The cone a fibre will accept

Shine a torch at the end of an optical fibre and most of the light does not go anywhere. What enters and stays entered is the part arriving within a certain cone, and the cone’s half-angle is fixed by two numbers: the refractive index of the glass in the middle and the index of the glass around it.

The derivation is two applications of Snell’s law and is worth doing because the cancellation at the end is the point. A ray entering the flat end face at angle $\theta$ to the axis refracts into the core at $\theta'$ with $\sin\theta = n_1\sin\theta'$. It then meets the wall at $90° - \theta'$, and it is trapped only if that angle exceeds the critical angle for the core against the cladding, $\arcsin(n_2/n_1)$. Putting the two together and simplifying,

$$\sin\theta_{\max} = \sqrt{n_1^2 - n_2^2} \equiv \mathrm{NA}.$$

The core index appears twice and cancels once. What is left contains no length at all — no diameter, no length of fibre, no wavelength — and it is the same for a fibre the width of a hair and a light pipe the width of an arm.

What the number is, in practice #

The index difference between core and cladding is tiny in the fibres that carry telephone traffic: a third of a per cent, giving an aperture of about 0.12 and an acceptance half-angle of seven degrees. That seems perversely small, and it is deliberate — the mechanism is the same one that gives a waveguide a cutoff, and a small aperture is what keeps the fibre to a single mode.

The contrast is the whole engineering trade. A wide cone is easy to fill and carries a pulse badly: the ray at the edge of the cone travels a longer path than the axial one by a factor of $n_1/n_2$, and over a kilometre that is tens of nanoseconds of spreading. A narrow cone is hard to fill and carries a pulse well. Everything about how a fibre is used follows from which of those matters.

That the two ends of the range are the same formula at two index steps is worth noticing. The plastic fibre and the telecommunications fibre are not different devices; they are one device at two settings of one number, and the number is a property of two materials rather than of any dimension.

Why a cladding at all #

The formula contains the cladding index, and a bare glass rod in air has a cladding — the air, with index one. That gives an aperture of $\sqrt{1.5^2 - 1} = 1.1$, which is greater than one and therefore means every ray entering the flat end is trapped. On the face of it a bare rod is the ideal fibre.

It is not, and the reason is the whole history of the subject. Total internal reflection at a glass–air surface is perfect only if the surface is perfectly clean and perfectly smooth. Touch the rod, lay it against another, let dust settle on it, and light leaves at the point of contact — because the evanescent field extending a fraction of a wavelength beyond the surface reaches whatever is there, which is the mechanism by which light escapes from a place the ray picture says it cannot be.

John Tyndall’s demonstration of 1854 — light following a jet of water out of a tank — worked for exactly this reason and could not be made into anything, since a jet cannot be handled. Bundles of bare glass fibres were tried in the 1920s and 1930s and leaked wherever they touched. The cladding, introduced in 1954, solves it by putting the reflecting surface inside the glass, where nothing can reach it: the guiding boundary is now between two solids that are permanently in contact, and the outside of the fibre can be handled, coated and bundled with no effect on what is guided.

The price is the aperture. Trading air for glass as the outer medium takes the acceptance cone from everything down to a few degrees, and the entire modern subject is spent managing what remains.

The ceiling on getting light in #

The last sentence of that caption is the useful one and it surprises people who have not met it. Putting a lens in front of a fibre feels like it should help: collect light over a large area, bring it to a small spot, and get more in. It does bring the light to a small spot. It also, by the same act, makes the light arrive over a much wider range of angles — and the fibre throws away everything outside its cone.

The reason is the invariance of the product of area and solid angle, which no arrangement of refracting or reflecting surfaces can increase. Demagnify by a factor $m$ and the area falls by $m^2$ while the solid angle rises by $m^2$; the product is fixed, and the fibre’s own etendue is the ceiling on what can pass. A lens can be necessary — a source whose light arrives over too small an angle wastes the fibre’s cone, and a lens fixes that — but it can never raise the ceiling.

That ceiling can be stated as a number a designer can use. The fibre’s etendue is its core area times $\pi\,\mathrm{NA}^2$; for a fifty-micrometre core at 0.22 that is about $3\times10^{-10}$ square-metre-steradians. A light-emitting diode of a square millimetre radiating into a hemisphere has an etendue of about $3\times10^{-6}$. The ratio is the best coupling obtainable by any optics whatever, and it is one part in ten thousand.

The same number, counting modes #

The ray account has a wave counterpart and the aperture appears in it too, which is what makes the number more than a geometrical convenience.

A guide of core radius $a$ at wavelength $\lambda$ supports a number of modes fixed by the combination $V = 2\pi a\,\mathrm{NA}/\lambda$, and for a large $V$ the count is about $V^2/2$. Everything about a fibre’s behaviour is decided by that one dimensionless group: below $V = 2.405$ exactly one mode is guided, and above it the count rises as the square.

Two features of that are worth drawing out. The aperture and the radius enter only as a product, so a fibre can be made single-mode either by making the core small or by making the index step small, and the two are interchangeable — telecommunications fibre does both a little rather than either a lot, because a core much below nine micrometres is hard to splice and an index step much below a third of a per cent guides too weakly to survive a bend. And the mode count is essentially the etendue in units of $\lambda^2$, which is the general relation between a ray-optical volume and a count of wave solutions, and the same relation that makes the number of standing waves in a box a volume divided by a wavelength cubed.

That last correspondence is worth stating plainly, because it explains why the etendue argument is a real constraint rather than a convention about ray bookkeeping. Etendue divided by $\lambda^2$ counts modes; modes cannot be created by rearranging anything; therefore etendue cannot be reduced. The optical invariant is a mode count wearing geometrical clothes.

Where this is the same problem as a solar dish #

The fibre problem and the concentrator problem are the same problem with the roles of the two quantities exchanged.

A concentrator takes light from a small solid angle — the Sun subtends about half a degree — and tries to compress it onto a small area. The invariant says the compression cannot exceed the ratio of the two solid angles, which is where the famous limit of about 46,000 suns comes from. A fibre takes light from a small area and a small solid angle both, and the invariant says the same thing about what can be pushed in.

Reading the two together makes the underlying statement clearer than either alone. What is conserved is not brightness in the everyday sense, and not intensity, but the volume a beam occupies in the space of positions and directions together. An optical system can reshape that volume — squeeze the positions and stretch the directions, or the reverse — and cannot compress it. That is a statement of the same kind as Liouville’s theorem in mechanics, which says the same thing about positions and momenta, and the resemblance is not an analogy: a ray’s direction cosines are its transverse momentum divided by its total, and the two theorems are one.

The practical consequence for anyone joining light sources to fibres is a rule that removes a whole class of hopeful design. The question is never “what lens will get more light in”. It is “is the source’s etendue smaller than the fibre’s” — and if it is not, the fraction lost is fixed before any optics is chosen.

What the cone is used for besides carrying signals #

A fibre’s aperture is a specification for communication and an instrument in its own right elsewhere, and the applications divide neatly according to which half of the etendue they exploit.

Where the area is what matters, fibres are used in bundles: an endoscope’s image guide is tens of thousands of fibres side by side, each carrying one picture element, and the resolution is the fibre count rather than anything optical. The aperture then sets how bright the image is and how much light the illumination bundle can deliver, and the design pressure is toward the largest aperture available, which is why such bundles use a large index step and accept the dispersion.

Where the angle is what matters, the fibre becomes a sensor. Bend it and the local angle of incidence at the wall changes; if the bend is tight enough some rays fall outside the cone and are lost, so transmitted power measures curvature. Warm it and both indices change, slightly differently, so the aperture changes; press it and the same happens through the stress-optic effect. A length of fibre is therefore a distributed detector of almost anything that deforms glass, and the read-out is a power rather than a phase — which makes it cheap, and much less sensitive than the interferometric alternative.

And where neither matters, the cone is a nuisance to be filled deliberately. A spectrometer fed by a fibre sees light filling the fibre’s full cone regardless of what was put in at the other end, because bends and imperfections redistribute the rays — a phenomenon with the ugly name of mode scrambling and the useful consequence that a fibre-fed instrument is insensitive to how its input was illuminated. That is the same insensitivity an integrating enclosure buys, obtained by a different route.

The wavelength a fibre does not smear, and the cone that decides it #

The aperture and the pulse are connected by one line of arithmetic, and it is worth doing because it explains the shape of the whole technology.

A ray at the edge of the acceptance cone travels inside the core at an angle whose cosine is $n_2/n_1$, so its path is longer than the axial ray’s by that factor. Over a length $L$ the two arrive apart by $L n_1(n_1 - n_2)/(c\,n_2)$, which for a plastic fibre at six per cent index step is about three hundred nanoseconds a kilometre. A pulse train faster than that arrives as a smear.

Reduce the index step and the smear falls in proportion, which is the first reason telecommunications fibre has an aperture of 0.12 rather than 0.5. Reduce it far enough to guide one mode and the path-length spread vanishes entirely, because there is only one path — and what limits the pulse then is the material’s own dispersion, a completely different mechanism with a completely different remedy.

So the acceptance cone is not merely a specification for how easy a fibre is to fill. It is the first term in the bandwidth, it is removed by making the fibre harder to fill, and the sequence of technologies — plastic fibre, graded index, single mode — is a sequence of retreats from a wide cone, each buying bandwidth at the cost of light.

Where the model stops #

The step-index picture is a ray picture and a fibre is a waveguide. The acceptance cone is exact for a fibre whose core is many wavelengths across, where the number of guided modes is large and the ray count is a good approximation to the mode count. A single-mode fibre has a core about nine micrometres across, carries one mode, and does not have an acceptance cone in this sense at all — what it has is a mode field whose overlap with the incoming light decides the coupling, and the aperture survives as a useful number rather than as an exact one.

The cladding is assumed thick and lossless. A ray at exactly the critical angle is not confined at all, and one just inside it has an evanescent tail reaching well into the cladding — so a fibre whose cladding is thin, or whose coating touches the guiding region, loses the outermost part of its cone. That is also how a fibre is deliberately tapped, and why a tight bend loses light: bending makes the local angle of incidence shallower and pushes the outer rays out of the cone.

And the aperture says nothing about what happens along the way. Light inside the cone is accepted; whether it arrives is a separate question, decided by absorption, by scattering and by the spread of path lengths. A fibre with a large aperture is easy to fill and disperses a pulse badly, which is why the numbers quoted in this essay run in the opposite direction from the numbers a communications engineer wants.

What the pictures cannot show #

The cone figure draws a meridional ray — one that crosses the axis — and there is a second family that does not. A skew ray spirals along the fibre without ever meeting the axis, and its confinement condition is different and weaker: skew rays can be trapped at angles outside the meridional acceptance cone. So the cone drawn is the acceptance cone for the rays that are easy to draw, and a real fibre’s acceptance is slightly larger and much harder to picture.

The coupling figure draws a fraction and hides what happens to the rest. The light outside the cone does not vanish: it enters the core, crosses to the cladding and leaves, sometimes after several metres of travelling in the cladding itself as a leaky mode. Measured a centimetre from the input, a fibre appears to be carrying far more than it will deliver, which is a standard trap in measuring what a fibre accepts.

Where the ladder goes next #

This ladder began with the brightness no lens can increase, which is the invariant stated for a beam. This rung applies it to a guide, where the acceptance is fixed by two indices and the ceiling by the product with the core’s area. The rungs after it: the mode count, where the ray picture is replaced by a count of solutions and the aperture reappears inside it; graded-index guiding, where the index falls smoothly and the ray paths become sinusoids that all arrive together; and the brightness theorem in a medium, where the invariant picks up a factor of the index squared and a source immersed in oil can therefore be brighter than the same source in air.

The habit worth carrying away is to look for what a formula does not contain. The acceptance angle has no length in it, and that absence says immediately that the cone cannot be improved by making anything bigger, that the same rule covers every scale of fibre, and that anybody proposing to get more light in by changing a dimension has misread which quantity is doing the limiting.