9.2.2 In-Band Linear Crosstalk

In-band crosstalk, resulting from WDM components used for routing and switching along an optical network, has been of concern since the advent of WDM systems [19]-[33]. Its origin can be understood by considering a static routing device such as a waveguide grating router (see Section 9.2 of LT1). For a router with N + 1 input and N + 1 outport ports, there exist (N + 1)² combinations through which a WDM signal with N + 1 wavelengths can be split. Consider the output at one wavelength, say,λ₀. Among the N(N + 2) interfering components that can accompany the desired signal, N components have the same carrier wavelength λ₀, while the remaining N(N + 1) belong to different carrier wavelengths and produce out-of-band crosstalk. The N interfering signals at the same wavelength originate from incomplete filtering by the routing device and produce in-band crosstalk. The total electrical field reaching the receiver can be written as [21]

where A₀ is the desired signal at the frequency ω₀ = 2πc/λ₀. The coherent nature of the in-band crosstalk is evident from Eq. (9.2.5).

To study the impact of in-band crosstalk on system performance, we consider the photocurrent generated at the receiver. Similar to the case of ASE discussed in Section 6.4.1, the receiver current I(t) = R_dE_r(t)², where R_d is the responsivity of the photodetector, contains interference or beat terms, in addition to the desired signal. One can identify two types of beat terms; signal-crosstalk beating resulting in terms like A^*₀A_n and crosstalk-crosstalk beating with terms like A^*_kA_n, where k ≠ 0 and n ≠ 0. The latter terms are relatively small in practice. If we ignore them, the receiver current is given by

where P_n = A_n² is the power and Φ_n(t) is the phase. In practice, P_n<< P₀because a WGR is built to reduce this kind of crosstalk.

Figure 9.4: Measured (a) probability densities as a function of N and (b) BER curves for several values of X when N = 16. (After Ref. [21]; ©1996 IEEE.)

Since bit patterns in each channel change in an unknown fashion, and phases of all channels are likely to fluctuate randomly, each term in the sum in Eq. (9.2.6) acts as an independent random variable. We can thus write the photocurrent as I(t) = R_d(P₀ + ΔP) and treat the crosstalk as intensity noise. Even though each term in ΔP is not Gaussian, their sum follows a Gaussian distribution from the central limit theorem when N is relatively large. Indeed, the experimentally measured probability distributions shown in Figure 9.4(a) indicate that ΔP becomes a nearly Gaussian random variable for values N as small as 8 [21]. The BER curves in Figure 9.4(b) were measured in the case of N = 16 for several values of the crosstalk level, defined as X = P_n/P₀, with P_nbeing constant for all sources of in-band crosstalk. Considerable power penalty was observed for values of X > -35 dB.

We can use the approach of Section 5.4.2 for calculating the power penalty. In fact, the result is the same as in Eq. (5.4.11) and can be written as

where

and X is assumed to be the same for all N sources of in-band crosstalk. An average over the phases in Eq. (9.2.6) was performed using (cos²θ) = ½. In addition, r²_x was multiplied by another factor of ½ to account for the fact that P_nis zero on average half of the times (during 0 bits). The experimental data shown in Figure 9.4(b) agree well with this simple model when polarization effects are properly included [21].

The impact of in-band crosstalk can be estimated from Figure 9.5, where the crosstalk level X is plotted as a function of N to keep the power penalty less than a certain value, while maintaining a BER below 10^-9 (Q = 6). To keep the penalty below 1 dB, r_x< 0.1 is required, a condition that limits XN to below -20 dB from Eq. (9.2.8). Thus, the crosstalk level X must be below -32 dB for N = 16 and below -40 dB for N = 100. Such requirements are relatively stringent for most routing devices. The situation is worse if the power penalty must be kept below 0.5 dB.

Figure 9.5: Crosstalk level X as a function of TV for several values of power penalty induced by in-band crosstalk.

The expression (9.2.7) for the crosstalk-induced power penalty is based on the assumption that the power fluctuations ΔP induced by in-band crosstalk ΔP can be assumed to follow a Gaussian distributions. If the contribution of crosstalk-crosstalk beating terms in Eq. (9.2.5) is included, ΔP does not remain Gaussian. A more accurate calculation uses the moment-generating function for finding the BER under such conditions [33]. The results show that BER is degraded further when all beating terms are included and that the optimum value of decision threshold at the receiver is also affected.

The calculation of crosstalk penalty in the case of dynamic wavelength routing through optical cross-connects (see Section 9.4 of LT1) becomes quite complicated because of a large number of crosstalk elements that a signal can pass through in such WDM networks [22]. The worst-case analysis predicts a large power penalty (>3 dB) when the number of crosstalk elements becomes more than 25 even if the crosstalk level of each component is only -40 dB. The crosstalk also depends on the topology used for an optical cross-connect [28]. Clearly, the linear crosstalk has the potential of becoming a limiting factor in the design of WDM networks and should be controlled. A simple technique consists of scrambling the laser phase at the transmitter end at a frequency much larger than the laser linewidth [34]. Both theory and experiments show that the acceptable crosstalk level exceeds 1% (-20 dB) with this technique [30].