OSNR, Dispersion & PMD Calculator

How much signal-to-noise survives a chain of amplifiers, how far a pulse spreads on the way, and why polarisation mode dispersion grows as a square root rather than a straight line.

The OSNR model here is the simplified one Identical spans, identical amplifiers, noise adding linearly, and no nonlinear penalty at all. Real launch power cannot be raised indefinitely: past an optimum, nonlinear effects degrade the signal faster than extra power improves it. This model does not show that turning point, so it is useful for comparing designs and for knowing roughly when to stop — not as a substitute for a vendor's planning tool.

OSNR over a chain of spans

The formula, with your numbers in it

The constant everyone writes as 58 is −10·log10(h·ν·Δν) in dBm, where Δν is the 0.1 nm reference bandwidth at the wavelength in question. Because it depends on wavelength it is not really a constant, so it is computed from Planck's constant rather than typed in.

The amplifier chain

SpanLoss dBGain dBMax out dBm
Leave the gain blank to match it to the span loss.

Chromatic dispersion

Polarisation mode dispersion

Why every amplifier costs you

An optical amplifier does not distinguish signal from noise. It amplifies what arrives and adds its own spontaneous emission on top, so each one improves the power and degrades the signal-to-noise ratio. A chain of amplifiers is a chain of small, permanent losses of quality.

The arithmetic is unforgiving in a specific way: doubling the number of spans costs exactly 3 dB of OSNR. Ten spans to twenty is 3 dB. Twenty to forty is another 3. That is why long-haul reach is discussed in terms of a budget being spent rather than a distance being covered.

It also means the first span is the expensive one to get wrong. A noise figure a decibel worse costs a decibel at the end regardless of how many spans follow, and a span two decibels lossier costs the same two decibels forever.

Where this model stops being true

In the arithmetic above, every decibel of extra launch power buys a decibel of OSNR, without limit. Reality does not work that way, and the reason is worth understanding rather than working around.

At high power the glass stops behaving linearly. Its refractive index varies slightly with intensity, so the signal modulates its own phase and that of its neighbours. Those effects grow rapidly with power while the noise benefit grows only linearly, so there is an optimum launch power — and past it, adding power makes the system worse.

This model has no nonlinear term, so it will happily tell you that +15 dBm gives a wonderful OSNR. Treat any figure much above a few dBm per channel as a number the model cannot support, and get the optimum from the equipment vendor's tool, which knows the fibre type and the channel plan.

Two kinds of spreading

Chromatic dispersion is deterministic. Different wavelengths travel at different speeds, a pulse contains a range of wavelengths, so it spreads — and it spreads linearly with distance. Because it is predictable it can be compensated, either with fibre of opposite dispersion or, in coherent systems, electronically at the receiver. That is why modern coherent transponders tolerate accumulated dispersion that would have been impossible fifteen years ago.

Polarisation mode dispersion is not deterministic, and that is what makes it awkward. The two polarisation modes travel at slightly different speeds, but the difference varies randomly along the fibre and changes with temperature and physical disturbance. The contributions partly cancel, so PMD grows with the square root of length rather than linearly — quadrupling the distance only doubles the PMD.

The figure you calculate is a statistical mean. The instantaneous delay follows a Maxwellian distribution about it, so a link can briefly exceed a budget it comfortably meets on average. That is why PMD problems present as intermittent errors that correlate with weather and with someone working near the cable, rather than as a steady degradation.