The wavelength a fibre does not smear

A prism separates colours because the index depends on the wavelength, and a doublet cancels the effect at two wavelengths by pairing two glasses. Both of those are about where a ray goes. This is about what the same dependence does to a pulse, which is the quantity a communications system is made of.

The quantity #

A pulse is a superposition of frequencies, and each travels at its own group velocity. What matters is not the velocity itself but how fast it changes with wavelength, and the standard measure is

$$D = -\frac{\lambda}{c}\frac{d^2 n}{d\lambda^2}$$

with units of picoseconds per nanometre of spectral width per kilometre of fibre. A source one nanometre wide, sent $L$ kilometres, arrives $D L \Delta\lambda$ wider than it left.

The second derivative is the operative thing, and it is worth being clear about why. A first derivative of the index would mean the whole pulse travels at a group velocity different from the phase velocity — which is true, and harmless, since a uniform delay does not change a shape. It is the variation of the group velocity across the pulse’s own band that pulls it apart.

That is the same distinction a wave packet meets when it spreads, and it is exactly the same mathematics: the second derivative of the dispersion relation is what broadens, in a fibre and in a quantum wave alike.

Total internal reflection is the mechanism that keeps light in the core at all, and everything here assumes it has already succeeded. Past the critical angle nothing crosses the boundary, so a ray launched inside the acceptance cone bounces indefinitely and a ray outside it leaks away within centimetres. That much is settled before this essay starts; the question is what happens to the pulse that got in.

Why silica has a zero #

Glass is transparent in a window between two absorptions: an electronic one in the ultraviolet and a vibrational one in the infrared. Each pulls the index in the opposite sense.

An absorption at shorter wavelength makes the index fall as the wavelength rises — normal dispersion, the ordinary case, the one a prism uses. An absorption at longer wavelength does the reverse. In the middle of the window the two contributions to the second derivative cancel, and the wavelength where they do is a property of the material’s absorption spectrum and nothing else.

For silica the zero sits at 1,273 nanometres, computed here from the Sellmeier coefficients rather than quoted, and the published value is 1,273. It is not adjustable: shifting it would require changing which absorptions the glass has.

The consequence for a system is that below 1,273 nm a fibre is normally dispersive — blue arrives first — and above it anomalously dispersive, with red arriving first. The sign matters, and the reason it matters is at the end of this essay.

The half that is engineering #

The other contribution has nothing to do with the glass.

A mode guided by a core does not travel entirely inside it: some of the field extends into the cladding, and how much depends on the wavelength. A longer wavelength spreads further out, sees more of the lower-index cladding, and therefore travels a little faster than the core’s index alone would suggest. That is a wavelength-dependent group velocity, so it is dispersion, and it is negative.

Its size is set by the core’s diameter and index step, both of which a manufacturer chooses. Adding it to the material curve moves the zero upward, and standard single-mode fibre is designed so that the total crosses zero at 1,310 nanometres.

Pushing further gives dispersion-shifted fibre, with the zero moved to 1,550 nm; flattening the waveguide contribution across a band gives dispersion-flattened fibre, with two zeros and a small value between them. All of them are the same glass.

That the two contributions are separable and only one is adjustable is the practical shape of the whole subject. The material sets what is available; the geometry decides where within it the device operates.

Why a shorter pulse is hurt more #

The refutation at the head of this essay is worth working through, because it is the fact that makes dispersion a limit rather than a nuisance.

A pulse of duration $\tau$ cannot be monochromatic. Its spectrum has a width of at least $1/\tau$, by the same reciprocal relation that makes a narrow resonance a long ringdown, and a pulse at the minimum is called transform-limited. So the spectral width in the broadening formula is not an independent quantity that a good enough laser could remove: it is set from below by the pulse the system wants to send.

Put the two together. The broadening of a transform-limited pulse over a length $L$ is proportional to $D L/\tau$, so a pulse half as long broadens twice as much in absolute terms and four times as much relative to its own width. Doubling the bit rate therefore costs a factor of four in the dispersion penalty, and the maximum rate–distance product of a link goes as the inverse square root of the length.

That scaling is why dispersion, and not loss, is what limits a modern link. Loss is dealt with by amplifiers, which can be strung along a route indefinitely; dispersion accumulates through every amplifier and has to be undone.

The reciprocal relation between a pulse’s duration and its spectral width is the constraint a system designer meets first. The fastest pulses are necessarily the broadest in frequency and therefore the most vulnerable to any variation of speed across that band — so doubling the bit rate halves the pulse and doubles the bandwidth, and the smearing per kilometre goes up by four. Dispersion penalises speed quadratically, which is why it became the binding limit exactly when the rates rose.

The wavelength that is used, and why it is not this one #

The loss of silica fibre has its own minimum, at 1,550 nanometres, of about 0.2 decibels a kilometre — a factor of ten better than anything else and low enough that a signal survives a hundred kilometres without amplification. That is where the world’s traffic runs.

At 1,550 nm the total dispersion of standard fibre is about 18 picoseconds per nanometre per kilometre. A source one nanometre wide over a hundred-kilometre span therefore arrives 1,800 picoseconds broader, which at ten gigabits a second is eighteen bit periods: the pulses have merged completely and the link does not work.

Four things are done about that and all four are in use.

Narrow the source. A distributed-feedback laser has a linewidth of a fraction of a picometre rather than a nanometre, which reduces the broadening by three orders of magnitude. This is the first and largest fix, and it is why fibre systems use single-frequency lasers rather than the cheaper broadband ones.

Compensate it. Splice in a length of fibre with the opposite sign of dispersion, chosen so the product of dispersion and length cancels. A dispersion-compensating module is a coil of specially designed fibre with a very large negative $D$, and every long span has one.

Move the zero. Use dispersion-shifted fibre, so the zero sits at the operating wavelength. This works beautifully for one channel and badly for many, for a reason in the next section.

Or use the dispersion. Which is the last section.

The other dispersion, which was solved first #

Before any of this mattered there was a much larger effect, and eliminating it is what made fibre a technology at all.

A fibre wide enough to support many modes gives each of them a different path length: a ray bouncing steeply down the core travels further than one running near the axis. The arrival times differ by a fraction of order the index contrast, which for an early step-index fibre was about one per cent — giving a spread of fifty nanoseconds per kilometre, four orders of magnitude worse than anything in this essay.

Two cures were found and both are still in use. Grading the index across the core, so that the steep rays spend their time in lower-index material and travel faster, cancels the difference to first order and brings the spread down by a factor of a hundred. The same profile that bends rays in a graded-index rod does it, and for the same reason.

Or make the core small enough that only one mode fits. A core of about nine microns at 1,550 nanometres supports a single mode, and modal dispersion is then not reduced but abolished — there is nothing for the light to take a different path in. Single-mode fibre is what every long link uses, and the residual chromatic dispersion this essay is about is what is left once the larger problem has been removed by construction.

Grading the index across the core was the first answer to modal dispersion and it is a partial one. A parabolic profile makes the steeper rays travel through a lower index and so faster, which compensates their longer path — but only to first order, and only for one wavelength. Making the core small enough to carry a single mode is the complete answer, and it is what leaves chromatic dispersion as the limit rather than one limit among several.

Why zero dispersion is not the right target #

The obvious plan — run at the zero — turns out to be a mistake as soon as a fibre carries more than one channel, and the reason is instructive.

Silica is very slightly nonlinear: its index rises by a tiny amount in proportion to the intensity. With several channels in one fibre, that nonlinearity mixes them, and one channel’s power can be transferred into another at a frequency given by the beat between them. This is four-wave mixing, and it is a phase-matching problem: it happens efficiently only when the interacting waves stay in step over a long distance.

Dispersion is what stops them staying in step. A little dispersion makes the channels travel at slightly different speeds, the interaction dephases within a few kilometres, and nothing accumulates. At exactly zero dispersion the phase matching is perfect and the crosstalk grows with the square of the length.

So a modern long-haul fibre is deliberately given a small non-zero dispersion — a few picoseconds per nanometre per kilometre — which is enough to spoil the phase matching and little enough to compensate at the end of the span. The optimum is not the extremum, which is a shape of answer that turns up whenever two effects have to be balanced rather than one minimised.

The case where the dispersion is the point #

The last option is the most elegant and it needs the anomalous sign.

Above the zero-dispersion wavelength, silica is anomalously dispersive: the longer wavelengths travel more slowly. The intensity-dependent index does the opposite thing to a pulse — it retards the peak relative to the wings, which shifts the leading edge to longer wavelengths and the trailing edge to shorter.

At one particular pulse shape and one particular power, those two effects cancel exactly, and the pulse propagates without changing shape at all. That is an optical soliton, it is a solution of the nonlinear Schrödinger equation, and it requires the anomalous sign — which is why it exists at 1,550 nanometres in ordinary fibre and not at 1,300.

Solitons were proposed in 1973, demonstrated in fibre in 1980, and carried real traffic in the 1990s. They are less used now than the flat arithmetic above suggests they should be, mostly because electronic compensation of dispersion turned out to be cheaper than maintaining the exact power a soliton needs. But the physics is the cleanest example in this collection of a nonlinearity being used to cancel a linear defect rather than being treated as one.

A packet spreads as it travels because its components move at different speeds, and everything in this essay is that picture inside a fibre. What a soliton does is arrange a second effect to undo it exactly — not to reduce it, not to compensate it at the far end, but to cancel it continuously at every point along the way, so the pulse has no history of having spread.

What the second derivative of the index controls is the ratio of the group speed to the phase speed, and a fibre’s operating point is chosen by looking for where that ratio stops changing with wavelength. That is a statement about a third derivative, which is why the zero-dispersion wavelength is a shallow crossing rather than a sharp one and why a source’s linewidth matters as much as its centre.

The measurement #

Dispersion is measured rather than assumed, and the standard method has a pleasing directness.

Send pulses at several wavelengths down the fibre and time their arrival relative to a reference. That gives the group delay against wavelength; differentiate it and the result is $D$. A hundred kilometres of fibre and a spread of ten nanometres gives delay differences of tens of nanoseconds, which is comfortable to measure, so the method works on installed links without disturbing them.

The subtlety is that the answer is an average over the length. A route made of spans from different manufacturers, spliced over decades, has a dispersion that varies along it, and only the total is measured. For linear compensation the total is all that is needed; for anything involving the nonlinearity — where the interaction depends on where along the fibre the dispersion is — the distribution matters and is much harder to obtain.

The other route is interferometric and works on short samples: put the fibre in one arm of an interferometer and measure the phase against wavelength directly. It gives the whole curve from a few metres of sample rather than the derivative of a delay, and it is what a manufacturer uses.

The fix that replaced the other three #

The four remedies above are the ones a system designer had until about 2010, and one development has since made most of them unnecessary — by changing what the receiver measures.

A conventional receiver is a photodiode, which responds to intensity. That throws away the phase, and dispersion is a phase effect: each frequency component is delayed by a different amount, so the information needed to undo it is exactly the information a photodiode discards. Once thrown away it cannot be recovered, which is why compensation had to be done optically, in the fibre, before detection.

A coherent receiver mixes the arriving light with a local laser and records both quadratures of the field, for both polarisations. What comes out is the complex field itself, sampled and digitised — and chromatic dispersion is then a linear filter with a known quadratic phase, which is inverted exactly by a digital filter with a few hundred taps. The pulses are put back together in a chip after the fact.

Two consequences follow and the second is the interesting one. Dispersion-compensating fibre disappeared from new builds: a coil of lossy fibre in every span, needing its own amplification, was replaced by arithmetic. And links are now deliberately built with the dispersion left in, because a large local dispersion makes the channels walk past one another quickly and so suppresses exactly the nonlinear mixing the section above describes. The optimum that used to be “a little dispersion, compensated span by span” has become “all of it, compensated once, in software”.

The same receiver also handles what this essay’s closing caveat says cannot be handled. Polarisation-mode dispersion drifts with temperature and with the cable being disturbed, so no fixed optical device can undo it — but a digital filter can be updated thousands of times a second, and adaptive equalisation tracks it continuously. A defect that was a hard limit became a routine one.

The same effect, used on purpose #

Dispersion is a nuisance in a fibre and an enabling tool in a laser, and the technique that made it one won a Nobel Prize.

The difficulty with amplifying a very short pulse is that its peak power is enormous: squeezing a modest energy into a hundred femtoseconds gives an intensity that damages the amplifier before the pulse leaves it. There is no way round that by building a tougher amplifier, because the damage threshold of glass is a property of glass.

The answer is to make the pulse long, amplify it while it is long, and make it short again. A strongly dispersive element — a pair of gratings, or a long length of fibre — stretches the pulse by a factor of a thousand or more, by sending its blue and red ends off at different speeds. The peak power falls by the same factor, so the pulse can be amplified by orders of magnitude without harming anything. A second dispersive element with the opposite sign then brings the components back together, and the pulse recompresses to something near its original duration with all the added energy in it.

That is chirped-pulse amplification, from 1985, and every high-intensity laser in the world uses it. It is what took laser peak powers from gigawatts to petawatts, and the same technique in a smaller form is what makes femtosecond surgical and machining lasers possible.

The engineering difficulty is exactly the subject of this page. Stretching and compressing must match not only in the second derivative but in the third and fourth, since a residue at any order leaves the recompressed pulse with wings that carry a substantial part of its energy. Designing a stretcher and a compressor whose dispersion curves agree to several orders over a wide band is most of the work, and it is the dispersion curve of this essay’s first figure being matched rather than minimised.

What it costs, and where the model stops #

The two contributions are only additive to first order. Treating material and waveguide dispersion as terms to be summed assumes the mode shape does not depend on the material’s dispersion, which is very nearly true for a weakly guiding fibre and less true for the high-contrast structures used in photonic-crystal fibre.

The waveguide term drawn here is a stand-in. It is a constant offset chosen to put the total zero at 1,310 nanometres, and a real waveguide contribution varies across the band. The material curve is computed and the total curve is illustrative — a distinction the figure states and worth restating, because it would be easy to read the dashed line as a measurement.

Higher orders matter at the zero. Where $D$ vanishes the slope of $D$ does not, and the broadening is then governed by the third derivative of the index. So a pulse at the zero-dispersion wavelength is not undistorted; it is distorted more slowly, and by a different functional form.

Polarisation supplies a dispersion of its own. A real fibre is not perfectly circular, so the two polarisations travel at slightly different speeds, and the difference wanders along the length. Polarisation-mode dispersion cannot be compensated by a fixed device because it changes with time, and it is the limit on the longest and fastest links.

And nothing here is about loss. Every curve is a delay, and the 0.2 dB per kilometre that decides the span length is a completely separate property with its own wavelength dependence. The two conspire only in the sense that they are minimised at different places.

The ladder from here #

Later rungs on this anchor: the third-order term and pulse distortion at the zero; polarisation-mode dispersion and why it is a statistical rather than a deterministic quantity; the nonlinear Schrödinger equation and the soliton’s stability against perturbation; dispersion management, where alternating spans of opposite sign give a link that is locally dispersive and globally flat; and photonic-crystal fibre, where the guidance mechanism is different enough that the dispersion can be made almost anything.

The neighbouring ladders are the rainbow’s angle, where the same material property separates colours in space, and the achromatic doublet, which cancels a first derivative with a second glass exactly as dispersion compensation cancels one with a second fibre. A packet spreading is the same second derivative acting on a quantum wave.