Four walls, and only one is stopping you

Dispersion tolerance falls with the square of the bit rate while loss and OSNR do not move at all. The same route, at four times the rate, reaches a sixteenth as far.

Only one limit is stopping you

An optical link can be stopped by four independent things, and at any given moment exactly one of them is doing it. The other three have slack. Money and effort spent on them produce nothing measurable.

That would be a simple problem if the answer stayed put. It does not: the four limits scale completely differently, and which one binds flips as the bit rate changes.

The four, and how each one scales

Loss is linear in distance. Transmit power minus receiver sensitivity, minus fixed losses and margin, divided by attenuation. It is the limit everyone learns first, and on a modern system it is often not the one that matters.

OSNR takes over the moment amplifiers appear, and it falls by 3 dB every time the span count doubles. Noise is permanent — nothing downstream can un-add it. Reach in an amplified system is a budget being spent, not a distance being covered.

Chromatic dispersion is linear in distance and quadratic in bit rate. That second part is what catches people.

PMD grows as the square root of distance, which makes it forgiving over long routes and almost useless as a lever — halving your PMD requires quartering the route.

The inverse square law

Dispersion tolerance falls with the square of the bit rate, because raising the rate both shortens the bit period and widens the signal's spectrum, and the two compound.

On standard single-mode fibre at 17 ps/nm/km, that produces this:

• 2.5 Gbit/s — tolerance 16,000 ps/nm, dispersion reach about 940 km. Loss binds first, at 88 km.
• 10 Gbit/s — tolerance 1,000 ps/nm, reach about 59 km. Dispersion now binds, ahead of loss.
• 40 Gbit/s — tolerance 62.5 ps/nm, reach about 3.7 km.
• 100 Gbit/s — a few hundred metres.

Nothing about the fibre changed between those rows. The route is identical. Four times the rate is one sixteenth of the tolerance, and a system that had comfortable margin becomes unbuildable.

This is why direct detection ran out of road, and why coherent receivers — which undo dispersion electronically after detection — were not an incremental improvement. They removed the wall.

Amplifiers fix loss and cost OSNR

The instinct when a link is loss-limited is to add amplification, and it works: optical power comes back. What comes with it is spontaneous emission noise, and unlike loss, noise cannot be recovered from.

Doubling the span count costs exactly 3 dB of OSNR. Ten spans to twenty is 3 dB; twenty to forty is another 3. It also means the first span is the expensive one to get wrong: a noise figure one decibel worse costs a decibel at the far end regardless of how many spans follow.

The span length nobody builds

Here is a result that falls straight out of the OSNR expression and surprises most people who see it.

For a fixed total distance, OSNR is maximised at a span length of 10/(α·ln 10) — about 20 km at typical attenuation. Not 80. Not 100. Twenty.

Almost nobody builds that, and they are right not to. Amplifier sites need buildings, power, access and maintenance, and four times as many of them is not a trade anyone makes for a few decibels. But it is a trade, and the number is worth knowing, because "our spans are 80 km" is a cost decision wearing the clothes of a technical one.

PMD is the one you cannot fix

Chromatic dispersion is deterministic: a fixed amount of opposite dispersion, or electronic compensation at a coherent receiver, cancels it exactly. PMD is not deterministic. The delay between polarisation modes varies randomly along the fibre and changes with temperature and physical disturbance, so there is no fixed correction to apply.

When PMD is the binding limit, the options are a lower bit rate, a different route, or new cable. There is no equipment upgrade. This is mostly a problem of older plant — fibre installed before the mid-nineties can be several times worse than modern cable, and it is exactly that fibre that operators try to run at higher rates.

The square-root growth also means it presents strangely. Because the figure is a statistical mean, with instantaneous delay varying around it, PMD problems appear as intermittent errors that correlate with temperature and with someone working near the cable — rather than as steady degradation that would be easy to diagnose.

How to use this

Find the binding limit before improving anything. It is the only one worth money.

Check how much room there is to the next one. If dispersion binds at 59 km and loss at 62, fixing dispersion buys three kilometres and is not worth doing alone.

Re-check at the rate you are actually going to run. A route qualified at 10 Gbit/s tells you very little about the same route at 100.

What a comparison like this leaves out

Nonlinear effects are absent, and they matter: they set the upper bound on launch power and are the reason more power stops helping past an optimum. Raman amplification, FEC coding gain, ROADM filter narrowing across a cascade, and interaction between channels in a loaded system are all missing too.

The dispersion tolerance is scaled from a stated reference point, which depends on modulation format and on how much penalty is acceptable. Change the format and you change the reference, not the law.

Frequently asked questions

Which limit actually stops an optical link?

Whichever of loss, OSNR, chromatic dispersion and PMD gives the shortest reach — and only that one. The other three have slack, and improving them produces nothing measurable. Find the binding limit before spending anything.

Why does the binding limit change with bit rate?

Because dispersion tolerance falls with the square of the rate while loss and OSNR do not move at all. On standard fibre, the same route is loss-limited at 88 km at 2.5 Gbit/s, dispersion-limited at 59 km at 10, and reaches under four kilometres at 40. The fibre is identical.

Why is dispersion quadratic in bit rate?

Because raising the rate both shortens the bit period and widens the signal's spectrum, and the two effects compound. Four times the rate leaves a sixteenth of the tolerance.

Why were coherent receivers such a big deal?

They undo chromatic dispersion in the electrical domain after detection, which removes the wall direct detection had run into. It is the reason hundred-gigabit transport over installed fibre exists.

If amplifiers fix loss, why not add more?

Because each one adds spontaneous emission noise, and noise is permanent — nothing downstream can remove it. Doubling the span count costs exactly 3 dB of OSNR, so amplified reach is a budget being spent rather than a distance covered.

Is there an optimal span length?

Yes: 10/(α·ln 10), about 20 km at typical attenuation. Almost nobody builds it, because amplifier sites need buildings, power, access and maintenance. That is a sound cost decision — but it is a trade, and worth knowing the size of rather than assuming 80 km spans are technically optimal.

Why can PMD not be compensated?

Because it varies randomly along the fibre and changes with temperature and disturbance, so there is no fixed correction. Chromatic dispersion is deterministic and cancels exactly; PMD does not. When PMD binds, the options are a lower rate, a different route, or new cable.

Why do PMD problems look intermittent?

Because the figure is a statistical mean and the instantaneous delay varies around it. A link can briefly exceed a budget it comfortably meets on average, so the symptom is errors correlating with temperature and with work near the cable, not steady degradation.