The delay that is a random variable

The wavelength a fibre does not smear treats dispersion as a property that can be computed, designed and cancelled: the glass fixes one part, the geometry of the core fixes the other, and a coil of fibre with the opposite sign — or a digital filter with a known quadratic phase — undoes the total. Near its end it names one dispersion that none of those remedies touches, because it changes with time.

That dispersion comes from something that sounds too small to matter. An ideal single-mode fibre has a perfectly circular core, and its two orthogonal polarisations travel at exactly the same speed. A real fibre’s core is elliptical by a fraction of a per cent, and the glass carries stresses frozen in when it was drawn and added when it was cabled, bent and laid. Each short piece of fibre is therefore a very weak version of a crystal that answers twice: it has a fast axis and a slow axis, and light polarised along one arrives a little before light polarised along the other.

If that were all, the delay would simply add up along the fibre and could be undone like any other fixed delay. It is not all, because the orientation of the ellipse does not stay put. It wanders along the fibre, changing direction on a scale of tens to hundreds of metres, and it does so at random.

A delay that grows as the square root of the length #

The way the pieces combine is the whole subject, and it needs the right picture of a polarisation state. Every state of polarisation of light — linear at any angle, circular of either hand, elliptical in between — is a point on the surface of a sphere, the Poincaré sphere, with the linear states round the equator and the two circular states at the poles. A waveplate acts on that sphere as a rotation: it turns every state about the axis through the two polarisations it leaves alone, by an angle equal to the phase delay it introduces.

The differential group delay of a piece of fibre can be drawn on the same sphere, as a vector pointing along the axis of its slow polarisation with a length equal to the delay. For two pieces in series the total is not the sum of the two vectors. The first piece’s delay vector is carried through the second piece, which rotates it, and only then is the second piece’s own vector added:

$$\vec\Omega_{\text{total}} = \vec\Omega_2 + R_2\,\vec\Omega_1.$$

The figure builds each fibre that way, one section at a time. The rotation angle of each section is its phase retardance, which at optical frequencies changes by many turns for a change of temperature too small to notice, so it is effectively random; the axis of each section is random too. So each step adds a vector of fixed length to a sum that has just been turned in a random direction.

That is a random walk, in three dimensions, and a random walk does not go anywhere in proportion to its number of steps. Its distance from the start grows as the square root, because the steps’ directions are uncorrelated and only their squared lengths add — the walk that comes home is the same arithmetic in its most familiar form. On the figure the rms delay of three hundred fibres grows with length to the power 0.50, and at 100 km its mean is 5.18 ps against 5.00 for a fibre whose coefficient is half a picosecond per root kilometre.

The dashed line is the same sections laid with their axes aligned, which is what an elliptical core that never turned would give. It grows in proportion to length, reaches 172 ps at 100 km, and leaves the frame within a few kilometres. The randomness of the fibre is what keeps its polarisation delay small. A perfectly uniform imperfection would be thirty times worse at 100 km, and the ratio grows as the square root of the length.

A distribution rather than a number #

Because the delay of a long fibre is the length of a sum of many random vectors, it has a distribution, and the distribution is known before any fibre is measured.

Each component of a long random walk’s end-point, by the central limit theorem, is Gaussian, and the three components are independent. The length of a vector with three independent Gaussian components has a definite distribution, and it is the one Maxwell wrote down in 1860 for the speeds of the molecules in a gas. The speeds in a still room are the length of a velocity vector whose three components are Gaussian; the polarisation delay of a fibre is the length of a delay vector whose three components are Gaussian. Nothing else about the two situations is alike, and the figure’s histogram is indistinguishable from the curve of a gas.

That changes what it means to state a fibre’s polarisation delay. A kilometre of fibre is specified by a coefficient in picoseconds per root kilometre, and the coefficient gives the mean of the distribution for a given length. On a particular day, at a particular wavelength, the delay is a draw from the distribution, and it may be well above the mean or well below it. Measuring it once tells a network operator very little about it tomorrow.

The distribution is sampled in two ways that turn out to be equivalent. Across many different fibres, as the figure does. Or across time in one fibre, because temperature changes, vibration from traffic on a bridge and a technician moving a patch cord all reshuffle the sections’ retardances and axes, and the fibre drifts from one member of the ensemble to another over minutes to days. The rate of the drift depends on how the fibre is installed — a buried cable drifts slowly, an aerial one on poles in the wind quickly — and the distribution it drifts through does not.

How often the delay is three times its mean #

A distribution with an unbounded tail means a link cannot be designed never to fail from polarisation delay. It can only be designed to fail rarely. The tail of a Maxwellian falls faster than an exponential, so the probability of exceeding a large multiple of the mean becomes small quickly, but it never becomes zero.

The number system designers use is the one marked. The delay exceeds three times its mean with probability $4.2\times10^{-5}$, which for a link watched continuously is about twenty-two minutes a year spent above that level. A receiver that tolerates three times the fibre’s mean delay is therefore expected to see those minutes as errors, and the budget is written as an outage probability rather than as a guarantee. Other engineering disciplines that design against a random load — a wind on a building, a flood on a river — do exactly the same thing, and the Maxwell tail here plays the part the hundred-year storm plays there.

One fibre, wandering across a band #

The same randomness that makes the delay differ between fibres makes it differ between wavelengths in one fibre. The retardance of each section is its birefringence times its length divided by the wavelength, so a small change of optical frequency changes every section’s rotation angle a little, and after enough sections the sum has been turned into a different random walk.

The width over which the delay stays correlated is set by the delay itself: two frequencies separated by $\Delta\omega$ see the accumulated delay vector rotated relative to one another by roughly $\Delta\omega$ times the delay, so the correlation is lost once that product reaches a radian or so — the reciprocal relation between a delay and a bandwidth that also makes a narrow line a long lifetime. For a mean delay of five picoseconds the scale is tens of gigahertz, and the simulated fibre loses half its correlation over 73. A dense wavelength-division system with channels 50 or 100 gigahertz apart therefore has channels whose polarisation delays are nearly independent.

That has a consequence that sounds unfair and is useful. On a given fibre on a given day, some channels sit on a peak of the drawn curve and suffer, and a few channels away others sit in a trough and do not. The outage probability of the previous figure applies to each channel separately, so a system carrying eighty channels is almost certain to have one of them in trouble at some moment, and almost certain never to have all of them in trouble at once. Spare capacity on another wavelength is a remedy that a fixed chromatic dispersion never offered.

The first-order picture #

On any one channel, over a narrow enough band, the fibre’s polarisation behaviour can be summarised simply. There are two special input polarisations — the principal states — such that light launched in either one arrives in a well-defined output polarisation, with no distortion, one of them delayed relative to the other by the differential group delay. Any other input is a superposition of the two principal states, and a pulse launched in it splits into two copies separated by the delay.

This is the first-order description, and it is what the delay vector’s direction and length encode: the direction on the Poincaré sphere is the principal state and the length is the delay. When the delay is a small fraction of a bit, the two copies overlap and the pulse is merely broadened a little; when it approaches a bit period, each one arrives on top of its neighbour’s partner and the signal is lost. The worst case is light launched halfway between the principal states, with equal power in both copies, and since the principal states wander with the fibre there is no way to arrange the launch to avoid it for long.

Higher orders matter when the band is wide compared with the correlation width: then the principal states and the delay themselves change across the signal’s own spectrum, and the pulse is not just split but distorted, in the way a packet whose components travel at a spread of speeds is. The frequency figure is that higher-order dependence drawn out across a band, and a signal wide enough to cover a noticeable fraction of the 73 gigahertz sees it.

The distance a bit rate allows #

Put the square root together with the bit period and a stark scaling follows. The tolerable delay is a fixed fraction of a bit, so it falls in proportion to the bit rate. The delay grows as the square root of the length. So the tolerable length falls as the inverse square of the bit rate: quadrupling the rate leaves a sixteenth of the distance.

That scaling is the history of the effect. Fibre installed through the 1980s and early 1990s was drawn and cabled with no attention to polarisation delay, because at the bit rates of the time it did not matter, and its coefficients are often near the top of the drawn range, around half a picosecond per root kilometre or worse. At 10 gigabits a second such a fibre was fine for hundreds of kilometres. At 40 gigabits a second the same fibre became unusable beyond a few tens, with nothing about the glass having changed. Fibre made since is spun as it is drawn — twisted, so that its residual ellipticity rotates rapidly along the length and averages out — and routinely reaches coefficients below a tenth of a picosecond per root kilometre.

The other half of the history is the one the wavelength a fibre does not smear ends with. A coherent receiver records both polarisations of the field, and a digital filter updated thousands of times a second can follow the principal states and undo the delay as it drifts. That turned polarisation-mode dispersion from a hard limit into a quantity to be tracked, and the tracking works precisely because the statistics are known: the equaliser is designed for the range of delays the Maxwell distribution makes likely and for the rate at which the fibre drifts through them.

Measuring a delay that will not hold still #

Because the delay changes across frequency in a known statistical way, the most direct measurement uses that change rather than fighting it. Put a polariser before the fibre and an analyser after it, and sweep the wavelength of the source. At each frequency the fibre carries the launched polarisation to some point on the Poincaré sphere, and the analyser passes the share of the light whose state lies towards its own axis. As the frequency changes, the output state wanders round the sphere at a rate set by the delay — a fibre with a large delay turns the state through many circuits over a narrow band — so the transmitted power rises and falls, and the number of maxima and minima in a band is proportional to the mean delay.

The constant connecting the count of extrema to the delay is itself a result of the random-walk statistics, which is the unusual feature of the method: the instrument is calibrated by the theory of the quantity it measures. A fibre whose sections were not randomly coupled would give a different count for the same delay, and the method would report the wrong number without any sign of trouble.

The more complete methods measure the fibre’s whole polarisation response at pairs of neighbouring frequencies — the rotation a retarder performs on the sphere, inferred by launching three or more known states — and extract the delay vector directly, direction and length. Repeated over days on one installed fibre, they trace out that fibre’s own Maxwellian, and the agreement between the distribution sampled in time and the distribution across an ensemble of fibres is the evidence that the statistical description is right rather than merely convenient.

Where this model stops #

The sections are identical, independent and uniformly oriented. Real fibre has a spread of section lengths and birefringences and a coupling length that varies with cabling, and the axes are not perfectly uniform in angle. None of that changes the square-root growth or the Maxwellian shape for a long fibre, which are consequences of the central limit theorem rather than of the particular sections, but it changes the coefficient and how short a fibre has to be before the random-walk description fails.

Short fibres are not in the random-walk regime. A fibre shorter than a few coupling lengths has a delay that grows in proportion to its length, as the aligned line does, and a Maxwellian only emerges after many sections. The crossover is near the coupling length, which is why that length is one of the two numbers that specify a fibre’s polarisation behaviour.

Polarisation-dependent loss is ignored. Components in a link — isolators, amplifiers, couplers — attenuate the two polarisations slightly differently, and the combination of that with polarisation delay produces effects the delay alone does not, including a loss of the orthogonality of the principal states.

And the fibre is linear. At the powers of long-haul systems the intensity-dependent index couples the two polarisations to each other and to neighbouring channels, and the clean separation between a random linear delay and everything else no longer holds.

What the pictures cannot show #

None of the figures shows the Poincaré sphere, and the whole mechanism lives on it. The delay vector is drawn only as its length, and its direction — the principal state, which is what an equaliser actually has to find — is discarded. Two fibres with the same delay and different principal states look identical on every figure here and demand completely different corrections. Nor can they show what the splitting does to the light itself: when the two copies of a pulse separate by more than the source’s coherence time, the light arriving is partly depolarised, and a fibre has turned polarised light into light with no direction of shaking without absorbing any of it.

Nor do the figures show time. The distributions are over fibres or over frequencies, and a real link drifts through the ensemble at a rate set by its installation, from hours for a buried cable to milliseconds for one struck by lightning or by a passing train. How fast the delay changes is a separate statistic from how large it is, and it is the one that decides whether a tracking equaliser keeps up.

Still open: whether dispersion has to be random to be harmless #

The random coupling is what keeps a fibre’s polarisation delay growing as the square root of its length rather than in proportion to it, and spinning fibre as it is drawn deliberately adds more randomness — or more precisely, more rapid rotation — to push the coupling length down. That raises a question the other dispersions do not: whether the arrangement that minimises a delay is always the most disordered one, or whether a designed, deterministic pattern of rotations could do better.

For some spin profiles the answer is known to be yes, over a limited band: a periodically varying spin can cancel the delay to first order at a design wavelength, much as two glasses cancel a derivative at two colours. What is not settled is how such a fibre behaves once it is cabled, bent and left to drift, when the frozen-in stresses of installation add a random part the drawing tower never saw.

The habit worth carrying away is the one the Maxwell distribution forces. When a quantity is built from many small contributions whose orientations are random, stop asking what its value is and ask what its distribution is. A fibre’s polarisation delay, the speed of a molecule and the distance of a random walker all look like numbers and are really lengths of random vectors, and the questions worth asking of them are about means, tails and how fast they forget.