9.3.1 Raman CrosstalkAs discussed in Section 4.4, SRS is generally not of concern for single-channel systems because of its relatively high threshold (about 500 mW near 1.55 µm). The situation is different for WDM systems because the transmission fiber can act as a Raman amplifier that is pumped by the multiwavelength signal launched into the fiber. Each channel is amplified by all shorter-wavelength channels as long as the wavelength difference is within the bandwidth of the Raman gain. The shortest-wavelength channel is most depleted as it can pump all other channels simultaneously. Variations in channel powers induced by Raman-induced interaction are one source of concern. However, this problem can be addressed in practice by adjusting the launch powers at the transmitter end or by employing a suitable optical filter at each amplifier. Even of more concern is the fact that the power transfer between any two channels is time-dependent because it depends on the bit patterns of those channels. Clearly, amplification can occur only when 1 bits are present in both channels simultaneously and pulses inside them overlap, at least partially. As bit patterns are pseudo-random in nature, power transferred to each channel through SRS fluctuates and acts as a source of noise during the detection process. Such Raman-induced crosstalk can degrade the performance of a WDM system, if left uncontrolled, and its impact has been considered in several studies [45]-[57]. Raman crosstalk can be avoided if channel powers are made so small that SRS-induced amplification is negligible over the entire fiber length. It is thus important to estimate the limiting value of the channel power. A simple model considers the depletion of the highest-frequency channel in the worst case in which 1 bits of all channels overlap completely [40]. The amplification factor for the mth channel is Gm = exp(gmLeff), where the Raman gain gmand the effective interaction length Leff are given by 
with Ωm = ω1 - ωm. For gmLeff << 1, Gm≈ 1 + gmLeff, and the shortest-wavelength channel at ω1 is depleted by a fraction gmLeff owing to the amplification of the mth channel. The total depletion for an M-channel WDM system can be written as 
The summation in Eq. (9.3.2) can be carried out analytically if the Raman gain spectrum (see Figure 4.10) is approximated by a triangular profile such that gR increases linearly for frequencies up to 15 THz with a slope SR = dgR/dv and then drops to zero. Using gR(Ωm) = mSRΔvch, the fractional power loss for the shortest-wavelength channel becomes [40] 
where CR= SRΔvch/(2Aeff). In deriving this equation, channels were assumed to have a constant spacing Δvch and the Raman gain for each channel was reduced by a factor of 2 to account for the random polarization states of different channels. A more accurate analysis should consider not only depletion of each channel because of power transfer to longer-wavelength channels but also its own amplification by shorter-wavelength channels. If all other nonlinear effects are neglected along with GVD, the evolution of the power Pnassociated with the nth channel is governed by the following equation (see Section 4.4.2): 
where α is assumed to be the same for all channels. This set of M coupled nonlinear equations can be solved analytically. For a fiber of length L, the result is given by [47] 
where Pt= ΣMm=1 Pm(0) is the total input power in all channels. This equation shows that channel powers follow an exponential distribution because of Raman-induced coupling among all channels. The depletion factor DR for the shorter-wavelength channel (n = 1) is obtained using DR = ( 1 - P1) /1, where 1 = P1(0)exp(-αL) is the channel power expected in the absence of SRS. In the case of equal input powers in all channels, Ptequals MPch in Eq. (9.3.5), and DR is given by 
In the limit M2CRPchLeff this complicated expression reduces to the simple result in Eq. (9.3.3). In general, Eq. (9.3.3) overestimates the Raman crosstalk. The Raman-induced power penalty is obtained using δR= -10 1og(l - DR) because the input channel power must be increased by a factor of (l - DR)-1 to maintain the same system performance. Figure 9.8 shows how the power penalty increases with an increase in the channel power and the number of channels. The channel spacing is assumed to be 100 GHz. The slope of the Raman gain is estimated from the gain spectrum to be SR = 4.9 x 10-18 m / (W-GHz) while Aeff = 50µ2 and Leff ≈ 1/α = 21.74 km. As seen from Figure 9.8, the power penalty becomes quite large for WDM systems with a large number of channels. If a value of at most 1 dB is considered acceptable, the limiting channel power Pch exceeds 10 mW for 20 channels, but its value is reduced to below 1 mW when the number of WDM channels is larger than 70.
Figure 9.8: Raman-induced power penalty as a function of channel number for several values of Pch. Channels are 100 GHz apart and are launched with equal powers.
The foregoing analysis provides only a rough estimate of the Raman crosstalk as it neglects the fact that signals in each channel consist of a random sequence of 0 and 1 bits. It is intuitively clear that such pattern effects will reduce the level of Raman crosstalk. A statistical analysis shows that the Raman crosstalk is lower by about a factor of 2 when signal modulation is taken into account [46]. The GVD effects that were neglected in the above analysis also reduce the Raman crosstalk since pulses in different channels travel at different speeds because of the group-velocity mismatch [48]. Both the pattern and walk-off effects can be included if we replace Eq. (9.3.4) with
where vgn is the group velocity of the nth channel and Pn(z,t) is the time-dependent channel power containing all pattern information. The set of equations (9.3.7) is not easy to solve analytically. Consider, for simplicity, power transfer between two channels by setting M = 2. The resulting two equations can be written as 
where dw= -  is the walk-off parameter in a frame in which pulses for channel 2 are stationary [39]. If we neglect pump depletion, Eq. (9.3.8) has the solution P1(L, t) = P1(0,t - dwz)e-αz. Using this solution in Eq. (9.3.9) and integrating over a fiber section of length L, we obtain P2(L, t)= P2(0,t)exp[x2(t) - αL], where 
governs the extent of Raman-induced power transfer. We can extend this approach for M interacting channels by adding contributions from all channels. Fluctuations in the power of the nth channel are then given by 
where dmn=  -  . Because of pseudo-random bit patterns in all channels, xn(t) fluctuates with time in a random fashion. When the number of channels is large, xn(t) represents a sum of many independent random variables and is expected to follow a Gaussian distribution from the central limit theorem. Since the channel power scales with xn(t) exponentially, it follows a log-normal distribution [51]. However, if powers are expressed in dBm units, channel power is related to xnlinearly, and its fluctuations obey a Gaussian distribution. From a practical standpoint, the first two moments of xnare most relevant. The average value µxrepresents the Raman-induced change in the average power. If channel powers are equalized at each amplifier, the crosstalk is governed by the variance σ2x. The ratio σx/µxis often used as a measure of the Raman crosstalk. These quantities can be calculated in an analytic form in some cases [46]. The preceding discussion applies to a single fiber segment. For a realistic WDM system, one must consider dispersion management and add the contributions of multiple fiber segments separated by optical amplifiers [57]. In the case of distributed amplification, the WDM signal is amplified within the same fiber where the signal is degraded through SRS. The periodic power variations can be included by replacing the factor e-αzin Eq. (9.3.11) with p(z), introduced first in Section 3.2.2 and obtained by solving Eq. (3.2.6). The details of the dispersion map enter into Eq. (9.3.11) through the walk-off parameter dmthat takes on different values in each fiber segment used to form the dispersion map. In general, crosstalk depends on details of the dispersion map and is reduced considerably when the dispersion is not fully compensated in each map period.
Figure 9.9: (a) Accumulated dispersion in one 80-km map period for four types of maps and (b) Raman crosstalk after 400 km for a WDM system whose 105 channels are separated by 200 GHz and launched with 6.3-mW power. (After Ref. [57]; ©2003 IEEE.)
Figure 9.9 shows calculated values of σx for a 105-channel (separated by 200 GHz) WDM system operating over a 400-km link with four types of dispersion maps. Each 40-Gb/s channel is launched with 6.3 mW of average power. Amplifiers are placed 80 km apart [57]. The type-1 map consists of a standard single-mode fiber (SMF) followed with a DCF. The maps of types 2 to 4 are designed using equal lengths of SMF and negative-dispersion fiber (NDF) but the map periods are 80, 40, and 20 km, respectively. For maps labeled type 1' and type 2', dispersion is not fully compensated (residual dispersion 130 ps/nm). The smallest crosstalk occurs for the type-1 map for which accumulated dispersion is high over most of the map period. It increases for the remaining three maps and becomes largest for the map with the shortest map period. Thus, dense dispersion management, although useful for several other reasons, makes the Raman crosstalk worse. This can be understood by noting that pulses in neighboring channels follow a zigzag path as they traverse from the SMF to the RDF section in a repetitive fashion. If the map period is small, two pulses that overlap initially never fully separate from each other. Clearly, Raman-induced power transfer is worst under such conditions. As seen in Figure 9.9, residual dispersion can be used to lower the level of Raman crosstalk.
Periodic amplification of the WDM signal can also magnify the impact of SRS-induced degradation. The reason is that in-line amplifiers add noise, which experiences less Raman loss than the signal itself, resulting in degradation of the SNR. Numerical simulations show that it can be reduced by inserting optical filters along the fiber link that block the low-frequency noise below the longest-wavelength channel [53]. Raman crosstalk can also be reduced using the technique of midspan spectral inversion [49].
Figure 9.10: Power penalty as a function of Raman crosstalk in four cases in which ASE noise follows a χ2 or Gaussian distribution and Raman-induced noise follows a log-normal or Gaussian distribution. (After Ref. [57]; ©2003 IEEE.)
How much Raman crosstalk can be tolerated in a WDM system? To answer this question, one must consider the BER at the receiver, assuming that the signal is corrupted both by amplified spontaneous emission (ASE) noise and Raman-induced noise. It should be kept in mind that the two noise sources may not follow the same statistics. The impact of ASE noise has been discussed in Section 6.4. If we assume that both noise sources are Gaussian in nature, one can simply add a third term of σ2SRS to the definition of σ1 in Eq. (6.4.3). A more precise treatment should follow the approach of Section 6.4.2 and employ the log-normal distribution associated with Raman crosstalk. In all cases, power penalty (increase in signal power required to maintain the same BER) can be calculated as a function of σx. Figure 9.10 shows this power penalty in four cases in which ASE noise follows a χ2 or Gaussian distribution and Raman-induced noise follows a log-normal or Gaussian distribution [57]. The combination of log-normal with χ2 distribution is the most accurate. It shows that the power penalty can be kept below 1 dB for σx < 0.5 dB. One can use this condition to find the maximum distance over which a system can operate in the presence of Raman crosstalk. The answer depends on the dispersion map, the number of WDM channels, and the power launched into each channel. For type-1 and type-2 dispersion maps in Figure 9.9, the distance exceeds 5,000 km even for a 70-channel WDM system (40 Gb/s per channel) if the channel power is kept below 2 mW. Thus, the problem of Raman crosstalk can be solved to a large extent by designing the WDM system appropriately.
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