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Digital communications
Digital communications

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2.5 Non-Linearity

A linear system can be defined in two ways: (1) one which obeys the principle of superposition, and (2) one possessing the frequency-preservation property.

If we consider an optical fibre with electromagnetic field as the input and output, then provided that the power level of the input signal is not too great (less than 1 mW, which is 0 dBm), the fibre may be well modelled by a linear system for most purposes.

When fibre is used for a single point-to-point link to convey a digital signal by on-off keying, it is not too serious if the fibre does not behave precisely as a linear system. When fibre is used in a more sophisticated way, however, linearity becomes important. When using wavelength division multiplexing (WDM), for example, superposition is important.

Activity 7

What might happen if the fibre carrying WDM signals does not behave as a linear system?


If superposition does not hold in the fibre, one consequence is that the presence of one signal on the fibre will affect the output of the other signal, so there will be crosstalk between the signals. There will also be new frequencies generated by the interaction between different frequencies on the fibre.

When the power levels in the fibre increase significantly above 0 dBm, nonlinear effects can become significant, and this is an important consideration in dense wavelength division multiplexing. Note that the degree of non-linearity depends upon the total power in the fibre, and adding more wavelength channels also increases the power and therefore increases the amount of non-linearity.

Dense wavelength division multiplexing, DWDM, is WDM with closely-spaced wavelengths.

Non-linearity is not always ‘bad’ however, and in fact non-linearity is exploited in optical amplifiers as discussed in Section 3.3.

There are several different types of non-linearity in optical fibre. You do not need to know much about them for this unit, but it is useful if you have met the names. You should appreciate that they all refer to non-linear effects in fibre, and that they become more significant at higher power levels.

In summary, the most important ones are:

  • self-phase modulation, SPM

  • cross-phase modulation, XPM

  • stimulated Brillouin scattering, SBS

  • four-wave mixing, FWM

  • stimulated Raman scattering, SRS

Of these, four-wave mixing (FWM) in particular was found to have potentially serious consequences for dense wavelength division multiplexing, and stimulated Raman scattering is important because it is the process behind Raman amplifiers (discussed later).

Four-wave mixing is an effect in which light at different wavelengths interacts to generate light at another wavelength. The effect is characterised in terms of the frequencies of the waves, and in general light at four frequencies are involved: f1, f2, f3 and fnew. The light with the three frequencies f1, f2 and f3 interact due to non-linearities in the refractive index and generate light at the new frequency, fnew where

However f2 can be the same as f1 so that four-wave mixing involving only three different frequencies is possible. In this case:

It would be misleading to call this ‘three-wave mixing’ even though there are only three frequencies involved, because the frequency relationship involves four terms. Notice that all four terms can be closely spaced (e.g. they could all be optical frequencies in the 1550 nm window). If you had only three terms, one of the frequencies would have to be substantially different from the other two. Thus if fnew = f1f3 and f1 is close to f3, then fnew is much smaller than either of f1 or f3. This effect, where a new, much lower, frequency is generated by the difference between two closely-spaced frequencies, is called beating. Beating can be a problem at receivers where two closely-spaced optical signals can generate a new electrical signal at the beat frequency. I do not wish to consider beating any further, however, and return now to the discussion of four-wave mixing.

Four-wave mixing is damaging to DWDM signals for two reasons: it attenuates the signals and it introduces new signals which can cause interference.

For FWM to be significant in fibre it is necessary for the phase relationship between the various light signals to remain constant over long distances. This occurs if the dispersion is low, so, ironically, FWM in the 1550 nm window is worse in G.653 dispersion-shifted fibre than in standard G.652 fibre.

Activity 8

There was another way of reducing the effects of dispersion – an alternative to using fibre with a low dispersion coefficient – that was described earlier. What was it?


It is possible to use dispersion compensation.

It is possible therefore to reduce pulse spreading due to dispersion as well as avoiding the problems of FWM by using standard single-mode fibre combined with dispersion compensation.

A solution which avoids the need for dispersion compensation is to use fibre which has low but not zero dispersion in the 1550 nm widow. This is the reason for another single-mode fibre specified by the ITU-T G.655 standard – non-zero dispersion-shifted fibre (NZ-DSF).

Raman scattering (C. V. Raman, Indian physicist (1888–1970)) is a process in which light is scattered by the material through which it propagates, but in the process it changes its wavelength – specifically, the wavelength gets longer. In stimulated Raman scattering (SRS) light at the longer wavelength is already present in the medium, and increases (stimulates) the scattering. The ‘pump’ light is injected at the amplifier to provide a source of power for the amplification process. This is considered further in Section 3.3. In fibres carrying DWDM, SRS has the effect of transferring power from channels at shorter wavelengths to channels at longer wavelengths.


What is dispersion-shifted fibre, and why has it become necessary in recent years when standard single-mode fibre was previously considered adequate?

Under what conditions is non-zero dispersion-shifted fibre preferable to dispersion-shifted fibre?


Standard single-mode fibre has the minimum dispersion in the 1300 nm transmission window. Dispersion-shifted fibre has the dispersion minimum shifted to the 1550 nm window.

Dispersion limits the bandwidth of single-mode fibre, so for high signalling rates low dispersion is required. The highest signalling rates can therefore be used at the wavelengths near the dispersion minimum.

Previously, the wavelengths used for transmission were in the 1300 nm window, which is therefore compatible with high signalling rates on standard single-mode fibre.

More recently the 1550 nm window has been used instead, because erbium-doped fibre amplifiers have been developed and these only operate around 1550 nm. Using standard single-mode fibre the higher dispersion would significantly limit the signalling rate at 1550 nm, so dispersion-shifted fibre is used instead.

Although low dispersion is desirable to minimise pulse spreading, it can increase the problems caused by non-linear effects in the fibre. In particular, when dense wavelength division multiplexing (DWDM) is used on a fibre, four-wave mixing (FWM) can be a significant problem if the dispersion is very low. DWDM uses the 1550 nm window, so the requirement for higher dispersion to reduce FWM can be achieved by going back to using standard single-mode fibre. The problem of high dispersion then returns, but that can be overcome through the use of dispersion compensation.

Dispersion compensation is complex to implement, so an alternative, preferable, solution is to use non-zero dispersion-shifted fibre. This has low enough dispersion in the 1550 nm window to allow high signalling rates without the need for dispersion compensation, but the dispersion is high enough to prevent problems with FWM.