The invariant that is a count

Three earlier arguments have treated étendue as geometry. The brightness no lens can increase states it for a beam, the cone a fibre will accept applies it to a guide, and the cone light has to find to get out follows it across a boundary. In each, the quantity is an area multiplied by a solid angle and the argument is about rays not being able to crowd.

That account has a gap in it that is easy to miss, and it is a gap of principle rather than of accuracy. Ray optics contains no length. A ray is a line, a line has no width, and nothing in the geometry says how small an area times a solid angle is allowed to be. Taken at its word, the invariant permits an étendue of zero — a beam with no width and no angular spread — and forbids only its increase.

There is a floor, and putting it in changes what kind of quantity étendue is.

Why the cell has an area and how big it is #

Draw the state of a one-dimensional optical system as a point in a plane: where a ray crosses the aperture, and what direction it is going in — measured as $n\sin\theta$, which is the quantity Snell’s law preserves across a boundary and which therefore behaves like a momentum.

A bundle of rays occupies a region of that plane, and the region’s area is the one-dimensional étendue. Every lens, mirror and refracting surface moves the region about and shears it — a lens is a shear in that plane, a stretch of empty space is a different shear — and none of them changes its area. That is the invariant, stated in the form that makes its conservation a theorem rather than a rule: it is Liouville’s theorem, and the pair of coordinates is a conjugate pair in the mechanical sense.

The floor comes from what a field can do rather than from what a ray can do. A field confined to a width $\Delta x$ at an aperture cannot have an arbitrarily narrow angular spread: the two are Fourier transforms of each other, so $\Delta x \cdot \Delta(n\sin\theta) \gtrsim \lambda$. That is the diffraction of a slit, restated as a statement about area in the phase plane, and it says that the plane has a grain: no field occupies a patch of area much less than $\lambda$.

So the étendue divided by $\lambda$, in one dimension — or by $\lambda^2$ in two — is a number, and the conservation of a real quantity has become the conservation of an integer.

It is worth pausing on how much that changes. A conserved real number can be made as small as one likes and forbids only an increase. A conserved count cannot go below one, and one is a state with its own properties: a single-mode field has a definite phase across the whole aperture, it can interfere with itself, and it can be amplified without the amplifier having to guess which of several patterns it is amplifying. None of that is available in the geometrical statement, and all of it follows from the count having a floor.

The relation the floor comes from is the uncertainty relation in all but name. Position and transverse momentum for a photon are conjugate exactly as they are for anything else, the phase-space cell is Planck’s constant for a particle and a wavelength for a wave, and the two statements differ only by which units the momentum is written in.

What every optical element does to the rectangle #

The phase plane makes the conservation a piece of linear algebra, and it is worth carrying out because it turns a rule into a property of a matrix.

A stretch of empty space leaves a ray’s direction alone and changes its position by the distance travelled times that direction: $x \to x + Ld$, $d \to d$. That is a shear along the position axis. A thin lens leaves a ray’s position alone and changes its direction by the position divided by the focal length: $x \to x$, $d \to d - x/f$. That is a shear along the direction axis. Every optical system is a product of those two, in some order, and a shear has determinant one.

So the region’s area is conserved because every element in the chain is represented by a unimodular matrix, and the invariant is the statement $\det = 1$. The ray-transfer matrices an optical designer uses have that determinant built in and it is usually presented as a convenient check on the arithmetic; it is the whole of the theorem.

The picture also says what an optical system can and cannot do, in a form that is immediately readable. Shears can turn a tall thin rectangle into a wide flat one, which is a telescope — collecting a wide beam of small angular spread and delivering a narrow one of large spread. They can rotate it, which is what a sequence of lenses at particular spacings does, and they can shear it into a parallelogram, which is a beam with a correlation between position and direction — a focusing or diverging beam, which is exactly what a sheared rectangle looks like. What no product of shears can do is shrink the area, and that is the one thing everybody wants.

The fan of plane waves inside every beam is the same decomposition from the wave side: a beam’s angular spectrum is its extent along the direction axis of this plane, and its transverse profile is its extent along the other. The two are Fourier partners, which is why the area has a floor and why squeezing one stretches the other.

The count, for real instruments #

A square millimetre of light-emitting diode radiating into a hemisphere has about twenty million spatial modes. A single-mode fibre has one, carrying two polarisations. Between them are eight decades and most of applied optics.

The count explains a fact that the geometrical invariant states without illuminating. A lens cannot gather a diode’s light into a single beam — and the reason, in the counting language, is that twenty million modes have twenty million independent phases, and no linear passive optical element reduces a number of modes. It can rearrange them: a fibre bundle takes the light of many modes and delivers it in many modes, a light pipe scrambles them, a homogeniser mixes them. Nothing merges them, because merging would mean deciding which phase the merged field has, and the information was never there.

The same statement runs the other way for a laser. A laser is bright not because it puts out much power but because it puts almost all of its power into one mode, so its étendue is at the floor and its radiance is enormous. A one-watt laser diode and a one-watt bulb deliver the same energy; the laser delivers it in one mode and the bulb in ten million, and the ratio of radiances is the ratio of mode counts.

The fibre, where the same count has a different name #

Waveguide theory counts modes constantly and does it with a quantity called the V-number, which is defined without any reference to étendue and is treated in a separate literature.

Counting phase-space cells in a fibre gives $2\pi^2a^2\mathrm{NA}^2/\lambda^2$ modes, from the core’s area, its acceptance cone and the two polarisations. Waveguide theory writes $V^2/2$ with $V = 2\pi a\,\mathrm{NA}/\lambda$. Expand the second and it is the first, exactly, with nothing approximated.

So the V-number is the square root of a mode count, which is why nothing about $V$ has an obvious meaning until it is squared and why the expression for it looks arbitrary. And the single-mode condition $V < 2.405$ — a number that comes from a Bessel function’s first zero and looks like a piece of special-function trivia — is the statement that a fibre’s étendue has fallen to about one square wavelength. The two derivations share no step and arrive at the same place.

The practical consequence is the size of a telecommunications fibre’s core. Nine micrometres across is not a manufacturing convenience and not a compromise: it is the diameter at which, for a numerical aperture of 0.12 and a wavelength of 1,550 nanometres, the core’s étendue is about $\lambda^2$. Above it the fibre carries many patterns which travel at different speeds and smear a pulse; below it there is nothing to smear, because there is only one. Why the count comes out as a set of bound solutions rather than a continuum is a question about a potential well rather than about geometry, and it is the same discreteness a string acquires by having ends.

The single-mode end, and the number every laser is sold by #

At the floor of the count the invariant acquires a third name, in a third community.

A beam’s waist radius multiplied by its far-field half-divergence is $M^2\lambda/\pi$, and $M^2$ is the number a laser’s specification sheet carries as “beam quality”. It is bounded below by one, it is one only for a beam in a single transverse mode, and for anything else it counts how many modes there are in each transverse direction.

So beam quality is a mode count, étendue is a mode count, and the V-number is the square root of one. Three communities measure the same conserved quantity, in three unit systems, under three names, and cite each other hardly at all.

The consequence that made this matter commercially is at the bad end. A high-power diode bar has an $M^2$ of order thirty in one direction: it emits from a strip a centimetre long and a micrometre thick, so its étendue in the long direction is enormous and in the short direction is at the floor. No lens focuses it to a small spot, because the étendue in the long direction has to go somewhere. The trick that made diode lasers useful for cutting metal is to use the light to pump a different laser, or to feed it into a fibre — in both cases discarding the incoming mode structure and re-emitting into a new set of modes chosen by a resonator, which is allowed because emission is not a passive rearrangement.

What a count buys that a geometry could not #

It is worth collecting the statements that only exist once the invariant is a number of states, because they are why this is not a restatement of the geometrical arguments.

There is a smallest beam. The geometrical invariant forbids concentration beyond the source’s own brightness and says nothing about a spot size. The count says the smallest étendue is $\lambda^2$, which for a beam focused through a cone of half-angle $\theta$ gives a spot of order $\lambda/\sin\theta$ — the diffraction limit, arrived at from the invariant rather than from a separate argument about apertures.

Coherence becomes a property with a number. A field in one mode has one phase; a field in $N$ modes has $N$ phases with nothing relating them. The number of modes is therefore the number of independent things going on, and a source’s coherence area is the area over which a single mode extends.

And the capacity of an optical channel becomes countable. How much information a beam of light can carry depends on how many independent channels it has, and the number of spatial channels is the mode count. That is why a fibre’s information capacity is discussed in terms of its modes, why space-division multiplexing means deliberately using a multimode fibre’s modes as separate channels, and why the whole subject has an upper bound set by an area and a solid angle.

The same count, in the cavity that started thermodynamics #

There is a place where this count was done long before anybody wrote it as an étendue, and setting the two beside each other shows that they are one calculation.

Counting the electromagnetic modes in a box — standing waves fitting between its walls — gives $8\pi\nu^2/c^3$ per unit volume per unit frequency, and that count is the whole of the classical blackbody calculation: multiply it by $k_BT$ per mode and the result is the curve that would not come down, which is the ultraviolet catastrophe. The number of modes was never in doubt; what was wrong was what each one holds.

That density of modes is this essay’s count, per unit volume rather than per beam. A cavity of volume $V$ within a bandwidth $\Delta\nu$ has $8\pi\nu^2V\Delta\nu/c^3$ modes, and a beam of étendue $G$ observed for a time $\tau$ within the same bandwidth has $2G\Delta\nu\tau/\lambda^2$ — the same expression with the volume written as a beam’s cross-section times the distance light travels in $\tau$. They are the same counting of the same states, done once by a thermodynamicist and once by an illumination engineer.

Which explains the agreement the concentration bound found and could not account for. The concentration limit met a thermodynamic bound derived from an entirely different argument, and the reason is that both arguments are counting the modes of the electromagnetic field. A blackbody at temperature $T$ fills every mode available to it with the same average energy; a concentrator cannot raise a beam’s radiance above that of the source because it cannot reduce the number of modes the light is in. Two subjects, one count.

It also settles what a thermal source is, in the language of mode counting: a source that fills an enormous number of modes at a low occupancy, as against a laser, which fills one at an enormous occupancy. The two can carry the same power. Everything that distinguishes them — coherence, focusability, the radiance each delivers — is the mode count and nothing else.

A factor of order one, and no photons anywhere #

The cell has area $\lambda$ in one dimension and $\lambda^2$ in two, up to a factor of order one. Which factor depends on how “the width of a beam” is defined — a second moment, a half-power width, an aperture edge — and the conventions differ by up to a factor of two. Every count here is exact in its scaling and approximate in its constant, which is why the mode count of a fibre is quoted as $V^2/2$ by one text and $V^2/4$ by another.

The counting is of spatial modes only. A real beam also has a spectrum, and the number of temporal modes is the bandwidth multiplied by the observation time. The total count is the product, and an argument about spatial modes that ignores the temporal ones underestimates a thermal source’s mode count by an enormous factor.

The phase-space picture is a paraxial one. Away from small angles the conjugate variable is $n\sin\theta$ rather than $n\theta$, which the figures use, but the shears that represent free propagation and lenses stop being exact shears and the region’s area is conserved while its treatment as a rectangle is not.

And nothing here counts photons. A mode is a field pattern, and how many photons are in it is a separate question with its own statistics. A single-mode beam can be intense or faint; the étendue is the same, and everything about how the light behaves when detected is different.

Why the modes are not the little boxes drawn #

The hero figure draws a rectangle tiled into cells and the tiling is the part that is not literal. Modes are not little boxes sitting side by side in phase space; they are a complete orthogonal set of field patterns, each of which is spread over the whole aperture and the whole angular range, and what they occupy is a phase-space volume in the sense of a sum rule rather than a partition. The tiling gets the count right and the picture of what a mode looks like wrong, and the second error is the one that leads a reader to expect that a mode has a location.

The mode-count figure places five systems on a line and cannot show that the five numbers mean different things in practice. A telescope’s hundred and forty modes are ones an astronomer would like to reduce to one, and adaptive optics is the attempt; a diode’s twenty million are ones nobody wants to distinguish; a fibre’s six hundred are a nuisance for communication and a resource for imaging. The count is the same quantity and what it is worth depends entirely on what is being done.

And the beam figure draws a product and hides what a beam looks like. Two beams with the same $M^2$ can have quite different profiles — one nearly Gaussian and slightly aberrated, one a superposition of several clean high-order modes — and they behave differently when focused through a small aperture even though their invariant is identical. $M^2$ is one number extracted from a field, and a field is not one number.

Still open: what the count is worth as energy #

Every result here counts, and a count is not yet an energy. The invariant says a system has so many modes; it does not say what can be done with the light in them.

The connection is thermodynamic and it is the question that comes next. A mode is a degree of freedom of the electromagnetic field, so a count of modes is a count of degrees of freedom, and a count of degrees of freedom is what an entropy is made of. That means the étendue of a beam is, within constants, its entropy — which is why the concentration limit met a thermodynamic bound derived from an entirely different argument, and why the agreement was not a coincidence.

What that costs in available work is a separate calculation and it has a number in it that is not the Carnot efficiency. Sunlight arrives with the spectrum of a six-thousand-kelvin body diluted over a solid angle a hundred thousand times smaller than the sky, and the dilution is entropy: the light has been spread over many more modes than a body at that temperature radiating into the whole sky would fill. How much of its energy any converter can keep in the limit depends on that comparison, and working it out is the argument that follows.

The habit worth carrying away is the move this whole essay is. When a conserved quantity has no natural unit, look for the constant that gives it one. Étendue has the dimensions of an area times a solid angle and there is exactly one constant available with those dimensions — $\lambda^2$ — and dividing by it turns a quantity with no scale into a number with a floor. The same move turns an action into a count of quantum states by dividing by $h$, and a volume of phase space into a count of microstates, and in each case what appears is not a tidier unit but a new fact: counts have least values, and real quantities do not.