The mode that will not turn a corner

A guided mode is a field that keeps its shape as it travels. What holds it together is that the part outside the core does not propagate: it decays, exponentially, and an exponentially decaying field carries nothing away.

Bend the guide and that stops being available. The mode must now travel round a curve while keeping its shape, which means every part of it has to have the same angular rate — so the field at a distance $x$ from the axis has to move at a speed larger by a factor $(1 + x/R)$. Far enough out, that required speed exceeds the fastest the surrounding medium permits, and the field there cannot be evanescent any more. It has to radiate — the same continuation from a decaying solution to a travelling one that governs the field beyond a totally reflecting surface and a chain driven above its top frequency.

That radius is a caustic, in the same sense the word carries elsewhere: a boundary beyond which a particular kind of solution does not exist. Everything about bend loss follows from where it is and how much field is out there.

Why the loss is exponential #

The amount of field at the caustic is the tail of the mode, and a mode’s tail is an exponential. So the leak is proportional to $e^{-2\gamma x_c}$, where $\gamma$ is the decay constant and $x_c$ is the caustic radius. And the caustic radius is proportional to the bend radius. Put the two together and the loss is exponential in the bend radius.

An exponential in a physical dimension is a sharp threshold, and this one is among the sharpest in guided optics. The loss crosses six decades over a few millimetres of radius, which means there is no gradual regime to work in: a coil of one size is perfect and a coil slightly smaller is a disaster.

That is worth stating as a practical rule because it is not what a smoothly-varying quantity would give. A fibre specified to tolerate a thirty-millimetre bend does not lose twice as much at fifteen; it loses something like a million times as much. Design margins on bend radius are therefore not margins in the usual sense — being twenty per cent inside the specification is a different situation entirely from being twenty per cent outside it.

Why the higher modes go first #

The two curves in that figure are separated by a large factor at every radius, and the separation is the useful part.

A higher-order mode has a larger transverse wavenumber inside the core, so it has a smaller decay constant outside — it is held less tightly, and its tail reaches further. Two consequences follow, and they compound. Its effective index is closer to the cladding’s, so its caustic sits closer in for a given bend. And its tail at that caustic is larger, because the decay is slower.

So a bend attacks the highest mode hardest, by a margin that grows as the bend tightens. A coil of a few turns at a well-chosen radius removes the higher modes almost completely and leaves the fundamental nearly untouched, which is a mode filter made of nothing but geometry.

That is used in earnest. High-power fibre lasers are built with a core large enough to be multimode — to keep the intensity below what the glass will stand — and are then coiled at a radius chosen to strip everything but the fundamental, so the output is single-mode from a guide that is not. The coil radius is a design parameter and the figure above is what it is chosen from.

Putting the two radii beside each other makes the mechanism’s economy plain. The mode is identical in both figures: the same profile, the same decay constant, the same everything. What differs is a single vertical line, and the loss is the field’s value where that line falls.

Because the field falls exponentially and the line moves linearly, the loss falls exponentially — and the sensitivity is entirely inherited from the profile. That is the sense in which bend loss is a measurement of the mode’s tail: the bend radius is a knob that reads off the field at a chosen distance, and the reading spans many decades because the thing being read is an exponential.

It also explains a fact that puzzles anyone measuring it. Bend loss depends far more strongly on the wavelength than anything else about a fibre — a factor of ten between 1310 and 1550 nanometres is ordinary — because the decay constant depends on how far the mode is from cutoff, and that changes quickly with wavelength. A fibre that is comfortably single-mode and tightly held at one wavelength is weakly held at a longer one.

What the ray picture gets wrong #

There is an older account of bend loss and it is worth saying exactly where it fails, because it is the natural one to reach for.

In the ray picture, light is guided because it strikes the core–cladding boundary beyond the critical angle. Bending the guide makes the angle on the outside of the bend shallower, and past some radius it falls below the critical angle and the ray escapes. That gives a threshold radius, and it predicts nothing at all above it.

Both halves of that are wrong. The threshold radius from the ray argument is far smaller than the radius at which loss is actually measured, and the ray argument predicts zero loss at any larger radius — where the truth is a loss that is small but exponentially sensitive, and which becomes fatal over a length of fibre.

The reason for the failure is the same one that makes the ray picture fail at a totally reflecting surface: there is field beyond the boundary, the ray picture says there is not, and every effect that involves that field is invisible to it. Bend loss is such an effect entirely. It is a property of the tail, and a ray has no tail.

The ray argument is not useless — it gives roughly the right answer for a large multimode guide, where the modes are many and the tails are short — and it is exactly wrong for a single-mode fibre, where the whole of the behaviour is in the part it denies exists.

The mode a guide is trying to hold #

The relation between cutoff and bend loss is worth making explicit, because it is the single number that predicts how a guide will behave.

A mode just above its cutoff is barely guided: its effective index is only just above the cladding’s, so its decay constant is small and its field extends a long way out. A mode far above cutoff is tightly held, with a short tail. The distance from cutoff is therefore what decides both how much field is out at the caustic and how far out the caustic sits, and it enters the loss twice.

That is why a guide’s cutoff is not merely a statement about whether a mode exists. It is the parameter that governs everything about how robustly it exists — how much a bend costs, how much a nearby object perturbs it, how far it couples into a neighbouring guide. A mode that is barely guided is barely guided in every respect at once.

The practical form of that is a rule of thumb: run a guide well above cutoff if it has to survive handling, and near cutoff only if something is meant to reach the mode from outside. Fibre sensors that measure by touching the evanescent field are deliberately operated near cutoff, and communication fibre deliberately is not.

What sets the radius in practice #

The caustic sits at $R$ times the fractional difference between the mode’s effective index and the cladding’s, so tightly-guided modes have distant caustics and weakly-guided ones have close ones. That gives the design rule: a fibre tolerates a tight bend in proportion to how strongly it guides, and how strongly it guides is the numerical aperture again.

The trade is the standard one and it runs the wrong way for communication. A large index step gives strong guiding and good bend tolerance, and it also gives a large acceptance cone, many modes and poor bandwidth. Standard telecommunications fibre has a very small index step precisely to be single-mode, and it pays for that with a bend tolerance of a few centimetres.

The commercial answer has been to redesign the cladding rather than the core. A trench of lowered index a little way out from the core leaves the mode’s shape almost unchanged near the centre and cuts its tail off before it reaches the caustic — so the mode is the same mode and there is much less of it out where the leaking happens. Bend-insensitive fibre made that way tolerates a five-millimetre radius where ordinary fibre needs thirty, with no change to the core, and it is a direct application of the picture in the first figure: the loss is what is at the caustic, so remove what is at the caustic.

The same argument for a whispering gallery #

The caustic argument is not about fibres and its cleanest instance is older.

A wave running round the inside of a curved wall — Rayleigh’s whispering gallery, in the dome of St Paul’s — is held against the wall by the same reflection that holds a guided mode, and it leaks by the same mechanism. There is a radius outside the wall at which the field would have to travel faster than the surrounding medium allows, and whatever field is out there radiates.

The difference from a fibre is that here the curvature is the whole of the confinement — there is no straight guide to compare with — so the loss cannot be made small by loosening the bend without also making the gallery bigger. What sets it is the number of wavelengths round the circumference, and the loss falls exponentially with that number for exactly the reason above.

The consequence is a resonator whose quality can be extraordinary. A glass sphere a hundred micrometres across holds a whispering-gallery mode with a radiation loss so small that the limit is set by absorption in the glass and by scattering from its surface rather than by leakage at all — quality factors above $10^9$, which is far beyond anything a mirror cavity of that size reaches. The whole of that performance is the exponential in this essay, evaluated at a large number.

Where the model stops #

The guide here is a slab and a fibre is round. The slab has the right structure — a core, a cladding, modes with tails, a cutoff — and its mode indices differ from a fibre’s by numbers of order one. Every scaling in this essay is the fibre’s; none of the absolute numbers is better than a factor of two.

The bend is treated as a static perturbation. A real bend has a transition at each end where the guide’s curvature changes, and light is lost there too, by a different mechanism: the mode of the straight guide and the mode of the bent guide are not the same shape, and what does not match is radiated. That transition loss does not depend on how long the bend is, so a single tight bend and a long gentle one are lost in different currencies.

And the loss coefficient’s prefactor is not computed here. Everything in the figures is in the exponent, and the prefactor is of order ten thousand per metre for a guide of this kind — a number that depends on the mode’s normalisation and on the geometry in a way the argument above does not fix. The exponential is the physics; the coefficient in front of it is a calculation.

Nothing here is about microbending. Pressing a fibre against something rough couples the guided mode to radiating ones at a spatial frequency the roughness supplies, which is a scattering problem rather than a caustic problem, has quite different scalings, and is closer in kind to what a periodic corrugation does to a guided mode. In practice it is often the larger loss, and it is the reason fibre is coated.

How much a bend costs, in practice #

Some numbers make the exponential concrete, and they are the ones anybody installing fibre has to know.

Standard single-mode fibre at 1550 nanometres loses about 0.1 decibels per turn at a bend radius of around fifteen millimetres, about 0.5 at ten, and several decibels at five. A single tight loop of thirty millimetres’ diameter — smaller than a coffee cup — is a fault. A cable coiled at a hundred millimetres is not.

Two consequences follow that are counter-intuitive until the exponential is believed. A long gentle bend costs almost nothing: the loss is per unit length at a given radius, so a hundred and eighty degrees at fifty millimetres is less than ten degrees at ten. And a fibre that has been over-bent does not degrade permanently unless the glass has been damaged — straighten it and the loss goes away entirely, which is why an intermittent fault that moves when the cable is moved is nearly always a bend.

The third consequence is the one that made bend-insensitive fibre a product. A fibre run inside a building is bent by whoever installs it, at radii nobody specified, and the older fibre could not survive being pushed round a corner in a wall. The trench design mentioned above was developed for that market, and its whole content is moving the caustic past where the field is.

What the pictures cannot show #

The profile figure draws a transverse cut through a straight guide and marks a radius computed for a bent one, which is a hybrid. In a genuinely bent guide the mode’s own profile shifts outward — the field is pushed toward the outside of the bend before it starts to leak — and that shift is a second effect the drawing does not contain. It matters most where the loss is already large.

Neither figure shows what happens to the radiated light. It leaves the guide, travels in the cladding, and in a real fibre is absorbed in the coating a few centimetres later — so a bent fibre is warm at the bend and dark a little way past it. In a fibre with no absorbing coating the light re-enters the core further along and interferes with what is still guided, producing an oscillating loss with bend radius that is measured and is not in this model at all.

Why the loss is quoted per turn #

A small point of bookkeeping is worth making explicit, because it is where the exponential and the geometry meet.

The loss coefficient — the fraction lost per unit length — depends only on the radius, not on how much of a circle the bend is. So a quarter turn at a given radius loses a quarter of what a full turn loses, and the natural unit is decibels per turn only because a turn at radius $R$ is a length $2\pi R$.

That has a slightly odd consequence. Loss per turn is the loss coefficient times $2\pi R$, so as the radius grows the coefficient falls exponentially while the length grows linearly — and the product is dominated by the exponential everywhere except at very large radii, where the linear factor makes the curve turn up again. In practice that upturn is at radii where the loss is already unmeasurable, so it never matters, but it is the reason the plotted curves are not straight lines on a logarithmic axis.

The other unit worth having is loss per metre of cable, which is what an installer needs. A cable with a bend somewhere in it has that bend’s loss once, regardless of the cable’s length, so quoting bend loss in decibels per kilometre — as attenuation is quoted — mixes two quite different things. A fibre link’s budget has an attenuation term proportional to length and a bend term proportional to the number of bends, and only the first improves when a shorter route is found.

Where the ladder goes next #

The guided-waves ladder began with the pipe that will not carry a low note, went through the channel with no walls and the wave a surface is enough to hold. This rung asks what a bend does. The rungs after it: coupling between guides, where two tails overlapping exchange power completely over a definite length; the transition loss at the ends of a bend, which is a mode-matching problem rather than a leakage one; and guides that confine by a band gap rather than by total internal reflection, which have no evanescent tail in the ordinary sense and fail at a bend for a different reason.

The habit worth carrying away is to ask where the small part of a field is. A quantity that is exponentially small is exponentially sensitive, and any mechanism that reaches out to where an evanescent tail lives will produce an effect that is unmeasurable and then catastrophic with nothing in between.