rather more.

The atmosphere is left out entirely. The 1,361 watts per square metre is the value above it; at the ground it is about a thousand on a clear day, with absorption bands cut out of the spectrum and a diffuse component that has been scattered into directions a concentrator cannot use. That last point matters more than the loss: diffuse light has a far larger étendue than direct light, so it cannot be concentrated at all, and a concentrating system in a hazy climate loses the diffuse fraction outright while a flat plate does not.

The absorber is taken as a perfect blackbody at every wavelength. A real receiver is coated to absorb strongly in the visible and radiate weakly in the infrared, which is a spectrally selective surface and which raises the optimum temperature and the efficiency substantially at low concentration. That is not a violation of anything — it is a device with a threshold in it, which is the same trick the cell uses.

And the detailed-balance calculation assumes every absorbed photon makes exactly one pair. A photon of twice the gap could in principle make two, and materials in which it does — by carrier multiplication — are an active subject; so are cells that use the carriers before they have given their excess energy to the lattice. Both would raise the single-gap ceiling, and neither has produced a working device near it.

Two beams whose only difference is a set of directions

The first figure draws two temperatures and cannot show what is different about the light at the two ends of the axis. Concentrated and unconcentrated sunlight have identical spectra, identical photon energies and identical polarisation; what differs is the range of directions the photons arrive from, which is not a quantity a plot against concentration can display and is the entire subject.

The efficiency curve draws an optimum at each concentration and hides how flat it is. Near the peak the efficiency changes by under a point for a hundred-kelvin change in absorber temperature, which is what makes such systems controllable at all and is invisible in a plot of the optimum alone.

And the detailed-balance curve draws an efficiency against a gap, which presents the choice as a one-dimensional one. A real material has a gap and also an absorption strength, a carrier lifetime, a surface recombination velocity and a cost, and the reason silicon dominates is nothing to do with the thirty per cent it shares with a dozen other semiconductors.

Still open: whether the entropy of a beam has a single right definition

Everything above treats radiation entropy operationally: what a converter can extract, computed from a balance. The more basic question — what the entropy of an arbitrary beam of light is — is less settled than the confident use of the word suggests.

For blackbody radiation the answer is standard and old: the entropy flux is four-thirds of the energy flux divided by the temperature, which is where the four-thirds in the Landsberg limit comes from. For diluted blackbody radiation there is an accepted expression too. For a general beam — with an arbitrary spectrum, an arbitrary angular distribution and an arbitrary degree of coherence — several definitions are in use, they disagree, and the disagreement is not merely about conventions: they assign different available work to the same beam.

The difficulty is the one the mode count named. A beam’s entropy depends on how many modes it occupies and how the photons are distributed among them, and “how many modes” is only unambiguous for a beam with a definite étendue and bandwidth. Partially coherent light, laser light with a structured mode content, and light that has been spectrally filtered all have entropies that depend on what is treated as known about them — which is the usual situation for an entropy and is unusually visible here.

The habit worth carrying away is the one the two curves of the first figure make together. When a beam is described by a temperature, ask which of its properties that temperature was fitted to. A spectrum has a temperature and a flux has a temperature, and for anything that has travelled they are different numbers. Quoting either alone is a description of the light; quoting the pair is a description of what can be done with it, and the ratio between them is the entropy that got in the way.

Part 5 of 6

This essay is one argument about Etendue. The others:

$The étendue, tiled. The phase space of a one-dimensional optical system: position across a twenty-micrometre aperture, against the optical direction cosine n·sinθ, for a system accepting ±0.1. The shaded rectangle is what the beam occupies, and its area — 4 micrometre-radians — is the one-dimensional étendue, the quantity no arrangement of lenses can reduce. Divided by a wavelength of 500 nanometres it is 8, and the rectangle is tiled here with exactly 8 cells of area one wavelength each. That is the whole content of the count: a cell of phase-space area λ is the smallest patch a field can be confined to, because squeezing it in position spreads it in direction by the same relation that gives a slit its diffraction pattern, so an étendue is a number of modes and its conservation is the conservation of a count. The cells are drawn four wide and two high, which is arbitrary: only a cell's area is fixed, not its shape — a beam may be confined tightly in position and loosely in angle or the other way round, and the trade is what an optical system is for.$ The invariant that is a count Part 4 — Étendue is an area times a solid angle, and ray optics gives it no floor — nothing in a ray has a size. Divide it by the square of the wavelength and it becomes a number of modes, its conservation becomes the conservation of a count, and the count has a least value of one. That is where the ray bound hands over to diffraction, and the handover is the same number written two ways.