206 CHAPTER 5. LIGHTWAVE SYSTEMS
In the case of multimode semiconductor lasers, the power penalty can be calculated
by following an approach similar to that of Section 4.6.2 and is given by [74]
δmpn=−5 log 10 ( 1 −Q^2 r^2 mpn), (5.4.5)wherermpnis the relative noise level of the received power in the presence of MPN.
A simple model for estimating the parameterrmpnassumes that laser modes fluctuate
in such a way that the total power remains constant under CW operation [75]. It also
assumes that the average mode power is distributed according to a Gaussian distribution
of RMS widthσλand that the pulse shape at the decision circuit of the receiver is
described by a cosine function [74]. Different laser modes are assumed to have the
same cross-correlation coefficientγcc, i.e.,
γcc=
〈PiPj〉
〈Pi〉〈Pj〉(5.4.6)
for alliandjsuch thati =j. The angular brackets denote an average over power
fluctuations associated with mode partitioning. A straightforward calculation shows
thatrmpnis given by [78]
rmpn=(k/√
2 ){ 1 −exp[−(πBLDσλ)^2 ]}, (5.4.7)where the mode-partition coefficientkis related toγccask=
√
1 −γcc. The model
assumes that mode partition can be quantified in terms of a single parameterkwith
values in the range 0–1. The numerical value ofkis difficult to estimate and is likely
to vary from laser to laser. Experimental measurements suggest that the values ofkare
in the range 0.6–0.8 and vary for different mode pairs [75], [80].
Equations (5.4.5) and (5.4.7) can be used to calculate the MPN-induced power
penalty. Figure 5.8 shows the power penalty at a BER of 10−^9 (Q=6) as a function of
the normalized dispersion parameterBLDσλfor several values of the mode-partition
coefficientk. For a given value ofk, the variation of power penalty is similar to that
shown in Fig. 5.7;δmpnincreases rapidly with an increase inBLDσλand becomes
infinite whenBLDσλreaches a critical value. Fork> 0 .5, the MPN-induced power
penalty is larger than the penalty occurring due to dispersion-induced pulse broaden-
ing (see Fig. 5.7). However, it can be reduced to a negligible level (δmpn< 0 .5dB)by
designing the optical communication system such thatBLDσλ< 0 .1. As an example,
consider a 1.3-μm lightwave system. If we assume that the operating wavelength is
matched to the zero-dispersion wavelength to within 10 nm,D≈1 ps/(km-nm). A
typical value ofσλfor multimode semiconductor lasers is 2 nm. The MPN-induced
power penalty would be negligible if theBLproduct were below 50 (Gb/s)-km. At
B=2 Gb/s the transmission distance is then limited to 25 km. The situation becomes
worse for 1.55-μm lightwave systems for whichD≈16 ps/(km-nm) unless dispersion-
shifted fibers are used. In general, the MPN-induced power penalty is quite sensitive
to the spectral bandwidth of the multimode laser and can be reduced by reducing the
bandwidth. In one study [83], a reduction in the carrier lifetime from 340 to 130 ps,
realized byp-doping of the active layer, reduced the bandwidth of 1.3-μm semicon-
ductor lasers by only 40% (from 5.6 to 3.4 nm), but the power penalty decreased from
an infinite value (BER floor above 10−^9 level) to a mere 0.5 dB.