Repeater Chain, OSNR and Reach

What is modelled

A long submarine fibre cable is split into spans of about 50 to 100 km. The light loses power in each span, and an optical repeater restores it, adding a little amplified spontaneous emission noise. These noise contributions add up, so the optical signal-to-noise ratio falls as 1/N. Raising the launch power helps against that noise, but the glass fibre itself becomes nonlinear and adds distortion that grows as the cube of the power, which gives an optimum.

Equations

• Noise of one repeater in the 12.5 GHz reference band PASE = NF · hν B G, with G equal to the span loss, and hν B = −57.95 dBm.
• OSNR = P / (N PASE), in decibels P + 57.95 − A − NF − 10 log N.
• Nonlinear noise η P3 per span, so SNR = P / [N (PASE + η P3)]; the optimum Popt = (PASE / 2η)1/3 has a nonlinear noise of exactly half the amplifier noise.
• QPSK error rate BER = ½ erfc(√(SNR/2)), with the symbol rate 32 GBd; the soft-decision FEC limit is 2 × 10⁻².

Checks run in the page

• Noise accumulation through the chain against the closed form.
• The optimum launch power found by scanning against the analytic one.
• The error-rate formula against Simpson integration of the Gaussian tail.
• The number of spans reached against the closed form.

Honest simplifications

The nonlinear noise follows a simplified Gaussian-noise scaling with a single coefficient, the number of spans does not change the nonlinear exponent, polarisation, chromatic dispersion, filtering and the transceiver penalties are left out, and one channel stands for the whole band. This is a teaching budget and not a design tool for a real cable system.

• Press Overdriven by 6 dB and see the nonlinear noise pull the signal-to-noise ratio below the optimum.
• Press Underpowered by 6 dB to see the opposite, noise-dominated, case.
• Press Trans-Pacific and watch the margin to the error-correction limit shrink to almost nothing.