Analysis of nonlinear phase noise in coherent fiber-optic systems based on phase shift keying

4722 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 27, NO. 21, NOVEMBER 1, 2009

Analysis of Nonlinear Phase Noise in CoherentFiber-Optic Systems Based on Phase Shift Keying

Shiva Kumar, Member, IEEE

Abstract—Analytical expressions for the phase variance in anonlinear fiber optic system based on phase-shift keying are de-veloped. The Gauss-Hermite functions are used as the orthogonalbasis to represent the noise field. Number of degrees of freedom(DOF) to accurately model the phase variance is estimated. Theamplifier noise excites higher order Gauss-Hermite noise modesand the nonlinear mixing of a signal pulse and higher orderGauss-Hermite noise mode leads to new noise fields which en-hance the nonlinear phase noise. The higher order noise modespropagate linearly and enhance the linear phase noise if thematched filter is not used at the receiver. Analytical expression forthe optimum launch power is developed taking into account thelinear and nonlinear phase noise.

Index Terms—Amplifier spontaneous emission (ASE), nonlinearoptics, optical fiber communication, optical Kerr effect, phasenoise, phase-shift-keying (PSK).

I. INTRODUCTION

T HE phase fluctuations induced by the coupling betweenKerr nonlinearity of the fiber and amplified spontaneous

emission (ASE) of inline amplifiers lead to performance degra-dation in fiber-optic systems based on phase-shift keying (PSK)or differential phase-shift keying (DPSK) [1]–[4]. The ampli-tude fluctuations caused by ASE is translated into phase fluctu-ations because of fiber nonlinearity. This is known as nonlinearphase noise. If the phase deviation exceeds , it leads to a biterror. This problem is first analyzed by Gordon and Mollenauer[1], and, hence, this noise is also called ”Gordon-Mollenauerphase noise.”

Gordon and Molleneuer pointed out that two degrees offreedom (DOFs) of the noise modes are of importance [1].These noise modes have the same form as the signal pulse. Oneof the noise modes is in phase with the signal and the other inquadrature. The in-phase component of the noise changes theamplitude of the signal pulse and, hence, leads to energy changewhile the quadrature component leads to a linear phase shift.The energy change is translated into an additional phase shiftdue to fiber nonlinearity. Gordon and Mollenauer argued thatthe noise modes other than the above mentioned modes haveless significant effects if the optical bandwidth is not too large.Mecozzi analyzed the nonlinear phase noise with arbitraryDOFs [5] and showed that if the signal bandwidth is equal tothe noise bandwidth, two DOFs are sufficient to describe the

Manuscript received December 16, 2008; revised May 02, 2009. This workwas supported by the NSERC discovery Grant. First published July 07, 2009;current version published September 10, 2009.

The author is with the Department of Electrical and Computer Engineering,McMaster University, Hamilton, ON L8S 4K1 Canada.

Digital Object Identifier 10.1109/JLT.2009.2026589

nonlinear phase noise, which corresponds to a matched filter. Ifthe filter bandwidth is larger than signal bandwidth, additionalDOFs are required. Later Ho [6], [7] and Mecozzi [8] derivedanalytical expressions for the probability density function ofnonlinear phase noise.

The analytical expressions of Gordon and Mollenauer [1],Mecozzi [5], [8] and Ho [6], [7] are valid only if the fiber disper-sion is zero. Attempts have been made to calculate the impactof nonlinear phase noise in the presence of dispersion [9]–[19].Green et al. [9] showed that variance of nonlinear phase noisebecomes quite small in dispersion managed transmission lineswhen the absolute dispersion of the transmission fiber becomeslarge. Green et al. assumed that the signal is CW and they usedthe approach typically used in the study of modulational insta-bility. Later in [10], the variance of nonlinear phase noise is cal-culated for a Gaussian pulse in a dispersion managed transmis-sion line and results showed that variance of nonlinear phasenoise due to single pulse self phase modulation (SPM) is quitesmall as compared to the case of no dispersion. McKinstrie etal. [11], [12] and Hanna et al. [13] developed analytical expres-sions for the nonlinear phase variance in soliton-based systems.Ho and Wang [14], [15] used first order perturbation techniqueto calculate the nonlinear phase noise variance due to self-phasemodulation (SPM) as well as intra-channel cross-phase mod-ulation (IXPM). Zhang et al. [16] used variational approachto calculate the phase variance in DPSK systems and showedthat the IXPM causes a partial correlation between adhacentpulses’ phase noise, leading to a decrease instead of an increasein differential phase noise as compared to the case of SPMacting alone. Serena et al. [17] used a parametric gain approachto study the nonlinear phase noise and they calculated the biterror rate (BER) using Karhunen-Loeve method. Their resultsshowed that the 3-dB advantage of DPSK systems over OOKsystems, based on linear considerations, is lost for large non-linear phases due to the larger sensitivity of DPSK to nonlinearphase noise. Demir [19] used a linearized perturbation theoryto characterize nonlinear phase noise and found that the ASEnoise which is initially white becomes colored because of severedisastrous amplification of ASE noise by the Kerr nonlinear ef-fects. An extensive review of the previous work on nonlinearphase noise can also be found in [19].

In this paper, we use a technique based on Gauss-Hermitebasis functions to calculate the variance of phase noise in acoherent system based on phase shift keying (PSK). In [10],the variance of nonlinear phase noise of a dispersive nonlinearfiber-optic system is calculated including only two degrees offreedoms (DOFs). Here, we extend the analysis to include ar-bitrary DOFs. DOF is the number of orthonormal set of func-

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KUMAR: ANALYSIS OF NONLINEAR PHASE NOISE IN COHERENT FIBER-OPTIC SYSTEMS 4723

tions to describe the noise field. In [1] and [10], two DOFs thatthat have the same form as the signal pulse are considered. Ina linear system, when a matched filter is used at the end of thetransmission line, these two DOFs are sufficient to describe thetotal noise field since the other noise modes are orthogonal tothe signal and do not contribute. In [1] and [10], it is assumedthat other DOFs are less significant when a matched filter is usedeven for the nonlinear systems. In this paper, we investigate theimpact of the other DOFs and find that the maximum error in-troduced by including only two DOFs is about 10% in our pa-rameter space when a matched filter is used. However, this errorcould increase for longer transmission distance/higher launchpower.

We represent the noise field as the superposition ofGauss-Hermite functions. Gauss-Hermite functions are theexact modes of the paraxial Helmholtz equation and they arewidely used in the description of optical resonators [20], [21].Lazaridis et al. [22] showed that the Gauss-Hermite functionsare the exact solutions of the linear part of the nonlinearSchrodinger (NLS) equation used to describe the temporalpulse propagation. Turitsyn and Mezentsev [23] used chirpedGauss-Hermite orthogonal functions for the description of thebreathing dynamics of dispersion-managed solitons. Lakobaand Kaup [24] represented a pulse in the strongly dispersionmanaged fiber as a linear superposition of Gauss-Hermite func-tions and they obtained the conditions for stationary nonlinearpulse propagation. If a th order Gauss-Hermite mode is excitedat the fiber input, its pulse shape remains invariant with distance(assuming nonlinear effects are absent), but the pulse width andchirp vary because of fiber dispersion. Therefore, Gauss-Her-mite functions provide a convenient set of orthogonal basisto represent both signal and noise fields at any transmissiondistance. We assume that the signal launched to the fiber is aGaussian pulse. The complex amplitude of the zeroth orderGauss-Hermite noise mode (which is also a Gaussian pulse)modifies the signal amplitude and is responsible for linear andnonlinear phase noise. The nonlinear mixing of the higher ordernoise modes with the signal pulse leads to new noise fields.Although the higher order noise modes are orthogonal to signalwhen matched filters are used, the noise fields generated bythe parametric process are not. In fact, some of them occupythe same spectral region as the signal pulse leading to non-linear amplitude and phase noise. As the signal power and/ortransmission distance increases, the amplitudes of the noisefields generated by the parametric process increases, too. Wehave also investigated the impact of electrical Gaussian filtersof arbitrary bandwidths at the end of the transmission line. Asthe filter bandwidth exceeds the signal bandwidth, higher ordernoise modes would not be orthogonal to the signal and theycontribute to the linear as well as nonlinear phase noise.

Gordon and Mollenauer showed that the total phase variancecan be minimized if the mean nonlinear phase shift is approxi-mately 1 rad. We have found that it holds true even for the disper-sive nonlinear systems, if the mean nonlinear phase of the signalsample after the correlator is approximately 1 rad. However, ourexpression for the mean nonlinear phase takes into account thedispersive effects. Based on this, an analytic expression for the

optimum launch power is developed assuming two DOFs andmatched filters.

In this paper, we make the following assumptions/approxi-mations so that the analytical expressions become simple, andscaling laws are clear. (i) The transmission system is operatingin the psuedo-linear regime and the nonlinear effects can betreated as a first order perturbation on the linear system. (ii) Thesignal power is much larger than the mean ASE power added byamplifiers. This approximation could break down if the launchpower is too low and/or amplifier noise figure and gain are quitelarge. (iii) Interaction between dispersion, SPM and ASE on asingle pulse is considered. The interaction between IXPM andASE, although it could be important in some system set up, isnot analyzed here.

The paper is organized as follows. In Section II, analyticaldescription of nonlinear phase noise is provided. A simple an-alytical expression for the phase variance is derived for arbi-trary dispersion maps when the DOF is two. Including the noisefields generated by the mixing of signal and higher order noisemodes, analytical expressions are obtained for the case of ar-bitrary DOFs. In Section III, the Monte-Carlo simulations ofthe NLS equation is carried out and the analytical results arecross-validated. Section IV contains the summary of this work.

II. NONLINEAR PHASE NOISE

The optical field envelope in an amplified transmissionsystem is governed by the nonlinear Schrodinger (NLS) equa-tion in the lossless form

(1)

where is the dispersion profile, is the nonlinear coeffi-cient, is the fiber loss/amplifier gainprofile. We assume that the fiber loss is exactly compensated byan inline amplifier. represents the noise field due to amplifi-cation, i.e.,

(2)

where is the location of an amplifier, is the number ofamplifiers, and is the noise field due to an amplifierlocated at . The mean and autocorrelation function of thenoise field are given by

(3)

(4)

and

(5)

where is the Amplified Spontaneous Emission (ASE) powerspectral density per polarization of an amplifier located at ,given by [1]

(6)

where is the gain of the amplifier, is spontaneous noisefactor, is Planck’s constant and is the mean optical car-


rier frequency. In the absence of nonlinear effects and amplifiernoise, if a Gaussian pulse is launched to the fiber, its propaga-tion is given by [25]

(7)

(8)

where is the pulse energy, , and are the in-verse pulse width, chirp and phase factor, respectively, given by

(9)

(10)

Here, is the half-width at - intensity point, and isthe accumulated dispersion

(11)

The peak power, and energy, are related by

(12)

where and is normalized such that

(13)

Expanding the optical field in a series, we have

(14)

where is the th order correction due to fibernonlinearity, and is the zeroth order linear solution.In this paper, we focus only up to the first order correction tothe optical field envelope. Substituting (14) in (1) (without theamplifier noise term) and collecting the terms proportional to ,we obtain

(15)

We will use (15) in Section II-A and B to calculate the impactof SPM on the signal and noise fields.

A. Two DOFs

Consider the optical field envelope immediately after an am-plifier located at . Focusing only on the impact of the noiseadded by this amplifier, the linear part of the optical field enve-lope at can be written as

(16)

where is the noise field added by the amplifierat . In this subsection, we consider only two noise modesgiven by

(17)

where and are the amplitudesof the in-phase component and quadrature components, respec-tively, and is the complex amplitude of the noise field. In(17), the in-phase and quadrature components are assumed tohave the same form as the signal pulse. Substituting (17) in (16),we find

(18)

Thus, the complex amplitude of the field envelope has changedbecause of the amplifier noise. Treating (18) as the initial con-dition, the zeroth order optical field envelope is described by

(19)

Substituting (19) in (15), the first-order correction due to SPMcan be written as

(20)

In practical systems operating in the psuedo-linear regime, thedispersion of the transmission fibers is fully compensated at thereceiver either in optical or electrical domain, i.e., ,where is the total transmission distance. Solving (20) withthe condition, , we find [26]–[28]

(21)

where

(22)

(23)

(24)

In (21), we have ignored the higher order terms such as andunder the assumption that the noise power is much smaller

than the signal power. Combining the first order and zeroth ordersolutions [(19) and (21)], total field envelope at the end of thetransmission line is

(25)

From (22) and (25), we see that the in-phase noise componentis responsible for energy shift and the consequent nonlinear

phase shift.The output of in-phase-quadrature (IQ) coherent receiver has

two output currents which are proportional to real and imaginaryparts of the optical field envelope. For simplicity, we assume thatthe constant of proportionality to be unity since it does not affectthe performance of a long-haul fiber-optic systems. The outputof the IQ receiver passes through the digital signal processing(DSP) unit after A/D converter and arbitrary filter shapes can berealized using DSP. In this section, we assume that the output ofthe IQ receiver passes through a matched filter, or equivalentlya correlator and the decision is based on the correlator output


(26)

where is the bit interval. Substituting (8) and (25) in (26), wefind

(27)

where

(28)

(29)

The phase of the correlator output is

(30)

where . In (30), we have ignored termsproportional to , , and . The first, second andthe last terms on the right hand side of (30) represent the deter-ministic nonlinear phase change, nonlinear and the linear phasechanges due to ASE of the amplifier located at , respectively.Therefore, the phase changes due to ASE of the amplifier lo-cated at are

(31)

Variance of energy shift is related to the variance of . From(4), (5) and (22), we have

(32)

(33)

Equation (33) holds because Ito calculus applies for the signaland noise set up considered here. Squaring and averaging (31),and using (32)–(33), we obtain

(34)

The first and the second terms in (34) represent the variance ofnonlinear phase noise and linear phase noise, respectively, dueto the amplifier located at . The in-phase component of thenoise field, and the quadrature component, are respon-sible for the nonlinear and linear phase noise, respectively. Sincethese components are statistically independent, total variance isthe sum of the variance due to linear and nonlinear phase noise,as given by (34).

Since the noise of amplifiers is statistically independent, vari-ance of phase noise due to all the amplifiers is

(35)

For -ary PSK signals, symbol error probability is deter-mined solely by the probablity density function (PDF) of thephase. Under the Gaussian PDF assumption, the variance cal-culated from (35) can be related to the symbol error probabilityfor MPSK signals [29]. Equation (35) is valid for arbitrary dis-persion maps with . To simplify (35) further andalso to make a direct comparison with [1] and [10], we considera transmission fiber consisting of two segments of equal lengthswithin an amplifier spacing. The dispersion of the first segmentis anomalous whereas that of the second segment is equal inmagnitude but opposite in sign. We assume that there is no pre-and postcompensation of dispersion. Since the amplifier spansare identical, , where is the am-plifier spacing, we can write

(36)

where

(37)

and (34) is modified as

(38)

where and . The first term in (38) issame as the expression for the nonlinear phase variance derivedin [10] except that the impact of matched filter on the first ordercorrection was not included in [10] and terms second order in

are ignored here. Adding contributions to the phase variancefrom all the amplifiers, we obtain the total variance as

(39)

When , (39) can be approximated as

(40)

The variance of nonlinear phase noise grows cubically withnumber of amplifiers while the variance of linear phase noise in-creases linearly for large . Therefore, nonlinear phase noisecould become dominant for ultra long haul transmission sys-tems. In Section III, we will use (39) and (35) to calculate thevariance of phase noise.

The optimum launch power is calculated by differentiatingwith respect to and setting it to zero. Since the peak

power, , we find the optimum launch power as

(41)

In a dispersion-free system, Gordon and Mollenauer showedthat the total phase variance is minimized when the mean non-


linear phase shift is approximately 1 rad. It holds true even forthe dispersive systems, if the mean nonlinear phase of the signalsample used for decision, is approximately 1 rad.

B. Arbitrary DOFs

In this section, we consider the impact of an electricalGaussian filter of arbitrary bandwidth. An exact solution ofthe linear part of NLS equation ( in (1)) is described byGauss-Hermite function of the form

(42)

(43)

If the initial optical field envelope at is ,the field envelope at any is given by [22] (Appendix A)

(44)where the evolution of , and are given by (9) and(10). In other words, the evolution of pulse width and chirp ofa th order Gauss-Hermite mode is same as that of a Gaussianpulse, (which is a zeroth order Gauss-Hermite mode)and the phase factor is times . Gauss-Hermitefunctions form an orthonormal set and arbitrary linear signalfield, can be expressed as the superposition of

(45)

where is the amplitude of the th order signal mode, given by

(46)

We assume that the pulse launched to the fiber is a Gaussianpulse of the form given by (8). In this case, and

. Next, as in the previous section, we considerthe noise field added by the amplifier located at . The noisefield can also be written as a superposition of

(47)

where is the amplitude of the th order noise mode given by

(48)

and . If we take only the first term ( )in the summation on the right hand side of (47), it correspondsto two DOFs ( and ) discussed in the previous subsection.We ignore the terms of the summation in (47) after , whichcorresponds to DOFs. Multiplying and and using(3)–(5) and (48), it follows that

(49)

(50)

where is a Kronecker delta function.The linear part of the optical field envelope for is

(51)

(52)

where

(53)

(54)

Nonlinear mixing of signal and noise modes leads to newnoise fields which can be calculated using the first order pertur-bation theory. Choosing the linear part of the optical field en-velope given by (52) as the zeroth order solution, the first ordersolution can be found out using (15). Using (52), the term onthe right hand side of (15) can be written as

(55)

Now we assume that the signal power is much larger than themean noise power, i.e.,

(56)

From (56), it follows that the terms proportional toare much smaller than the terms proportional to . Inother words, only the terms proportional to andin (55) contribute significantly (after averaging) and the otherterms can be ignored. Now, (15) can be written as

(57)

Here, represents the new noise fields generated by themixing of signal and noise fields. Since (57) is a linear equation,total solution, can be expressed as the superpositionof responses due to different forcing functions on the right handside of (57), i.e.,

(58)

where is the noise field generated by the nonlinearmixing of the signal (with energy ) and the th order noisemode. Substituting (58) in (57), we obtain

(59)

and


(60)

The solution of (59) is given by (21) of Section II-A. Equation(60) is solved in Appendix B. In this subsection, we assumethat the filter bandwidth is arbitrary. For the sake of computa-tional ease, we choose the electrical Gaussian filter at the re-ceiver given by

(61)

When , the sample of the filter output atcorresponds to the output of the correlator discussed in the

Section II-A. Solving (60) and after passing it through the filter,the sample of the nonlinearly generated noise fields at is(Appendix B)

(62)

where and are defined in Appendix B. From (111), (117),(116), (124), and (123), we find that and are zero when

is odd which means that the energy transfer from the initiallylaunched fundamental signal mode (Gaussian) to a higher orderodd noise mode does not occur within the first order approxi-mations.

The higher order noise modes given by the terms within (52) also pass through the filter, and, therefore, the corre-sponding filter output at is

(63)

where

(64)where denotes the Fourier transform. Substituting (42) in(64), is calculated as

(65)

where

(66)

When a matched filter is used, , and for.

The zeroth order solution at after passing through thefilter is

(67)

When , the first order correction is given by (21) and afterpassing through the Gaussian filter, the sample at is givenby

(68)

where

(69)

When a matched filter is used ( ), (69) and (28) are iden-tical. Combining (62), (63), (67), and (68), total filter output at

is

(70)

The output phase is

(71)

where

(72)

, and . In (71), thefirst term ( ) represents the deterministic nonlinear phasechange due to SPM. The phase changes due to ASE introducedby the th amplifier can be written as

(73)

where and are the linear and nonlinear phasechanges, respectively, and are given by

(74)

(75)

where


(76)

(77)

From (75), we see that contributions to the nonlinear phase noisecomes not only from the real part of but also from imaginarypart of whereas in the case of two DOFs (Section II-A), onlythe real part of is responsible for nonlinear phase noise. Thecontribution to the linear phase noise comes only from the imag-inary part of in both cases. Therefore, the linear and nonlinearphase noises are not statistically independent unlike the case inthe Section II-A. Squaring and averaging (73), and using (50),we obtain

(78)

The first term ( ) on the right hand side of (78) represents thenonlinear phase noise, the second term represents linear phasenoise, and the last term represents the correlation between linearand nonlinear phase noise which is absent when . Thevariance of phase noise due to all the amplifier is given by (35).

As before, we consider the simple periodic dispersion mapwith two types of fiber segments and the average dispersion be-tween amplifiers is zero. In this case, we have

(79)

(80)

where

(81)

(82)

and are defined in Appendix B. Using (78), (35),(79), and (80), the total phase variance is obtained as

(83)

where

(84)

(85)

(86)

Comparing the variance of linear phase noise for the case oftwo DOFs (second term of (39)) and that for the case ofDOFs (second term of (83)), we see that the variance is en-hanced due to the excitation of higher order noise modes. How-ever, when a matched filter is used, for and thesetwo expressions for the linear phase variance become identical.Hence, for the linear case, it is sufficient to include two DOFswhen a matched filter is used. However, nonlinear phase noisedue to the mixing of signal and higher order noise modes (de-scribed by terms and ) does not vanish even when thematched filter is used. Equation (83) is used in Section IV tocalculate the phase variance.

III. RESULTS AND DISCUSSION

To test the validity of the approximations done in ob-taining (39) and (83), numerical simulations of the nonlinearSchrodinger (NLS) equation by the split-step Fourier techniqueis carried out. We assume the following parameters throughoutthe paper: W km ,

dB km, Gb s,which corresponds to a noise figure of 3 dB, and

km. We assume thata Gaussian pulse with full width half-maximum (FWHM) of12.5 ps is launched to the fiber link so that ps. Thecomputational bandwidth is 320 GHz and ASE is propagatedover the entire computational bandwidth. A Gaussian filter ofarbitrary bandwidth is used in electrical domain and no opticalfilter is used. 4000 runs of NLS equation are carried out and thephase variance of the decision variable is calculated. In Fig. 1,the matched filter is used at the end of the transmission linewith . For Figs. 1–4, two types of fibers are usedbetween inline amplifiers, the first one is an anomalous disper-sion fiber of length 40 km and the second one is the normaldispersion fiber of the same absolute dispersion and the samelength. The ’+’ marks in Fig. 1 shows the numerical simulationresults and the solid line shows the analytical results calculatedusing (83) with . As the dispersion increases, the varianceof nonlinear phase noise due to SPM decreases consistentwith the results of [9] and [10]. The nonlinear phase variancegrows cubically with distance and, therefore, the differencebetween the variances for the case of ps nm kmand ps nm km increases significantly for longertransmission lengths.

To estimate the number of DOFs required when a matchedfilter ( GHz) is used, in Fig. 2, we have plotted thephase variance as a function of length of transmission line forvarious DOFs using (83). From Fig. 2, we see that the phasevariance does not change as the number of DOFs is changedfrom 6 ( ) to 14 ( ). However, there is about 10%change in variance as the number of DOFs is changed from 2to 6 when ps nm km and km, and thecorresponding change in variance when ps nm km is6%. In Fig. 3, a Gaussian filter with GHz which hasa bandwidth twice that of a matched filter is used at the receiver.In this case, we see that two DOFs are not sufficient to describethe impact of noise on the phase variance. The errors introducedby using 2, 6, and 10 DOFs are 30%, 4% and 1%, respectivelyfor ps nm km and km. As the filter


Fig. 1. Phase variance dependence on the total length of the transmission line.�� mW. Solid line and + marks show the analytical and nu-merical simulation results, respectively. The dotted line shows the analyticalresults when fiber nonlinearity is absent, which is independent of dispersion.�� is used for analytical results.

Fig. 2. Dependence of variance on the DOFs with a matched filter. Dotted line,circles, +, and solid line show the analytical results with DOF 2, 6, 10, and 14,respectively. Other parameters are same as that of Fig. 1.

bandwidth increases, higher order noise modes and noise fieldsdue to nonlinear mixing of the signal and higher order noisemodes occupy the pass band of the filter. Therefore, as the filterbandwidth increases, the variance of linear phase noise as wellas nonlinear phase noise increases.

Fig. 4 shows the dependence of phase variance on the launchpower. When the launch power is low, the linear phase noisedominates (because of dependence in (83)). At high launchpower, nonlinear phase noise becomes significa [because ofdependence in (83)]. The optimum launch power is calculatedto be 1.8 mW using (41) which is in agreement with numer-ical simulations. At high launch powers ( mW), there is asmall discrepancy between the analytical results and simulationresults which is due to the fact that we have ignored the termscontaining and higher. The first order perturbation theory

Fig. 3. Dependence of variance on the DOFs with a Gaussian filter with �

�� GHz. Dotted line, circles, +, and solid line show the analytical resultswith DOF 2, 6, 10, and 14, respectively. Other parameters are same as that ofFig. 1.

Fig. 4. Dependence of phase variance on peak launch power. Matched filter isused. Solid and + show the analytical and numerical simulation results, respec-tively. � �� km and �� ps�nm � km. �� is used foranalytical results.

is known to become inaccurate at large launch powers and/orlonger transmission distance. It may be possible to increase theaccuracy of the calculations using the multiple-scale approachesof [23], [24], and [30] when the dispersion map is periodic.Alternatively, a second-order perturbation theory [28] could beused which is shown to be quite accurate for the description ofSPM and XPM for the range of launch powers and transmissiondistances of practical interest.

Next, we consider a dispersion map with two types of trans-mission fibers within an amplifier spacing. Let and bethe dispersion parameters of these fibers and, and be theirrespective lengths. The average dispersion of these fibers is

(87)

The dispersion of the transmission fibers is compensated bypre- and postcompensating fibers. The dispersion coefficientsand lengths of pre- and postcompensating fibers are so se-lected that the total accumulated dispersion before decision is


Fig. 5. Dependence of � �� on the average dispersion, � andthe local dispersion, � . Matched filter is used. Total transmission dis-tance, � �� km,�� mW, location of the first inline amplifier, � �� and �� .

zero. Fig. 5 shows the dependence of on the averagedispersion, for different values of . is thereal part of defined in (28) which is proportional tothe nonlinear phase acquired due to propagation from to

. We have assumed the following parameters in Fig. 5.The dispersion parameter of the pre- and postcompensatingfiber, ps nm km, km,

km, transmissiondistance (excluding lengths of pre- and postcompensationfibers), km and mW.50% of the total accumulated dispersion of the transmissionlink is compensated using the precompensating fiber. As canbe seen in Fig. 5, decreases as or increases. Solidline in Fig. 6 shows the phase variance calculated from (78) and(35) and + shows the numerical simulation results. The phasevariance also shows the similar dependence on and as

. As and/or increases, the nonlinear contributionto the phase variance becomes quite small. However, in thiscase, pulses significantly broaden and overlap with neighboringpulses and it is likely that the ASE-induced nonlinear phasenoise due to IXPM becomes important. This would be thesubject of a future investigation.

IV. CONCLUSION

We have developed analytical expressions for the linear andnonlinear phase variance due to SPM. The noise field is rep-resented using Gauss-Hermite orthonormal set. In the first part,only two degrees of freedom (DOFs) of the noise field is consid-ered which leads to simple analytical expressions for the phasevariance. The in-phase component of the noise mode leads to en-ergy fluctuations and consequent nonlinear phase fluctuations,and the quadrature component leads to linear phase noise con-sistent with [1]. In the second part, the results are extended toarbitrary DOFs and Gaussian filters with arbitrary bandwidthsare inserted at the end of the transmission line. Nonlinear mixing

Fig. 6. Dependence of phase variance on the average dispersion, � and thelocal dispersion � . Solid line and + show the analytical (with ! � ") and nu-merical simulation results, respectively. Dotted line shows the analytical resultsfor the case of � � � with ! � ". The other parameters are same as that ofFig. 5.

of higher order noise modes with the signal pulse leads to newnoise fields which enhance the nonlinear phase noise. Higherorder noise modes propagate linearly and contribute to the linearphase noise. However, when a matched filter is used, higherorder modes are orthogonal to the signal and linear phase noiseis not enhanced. We have found that the maximum error intro-duced by including only two DOFs is about 10% in our param-eter space when a matched filter is used. However, this errorcould increase for longer transmission distance/higher launchpower. When the filter bandwidth exceeds the signal bandwidth,higher order noise modes become more important which con-tribute to linear as well as nonlinear phase noise. We have alsodeveloped an analytic expression for the optimum launch powerwhich is in good agrement with numerical simulations.

APPENDIX AGAUSS-HERMITE SOLUTIONS

Consider the Schrodinger equation given by (1) withand . Using the lens-like transformation [31]

(88)

in (1) with leads to [32]

(89)

where

(90)

, and “.” denotes the differentiation with respect tothe argument. Dividing (89) by , we get

(91)


where(92)

(93)

If we choose

(94)

and

(95)

we obtain

(96)

Let

(97)

Substituting (97) in (96), we obtain

(98)

The above equation is the eigenvalue equation corresponding tothe quantum harmonic oscillator. The eigenfunction equalsGauss-Hermite functions of (42) and the eigenvalue is givenby

(99)

The constraints imposed in (94) and (95) on and withthe initial condition and lead to thefollowing solution:

(100)

Note that (100) is same as (9) which is obtained using Fouriertransform technique. Substituting (100) in (92), we find

(101)

Combining (88), (97) and (99), we obtain the final solution

(102)To our knowledge, the above solution was first obtained byLazaridis et al. [22] for temporal pulse propagation using adifferent approach.

APPENDIX BSOLUTION OF (60)

Consider the terms on the right hand side of (60). They canbe written as

(103)

where

(104)

and

(105)

We take the Fourier transform of (60) to obtain

(106)

where

(107)

(108)

Noting that

(109)

can be simplified as

(110)

where

(111)

and

(112)

(113)

Considering only the term containing in (106), it canbe solved to obtain


(121)

(114)

Multiplying by the filter transfer function((61)) and inverse Fourier transforming, we obtain the sampleat as

(115)

where

(116)

(117)

(118)

(119)

Next, let us consider the second term on the right hand sideof (106). Comparing (103) and (105), we see that they are thesame except for the phase factor, i.e.,

(120)

Therefore, the solution of (106) considering only the secondterm on the right hand side is shown in (121), at the top of thepage. Proceeding as before, the sample at of the filteroutput is

(122)

where

(123)

(124)

(125)

The complete solution is

(126)

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Shiva Kumar (M’03) received the B.E. degree in electronics and communica-tions from Mysore University, India, in 1988, the M.S. degree in 1990, the Ph.D.degree in 1994 in electrical communication engineering from the Indian Insti-tute of Science, Bangalore, and the Ph.D. degree in 1997 in communicationsengineering from Osaka University, Osaka, Japan.

He was a Postdoctoral Fellow at the University of Jena, Germany, supportedby the Alexander von Humboldt Foundation from 1997–1998. He was withCorning, Inc., New York, as a Senior Research Scientist (1998–2002) and asa Research Associate (2002–2003). He joined McMaster University, Hamilton,ON, Canada, in 2003. Currently, he is an Associate Professor in the Departmentof Electrical and Computer Engineering, McMaster University. His current re-search interests include optical communication, solitons, nonlinear optics, andphotonic devices.

Analysis of nonlinear phase noise in coherent fiber-optic systems based on phase shift keying

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