0, 2 V
22Figure 4.16: Schematic of regeneration configuration
Figure 4.16 presents the configuration of the DPSK system with regeneration. When
signals with complex Gaussian noise at point 1 can be represented as u(t) = A(t) + n1(t), where A(t) = ±A0 is the DPSK signal, and n1 is complex Gaussian noise with the vari-ance of 2σ21. To better understand the noise suppression capability of a regenerator, the inherent noises of the regenerator can be included after the regenerator, and therefore, all regenerators discussed in this work are assumed to be noiseless and lossless (or gain
= 1). With phase-preserving regeneration, the amplitude-regenerated signals become AR{u(t)} = ±p
A20+ 2σ12 × exp(jϑ), where AR{·} indicates the phase-preserving am-plitude regeneration without changing average power and ϑ = ](A0 + n1) is the PN.
Identical to Eq. (3.76), the pdf of ϑ can be expressed as a Fourier series,
pΘ(ϑ) = 1 2π
X∞ m=−∞
√πρ1
2 e−ρ12 Πm(ρ1)e−jmϑ, (4.24) where ρ1 = A20/(2σ21) is the SNR at point 1. When complex Gaussian noise with a variance of 2σ22 is added between points 1 and 2, as shown in Fig. 4.16, the pdf of differential PN at point 1 and point 2 are,
p∆Θ(∆ϑ) = ρ1e−ρ1 8
X∞ m=−∞
Π2m(ρ1)e−jm∆ϑ, (4.25)
p∆Θ,pp(∆ϑ) = πρ1ρppe−(ρ1+ρpp) 32
X∞ m=−∞
Π2m(ρ1)Π2m(ρpp)e−jm∆ϑ, (4.26)
where ρpp = (A20+ 2σ12)/(2σ22) = ρ2(1 + ρ−11 ) and ρ2 = A20/(2σ22). Eqs. (4.25) and (4.26) are derived based on the fact that the characteristic function of the sum of independent random variables is the product of individual characteristic functions. Then, the error probability of signals with phase-preserving regeneration is,
pe,pp = 1 − Z π
2
−π2
p∆Θ,pp(∆ϑ)(∆ϑ) d∆ϑ (4.27)
1 πρ1ρppe−(ρ1+ρpp) X∞ (−1)m 2 2
ation, according to Eq. (3.53), is,
pe,0 = e−ρ0
2 , (4.28)
where ρ−10 = ρ−11 + ρ−12 is the final SNR at the receiver. Furthermore, an ideal coherent regenerator, such as an ideal coherent PSA [18], is assumed to be able to completely eliminate both AN and PN. The error probability determined by a coherent beam at the regenerator is identical to the case of coherent detection, erfc(√
ρ1)/2, where erfc{·} is the complementary error function [48]-[50]. Because the regenerator is lossless, the error probability at the direct-detection receiver is exp(−ρpp)/2. With negligible differences the cross term can be omitted, and the final error probability with coherent regeneration is,
pe,co = 1 2
¡erfc (√
ρ1) + e−ρpp¢
. (4.29)
PPAR PPAR
( ) u t ( )
u t
Figure 4.17: Setup of the PN-averaged regenerator. PPAR: phase-preserving amplitude regenerator.
Moreover, phase-preserving amplitude regenerators can simultaneously realize AN elimination and PN averaging [37]. As shown in Fig. 4.17, a DI converts the DPSK signals into two phase-modulated OOK signals: DB, [A(t) + A(t − T ) + n1(t) + n1(t − T )]/2, and
eliminate AN of both marks and spaces, i.e., the output power of both OOK signals is either 0 or A20 + σ12 and the signals become ¯u(t) = AR{A(t) + [n1(t) ± n1(t − T )]/2}, where 3-dB loss is neglected and ± depends on A(t) and A(t − T ) being in-phase or out-of-phase. Since ϑ(t) ∼= ={n1(t)}/A(t), where ={·} is the imaginary part, the PN of ¯u(t) can be approximated as [ϑ(t) + ϑ(t − T )]/2, which turns out to be the averaged PN. Although cascading another DI after the PNA regenerator can further increase the correlation between PNs of adjacent pulses, it also induces additional AN, and this is beyond the scope of this discussion. Owing to ¯u(t) and ¯u(t − T ) influenced by identical noise n1(t − 2T )/2 as well as independent noise n1(t)/2 and n1(t − 2T )/2, the pdfs of differential PN at points 1 and 2 are (Appendix B)
p∆Θ(∆ϑ) = e−4ρ1 8
X∞ m=−∞
Hm(1, 4ρ1)e−jm∆ϑ, (4.30)
p∆Θ,pna(∆ϑ) = πρpnae−(4ρ1+ρpna) 32
X∞ m=−∞
Hm(1, 4ρ1)Π2m(ρpna)e−jm∆ϑ, (4.31)
where
Hm(ξ, ρ) = Z ∞
0
ye−(1+ξ)yI0
³ 2p
ξρy
´
Π2m(y) dy , (4.32)
and ρpna = (A20 + σ12)/(2σ22) = ρ2[1 + (2ρ1)−1]. Moreover, since the ideal amplitude regeneration of OOK signals must make a binary decision on signals, the error probability determined at the PNA regenerator, pe,1, depends on the criterion of the binary decision.
While the amplitude regenerators in Fig. 4.17 are steplike and mutually independent, the error probability can be approximated as double that of the OOK signals: pe,1 = exp(−ρ1/2) [48]. Otherwise, when the decision is made by comparing the power of two OOK signals [35][36], it is similar to direct-detection DPSK: pe,1= exp(−ρ1)/2. Therefore, by integrating Eq. (4.31) the error probability at the receiver is,
The first term in Eq. (4.33) is pe,1 with the best decision criterion. Even though the amplitude regenerators in Fig. 4.17 only remove the AN of spaces with marks left un-changed, the PN of adjacent bits have been simultaneously averaged. The regenerated signals become ¯u0(t) = A(t) + [n1(t) ± n1(t − T )]/2 of which the phase is identical to that of ¯u(t) but residual AN still exists. Hence, neighboring pulses at point 2 contain identical and mutually independent noises with variances of σ12/2 and σ21 + 2σ22, respectively, and the pdf of differential PN and the error probability become,
p∆Θ,pna0(∆ϑ) = (4ρ1+ ρ2)e−4ρ1
Figure 4.18(a) plots the pdfs of PN, differential PN, and averaged differential PN with-out n2, which are described by Eqs. (4.24), (4.25) and (4.30) with ρ1 = 14 dB. The tail of differential PN distribution, which is the main contributor to the error probability, is effectively suppressed by PNA regeneration. Moreover, the variance of these phase distri-bution can be simply calculated throughRπ
−πθ2pX(θ) dθ, where pX(θ) is the corresponding pdf, and they are,
PN : π2 Differential PN : π2
3 + πρ1e−ρ1 X∞ m=1
(−1)m
m2 Π2m(ρ1) Averaged differential PN : π2
3 + πe−4ρ1 X∞ m=1
(−1)m
m2 Hm(1, 4ρ1)
The results plotted in Fig. 4.18(b) indicate that the configuration of Fig. 4.17 can realize PN averaging, since the variance of averaged differential PN is about the quarter of that
After loading n2, the solid curves in Fig. 4.19 show the analytical results of pe,0, pe,co, pe,pp, pe,pna and pe,pna0, corresponding to Eqs. (4.28), (4.29), (4.27), (4.33) and (4.35), respectively. The error floors for coherent regeneration and PNA regeneration are deter-mined by n1, and it indicates that any regenerator can only improve signal performance but cannot correct existing errors. In addition, the error probabilities indicated by mark-ers in Fig. 4.19 are assessed by the brute-force Monte Carlo method, performed with 109 bits, and the results closely agree with each other.
The SNR required to achieve the BER of 10−9, shown in Fig. 4.20, indicates that PNA regeneration can remove most penalty induced by PN, because the difference between its SNR and that of perfect coherent regeneration is less than ∼0.3 dB. However, if steplike independent amplitude regenerators are adopted in PNA regeneration (dashed curve in Fig. 4.20) then regeneration fails when ρ1 is less than 16.2 dB. Accordingly, the amplitude regenerators shown in Fig. 4.17 dominate the performance of PNA regeneration if the input SNR is low. Furthermore, Fig. 4.20 demonstrates that the improvement over phase-preserving regeneration vanishes when SNR is lower than 15 dB, because the PN induced by n1 dominates the whole PN. Similarly, since averaging PN suppresses most PN-induced penalties, the signals with PNA residual AN regeneration can outperform those with phase-preserving regeneration, as ρ1 is low.
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 10−10
10−5 100
Phase ( π )
Probability density function
(a)
6 8 10 12 14 16 18 20
10−2 10−1
ρ
1(dB) Variance (rad
2)
(b)
PN Diff. PN
Averaged Diff.PN
PN Diff. PN
Averaged Diff. PN
Figure 4.18: (a) pdf of phase distributions with ρ1 = 14 dB, and (b) variance of phase distributions as a function of ρ1
10-4
Figure 4.19: Error probabilities of DPSK signals with various regeneration schemes (curves: analytical results and markers: Monte Carlo method)
12.5 13
SNR required to reach error probability of 10-9
,pp
Chapter 5 Conclusions
In this dissertation research, we have studied a number of key all-optical signal pro-cessing issues that are vital to the high bit-rate, high performance optical communications applications.
5.1 Wavelength conversion
For wavelength conversion, our work has proposed a novel wavelength conversion scheme called DXPoM by simply adding an extra birefringence delay line in the stan-dard XPoM. The delay line adjusts the time differential between two orthogonal modes to overcome the limitation of the carriers recovery time in an SOA. Both the large-signal simulation based on TMM and experimental results indicate that the proposed method significantly improves the wavelength conversion performance. The new technique ex-hibits an error free wavelength conversion operation with more than 7-dB power penalty improvement over XPoM. The DXPoM scheme shows a pulse reshaping function, an im-provement of more than 300% in rise time, reduced timing jitter and a higher ER. The small-signal model of XPoM is presented for the first time. Nevertheless, the modulation bandwidth of XPoM is identical to XGM, and is limited by the carrier’s recovery time.
By considering the time delay between two orthogonal modes, the corresponding transfer function of DXPoM shows apparent bandwidth improvement. Through the analytical expression of modulation bandwidth of DXPoM, we can readily observe the relations be-tween the bandwidth improvement of DXPoM and several parameters, such as lifetimes,
confinement factors, gain coefficients, and operating points.
5.2 Format conversion
The work of format conversion was done at UMBC during the period of my exchange program. We used the XPM effect in a nonlinear PCF to carry out the conversion of RZ-OOK-to-RZ-DPSK at 40 Gb/s for the first time. Because the PCF has high linear birefringence, and its birefringent axes remain for entire length, launching the probe at 45◦ relative to the birefringence axes of the PCF and having the PPD greater than ∼ 6 nm can realize polarization-insensitive XPM-based conversion. While the OOK pump and probe are launched at the same birefringent axis, the converted DPSK probe performed better than the baseline of the OOK signal. However, after the pump is polarization scrambled, an error floor was observed. Once the probe was adjusted to be at 45◦ with respect to the PCF birefringent axes, error-free BER was achieved, but it shows ∼ 7-dB penalty compared with the baseline of the DPSK signal. Moreover, this part theoretically analyzes XPM effect in a highly birefringent HNLF with an arbitrary polarized pump beam by the nonlinear Schr¨odinger equation. When the probe beam is launched at an axis and 2γP1L = π, the optimized threshold and penalties of a DPSK probe are 0 ∼ 76.2% and 0 ∼ 14.1 dB. By contrast, if the probe beam with ∆BL = 2π is launched at 45◦ and 2γP1L = π, the variations can be reduced to 31.5% ∼ 42.8% and 3.7 dB ∼ 5.6 dB, and they become 23.2% ∼ 41.8% and 2.5 dB ∼ 5.4 dB by setting 2γP1L = 1.1π. Consequently, launching the probe at 45◦ can decrease the variations of both threshold and sensitivity caused by arbitrary polarized OOK signals, but the penalty is never zero. Furthermore, since optical amplifiers are required to compensate transmission loss or increase nonlinear
must have AN. Through theoretical investigation, the SNR of > 29 dB is required to suppress the correspond penalty of the DPSK probe less than 1 dB.
5.3 Phase regeneration
As to the regeneration of the DPSK format, we propose a novel all-optical PSA-based PNA that does not require a complicated optical coherent pump beam with optical phase-locking. The PNA can increase the correlation between PN of neighboring bits.
Theoretical analyses confirm that inserting PNAs after in-line amplitude regenerators can effectively reduce the differential PN. Since the PN is suppressed by increasing its correlation, the multiple PN averaging effect dependents on the characteristics of PN.
According the characteristics, PN can be divided into two categories, random and deter-ministic PNs, and the later one is caused by signals and fiber nonlinearity. In a high-speed amplitude managed DPSK transmission system, the main PNs of two categories are linear PN and IFWM-induced PN, respectively. For linear PN, the total differential PN is lim-ited to less than that before the first averager and is independent of the number of DPSK spans. For IFWM-induced PN, a comprehensive theoretical model that incorporates the nonlinear effect of DCF is established to investigate the property of IFMW-induced PN, and the results indicate that the PN induced in DCF cannot be neglected. The multiple PN averaging effect on IFWM-induced PN is examined by both theoretical analysis and numerical simulation, and a strong agreement between them confirms the convergence of IFWM-induced PN with periodic PN averaging. Furthermore, the semianalytical method is used to compute BER and shows that PN averaging can obviate the nonlinear penalty.
Moreover, the numerical simulation of a practical 40 Gbps DPSK system confirms that inserting PNAs after in-line amplitude regenerators can effectively reduce the differential
PN even after the transmission of 2800 km. Compared with the system without regener-ation, periodic PNAs can reduce the STD of DPN by ∼ 75%. Moreover, by considering only linear PN, we analytically derives the BERs of amplitude-regenerated DPSK signals with no additional phase processing, PN averaging, and ideal phase regeneration. The SNR difference at the BER of 10−9 between PNA regeneration and ideal phase regenera-tion is less than 0.3 dB, indicating that incoherent PNA regeneraregenera-tion eliminates most of the PN-induced penalty, even though PN is statistically reduced by less than 30%.
5.4 Future works
In order to increase the spectral efficiency to more than 1 b/s/Hz without polarization-division multiplexing, multilevel signals, such as DQPSK and quadrature-amplitude mod-ulation (QAM), must be adopted to transmit more than one bit per symbol. While the DQPSK format are transmitted with polarization-division multiplexing, the spectral ef-ficiency is more than 2 b/s/Hz [64]. Recently, the interest in the DQPSK format has increased because of its high tolerance to fiber nonlinearities, similar to the DPSK format, and several all-optical signal processing schemes related to the DQPSK format have been proposed, such as wavelength conversion [65] and format conversion [14][66]. However, compared with DPSK signals, DQPSK signals have lower tolerance to PN [67]. Although phase-preserving amplitude regeneration may be applied to decelerate the accumulation of the nonlinear PN in DQPSK systems, the residual PN can be much more harmful to signal performance. Consequently, the all-optical phase regeneration of DQPSK would be an important research topic, and it has not been investigated. Actually, cascading the PN-averaged regenerators shown in Fig. 4.17 is able to average the PN of DQPSK signals.
bits whose phase difference is π/2, but leave those with differential phase of π/4 or 3π/4 alone. Then, by adding phase shift of π/4 in the second DI, the second regenerator can averaged the PN of unchanged bits. However, the key issue would be choosing proper amplitude-preserving amplitude regenerators, which can only generate negligible PN.
PPAR
Reg.
PPAR( ) u t
PPAR
PPAR
u t ( )
S
4Figure 5.1: Cascading PN-averaged regenerators