5 10 15 20 25
10 20 30 40 50
∆Θ w/ PNA
∆Θ w/o PNA
∆ϑ
(M)1,nFigure 4.8: The variances of differential linear PN as functions of the number of PNAs
where ˆCk is the correlation coefficient and σ2d is the variance of deterministic PN. Due to ϑN,n= ϑn, Eq. (4.12) can be rewritten as,
∆Θn=
M +1X
k=0
à M X
N =1
aN,k
!
· ϑn−k. (4.18)
As a result, the variance of total deterministic PN is,
∆Θ2®
=
ÃX
k
Bk× ˆCk
!
× σr2, (4.19)
where
=
The last three terms in Bk could be viewed as the binomial distributions centered at k = 0, N/2 and −N/2, with the variances of (N + 1)/2, (N + 2)/4 and (N + 2)/4, respectively. Then h∆Θ2i is expected to approach a maximum value, when the first binomial distribution contributes the most to the product in Eq. (4.19), but the second and the third terms contribute little. Moreover, if two pulses are far from each other in the time domain, the correlation between their PNs should be zero. Consequently, as M is large, the main contribution to the differential PN is 2δ0,k+ δ1,k+ δ−1,k of Bk. That is, h∆Θ2i converges to 2(1 + ˆC1)σd2 as M → ∞. If PNAs are not inserted, h∆Θ2i becomes 2M2(1 − C1)σd2, which accumulates at an accelerated rate due to coherent characteristics of PN and is not improved by inserting phase-preserving amplitude regenerators.
Moreover, with respect to high-speed transmissions (> 10 Gbps) in SSMF with dis-persion of 17 ps/nm/km, the inter-channel nonlinearity affects systems much less than intra-channel nonlinearity due to fast walk-off between channels and pulse broadening.
Among intra-channel effects, isolated pulse SPM (ISPM) and intra-channel XPM (IXPM) only induce identical deterministic nonlinear PN to each pulse, owing to the same power of pulses. Hence, in amplitude-managed highly dispersed RZ-DPSK systems, the nonlin-ear PN induced by intra-channel FWM (IFWM) is the main contribution to deterministic PN, and the correlation coefficient of IFWM-induced PN is necessary to understand in detail the effect of periodic PN averaging on deterministic PN. To simplify the analysis, the nonlinear effect is treated as a perturbation, and the optical field of n-th pulse of
RZ-DPSK singal is assumed to be,
where 1.66τ is the FWHM. In previous works [61][62], nonlinearity of DCF was neglected.
However, it significantly influences the characteristics of IFWM-induced PN, and espe-cially the correlation between the PNs of neighboring bits. Accordingly, the analysis is extended to the nonlinear effects from both SSMF and DCF. Following some manipula-tion, the nonlinear PN can be represented as,
ϑn =X is required for complete dispersion compensating, and Fl,m(L, α, β00) is,
Fl,m(L, α, β00) = <
giving the strength of the nonlinear effect from l-th, m-th and (l + m)-th pulses. The first and second terms of Eq. (4.21) represent the nonlinear effects occurring in SSMF and DCF, respectively.
To investigate the IFWM-induced PN of a specific system, 40 Gbps RZ-DPSK signals with a 33% duty cycle are considered; the length of in each span is 80 km; the nonlin-ear coefficients, the loss and the GVD parameters of SSMF and DCF are 1.3 and 5.4 W−1km−1, 0.2 and 0.65 dB/km, and -21.7 and 127.6 ps2/km, respectively. The launch
where Li,eff is the effective length per span and Pi,ave = √
πPiτ /T is the average launch power. Since the pulses are highly broadened when sent into DCF, two pulses will in-teract even though these pulses are far away from each other. Hence, all distributions of −22 ≤ l, m, l + m ≤ 22 in Eq. (4.21) are considered to fully capture pulse-to-pulse interactions, and l 6= n and m 6= n are set to exclude ISPM and IXPM effects. With a De Bruijn sequence of 216 bits and Φnl= 1 rad after 40 spans, Figs. 4.9(a)-(c) depict the dis-tributions of nonlinear PN caused by nonlinearities of both SSMF and DCF, only SSMF, and only DCF, respectively. These figures clearly show that the strength of nonlinear effect in DCF is not such small to be able to be neglected. The correlation coefficients of IFWM-induced PN are also plotted in Fig. 4.10, and the nonlinearity of DCF could increase the correlation of PNs of remote pulses but decrease that of adjoining pulses.
Based on the results of Fig. 4.10 and Eq. (4.17) and the setup of Fig. 4.6 with K = 1 and without amplitude regenerators, Fig. 4.11 analytically plots the variance of differential PN as a function of the number of the spans. The variance is maximal near 20 spans, and converges to about 3σ2d very slowly. Figure 4.11 also plots the results of a numerical simulation to verify the effect of PN averaging. All of the parameters of the simulation are identical to those in Fig. 1, and ASE noise is neglected to focus on the pattern effect. When the number of spans is less than 20, the theory agrees excellently with the simulation. Since the amplitude of the signals is not regenerated in the simulation, the IFWM-induced AN generates additional nonlinear PN increasingly. Therefore, for N >
35, h∆Θ2i with PNAs begins to increase rather than decrease. However, if ideal phase-preserving amplitude regenerators are inserted behind each span, then the simulation results would be identical to the theoretical results. Furthermore, both the results without PNA in Fig. 4.11 agree that increases as the square of the distance.
−5 0 5
Figure 4.9: The distributions of IFWM-induced PN, ϑn and ϑn−1, generated in (a) SSMF and DCF (b) only SSMF, and (c) only DCF
To further investigate the effect of PN averaging on reducing nonlinear impairment, a semi-analytical method is used to compute the error probability and the nonlinear penalties. Based on Eq. (3.54) and previous work [62], the error probability related to an SNR, ρs, and the nonlinear differential PN is equal to,
pe=
The error probability curves as functions of SNR with different IFWM-induced PNs are