Convergence of phase fluctuation induced by
intrachannel four-wave mixing in differential
phase-shift keying transmission systems via phase
fluctuation averaging
Chia Chien Wei and Jason (Jyehong) ChenInstitute of Electro-Optical Engineering and Department of Photonics, National Chiao-Tung University, 1001 Ta Hsueh Road, Hsin-Chu, Taiwan, 300
Received January 9, 2007; revised February 11, 2007; accepted February 12, 2007; posted February 16, 2007 (Doc. ID 78828); published April 17, 2007
This work investigates the effect of phase fluctuation averaging on phase fluctuation induced by intrachan-nel four-wave mixing (IFWM) in highly dispersed differential phase-shift keying transmission systems. Through repeatedly averaging the phase fluctuations of adjacent pulses, a simple analytical model and nu-merical simulation revealed that the IFWM-induced differential phase fluctuation is suppressed and con-vergent, even after an ultralong transmission. The influence of averaging the phase fluctuations on the bit error rate is also evaluated by the semianalytical method. © 2007 Optical Society of America
OCIS codes: 060.2330, 060.4370.
In recent years, the return-to-zero (RZ) differential phase-shift keying (DPSK) format has become a promising alternative to the traditional on–off keying format, especially for long-haul high-speed optical communication systems.1 However, since DPSK sig-nals are demodulated via a delay interferometer in front of a receiver, both amplitude fluctuation (AF) and phase fluctuation (PF) will degrade the perfor-mance of received signals, and the regeneration of the DPSK format should be able to reduce AF and/or PF to extend the transmission distance. Although a phase-sensitive amplifier could eliminate both AF and PF simultaneously, a complicated and impracti-cal optiimpracti-cal phase-locking pump beam is required.2 Nevertheless, the AF of DPSK signals can be effec-tively eliminated by several phase-preserving ampli-tude regenerators.3,4 Thus, in the amplitude-managed DPSK systems with these regenerators, accumulated PF is mainly responsible for the limit on the reach of signals. Moreover, with respect to dispersion-managed high-speed transmissions, the nonlinear phase noise generated from the beating be-tween the amplified spontaneous emission (ASE) noise and the signal could be reduced through noise suppression4and pulse broadening.5Accordingly, the accumulated PF in amplitude-managed highly dis-persed RZ DPSK systems is due mostly to the linear phase noise caused by the ASE noise and the nonlin-ear PF induced by intrachannel four-wave mixing (IFWM).
A novel phase noise averager (PNA) was recently proposed to average the PFs of two adjacent pulses,6 and the differential PF (DPF) between neighboring bits converges even after an ultralong distance. How-ever, the earlier work considered the PFs of neighbor-ing bits to be uncorrelated.6 For a high-bit-rate 共⬎40 Gbits/s兲 system, since IFWM leads to partial correlation between the PFs of adjoining bits,7the ef-fects of PNAs must be reexamined. Extending the
previous work,6this Letter discusses the effect of PF averaging on IFWM-induced PF. Even though a cor-relation exists between the PFs, this work confirms that IFWM-induced DPF remains suppressed and convergent in DPSK transmission systems with peri-odically inserted PNAs. By estimating the bit-error rate (BER) with a semianalytical method, the effect of PF averaging is corroborated with the estimations of nonlinear penalties. To our best knowledge, the IFWM-induced PF is effectively eliminated for the first time.
Although the IFWM-induced PF is caused by the beating of broadened optical pulses under highly dis-persive conditions, and therefore its strength is de-terministic, it shows almost random statistics due to the random data of optical pulses.7 However, the characteristics of IFWM-induced PF differ from those of ASE-related phase noise. First, if fiber spans in-cluding standard single-mode fiber (SSMF) and cor-responding dispersion-compensating fiber (DCF) are employed repeatedly one after the another, the PFs generated in all spans are identical and add coher-ently span after span as the worst case.8 Hence, while the variance of the PF caused in each span is
2, then the variance becomes N22after N spans, in-stead of N2. Second, the PFs of adjacent bits are cor-related, and this correlation is given by 具mn典 =2C
兩m−n兩, where具·典 denotes an expectation value;n
is the IFWM-induced PF of the nth bit in each span, and Ckis the correlation coefficient between the PFs
of two pulses away from k bits. As a result, after N spans, the total DPF,⌬⌽, is N共n−n−1兲, and its
vari-ance is具⌬⌽2典=2N22共1−C1兲.
Since a PNA can transform PF, n, to become
n共1兲=共n+n−1兲/2, when the signals are sent
through PNAs M times, the PF becomes n共M兲 = 2−M兺
k=0M 共 M
k兲n−k,6 where 共Mk兲 is defined as ⌫共M
+ 1兲/关⌫共k+1兲⌫共M−k+1兲兴 and ⌫共·兲 is the gamma
func-May 15, 2007 / Vol. 32, No. 10 / OPTICS LETTERS 1217
tion. Consequently, when a PNA is inserted behind every span, the total PF after N spans is ⌽n =兺M=1N n共M兲=兺k=0N 关兺M=1N 共Mk兲/ 2M兴
n−k. Note that this
re-sult is derived from the characteristics of the PF:n
are identical in all spans. Therefore the total DPF can be written as ⌬⌽=⌽n−⌽n−1=兺k=0N+1akn−k, where the coefficients ak are 兺M=1N 关共
M k兲−共
M
k−1兲兴/ 2M=␦0,k+␦1,k −共Nk+1兲/ 2N and ␦
i,j is the Kronecker delta function.
Sincenare mutually correlated, the variance of the total DPF is not 共兺k=0N+1ak2兲2. This correlation is con-sidered by writing the variance as
具⌬⌽2典 =
冉
兺
k
CkAk
冊
2, 共1兲where Akis the autocorrelation of ak: Ak= ak丢a−k= 2␦0,k+␦1,k+␦−1,k + 4
冤
冉
2N + 2 N + k + 1冊
22N+2 −冉
N + 2 k + 1冊
2N+2 −冉
N + 2 − k + 1冊
2N+2冥
共2兲 and 丢 denotes convolution. The last three terms in Eq. (2) could be viewed as the binomial distributions,Pp共k兩Z兲=共kz兲⫻pk共1−p兲z−k with p = 0.5, centered at k
= 0, N / 2, −N / 2, with variances of共N+1兲/2, 共N+2兲/4, and共N+2兲/4, respectively. Then 具⌬⌽2典 is expected to approach a maximum value, when the first binomial distribution contributes the most to the product in Eq. (1), 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 PFs should be zero. Consequently, as N is large, the main contribution to the DPF is 2␦0,k+␦1,k+␦−1,k in Eq. (2). That is,具⌬⌽2典 converges to 2共1+C1兲2. How-ever, the correlation coefficients are necessary to de-termine in detail the variation of the DPF with peri-odic PF averaging.
For simplicity to calculate Ck, the optical field of
the nth pulse of RZ DPSK signals is assumed to be
un共t兲=sn
冑
P exp关−共t+nT兲2/共22兲兴, where sn= ± 1indi-cates binary data encoded by a phase shift of either 0 or; T is the bit period, and 1.66is the full width at half-maximum. Moreover, the nonlinear effect is treated as a perturbation to determine theoretically the distribution of the IFWM-indcued PF. However, in previous work,7,8 nonlinearity in DCF was ne-glected. Actually, it significantly influences the char-acteristics of IFWM-induced PF, and especially the correlation between the PFs of neighboring bits. Ac-cordingly, our analysis is extended to the nonlinear effects in both SSMF and DCF. From the previous work7,8 and considering a complete postdispersion compensating scheme, the nonlinear PF can be rep-resented as
n=
兺
l,mslsmsnsl+m−n关␥1P1Fl,m共L1,␣1,1
⬙
兲−␥2P2e−␣2L2Fl,m共− L2,␣2,2
⬙
兲兴, 共3兲 where ␥i, Pi, Li, ␣i, and i⬙
are the nonlinearcoeffi-cients, the launch peak power, the length, the loss,
and the group-velocity dispersion of SSMF共i=1兲 and DCF 共i=2兲, and Fl,m共L,␣,
⬙
兲 is Fl,m共L,␣,⬙
兲 = R冋
冕
0 L e−␣z冑
1 + 2j⬙
z/2+ 3共⬙
z/2兲2 ⫻ exp再
−冉
T 冊
2冋
3lm 1 + 3j⬙
z/2 + 共l − m兲 2 1 + 2j⬙
z/2+ 3共⬙
z/2兲2册
冎
dz册
, 共4兲giving the strength of the nonlinear effect from the
lth, mth, and 共l+m兲th pulses. The first and second
terms of Eq. (3) represent the nonlinear effects occur-ring in SSMF and DCF, respectively. Throughout this Letter, 40 Gbit/ s RZ DPSK signals with a 33% duty cycle are considered; the length of SSMF in each span is 80 km; the nonlinear coefficients, the loss, and the group-velocity dispersion 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 power of DCF is 7 dB lower than that of SSMF, and this power is chosen by setting the mean nonlinear phase shift ⌰nl= N␥1P1,aveL1,eff+ N␥2P2,aveL2,eff to a specific value, where Li,eff is the effective length per span and Pi,ave=
冑
Pi/ T is the average launch power. Sincethe pulses are highly broadened when sent into DCF, two pulses will interact even though these pulses are far away from each other. Hence, all distributions of −22艋l, m,l+m艋22 in Eq. (3) are considered to fully capture pulse-to-pulse interactions, and l⫽n and m ⫽n are set to exclude self- and intrachannel cross-phase modulation effects. With a De Bruijn sequence of 216bits and⌰nl= 1 rad after 40 spans, Fig. 1 plots the correlation coefficients of PFs contributed to by both SSMF and DCF, SSMF only, and DCF only, and the corresponding distributions ofnandn−1caused
in each span are shown in insets. These figures clearly show that the nonlinear effect in DCF cannot
Fig. 1. (Color online) Correlation coefficients of the IFWM-induced PFs. Insets, corresponding distributions ofnand
n−1.
be neglected, and it increases the correlation of PFs of remote bits but decreases that of adjoining bits.
Based on the results of Fig. 1 and Eq. (1), Fig. 2 analytically plots the variance of DPF as a function of the number of the spans. The variance is maximal near 20 spans and converges to about 32 very slowly. Figure 2 also plots the results of a numerical simulation using commercially available software to verify the effect of PF averaging. All of the param-eters of the simulation are identical to those in Fig. 1, and ASE noise is neglected to focus on the pattern ef-fects. When the number of spans is smaller than 20, the theory agrees excellently with the simulation. Since the amplitude of the signals is not regenerated in the simulation, the IFWM-induced AF generates additional nonlinear PF increasingly. Therefore, for
N⬎35, 具⌬⌽2典 with PNAs begins to increase rather than decrease. However, if ideal phase-preserving amplitude regenerators are inserted behind each span, then the simulation results will be identical to the theoretical results. Furthermore, both results without PNA in Fig. 2 agree that具⌬⌽2典 increases as the square of the distance.
To further investigate the improvement by PF av-eraging, a semianalytical method is used to compute the BER and the nonlinear penalties. The BER re-lated to a signal-to-noise ratio (SNR),s, and the non-linear DPF is equal to8,9 pe=
冓
Q1冉
冑
2s冏
sin⌬⌽ 2冏
,冑
2s冏
cos ⌬⌽ 2冏
冊
− e−s 2 ⫻ I0共s兩sin ⌬⌽兩兲冔
, 共5兲where Q1共·, ·兲 is the Marcum Q function and I0共·兲 is the modified Bessel function of the first kind. The BER curves as functions of the SNR with the IFWM-induced PFs are calculated based on Eqs. (3)–(5), and the conditions of ⌰nl= 1 , 3 , 5 rad after 40 spans with and without PNAs are both plotted in Fig. 3. For comparison, the baseline shown in Fig. 3 is the BER without the PFs of e−s/ 2. Note that all the BER
curves with periodic PF averaging overlap the base-line, and this fact indicates that the nonlinear pen-alty is negligible. Moreover, both increasing the transmission distance and the launch power will in-crease and therefore degrade DPSK signals increas-ingly. In addition, longer sequences (up to 224) are tested in calculating both Ck and the BER, and the
results do not change significantly with the length of the pulse train as it exceeds 216.
This work demonstrates that periodically inserting PNAs into RZ DPSK transmission systems can effec-tively suppress IFWM-induced PF. A comprehensive theoretical model that incorporates the nonlinear ef-fect of DCF is established, and the results indicate that the PF induced in DCF cannot be neglected. A strong agreement between theoretical analysis and numerical simulation is shown, confirming the con-vergence of IFWM-induced PF with periodic PF
aver-aging. Further, the semianalytical method is used to compute BER and shows that PF averaging can obvi-ate nonlinear penalties.
The authors thank the reviewers for valuable com-ments. This research was sponsored by the National Science Council, Republic of China, grants NSC 96-2752-E-009-004-PAE, 2221-E-009-225 and 95-2221-E-009-349. C. C. Wei’s e-mail address is mgyso.wei@gmail.com.
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Fig. 3. (Color online) BER curves as functions of the SNR. May 15, 2007 / Vol. 32, No. 10 / OPTICS LETTERS 1219
