2544 OPTICS LETTERS / Vol. 31, No. 17 / September 1, 2006

Power-weighted dispersion distribution functionfor characterizing nonlinear properties of

long-haul optical transmission links

Xing WeiBell Laboratories, Lucent Technologies, Murray Hill, New Jersey, 07974

Received April 13, 2006; revised June 8, 2006; accepted June 9, 2006;posted June 13, 2006 (Doc. ID 69982); published August 9, 2006

Based on the first-order perturbation theory, we introduce the power-weighted dispersion distribution(PWDD) function to characterize nonlinear properties of fiber-optic transmission links. This technique offersa new perspective on dispersion management for pseudolinear transmission. A rectangular PWDD model isused to describe the formation of ghost pulses analytically in the time domain without relying on the as-sumption that the pulse shape is Gaussian. © 2006 Optical Society of America

OCIS codes: 060.2330, 190.4370, 190.4380, 260.2030.

The adverse effects of Kerr nonlinearity in high-capacity long-haul optical transmission systems havebeen the focus of extensive studies for many years.Depending on the dispersion maps, the systems canbe roughly categorized into the low-dispersion(soliton-like) regime and the high-dispersion (pseudo-linear) regime.1 The dominating nonlinear effects, aswell as the associated transmission penalties, appearto be very different in these two distinct regimes. Re-cently, Louchet et al.2 showed that these disparatenonlinear effects can all be characterized by a singlenonlinear transfer function of the transmission link,provided that the nonlinearity can be considered afirst-order perturbation. This approach allows us totreat all the nonlinear effects within the same frame-work and offers simple guidelines for system designand optimization.

In this Letter we characterize the nonlinear prop-erty of an arbitrary dispersion map by introducingthe power-weighted dispersion distribution (PWDD)function. We show that the nonlinear transfer func-tion of the transmission link is simply the Fouriertransform of the PWDD function. We further derivethe time-domain nonlinear transfer function and di-rectly reproduce some important features of intrac-hannel nonlinear interactions regardless of the pulseshape.

We start with the nonlinear Schrödinger equation

� �

�z+

i

2���z�

�2

�t2 +��z� − g�z�

2 �A�z,t�

= i��A�z,t��2A�z,t�, �1�

where A�z , t� is the complex optical waveform in re-tarded time, ���z� is the group-velocity dispersion,��z� and g�z� are the loss and gain coefficients alongthe link, and � is the fiber nonlinearity coefficient.The amplified spontaneous emission noise from theoptical amplifiers is not considered in this analysis.For most of the discussions in this Letter, we considerA�z , t� to be a single-channel signal, although theanalysis is applicable to multiple channels as well. In

the absence of nonlinearity ��=0�, Eq. (1) has a

0146-9592/06/172544-3/$15.00 ©

simple solution, which is best described in the fre-quency domain

A�z,�� = �P0 exp�G�z� + iC�z��2

2 �u�0����, �2�

where P0 is the average launch power, G�z�=�0z�g�z��

−��z��dz� is the logarithmic loss/gain profile of thelink, C�z�=�0

z���z��dz� is the cumulative dispersion,and u�0����= �1/2���−�

+�u�0��t�exp�i�t�dt is the Fouriertransform of the transmitted waveform u�0��t� [nor-malized such that the average �u�0��t��2�=1]. To takethe nonlinearity into account, a first-order perturba-tive term is inserted into Eq. (2):

A�z,�� = �P0 exp�G�z� + iC�z��2

2 �� �u�0���� + u�1��z,��. �3�

The first-order nonlinear term u�1��z ,�� at the end ofthe transmission �z=L� can be expressed as2,3

u�1��L,�� = i�P0Leff�−�

+�

d�1�−�

+�

d�2���1�2�

�u�0��� + �1�u�0��� + �2�u�0�*�� + �1 + �2�,

�4�

where Leff=�0Lexp�G�z�dz is the effective length, and

���1�2� is the dimensionless nonlinear transfer func-tion

��� =1

Leff�

0

L

exp�G�z� − iC�z�dz, �5�

which fully characterizes the transmission system’snonlinear property within the first-orderapproximation.2 The same spectral domain perturba-tive approach has also been used before to describethe nonlinear transformation over one period of thedispersion map, for example in Refs. 4 and 5. In Eq.(4), ���1�2� is the coefficient for the four-wave mixing

(FWM) process ��+�1�+ ��+�2�− ��+�1+�2�→�.

2006 Optical Society of America

September 1, 2006 / Vol. 31, No. 17 / OPTICS LETTERS 2545

Viewing the nonlinear interactions in the spectral do-main has certain advantages even for a single chan-nel signal. For example, the effect of using shorterpulses to suppress nonlinear timing and amplitudejitters6 is apparent in Eq. (4), since the power spec-tral density is lower with shorter pulses.

We note that much insight can be gained by carry-ing out the integration in Eq. (5) over the cumulativedispersion C instead of over z. This can be done bycollapsing C�z� and G�z� into one function J�C�,

J�C� =1

Leff n

exp�G�zn�C���dzn�C�

dC � , �6�

where zn�C� is the inverse function of C�z� for the nthfiber span. We shall call J�C� the power-weighted dis-persion distribution (PWDD) function of the trans-mission link. Note that J�C� is normalized such that�−�

+�J�C�dC=1. We can then rewrite Eq. (5) in theform of a Fourier transform:

��� = �−�

+�

J�C�exp�− iC�dC. �7�

Figure 1 shows an example of a multiple-spantransmission system. It contains 20 identical fiberspans with a span length of 100 km, making the totaldistance L=2000 km. The fiber chromatic dispersionis 17 ps/ �nm km� (���−22 ps2/km at the wavelengthof 1.55 m), the fiber loss is 0.21 dB/km, the residualdispersion per span is 78 ps/nm with in-line disper-sion compensation, and the dispersion precompensa-tion is −1015 ps/nm [or C�0��1295 ps2]. Lumpedamplification is used at the end of each span to com-pensate for the fiber loss. Postcompensation is usedto bring the net residual dispersion at the end of thelink to zero. The calculated PWDD function of thissystem is shown in Fig. 1(c). Here each span contrib-utes to J�C� a single-sided exponential decay func-tion, and these individual functions are staggered toform a broader function J�C�. In this example thevalue of the precompensation has been chosen suchthat the positive and negative portions of C have

Fig. 1. Example of a multiple-span transmission system.(a) Power profile, (b) cumulative dispersion, and (c) power-weighted dispersion distribution (PWDD). The contribu-

tions from a few individual spans are also shown in (c).

roughly equal weight, or �−�0 J�C�dC��0

+�J�C�dC foroptimal nonlinear performance.6–8 The model systemhere is only a simplified example, and the generalPWDD concept applies to more realistic systems withspans of arbitrary lengths, losses, gains, dispersions,and even different nonlinear coefficients.

The corresponding ��� of this example system isplotted as the solid curves in Fig. 2(b). The main fea-ture of ��� is a peak centered at =0. The width ofthis peak is proportional to the square of the FWMphase-matching bandwidth, also known as the non-linear diffusion bandwidth.2 The sawtoothlike compo-nent in J�C� produces only negligibly small peaks at� ±0.06 ps−2 (not shown) and even smaller harmon-ics.

The Fourier transform relation between the PWDDfunction J�C� and the nonlinear transfer function��� is the key concept of this Letter. It provides newinsight into the effect of dispersion mapping inpseudolinear transmission. An important aspect ofdispersion management is to create a wide and yetreasonably smooth J�C� to shrink the nonlineartransfer function ���. A relatively large residual dis-persion per span is therefore desirable because it in-creases the overall width of J�C�. On the other hand,one should also avoid using a residual dispersion perspan that is too large, because this would break J�C�into sharp spikes and enhance the unwanted compo-nents of ��� at larger ��.

The simplest approximation of J�C� for studyingthe system’s nonlinear characteristics is probably arectangular function illustrated by the dashed line inFig. 2(a). The rectangular PWDD model yields nearlythe same center peak of ���, which explains why it isoften a fairly good approximation. The rectangularPWDD model represents a hypothetical transmissionsystem employing lossless fiber (uniform power pro-file) with constant dispersion �� and no in-line dis-persion compensation (the entire link dispersion iscompensated with predispersion or postdispersioncompensation, or a combination of the two). Thismodel has been used to study intrachannel nonlinearinteractions in long-haul transmission systems.9–11

A well-known nonlinear phenomenon in high-speed(e.g., 40 Gbit/s) pseudolinear transmission is theghost pulse generation due to intrachannel FWM(IFWM). In this process, three optical pulses locatedat T1, T2, and T3 interact and generate a ghost pulseat t�T1+T2−T3. Almost all the analytical studies ofIFWM6,8–14 have assumed a Gaussian shape for the

Fig. 2. (a) PWDD function J�C� and (b) real and imaginaryparts of the nonlinear transfer function ���. The dashedcurves correspond to the rectangular PWDD model.

optical pulses. In the following, we show that the

2546 OPTICS LETTERS / Vol. 31, No. 17 / September 1, 2006

Gaussian assumption for the pulse shape can bedropped with the nonlinear transfer function andPWDD method. To study IFWM, it is more intuitiveto express the nonlinear interactions in the time do-main. Interestingly, Eq. (4) has a direct time-domaincounterpart5:

u�1��L,t� = i�P0Leff�−�

+�

dt1�−�

+�

dt2h�t1t2�

�u�0��t + t1�u�0��t + t2�u�0�*�t + t1 + t2�,

�8�

where h�t1t2� is the time-domain nonlinear transferfunction. Similar to ���1�2�, h�t1t2� is also directly re-lated to the function J�C�:

h�t1t2� = �−�

+� J�C�

�C�exp�i

t1t2

C �dC. �9�

Consider a pulse waveform p�0��t� that is nonzero onlyin the vicinity of t=0. Three interacting pulseslocated at T1, T2, and T3 are then p�0��t−T1�,p�0��t−T2�, and p�0��t−T3�. Substituting p�0� for u�0� inEq. (8), we obtain the corresponding ghost pulse

p1+2−3�1� �L,t� = i�P0Leff�

−�

+�

dt1�−�

+�

dt2h�t1t2�

�p�0��t − T1 + t1�p�0��t − T2 + t2�p�0�*�t − T3

+ t1 + t2�. �10�

Since the product of the three pulse waveforms in theabove integral is nonzero only near t=T1+T2−T3, t1=T3−T2, and t2=T3−T1 (assuming very short pulses),we can use the approximation h�t1t2��h��T1−T3��T2−T3� and bring it outside the integral. The positionand shape of the ghost pulse are then determined bythe remainder of the integral, which is equal to theconvolution of three functions p�0��t−T1�, p�0��t−T2�,and p�0�*�−t−T3�. If p�0��t� is a Gaussian pulse, the re-sult of the convolution is still Gaussian but broad-ened by a factor of �3.9,13 This broadening factor,however, does not apply to other pulse shapes in gen-eral. For example, if p�0��t� were a sinc function, thewidth of the resulting ghost pulse would be the sameas p�0��t�. The scaling of the ghost pulse amplitudeh��T1−T3��T2−T3�, on the other hand, is more gen-eral and does not require the Gaussian pulse shape.If we consider a rectangular PWDD model in Eq. (9)with C ranging from 0 to ��L (nonsymmetric disper-sion map), we can derive h��T1−T3��T2−T3� analyti-cally:

h��T1 − T3��T2 − T3�

=1

����LE1�− i

�T1 − T3��T2 − T3�

��L � , �11�

� −t

where E1�x�=�x �e / t�dt is the exponential integral

function. Here we have reproduced the result in Refs.9 and 11 without using the Gaussian approximationfor p�0��t�.

The above analysis can also be used to study intra-channel cross-phase modulation (IXPM), althoughspecial care should be taken to avoid the divergenceat C=0 in Eq. (9). The general concept of PWDD canbe further extended to address many other issues inthe analysis and modeling of the nonlinearSchrödinger equation. For example, it provides asimple explanation for the spurious FWM tones ob-served in split-step simulations using an improperstep size.15 The split-step method essentially approxi-mates the function J�C� with a series of delta func-tions. If these delta functions are equally spaced in C,artifacts in the FWM efficiency ����1�2��2 may arisefrom interference.

In conclusion, we have shown that the PWDD func-tion is a versatile tool to characterize and visualizenonlinear properties of long-haul optical transmis-sion systems. The only intrinsic assumption in thismethod is the first-order perturbation. For systems inthe strong perturbation regime, higher-order pertur-bations may be necessary.16

X. Wei’s e-mail address is xingwei@ieee.org.

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