208 CHAPTER 5. LIGHTWAVE SYSTEMS
power. Furthermore, the two modes are anticorrelated in such a way that the main-
mode power drops belowP ̄m/2 whenever side-mode power exceedsP ̄m/2, so that the
total power remains nearly constant [85]. Thus, an error would occur even for “1” bits
wheneverPs>P ̄m/2. Since the two terms in Eq. (4.5.2) make equal contributions, the
BER is given by [84]
BER=
∫∞P ̄m/ 2p(Ps)dPs=exp(
−
P ̄m
2 P ̄s)
=exp(
−
Rms
2)
. (5.4.9)
The BER depends on the MSR defined asRms=P ̄m/P ̄sand exceeds 10−^9 when MSR<
42.
To calculate the MPN-induced power penalty in the presence of receiver noise,
one should follow the analysis in Section 4.5.1 and add an additional noise term that
accounts for side-mode fluctuations. For ap–i–nreceiver the BER is found to be [85]
BER=
1
2
erfc(
Q
√
2
)
+exp(
−
Rms
2+
R^2 ms
4 Q^2)[
1 −
1
2
erfc(
Q
√
2
−
Rms
Q√
2
)]
, (5.4.10)
where the parameterQis defined by Eq. (4.5.10). In the limit of an infinite MSR, Eq.
(5.4.10) reduces to Eq. (4.5.9). For a noise-free receiver (Q=∞), Eq. (5.4.10) reduces
to Eq. (5.4.9). Figure 5.9 shows the BER versus the power penalty at a BER of 10−^9 as
a function of MSR. As expected, the power penalty becomes infinite for MSR values
below 42, since the 10−^9 BER cannot be realized irrespective of the power received.
The penalty can be reduced to a negligible level (<0.1 dB) for MSR values in excess
of 100 (20 dB).
The experimental measurements of the BER in several transmission experiments
show that a BER floor above the 10−^9 level can occur even for DFB lasers which ex-
hibit a MSR in excess of 30 dB under continuous-wave (CW) operation [88]–[91].
The reason behind the failure of apparently good lasers is related to the possibility of
side-mode excitation under transient conditions occurring when the laser is repeatedly
turned on and off to generate the bit stream. When the laser is biased below threshold
and modulated at a high bit rate (B≥1 Gb/s), the probability of side-mode excitation
aboveP ̄m/2 is much higher than that predicted by Eq. (5.4.8). Considerable atten-
tion has been paid to calculate, both analytically and numerically, the probability of
the transient excitation of side modes and its dependence on various device parame-
ters [87]–[98]. An important device parameter is found to be thegain marginbetween
the main and side modes. The gain margin should exceed a critical value which de-
pends on the bit rate. The critical value is about 5–6 cm−^1 at 500 Mb/s [88] but can
exceed 15 cm−^1 at high bit rates, depending on the bias and modulation currents [93].
The bias current plays a critical role. Numerical simulations show that the best perfor-
mance is achieved when the DFB laser is biased close to but slightly below threshold
to avoid the bit-pattern effects [98]. Moreover, the effects of MPN are independent of
the bit rate as long as the gain margin exceeds a certain value. The required value of
gain margin exceeds 25 cm−^1 for the 5-GHz modulation frequency. Phase-shifted DFB
lasers have a large built-in gain margin and have been developed for this purpose.