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OPTICS LETTERS / Vol. 28, No. 23 / December 1, 2003

Analysis of intrachannel four-wave mixing

in differential phase-shift

keying transmission with large dispersion

Xing Wei

Bell Laboratories, Lucent Technologies, Murray Hill, New Jersey 07974

Xiang Liu

Bell Laboratories, Lucent Technologies, Holmdel, New Jersey 07733

Received May 23, 2003

Intrachannel four-wave mixing (IFWM) in highly dispersed differential phase-shift keying (DPSK) transmis-

sion systems is studied with a simple analytical model and numerical simulation.

effect in DPSK systems is smaller and less dependent on the bit pattern than in on–off-keying systems with

the same average power. The reduced pulse energy in DPSK and correlation between the nonlinear phase

shifts of two adjacent pulses contribute to the robustness of DPSK versus IFWM.

America

OCIS codes:

060.2330, 190.4370, 190.4380, 260.2030.

It is found that the IFWM

© 2003 Optical Society of

Intrachannel nonlinear effects, such as intrachannel

cross-phase modulation (IXPM) and intrachannel

four-wave mixing (IFWM), are the dominating sources

of nonlinear penalties in the so-called pseudolinear

high-speed transmission systems in which the optical

pulses are highly dispersed and overlap strongly

during transmission.These nonlinear effects have

been studied extensively.1–7

IXPM- and IFWM-induced timing and amplitude jit-

ters of the ones can be greatly reduced in a symmetric

dispersion-managed link, whereas the IFWM-induced

ghost pulses on the zeros do not cancel.6

with proper dispersion management, the ghost pulses

are the major remaining nonlinear impairment.

eral methods to suppress the ghost pulses have been

proposed.8,9

Recently, superior transmission performance has

been demonstrated with return-to-zero (RZ) differen-

tial phase-shift keying (DPSK) in highly dispersed

40-Gbit?sultralong-haul

major advantage of DPSK over on–off keying (OOK) is

the improved receiver sensitivity in the linear regime.

In addition, DPSK greatly reduces the nonlinear

IXPM effect in highly dispersed transmissions by use

of pulses of equal energy in all time slots.

IFWM introduces pattern-dependent nonlinear phase

shifts to the DPSK signal.12

performance degradation that is due to IFWM in

DPSK systems has not been investigated in detail.

In this Letter we study this effect by theoretical

analysis and numerical simulation.

IFWM causes less performance degradation in DPSK

than it does in OOK with the same average power

because of the lower peak power of DPSK and a partial

cancellation of the nonlinear phase shifts.

We use a simple model similar to what has been

previously used for the study of IFWM in RZ OOK

It is known that the

Therefore,

Sev-

transmissions.10,11

One

However,

To our knowledge, the

We show that

systems3,6and extend the analysis to RZ DPSK.

consider the transmission of a 40-Gbit?s signal over

L ? 100 km of standard single-mode fiber with a

dispersion of D ? 17 ps?nmkm at the wavelength of

1550 nm (or b00? 222 ps2?km).

ficient of the single-mode fiber is g ? 1.2 rad?Wkm.

Here we neglect the fiber loss and focus on the non-

linear effects.For simplicity the fiber birefringence

and the nonlinearity of the dispersion-compensating

fiber are also neglected.The predispersion compen-

sation is 2850 ps?nm, exactly one half of the total

dispersion of the transmission fiber.

dispersion map fully symmetric to minimize the IXPM

and IFWM effects.6

In this study we also neglect

the nonlinear interaction between the signal and the

amplified spontaneous emission (ASE) noise.

interaction could produce additional phase noise in

DPSK, which is known as the Gordon–Mollenauer ef-

fect.13

Fortunately, this is not critical here since this

self-phase modulation- (SPM-) induced phase noise

is suppressed in the highly dispersed transmission

regime.12

In our model we consider a RZ DPSK optical pulse

train that consists of 2N Gaussian pulses.

tical field of each pulse before transmission can be

expressed as

We

The nonlinear coef-

This makes the

Such

The op-

uk?0,t? ? Akexp?2?t 2 kT?2?2t2?,

(1)

where k ? 1,2,...2N denotes the kth pulse, T ? 25 ps

is the bit period, t ? 5.0 ps corresponds to an 8.3-ps

width (FWHM) of the pulse intensity profile, and Ak?

6A0 is the complex amplitude of the DPSK signal.

Here, A0 is a constant related to the average power

P0 through P0 ?pp jA0j2t?T.

tive approach, Mecozzi et al.6showed that nonlinearity

produces the following term in the center of the kth

Using the perturba-

0146-9592/03/232300-03$15.00/0© 2003 Optical Society of America

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December 1, 2003 / Vol. 28, No. 23 / OPTICS LETTERS

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time slot:

Duk?L,t ? kT? ?

X

l1m2n?k

Z L?2

2L/2dz

igAlAmAn?

?1 2 2ib00z?t21 3?b00z?t2?2?1?2

Ω

t2?1 1 3ib00z?t2?

?l 2 m?2T2

t2?1 2 2ib00z?t21 3?b00z?t2?2?

3

3 exp

23?l 2 n??m 2 n?T2

2

æ

,

(2)

where ?l,m,n? are the indices of the IFWM contribut-

ing pulses.In this study, terms with l ? k or m ? k

are excluded because they are from IXPM and SPM

and produce only a constant phase shift to all pulses

in DPSK.We note here that jb00jL?2t2? 43, therefore

we can use the approximation based on jb00jz?t2¿ 1

to obtain the following much simplified expression:

Duk? 2i

2gt2

p3jb00j

3 Ci

X

l,m

AlAmAl1m2k?

∑2?l 2 k??m 2 k?T2

jb00jL

cos t

t

dt is the cosine integral func-

It should be noted that the IFWM-induced pulse

might not be exactly centered in the kth time slot, and

it might also have a larger pulse width.

however, are neglected in the following analysis since

our goal is to determine general features of the IFWM

effect in DPSK and compare them with OOK.

The fact that Duk has no real part, but only an

imaginary part (assuming A0 is real) as shown in

approximation (3), indicates that the IFWM distorts

mainly the phase of the original light pulse.

mation (3) allows us to compute efficiently such a

nonlinear phase shift of the kth pulse:

∏

,

(3)

where Ci?x? ? 2R`

tion.

x

These details,

Approxi-

Dfk? Duk?Ak.

(4)

For each bit pattern we can then calculate the dif-

ferential phase error DfN2 DfN11for the Nth and

the ?N 1 1?th pulses positioned in the middle of the

pulse train and define the following differential phase

eye-closure factor (ECF):

ECFDPSK?2jDfN2 DfN11j

p

.

(5)

This is used to quantify the penalty in DPSK based on

the observation that the differential phase eye14closes

(ECFDPSK? 1) when the jDfN2 DfN11j reaches p?2.

We use approximation (3) and Eqs. (4) and (5) to

compute ECFDPSK for a pulse train that contains

14 pulses (N ? 7).Since only the relative phase

matters, we choose A1 to be positive and exhausted

all 213? 8192 polarity (phase) combinations for the

remaining 13 pulses. The optical power used in

this study is P0? 5 mW.The calculated ECFs are

presented in Fig. 1(a) as a histogram on a logarithmic

scale. The histogram appears to decay rapidly and

the worst-case ECFDPSK is less than 0.2.

increase of N (and the number of patterns) in the

calculation does not change the result significantly,

which indicates that N ? 7 is sufficient for this study.

We perform the same calculation for RZ OOK.

Approximation (3) is still valid for OOK.

difference is that Akis now either 0 orp2A0to main-

tain the same average power P0.

bit to be a zero (AN? 0) and exhaust all the 22N21

combinations for the other 2N 2 1 time slots.

ECF for OOK is defined in the field domain (we do so

because the ASE noise can be better approximated by

a Gaussian random variable in the field domain than

in the intensity domain), i.e.,

Further

The only

We choose the Nth

The

ECFOOK?

jDuNj

p2jA0j

.

(6)

Here we have neglected the fluctuations on ones.

result for N ? 7 is shown in Fig. 1(b).

trast to DPSK, the histogram for OOK has a long tail

that extends beyond 0.4.

to the well-known worst-case ghost pulse problem, i.e.,

a single zero surrounded by successive ones on both

sides.5

These worst-case ghost pulses could give rise

to error floors in transmission experiments when long

pseudorandom bit sequence (PRBS) patterns are used.

There are two main reasons that the ECF in DPSK

is smaller and also less dependent on the bit pattern.

The first reason is obvious.

power, each pulse that represents a one in OOK car-

ries twice as much energy compared with a pulse in

DPSK. Approximation (3) shows that the magnitude

of each contributing term for Duk is proportional to

jAlAmAn?j.

OOK can be larger than its counterpart in DPSK

by a factor of 2p2 if l, m, and n are all ones for

most combinations, which corresponds to the worst

case of OOK.The second reason is more subtle,

that is, the phase shifts DfN and DfN11 are par-

tially correlated.This can be clearly seen in Fig. 2.

We can prove that IFWM processes that involve

both the Nth and the ?N 1 1?th pulses, for ex-

ample, ?l? 1 ?N 1 1? 2 ?l 1 1? ! ?N? and ?l 1 1? 1

?N? 2 ?l? ! ?N 1 1?, contribute an identical phase

shift to DfN and DfN11 and cancel out in the

The

In strong con-

This long tail corresponds

With the same average

Therefore, each contributing term in

Fig. 1.

sampled over 8192 different bit patterns.

ECF histograms for (a) DPSK and (b) OOK, each

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OPTICS LETTERS / Vol. 28, No. 23 / December 1, 2003

Fig. 2.

calculated for the 14-bit-long DPSK pulse train with

8192 different bit patterns.

Distribution of the IFWM-induced Df7 and Df8

Fig. 3.

and the differential phase error Df72 Df8.

curves are simply guides to the eye.

Histograms of the nonlinear phase shift Df7

The fitting

Fig. 4.

DPSK (left) and amplitude (square root of intensity) eye

diagram for RZ OOK (right).

Simulated differential phase eye diagram for RZ

The PRBS length is 292 1.

differential phase error regardless of the bit pattern.

Figure 3 shows histograms of the calculated Df7and

Df72 Df8 for the DPSK pulse train that contains

14 pulses.If Df7 and Df8 were uncorrelated, the

standard deviation of Df72 Df8would be

larger than that of Df7. Because of the correlation,

the differential phase error appears to have the same

distribution width as the IFWM-induced phase shift

of a single pulse.

To verify the effectiveness of the simple model we

perform numerical simulations using a PRBS of length

292 1 at 40 Gbits?s with the same transmission pa-

rameters that we used in the above analysis.

nonlinear effects, not only IFWM but also IXPM and

p2 times

All the

SPM, are included in the simulation.

still neglected. The results shown in Fig. 4 are in good

agreement with our model prediction.

In conclusion, our model analysis and numerical

simulation show that IFWM introduces phase noise

in strongly dispersion-managed RZ DPSK systems,

but the penalty appears to be smaller than that in RZ

OOK systems with the same average power because

the worst cases are significantly suppressed with

DPSK.We caution, however, that this analysis is

mainly to understand the underlying physics, and the

model used here is much too simplified compared with

practical systems. Nonetheless, this simplified model

might provide useful guidance for the design of future

high-speed optical transmission systems.

ASE noise is

We thank A. H. Gnauck, A. R. Chraplyvy, R. E.

Slusher, and R. C. Giles for inspiring discussions.

Xing Wei’s e-mail address is xingwei@lucent.com.

References

1. R.-J. Essiambre, B. Mikkelsen, and G. Raybon, Elec-

tron. Lett. 35, 1576 (1999).

2. P. V. Mamyshev and N. A. Mamysheva, Opt. Lett. 24,

1454 (1999).

3. A. Mecozzi, C. B. Clausen, and M. Shtaif, IEEE Photon.

Technol. Lett. 12, 392 (2000).

4. R. I. Killey, H. J. Thiele, V. Mikhailov, and P. Bayvel,

IEEE Photon. Technol. Lett. 12, 1624 (2000).

5. M. J. Ablowitz and T. Hirooka, Opt. Lett. 25, 1750

(2000).

6. A. Mecozzi, C. B. Clausen, M. Shtaif, S.-G. Park, and

A. H. Gnauck, IEEE Photon. Technol. Lett. 13, 445

(2001).

7. P. Johannisson, D. Anderson, A. Berntson, and J.

Mårtensson, Opt. Lett. 26, 1227 (2001).

8. P.Johannisson, D.Anderson,

Berntson, M. Forzati, and J. Mårtensson, Opt. Lett.

27, 1073 (2002).

9. X. Liu, X. Wei, A. H. Gnauck, C. Xu, and L. K.

Wickham, Opt. Lett. 27, 1177 (2002).

10. A. H. Gnauck, G. Raybon, S. Chandrasekhar, J.

Leuthold, C. Doerr, L. Stulz, A. Agarwal, S. Banerjee,

D. Grosz, S. Hunsche, A. Kung, A. Marhelyuk, D.

Maywar, M. Movassaghi, X. Liu, C. Xu, X. Wei, and

D. M. Gill, in Optical Fiber Communication Confer-

ence, Vol. 70 of OSA Trends in Optics and Photonics

Series (Optical Society of America, Washington, D.C.,

2002), postdeadline paper FC2.

11. C. Rasmussen, T. Fjelde, J. Bennike, F. Liu, S. Dey, B.

Mikkelsen, P. Mamyshev, P. Serbe, P. van der Wagt,

Y. Akasaka, D. Harris, D. Gapontsev, V. Ivshin, and

P. Reeves-Hall, in Optical Fiber Communication Con-

ference, Vol. 86 of OSA Trends in Optics and Photonics

Series (Optical Society of America, Washington, D.C.,

2003), postdeadline paper PD18.

12. X. Liu, C. Xu, and X. Wei, in Proceedings of the Eu-

ropean Conference on Optical Communication (EEOC,

Copenhagen, Denmark, 2002), paper 9.6.5.

13. J. P. Gordon and L. F. Mollenauer, Opt. Lett. 15, 1351

(1990).

14. X. Wei, X. Liu, and C. Xu, IEEE Photon. Technol. Lett.

15, 1636 (2003).

M.Marklund, A.