The same cone, and a different arrival

The cone a fibre will accept is fixed by two refractive indices and nothing else, and multiplying it by the core’s area gives the étendue — all the light the guide will ever carry. That is a complete answer to the question of how much gets in.

It is silent on a second question that turns out to matter as much. Of the light that gets in, does it all arrive at the same time?

For a step-index fibre the answer is emphatically no. A ray bouncing at the steepest angle the guide accepts travels a longer path than one going straight down the axis, by a factor of one over the cosine of that angle, and for a numerical aperture of 0.21 in a core of index 1.47 that is one per cent. Over a kilometre, one per cent of five microseconds is fifty nanoseconds, and a pulse fifty nanoseconds wide cannot be sent more often than about ten million times a second. Note which quantity this is and which it is not: the fibre still carries every photon it accepted, so nothing about the count of modes has changed, and the loss is entirely in when they arrive. Early multimode fibre was limited to a few tens of megahertz over a kilometre by this alone, and the loss was not the problem.

Making the long way round the fast way round #

The fix is to notice what is being compared. A ray far from the axis has further to go. Give it somewhere faster to go, by making the index fall with radius, and the two effects can be set against each other.

A ray in such a medium does not bounce. It curves continuously, bending back towards the axis wherever it strays, and the path is a sinusoid — the same continuous bending a gradual change of medium produces instead of a reflection, applied along the whole length of the guide rather than at a boundary. What makes the parabola the right profile is that the restoring effect is proportional to the displacement, exactly as it is for any minimum examined closely enough — so the ray is a harmonic oscillator in the radial coordinate, and a harmonic oscillator’s period does not depend on its amplitude.

That is the whole of the result, and it is worth stating in the form that makes it surprising. The pitch is the same for every ray, so every ray returns to the axis at the same place, and a pulse that entered as a set of rays at different angles reassembles itself every millimetre or so along the fibre. The equalisation of arrival times is a consequence of isochronism, and isochronism is a property the parabola has and no other profile does.

The idea is old — it is the graded-index lens, worked out in the nineteenth century and made practical in fibre in the early 1970s — and the reason it took so long is the second figure.

It is also worth noticing what the parabolic core has in common with a quite different object. A ray in it obeys the same equation a mass on a spring obeys, so the fibre is a lens that is being applied continuously rather than at a surface, and the sinusoid is the trajectory rather than the wave. The fan of plane waves inside every beam is the wave-side description of the same thing, and in that language the graded core is a medium whose modes have equally spaced propagation constants — which is the isochronism restated, and the reason the pulse reassembles.

How sharp the optimum is #

The cancellation is exact only to first order in the index contrast. At second order there is a residual, and it can be reduced by leaning the profile slightly away from a parabola.

Two numbers come out of that and both are worth having.

The first is the improvement: a factor of five hundred and seventy. A fibre limited to ten megahertz over a kilometre becomes one limited to several gigahertz, on the same glass, with the same core diameter, accepting light from the same cone. Nothing about the invariant has been touched.

The second is how narrow the minimum is. An exponent wrong by a tenth — five per cent — costs a factor of ten in delay spread. There is no forgiving region: the profile has to be right to a per cent or the benefit collapses, which is why graded fibre could not be made until vapour-deposition processes could lay down a hundred or more layers of controlled composition, and why the profile of every preform is measured rather than inferred from the recipe.

It is also why the optimum sits at 1.96 rather than at 2. The value depends on the contrast, on the wavelength through the material’s own dispersion, and on which rays are counted, so a fibre optimised at 850 nanometres is not optimised at 1,300 — and a real graded fibre is specified for a wavelength band rather than in general. The residual after optimisation is second order in the contrast, which is why a low-contrast fibre does better in relative terms and worse in absolute light-gathering.

What the tracing had to prove about itself #

The number 1.96 is a measurement rather than a quotation, and measuring it required a check that is worth naming because it nearly went wrong.

Integrating a ray over a kilometre of fibre means following about nine hundred thousand oscillations, which no straightforward integrator does without accumulating error. The figure instead traces twenty millimetres — some eighteen pitches — with a step of a tenth of a micrometre, and scales the result, which is legitimate because the delay difference accumulates linearly with distance.

Even then the answer moves. At a hundred and twenty thousand steps the minimum is found at 1.88 and the spread there is ten times too large; at two hundred thousand it settles at 1.96, and it stays there at four, eight and sixteen hundred thousand. The integration error was larger than the effect being measured until the step was small enough, and a calculation stopped one setting earlier would have produced a plausible curve with a minimum in the wrong place. The figure refuses an exponent outside 1.9 to 2.0 for that reason, which is a check on the arithmetic rather than on the physics.

The lens the same profile makes #

The same gradient does something else that is worth a section, because it is where most people have met it without knowing.

Cut a short length of parabolic-index rod — a quarter of a pitch — and it is a lens. Rays entering parallel to the axis are sinusoids that reach their first crossing after exactly a quarter period, so they all meet at the flat end face: the rod images without a curved surface anywhere on it. A half pitch relays an image at unit magnification; a full pitch returns it unchanged.

That is a gradient-index rod lens, and it is what a photocopier’s scanner bar, an endoscope’s relay, a fibre collimator and a laser-diode coupler are made of. The reason it is used rather than a conventional lens is a manufacturing one and it has the shape of this whole essay: a flat-ended cylinder can be butted against a fibre and glued, where a curved lens has to be positioned in three dimensions, and the alignment tolerance is what dominates the cost of an optical assembly.

Its focal length has the pitch in it and therefore has no curvature in it, so the lens is corrected differently from a curved one: it has no spherical aberration from a surface because it has no surface, and it acquires aberrations instead from the profile departing from a parabola at large radius. The same per cent that decided a fibre’s bandwidth decides a rod lens’s image quality, and it is the same measurement on the same preform.

What it costs #

Something has to be given up, and what is given up is exactly half the light.

The acceptance at a radius $r$ is set by the difference between the index there and the cladding’s, and in a graded core that difference has already been partly spent. At the core’s edge it is nothing at all: a graded fibre accepts no light whatever at the outer rim of its core, where a step-index fibre accepts its full cone.

Integrating gives $\alpha/(\alpha+2)$, which is a half at $\alpha = 2$ and approaches one as the profile approaches a step. So the trade is explicit: half the étendue for five hundred times the bandwidth, on the same glass.

Nothing about that violates the invariant, and it is worth being careful about why. The invariant says what a given acceptance can gather and forbids concentrating beyond it. It does not say that a guide must use all the acceptance its materials would allow. The graded fibre has simply been built with a smaller acceptance than its peak index permits, and has been paid for it in a currency the invariant does not price.

The same trick everywhere a path length has to be equalised #

Equalising arrival times across a bundle of paths is a general problem, and the parabolic fibre’s solution is one of a family worth recognising.

An achromatic doublet equalises focal lengths across a range of colours by combining two glasses whose dispersions differ, which is cancelling a derivative rather than a value — the same move as leaning the exponent below two. A phased array equalises arrival times across an aperture by delaying each element deliberately. A chirped mirror equalises the delay of the colours in an ultrashort pulse by sending the red ones deeper into the stack than the blue.

What they share is the structure of the fix. In each case a spread exists because two quantities that ought to have compensated do not, the compensation is arranged by making one of them depend on position, and the result is exact to first order and leaves a second-order residual that then sets the performance. The residual is where all the engineering is, and a design that only achieves the first-order cancellation is usually easy and usually not good enough.

The graded fibre is the cleanest member of the family because the compensation is complete rather than approximate for the idealised profile: it is not that the path-length difference has been reduced, it is that a harmonic oscillator’s period does not depend on its amplitude, and isochronism is exact.

Why a fibre that was superseded is what the building is wired with #

The delay spread a graded fibre fights is a property of having several modes. A fibre with one mode has none of it, and by the early 1980s single-mode fibre — a core small enough that its étendue is about one square wavelength — had removed the problem rather than solving it, along with most of what remains of chromatic dispersion once the wavelength is chosen well. Every long-distance link in the world is single-mode.

Every data centre is not. The links between racks, and most of the links inside buildings, are graded-index multimode, and the reason is an étendue argument about the source rather than about the fibre.

A single-mode core has an étendue of about $\lambda^2$, so only a source whose own étendue is that small can be coupled into it efficiently. That means a single-mode laser with a carefully aligned lens, and the alignment tolerance scales with the core — which is the same statement as a beam’s waist and divergence trading against each other at constant product: a nine-micrometre core needs submicrometre placement and active alignment during manufacture. A fifty-micrometre graded core has thirty times the diameter and about six hundred times the étendue, so it can be fed by a cheap vertical-cavity laser through a moulded lens with a tolerance of several micrometres, and the connector on the end can be a plastic moulding rather than a polished ceramic ferrule.

So the choice is a comparison of two costs. Over ten kilometres the delay spread of a multimode fibre is fatal and the cost of a precise connector is negligible per kilometre. Over a hundred metres the delay spread is a picosecond and the connector cost dominates, because there are thousands of them. The crossing is around a few hundred metres, which is where the standards put it, and it is set by the étendue of a laser diode rather than by anything about glass.

That is a satisfying place for the invariant to arrive. The invariant was introduced as a bound on concentration, then as a count of modes, then as an entropy; here it decides which of two technologies is used in which building, through the cost of aligning a lens.

Rays without phase, and no skew rays at all #

The pulse is treated as a set of rays with no phase. A short pulse also has a bandwidth, and a fibre’s material dispersion smears it independently of anything modal — an effect that is negligible over a hundred metres of multimode fibre and dominant over a hundred kilometres of single-mode, and which has a wavelength at which it vanishes. The two kinds of spreading add in quadrature and the figures here count only one.

Every ray here is meridional. Rays that spiral round the axis without crossing it — skew rays — are a large fraction of the light in a real multimode fibre, they follow helical paths, and their delays differ from the meridional ones. Including them broadens the delay spread and shifts the optimum exponent, and a full treatment is a mode calculation rather than a ray one.

The ray picture is used where a fibre has a few hundred modes, which is enough for rays to be a fair description and not enough for them to be exact. The true answer is the group delay of each guided mode, obtained from the wave equation, and the ray result is its large-mode-number limit. For the highest-order modes, which are the ones setting the spread, the approximation is at its worst.

The guide is treated as a well rather than as a boundary. A graded core guides by refraction rather than by reflection, which is the difference between a channel with walls and one with none and is why the cutoff behaves differently: a step-index guide has a sharp condition and a graded one has a continuum of turning points, each mode reaching the radius where its own transverse momentum runs out.

Mode coupling is left out entirely. Light in a real fibre transfers between modes at bends, at splices and at microscopic irregularities, so a ray does not keep its launch angle for a kilometre. The effect is to average the delays, which makes the measured spread grow as the square root of the length rather than in proportion to it beyond some distance — a large correction, and one that improves matters rather than worsening them.

Loss is ignored throughout. A real fibre attenuates, and the attenuation is not the same for every mode — the highest-order modes run closest to the core boundary, are scattered most and leak at bends — so a long fibre preferentially removes the very modes that set the delay spread. The measured bandwidth of a long multimode fibre is therefore better than a lossless calculation predicts, for a reason that is a loss.

And the profile is taken as exactly $\alpha$-shaped. A real preform has a dip at the centre from the deposition process, discrete steps between layers, and a profile that varies along its length; each of those costs more than the difference between 1.96 and 2.

A helix drawn as a line in a plane #

The ray paths are drawn in a plane and a fibre is round. What the figure shows is the projection of a helix for every ray that has any angular momentum about the axis, and the rays with the most of it — the skew rays the model omits — are exactly the ones whose projection would look most misleading.

The delay figure draws a spread between the extreme rays, which is a range rather than a distribution. What a receiver sees is a pulse shape, built from the delays of hundreds of modes weighted by how much power each carries, and its width is not the range. A fibre’s bandwidth is specified from the measured pulse response for that reason, and the relation between it and the figure’s quantity depends on how the fibre was launched into.

And none of these figures can show the thing the whole subject turns on, which is that the profile is a material gradient — a continuous change of composition through the glass, a few per cent of germanium at the centre falling to none at the edge, laid down as vapour and fused. The curve on the third figure is a recipe, and its sharpness is a statement about a chemical process rather than about optics.

Still open: how much of the invariant survives when the guide is not a guide #

Every argument so far has been about light in free space or in a guide large compared with a wavelength, where a ray is a fair description and the invariant is a statement about geometry the wave picture then refines.

What happens when the structure is comparable with the wavelength is genuinely different. A photonic crystal, a plasmonic waveguide, a metasurface and a subwavelength antenna all confine light in ways the ray picture does not describe, and several of them appear to concentrate it beyond what the geometrical bound permits — a plasmonic tip focuses light into a region far smaller than $\lambda^2$, which is an étendue below the floor the mode count established.

The resolution in each known case is that the bound was being applied to the wrong thing: the field at a plasmonic tip is not a propagating mode, it is bound to the metal and decays away, and the étendue theorem is about propagating fields. But the boundary between the two is not sharp, near-field and far-field shade into one another, and there is an active literature on what the correct statement of the invariant is for structured media — including whether the classical bound can be exceeded for a propagating field over a narrow band, which some thin-film light-trapping designs claim and which is a live question rather than a settled one.

The habit worth carrying away is the one the first two figures make together. When a bound fixes one quantity, check whether the thing being optimised is that quantity. The étendue of a fibre was fixed and could not be improved, and the entire useful life of multimode fibre came from noticing that the quantity anyone actually cared about — how fast a pulse could be sent — was not it, and was nearly independent of it. A conservation law forecloses one line of attack and says nothing at all about the others.