Optical Fiber Technology 19 (2013) 75–82

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XPM and SPM induced crosstalk in WDM system employing distributed Ramanamplifier for DPSK and OOK modulation format

Anamika ⇑, Vishnu PriyeElectronics Engineering, Indian School of Mines, Dhanbad 826 004, India

a r t i c l e i n f o

Article history:Received 24 July 2012Revised 9 November 2012Available online 31 December 2012

Keywords:Cross phase modulationSelf phase modulationDifferential phase-shift keyingON–OFF keyingWavelength divison multiplexedOptical communication

1068-5200/$ - see front matter � 2012 Elsevier Inc. Ahttp://dx.doi.org/10.1016/j.yofte.2012.11.003

⇑ Corresponding author.E-mail address: anamika.ece.ism@gmail.com ( Ana

a b s t r a c t

Cross-phase modulation (XPM) and self-phase modulation (SPM) induced nonlinear crosstalk has beenstudied analytically for coherent DPSK and OOK signal in NRZ- and RZ-modulation format for WDM sys-tem employing distributed Raman amplifier. The study shows that 40 Gb/s RZ-DPSK signal with 33.3%duty cycle experiences minimum XPM and SPM induced crosstalk. The results also reveal that minimumcrosstalk was induced in backward pumped DRA among the three pumping schemes i.e. forward, back-ward and bi-directional. Results assume importance for minimizing deleterious XPM and SPM effects inoptical communication system.

� 2012 Elsevier Inc. All rights reserved.

1. Introduction

Optical communication systems can support Tb/s capacitiesover long distances making them an ideal technology for highcapacity wireline networks. With the growing bandwidth demand,there is a tremendous interest in increasing the transport capacityand transmission distance of WDM system. Expanding networkfunctionality into the optical domain is another aim of fiber-opticcommunication research. Optical networks with high spectral effi-ciency are being designed to have higher per fiber transport capa-bilities as well as low cost per transmitted bits. Modulation formatsuch as intensity and phase modulation formats is a key technol-ogy that enables the design of such networks. ON–OFF keying(OOK), carrier suppressed return to zero (CSRZ), Modified duo-binary (MDB), Alternate mark inversion (AMI) are some examplesof intensity modulation format. Similarly, Differential Phase shiftkeying (DPSK), Quadrature phase shift keying (QPSK) and differen-tial quadrature phase shift keying (DQPSK) are some examples ofphase modulation formats. In this research article, analyticalexpressions have been derived for XPM and SPM induced crosstalkfor two modulation formats-one intensity modulation format i.e.OOK and other phase modulation format i.e. DPSK. The reason forchoosing the two formats stem from the fact that in literature anumber of research work is available that deals with the perfor-mance of the two formats [1–4]. Similar study for other modula-

ll rights reserved.

mika).

tion formats such as CSRZ, MDB, AMI, QPSK and DQPSK will bepursued in future.

Coherent detection system has gained considerable attentionover past few years because of improved receiver sensitivity andincreased spectral efficiency of wavelength division multiplexed(WDM) system compared to Intensity Modulation Direct Detection(IM/DD) systems [5]. The advantage of coherent detection tech-niques is that the amplitude and phase of the detected optical sig-nal can be measured and hence information can be transmitted inthe form of amplitude, phase or frequency of optical signal. WhenOOK and DPSK modulation formats are implemented for coherentsystems, it is required that phase of the electrical field associatedwith optical signals remains constant as the detector response de-pends on the phase of the received signal. When large number ofsignals simultaneously propagates in wavelength division multi-plexed (WDM) system, due to high power confinement, nonlineareffect comes into play. Cross-Phase modulation (XPM) and Self-Phase modulation (SPM) are two such nonlinear effects also knownas optical Kerr effects [6–10]. The change in phase of channel isproportional to its own intensity in SPM and to the intensity ofother channels in XPM. The phase of the signal gets affected dueto XPM and SPM is denoted as crosstalk or nonlinear phase shiftand hence the detector response changes in coherent detectionsystem.

Chromatic dispersion is referred to broadening of input signal asit travels down the fiber length. It is the second derivative of opti-cal phase with respect to optical frequency. The interaction be-tween nonlinearity and dispersion is an important issue in thedesign of Lightwave system. Phase modulation of signals in

76 Anamika, V. Priye / Optical Fiber Technology 19 (2013) 75–82

WDM system due to SPM and XPM gets converted to intensitymodulation through dispersion and thus results in waveform dis-tortions. Depending on fiber chromatic dispersion and its manage-ment, XPM induced nonlinear phase shift may become verydetrimental for WDM signals [11–13]. In WDM transmission sys-tems, XPM induces a broadening of the signal spectrum and sowider optical filter bandwidth is required at the receiver. This de-grades the system performance, because more spontaneous emis-sion noise enters the receiver [14]. Moreover, the damagingeffects of dispersion become more dominant on wider spectrumof signal and hence system performance degrades further. It hasbeen found that XPM-induced signal broadening is similar to theone induced by laser phase noise. In the decision circuit of the re-ceiver [5], the phase fluctuations cause error in decision making.The phase fluctuation due to laser phase noise is minimized byusing semiconductor laser whose line width is a small fraction ofbit rate. But phase fluctuation due to XPM induced signal broaden-ing still remains and causes error in the signal detection.

Electronic predistortion (EPD) of chromatic dispersion usingdigital signal processing is a cost-effective alternative to conven-tional optical dispersion compensation (ODC) using inline disper-sion compensating fiber [15]. The inline dispersion compensationin ODC systems is replaced by an EPD transmitter that pre-compensates the individual WDM channels for the chromaticdispersion of the entire transmission distance. EPD has been exper-imentally demonstrated for 10 Gb/s single channel and WDMsystem [16,17]. Research experiments [18,19] and simulations[20] have demonstrated that EPD systems are strongly degradedby SPM and XPM compared to ODC systems.

Optical amplifiers (OA) such as Erbium-doped fiber amplifiers(EDFA) and distributed Raman amplifier (DRA) are used to com-pensate attenuation of signals which is a major limiting factor inoptical communication. In presence of OA, the nonlinear effectsXPM and SPM are greatly enhanced. In DRA high power pumpco- and counter propagate with the signal providing continuousgain all along the transmission line. Raman pumps also interactamong themselves but the spacing between them is in the presentwork has been considered to be around 10 nm. At such large sep-aration only stimulated Raman scattering (SRS) will be the domi-nant nonlinear effects and will cause power transfer between thepumps. XPM dominate at smaller interchannel spacing and willnot be a detrimental effect. Another nonlinear effect FWM betweenpump–pump and pump-signal causes generation of new wave-lengths under the condition of phase matching. These new wave-lengths can be in the range of both signal wavelengths and pumpwavelengths. But appropriate selection of fiber dispersion helpsin minimizing FWM. Moreover, the effect has been reported tobe more dominant in broad band amplifiers made from NZDF withzero dispersion wavelength located between pump and signalbands [21–24].

It has been found in research that when multiple pumps at dif-ferent wavelength are launched into the fiber, the gain at shorterwavelength decreases whereas at longer wavelength increases. Inpresent analysis this difference in gain is simplified by assumingequal gain for all pumps. Similar approach can be found in someprevious works [25–29] in which when distributed Raman ampli-fier was analyzed, undepleted pump approximation was assumedas the intensity of pump is much higher than that of signal. Thisassumption is generally true for many practical cases and allowsone to solve the analytical equation of evolution of pump powerand signal power for a variety of pumping configuration. In thepresent work, undepleted pump approximation has been assumedfor the derivation of analytical expression of crosstalk.

In our analysis noise power evolution and interpulse collisionhas not been considered. In Ref. [24], the author gives a detaileddiscussion of different types of noise occurring in DRA such as

amplified spontaneous noise, signal-spontaneous beat noise, mul-ti-path interference (MPI) noise, transfer of relative intensity ofnoise (RIN) from pump to signal. Some work has been done in thisfield such as Ref. [25] that gives an analytical study of RIN, Ref. [26]where MPI in DRA has been discussed. Again in Ref. [30], theauthor gives an analytical characterization of evolution of signalpower and noise figure in forward pumped DRA. As can be seenin literature, consideration of noise requires detailed analysis andhas hence been deferred by the authors for future. It has beenfound in research that for complete collisions, the collision-induced frequency shift of a pulse is negligible, whereas positionshift is significant [31,32]. For strong dispersion management ithas been found that incomplete collisions can be neglected,whereas for weak dispersion management system the contributionof the incomplete collisions can be significant [31,32]. As the colli-sion induced frequency shift is negligible, there is no significantshift in frequency of signal. Moreover dispersion management isnot considered in the analysis hence the impact of collision rateon crosstalk performance is neglected without loss of significantchange in result.

There has been considerable research work on study of nonlin-ear effects and its effect on signal degradation [33–35] and inter-play between them [36–40]. XPM effect in WDM system for OOKand DPSK modulation format has been studied theoretically [41],experimentally [42] and numerically [43]. Due to random temporalwaveform alignments of WDM channels, large variations of XPMdegradation have been predicted for OOK and DPSK signals[4,42]. The effect of nonlinear stimulated Raman scattering hasbeen previously studied by the present authors in WDM systememploying EDFA [44] and DRA [45]. In continuation with the pre-vious work, in this paper using statistical methods a comprehen-sive study of XPM and SPM induced crosstalk or nonlinear phaseshifts has been done for WDM system employing DRA for DPSKand OOK signal in NRZ- and RZ-modulation format. On the basisof the detailed study we have shown that 40 Gb/s RZ-DPSK signalwith 33.3% duty cycle is most tolerant to XPM and SPM inducedcrosstalk in coherent detection system. In WDM system employingDRA, backward pumped DRA is most tolerant to XPM and SPM in-duced crosstalk among the three pumping scheme i.e. backward,forward and bi-directional.

The paper is organized as follows. In Section 2 closed form for-mulae are derived to study XPM and SPM induced crosstalk inWDM system employing DRA. Section 3 uses the closed formedformulae to evaluate crosstalk in a typical configuration of WDMsystem. Performance of different modulation formats at differentdata rates and pulse shape used in optical data transmission sys-tem is investigated. After choosing the best performing pulse shapeand data rate, crosstalk has been investigated for the three pump-ing scheme of DRA i.e. forward, backward and bi-directional. Sec-tion 4 concludes the paper.

2. Theory

In N-channel WDM systems employing distributed Ramanamplifier for bi-directional pumping scheme, the phase modula-tion of mth channel induced by nth channel due to XPM is givenas [46]

umnðtÞ ¼ 2cZ L

0Pnð0; t � dmnz0Þe�az0gFðz0ÞgBðz0Þdz0 ð1Þ

where Pn(z, t) is the power of nth channel as a function of length zand time t. Pn(0, t) is the launched power at the transmitter, c isthe nonlinear co-efficient, a is the fiber attenuation co-efficient,dmn(=1/vn � 1/vm) is propagation time difference between the twodifferent channels (m, n) during a unit length transmission. In Eq.

Anamika, V. Priye / Optical Fiber Technology 19 (2013) 75–82 77

(1) it has been assumed that fiber dispersion causes just pulse walk-off and no pulse distortion [46–48].

In DRA, forward amplification gain [49] is

gFðz0Þ ¼ expgP

kskp Pso þ PPo

� �Aeff a

ð1� e�az0 Þ

0@

1A ð2Þ

where peak Raman gain g = 6.5 � 10�14 m/W, Ppo and Pso are the ini-tial pump and signal power respectively, ks (kp) is Signal (Pump)Wavelength.

Backward amplification gain [49] is

gBðz0Þ ¼ expgðP

kskp Pso þ PPLÞAeff a

ðeaðz0�LÞ � 1Þ !

ð3Þ

where PPL is initial backward pump power, L is the length of DRAand other parameters are same as (2).

In the expression of forward and backward gain followingassumptions has been made:

(a) The attenuation constant of pump and signal wavelength issame and equal to 0.2 dB/km. Variation in attenuation con-stant with wavelength has been neglected.

(b) In forward gain pump depletion has been taken into consid-eration since the optical signal at the Stokes frequency existsfrom the beginning.

(c) In backward gain pump depletion is neglected as the systemis designed not to induce pump depletion because the signallevel will fluctuate if pump depletion occurs.

Power spectral density of umn is found by taking the Fouriertransform of auto-correlation function and is given as

SumnðxÞ ¼ SPmn ðxÞjHmnðxÞj2 ð4Þ

where SumnðxÞ is the power spectral density of umn(t) and SPmn ðxÞ is

the power spectral density of Pn(0, t) which depends on pulse shapep(t).

A rectangular NRZ pulse is given as

pðtÞ ¼2P0 jtj < T

2

� �0 jtj > T

2

� �( )

ð5Þ

Fourier transform of p(t) is given as

jPðxÞj ¼ 2P0Tsin xT

2xT2

ð6Þ

Similar representation for RZ pulse of duty cycle sb is obtainedby replacing T with T

sbin the expression for jP(x)j, where sb as-

sumes value 2 and 3 for 50% and 33.3% duty cycle respectively.Hence power spectral density

SPmn ðxÞ ¼1

4TjPðxÞjð2Þ þ 1

4T2

X1k¼�1

PkT

� ���������2

d x� kT

� �ð7Þ

For Eq. (4), the transfer function is obtained as

HmnðxÞ ¼ 2cZ L

0e�jdmnz0xe�az0eK 0 ð1�e�az0 ÞeðK

00 ðeaðz0�LÞ�e�aLÞÞ dz0 ð8Þ

where K 0 ¼ cP

kskpPsoþPPoð ÞaAeff

and K 00 ¼ cP

kskpPsoþPPLð ÞaAeff

.Substituting z0 = ln x.The equation becomes

HmnðxÞ ¼ 2ceK 0e�K 00e�aLZ eL

1

x�ðaþjdmnxÞ

xe�K 0xþK 00xe�aL

dx ð9Þ

The exponential e�K 0t�aþK 00tae�aL is binomially expanded to be thesum of two terms (1 � K0t�a + K00tae�aL) and the higher order termsare neglected which contribute insignificantly in the seriesexpansion.

HmnðxÞ ¼ 2c1� e�ðaþjdmnxÞL

aþ jdmnx

� 2cK 0

1� e�ð2aþjdmnxÞL

2aþ jdmnx

þ 2cK 00e�aL 1� e�ðjdmnxÞL

jdmnx

ð10Þ

2.1. DPSK modulation format

At DPSK [46] receiver, the differential nonlinear phase shiftDumn(L, t) = umn(L, t) � umn(L, t � T) adds to the differential phaseof the signal. Here T is the bit period. Hence power spectral densityof Dumn(L, t) is

SDumnðxÞ ¼ 4SPmn ðxÞjHmnðxÞj2sin2ðxT=2Þ ð11Þ

The variance of XPM induced crosstalk is given by [13,46]

r2XPMðm;nÞ ¼

12p

Z 1

�1SDumn

ðxÞdx ð12Þ

Substituting the expression of power spectral density of DPSK-NRZ signal i.e. SPmn ðxÞ in the expression of SDumn

ðxÞ and performingthe infinite integration of Eq. (12) using Tables of Integral and Ser-ies [50] and is given in Appendix A.

2.2. OOK modulation format

At OOK receiver, the nonlinear phase shift umn(L, t) adds to thephase of the signal. Hence power spectral density of umn(L, t) is

SumnðxÞ ¼ SPmn ðxÞjHmnðxÞj2 ð13Þ

The variance of XPM induced crosstalk is given by

r2XPMðm;nÞ ¼

12p

Z 1

�1SumnðxÞdx ð14Þ

Substituting the expression of power spectral density of OOK–NRZ signal i.e. SPmn ðxÞ in the expression of Sumn

ðxÞ and performingthe infinite integration of Eq. (14) using Tables of Integral and Ser-ies [50] and is given in Appendix A.

In N-channel WDM system, each channel is modulated inde-pendently by random data. Thus, the variance can be approximatedas

r2XPMðmÞ ¼

XN

n¼1

r2XPMðm;nÞðm – nÞ ð15Þ

XPM and SPM are nonlinear Kerr effects that originate fromintensity dependence of the refractive index. While SPM refers toself induced phase shifts experienced by an optical signal duringits propagation in optical fiber, XPM refers to nonlinear phase shiftof optical field induced by co-propagating field at different wave-length. From the equation of nonlinear phase shift [5], it can besaid that for equally intense optical field, the contribution ofXPM to the nonlinear phase shift is twice compared to that ofSPM. The variance of SPM induced nonlinear phase shift can be cal-culated directly from the expression of XPM by making inter-channel separation equal to zero as is given in Ref. [46] (Eqs. (5)and (6)). This is due to the fact that in SPM change in phase ofthe signal is proportional to its own intensity. Though SPM causesXPM induced PM (Phase Modulation) to PM conversion and IM

Fig. 2. Variation of crosstalk standard deviation (dB) with signal wavelength (nm)for OOK at bit rate (B) = 2.5 Gb/s, 10 Gb/s, 40 Gb/s.

78 Anamika, V. Priye / Optical Fiber Technology 19 (2013) 75–82

(Intensity modulation) conversion, the analytical characterizationof which has been done in Ref-[38], this conversion is negligiblefor XPM characterization in WDM system. In the final calculation,the total nonlinear phase shift of a signal in an N-channel WDMsystem is calculated by the sum of XPM induced phase shift fromremaining N-1 channels and SPM induced phase shift from thesame channel. The total variance of nonlinear effect is given bythe sum of variance of XPM and SPM and is in accordance with ear-lier approach [7, Eq. (2)].

The variance of SPM induced crosstalk is calculated by takingthe interchannel separation (Dk) equal to zero in the expressionof variance of XPM [46]. Hence, for Dk = 0,

r2SPMðmÞ ¼

14r2

XPMðmÞ ð16Þ

The factor of 1/4 comes from the fact that phase shift inducedby XPM is twice as large as SPM for the same intensity of the signal[46].

Thus total variance of XPM and SPM induced crosstalk in mthchannel is given as [7]

r2x ðmÞ ¼ r2

SPMðmÞ þ r2XPMðmÞ ð17Þ

The crosstalk standard deviation is evaluated on decibel scaleby using the formulae

rxðdBÞ ¼ �10log10e�rx ¼ rx � 10log10e ð18Þ

3. Results and discussion

Fig. 1 shows the DRA configuration considered for our analysis.The N-channels of WDM system are assumed to transmit 0 dBmpower in wavelength range of 1515–1575 nm with inter–channelseparation of 1 nm. As the gain spectrum of Raman amplifier de-pends on pump wavelength, it gives flexibility to achieve gain atany desired signal wavelength. The multi-pumps consisting of6 pumps are in wavelength range of 1420–1470 nm, separated by10 nm and each having power of 20 dBm yielding wideband andflat gain spectrum in signal wavelength range of 1515–1575 nm.This is done in order to achieve broadband and flat gain which isknown as ‘‘WDM pumping’’ scheme [51,52]. Each pump thus pro-vides peak Raman gain to signal at a shifted wavelength of 100 nmfrom pump wavelength and has a gain spectrum of 10 nm.

The generalized formulae are derived for WDM system employ-ing bi-directional pumped DRA. The forward and backward pump-ing are considered to be special cases of bi-directional pumping bytaking into account the launched pump power. Furthermore, theinteraction among pumps is neglected in current analysis and

Fig. 1. Schematic diagram of bi-di

spotlight is on crosstalk between signals as they are continuouslypumped while propagating in the fiber. For analysis purpose stan-dard SMF fiber is used with following parameters: c (Nonlinear Co-efficient) = 1.18 W�1 km�1; Aeff (Effective Area) = 80 lm2; Zero Dis-persion Wavelength = 1265.5 nm; Dispersion Slope = 0.058 � 103

ps/nm/nm/km; a (attenuation co-efficient) = 0.2 dB/km; L(length) = 80 km.

First the effect of bit rate on XPM and SPM induced crosstalk hasbeen investigated. Figs. 2 and 3 show the variation of XPM and SPMinduced crosstalk with signal wavelength for bit rate of 2.5 Gb/s,10 Gb/s and 40 Gb/s for NRZ-OOK and NRZ-DPSK signal. We havenot included the results for 100 Gb/s as at such high bit rates intra-channel XPM and FWM i.e. IXPM and IFWM becomes dominantand ignoring such effects might result in large inaccuracy in re-sults. The authors propose to include IXPM in future works to takeinto account intrachannel nonlinearities. It can be seen from thefigure that with the increase in bit rate of the signals, crosstalk de-creases and minimum crosstalk is observed for 40 Gb/s signal. Thisis because walk-off length Lw (=T/jdmnj), the distance at which twomth and nth pulses of length T becomes completely walkoff [46],varies inversely with fiber dispersion co-efficient and bit rate ofthe system. Neglecting the impact of collision rate between thepulses and having only considered walk-off length (which is actu-ally the inter collision distance), it can be said from the analysisthat in DRA variance of crosstalk varies with Lw. Thus, crosstalkdecreases with the increase in bit rate of the system, otherparameters remaining constant. We have neglected the impact of

rectional multi-pumped DRA.

Fig. 3. Variation of crosstalk standard deviation (dB) with signal wavelength (nm) for DPSK at bit rate (B) = 2.5 Gb/s, 10 Gb/s, 40 Gb/s.

Fig. 4. Variation of crosstalk standard deviation (dB) with signal wavelength (nm) for NRZ-OOK, RZ-OOK (50% duty cycle) and RZ-OOK (33% duty cycle).

Fig. 5. Variation of crosstalk standard deviation (dB) with signal wavelength (nm) for NRZ-DPSK, RZ-DPSK (50% duty cycle) and RZ-DPSK (33% duty cycle).

Anamika, V. Priye / Optical Fiber Technology 19 (2013) 75–82 79

collision rate between the pulses and have only considered walk-off length (which is actually the inter collision distance).

After selecting the best performing bit rate, effect of variation induty cycle of pulse was investigated. Figs. 4 and 5 show the varia-tion of XPM and SPM crosstalk with signal wavelength for differentduty cycle of the pulse for both OOK and DPSK signal at 40 Gb/s

data rate. Other parameters assumed are same as the previousanalysis and configuration of fiber link is as shown in Fig. 1. Itcan be seen from the figures that with the decrease in duty cycleof the pulse, XPM and SPM crosstalk decreases for both OOK andDPSK signal. The reason for RZ-pulse suffering less crosstalk com-pared to NRZ-pulse is because in RZ-pulse, the ‘1’ bit occupy only

Fig. 6. Variation of crosstalk standard deviation (dB) with signal wavelength (nm) for RZ-DPSK signal (33% duty cycle) at 40 Gb/s for forward, backward and bi-directionalpumped DRA.

80 Anamika, V. Priye / Optical Fiber Technology 19 (2013) 75–82

a fraction of the time slot. Hence the probability of overlap of‘1’ bitin neighbouring channels decreases. On comparing Figs. 4 and 5, itis found that for both RZ- and NRZ-modulation format there is adecrease of almost 20% in crosstalk from OOK to DPSK signal.The reason for the DPSK signal suffering less crosstalk stems fromthe fact that in the DPSK signal only differential phase of the signalis affected. On the contrary in OOK signal the total phase of the sig-nal gets affected due to XPM induced phase shift. Thus detector re-sponse gets more affected in case of OOK signal compared to DPSKsignal. Hence RZ-DPSK signal with 33.3% duty-cycle suffers mini-mum crosstalk compared to NRZ-DPSK signal, NRZ-OOK signal,RZ-DPSK and RZ-OOK signal with 50% duty cycle.

On the basis of above analysis it can be observed that 40 Gb/sRZ-DPSK signal (33.3% duty cycle) minimum SPM and XPM cross-talk is induced in WDM system. Using the above observation, var-iation of SPM and XPM induced crosstalk with signal wavelength isstudied for 40 Gb/s RZ-DPSK signals (33.3% duty cycle) for differentpumping scheme. The expression of variance of SPM and XPM in-duced crosstalk was used to investigate crosstalk in forwardpumped DRA and backward pumped DRA by taking into accountthe launched pump power i.e. by making backward pump PPL

strength equal to zero in the generalized formulae of bi-directionalpumped DRA for analysis of forward pumped DRA and vice versa. Itcan be observed from Fig. 6 that backward pumping scheme suffersminimum crosstalk. Hence it can be said that backward pumpedDRA with RZ-DPSK signal (33.3% duty cycles) at 40 Gb/s suffersfrom minimum crosstalk and is apposite for opticalcommunication.

4. Conclusion

XPM and SPM induced crosstalk in WDM system employing bi-directional pumped DRA has been evaluated using closed form for-mulae for DPSK and OOK signal in RZ and NRZ modulation format.It is found that 40 Gb/s RZ-DPSK signal has best crosstalk perfor-mance compared to 10 Gb/s NRZ DPSK, 40 Gb/s RZ–OOK and40 Gb/s NRZ–DPSK signal. Among the three pumping schemes,minimum crosstalk is observed for WDM system employing back-ward pumping scheme. The results based on the statistical analysisshows that a 40 Gb/s RZ-DPSK signal (33.3% duty cycle) offers the

2ReðMO�Þ ¼ ð2cÞ2K 00e�aL

a2 þ d2mnx2

� �d2

mnx2� � ð1þ e�aLÞd2

ijx2ð1� cos dmnxLÞ þ

n24

superlative crosstalk performance in a WDM communicationsystem.

Acknowledgment

One of the authors, Anamika would like to acknowledge ISMDhanbad for encouragement and financial support for conductingthe present research work.

Appendix A

jHijðxÞj2 ¼ jM þ N þ Oj2 ¼ ðM þ N þ OÞ � ðM þ N þ OÞ�

¼ jMj2 þ jNj2 þ jOj2 þMN� þM�N þMO� þM�Oþ ON�

þ O�N

¼ jMj2 þ jNj2 þ jOj2 þ 2� ReðMN�Þ þ 2� ReðMO�Þ þ 2

� ReðNO�Þ

Substituting the values of M, N and O from equation of Hmn(x)and simplifying we get the following terms

jMj2 ¼ ð2cÞ2

a2 þ ðdmnxÞ2ð1� e�aLÞ2 þ 4e�aL sin2 dmnxL

2

� � ðA:1Þ

jNj2 ¼ ð2cÞ2K 02

4a2 þ ðdmnxÞ2ð1� e�2aLÞ2 þ 4e�2aL sin2 dmnxL

2

� � ðA:2Þ

jOj2 ¼ ð2cÞ2K 002e�2aL

ðdmnxÞ24 sin2 dmnxL

2

� � ðA:3Þ

2ReðMN�Þ ¼ð2cÞ2K 0 2a2 þ d2

mnx2� �

a2 þ d2mnx2

� �4a2 þ d2

mnx2� �

� 1þ e�3aL � ðe�2aL þ e�aLÞ 1� 2 sin2 dmnxL2

� �� �

þ 2C2K 0admnx sin dmnxL

a2 þ d2mnx2

� �4a2 þ d2

mnx2� � e�aL � e�2aL

� �ðA:4Þ

admnx sin dmnxLð1� e�aLÞo35 ðA:5Þ

2ReðNO�Þ ¼ ð2cÞ2K 0K 00e�2aL

4a2 þ d2mnx2

� �d2

mnx2� � 1þ e�2aL

� �d2

mnx2ð1� cos dmnxLÞ þ 2admnx sin dijxLð1� e�2aLÞ

n o24

35 ðA:6Þ

Anamika, V. Priye / Optical Fiber Technology 19 (2013) 75–82 81

The expression of HmnðxÞ and SPmn ðxÞ (assuming rectangularNRZ pulse shape) is substituted in equation of variance r2

x

� �in

Eqs. (9) and (11). On performing the infinite integration, we getthe expression of variance r2

x

� �for OOK and DPSK signal.

For OOK, the expression of variance is

r2XPM ¼ ðFðTÞ þ F1ðTÞ þ F2ðTÞ þ F3ðTÞ þ F4ðTÞ þ F5ðTÞÞ

For DPSK, the expression of variance is

r2XPM ¼ ðFðTÞ þ F1ðTÞ þ F2ðTÞ þ F3ðTÞ þ F4ðTÞ þ F5ðTÞÞ � ð1=4Þ

� ðFð2TÞ þ F1ð2TÞ þ F2ð2TÞ þ F3ð2TÞ þ F4ð2TÞ þ F5ð2TÞÞ

where

FðTÞ ¼ C2P20

a3Lwfð1� e�aLÞ2P1ðaÞ þ e�aLP2ðaÞg ðA:7Þ

F1ðTÞ ¼C2K 02P2

0

8a3Lwfð1� e�2aLÞ2P1ð2aÞ þ e�2aLP2ð2aÞg ðA:8Þ

F2ðTÞ ¼8C2K 02P2

0e�2aL

3d2mnT

� minT2

4;d2

mnL2

4

!3 max

T2;dmnL

2

� ��min

T2;dmnL

2

� � �

ðA:9Þ

F3ðTÞ¼�C2K 0P2

0

a3Lw1þe�3aL�e�2aL�e�aL� � P1ðaÞ

3þ2P1ð2aÞ

3

� �

þ12ðe�aLþe�2aLÞðP2ðaÞþ

2P2ð2aÞ3

Þþ C2K 0P2

0

24a3Lw

P3ð2aÞ4�P3ðaÞ

� �ðA:10Þ

F4ðTÞ ¼C2K 00e�aLP2

0

a3Lwð1þ e�aLÞP2ðaÞ � ð1� e�aLÞP3ðaÞ� �

þ C2K 00e�aLP20T

admnð1� e�aLÞP4ðT; dmnLÞ� �

ðA:11Þ

F5ðTÞ ¼C2K 0K 00e�2aLP2

0

8a3Lwð1þ e�2aLÞP2ð2aÞ � ð1� e�2aLÞP3ð2aÞ� �

þ C2K 0K 00e�2aLP20T

2admn½ð1� e�2aLÞP4ðT; dmnLÞ�

ðA:12Þ

AndC ¼ 2c

P1ðaÞ ¼ ðe�aLw þ aLw � 1Þ

P2ðaÞ ¼ 2ðe�aLw þ e�aL�1Þ� ðe�ajLþLw j þ e�ajL�Lw jÞþaðjLþ Lwj� jL� LwjÞ

P3ðaÞ ¼ ½signðdmnLÞð2� 2e�aLÞ þ signðT � dmnLÞð1� e�ajL�Lw jÞ� signðT þ dmnLÞð1� e�ajLþLw jÞ�

P4ðT;dijLÞ ¼ signðdmnLþ2TÞ ðdmnLÞ2

8þðdmnLÞT

2þ T

2

!� signðdijLÞ

ðdmnLÞ2

4

þ signð�dmnLþ2TÞ ðdmnLÞ2

8þðdmnLÞT

2� T2

2

!

Lw ¼Tjdmnj

The integration has been performed using formula 3.826.1,3.824.1, 3.725.1 and 3.725.3 of Ref [50] and basic trigonometricand algebraic functions.

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