Evaluation of BER degradation and power limits in WDM. networks due to Four-Wave Mixing by Monte-Carlo. simulations

Transcription

1 Evaluation of BER degradation and power liits in WDM networks due to Four-Wave Mixing by Monte-Carlo siulations John Neokosidis, Thoas Kaalakis, Aris Chipouras and Thoas Sphicopoulos Departent of Inforatics and Telecounications, University of Athens Panepistiiopolis Ilissia, Athens, Greece, GR-5784 Abstract: Fiber non-linearities can degrade the perforance of a Wavelength Division Multiplexing optical network. For high input power, low chroatic dispersion coefficient or low channel spacing the severest penalties are due to Four Wave Mixing (FWM). To copute the Bit Error Rate (BER) due to the FWM noise, the probability density function (pdf) of both the space and ark states ust be accurately evaluated. In this paper, an accurate evaluation of the pdf of the FWM noise in the space state is given for the first tie using Monte-Carlo (MC) siulations. Additionally, it is shown that the pdf in the ark state is not syetric as assued in previous studies. Diagras are presented which allow the estiation of the pdf given the nuber of channels in the syste. The accuracy of the previous odels is also investigated and finally the results of this study are used to estiate the power liits of a WDM syste. 003 Optical Society of Aerica OCIS codes: ( ) Multiplexing; ( ) Nonlinear optics, four-wave ixing

2 I. INTRODUCTION Wavelength Division Multiplexing (WDM) is a proising technology for the realization of all-optical networks, allowing the full utilization of the fiber bandwidth. However, the perforance of a WDM syste is strongly influenced by both linear and non-linear phenoena deterining the signal propagation inside the optical fiber. Although linear propagation effects ay be copensated, using optical aplifiers and chroatic dispersion copensators, there is a class of non-linear effects, such as Self Phase Modulation (SPM), Cross Phase Modulation (XPM) and Four Wave Mixing (FWM) that pose additional liitations in dense WDM systes. Both XPM and FWM cause interference between the different wavelength channels resulting in an upper power liit for each WDM channel. However, the severest probles are iposed by FWM since the power of the FWM product is inversely proportional to the square of the channel spacing while the influence of the XPM is approxiately inversely proportional to the channel spacing []-[]. In this paper an accurate statistical description of the FWM noise will be given and its ipact on the syste perforance will be investigated. The properties of the FWM noise are investigated using nuerical Monte-Carlo siulations both for the ark and the space state. The results show that in the case of the space state the pdf of the FWM-FWM beating noise, which constitutes the ain noise source of the syste, exhibits an exponential decay. The coputed pdf for the ark state is shown to be asyetric, a fact which is confired theoretically as well. Estiations of the pdf for the ark and the space states is accoplished by relevant diagras in ters of the nuber of channels within the syste. A coparison is also carried out between the values of the error probabilities obtained by the present ethod and the two other ethods previously proposed [3]-[4]. The first one assues

3 a Gaussian statistics for the FWM noise in the optical doain, allowing the coputation of the syste s Bit Error Rate (BER) in closed for [3]. Although this ight provide a first insight on the iplications of the FWM noise, its validity ust be exained since the assuption of Gaussian statistics is soewhat arbitrary. Indeed, the FWM noise is a su of a large nuber of coponents, dependent on each other. Hence, the central liit theore [5], on which the Gaussian approxiation is based, ay not be valid in this case. In the other approach, the probability density function (pdf) of the FWM noise in the ark state can be approxiated with a syetrical double-sided exponential distribution [4]. However, as entioned before the distribution of the FWM noise in the ark state is asyetric and hence the syetrical exponential approxiation ay prove inaccurate. It is shown that in any cases, there are significant differences between the two approaches. Finally, the proposed odel is used to assess the iplications of the FWM in a WDM syste eploying nonzero dispersion or standard single ode fibers as well. The rest of the paper is organized as follows: In section II.A, soe basic considerations are given concerning the origin of the FWM phenoenon which will be used in section II.B to derive an expression for the photocurrent at the receiver. This expression will be used in section III in order to copute nuerically the pdfs of the FWM noise in the ark and space states. The asyetry of the pdf in the ark state, observed by the siulations, is also theoretically justified. The accuracy of the syetrical double-sided pdf is investigated in section IV.A while the validity of the Gaussian approxiation is exained in section IV.B. The iplications of the FWM effect in a WDM syste is discussed in section V. In section VI, guidelines for the incorporation of the other noises (theral and Aplified Spontaneous Eission ASE) are given. The work is concluded in section VII. 3

4 II. BASIC ASSUMPTIONS AND THEORETICAL BACKGROUND A) Four Wave Mixing In this work, a WDM syste with equally spaced channels and ASK odulation is considered which is the ost frequently used odulation schee. All signals are assued co-polarized and synchronized, which represents a worst-case scenario [6]. Only the pdf of the photocurrent for the central channel has to be derived for the calculation of the BER, since, in this case the energy conservation requireent is satisfied by the largest nuber of frequency cobinations [7]. Provided that the photocurrent depends on the bit values and the optical phases of all channels in a rather coplicated rando anner, a closed for of the pdf of the FWM noise is not possible. Consequently, Monte-Carlo (MC) siulations are used for its deterination. The origin of the FWM effect is the existence of the third-order non-linear polarization vector W NL. Considering optical waves oscillating at frequencies ω i and linearly polarized along the sae axis x, W NL is expressed in the following for: [ j( ki z ωit )] jθi W NL = xˆ Wi e + cc = xˆ Wie + cc () where W i is the aplitude of the third order polarization at the frequency ω i. For i=n, where n is the assued channel, it can be shown that [8]: W n 3ε o = χ 4 3ε o + χ (3) xxxx E (3) xxxx n p E n q 3ε o + χ 4 r (3) xxxx j( θ p + θ q θ r θ n ) E E E e +... [ ( E + + ) p Eq Er En ] + () (3) where E i is the electric field at frequency ω i, ε o is the vacuu perittivity and χ xxxx is the thirdorder nonlinear susceptibility. The third su is the contribution of the FWM noise. This su 4

5 runs for all integers p, q, r that satisfy the conditions p+q-r=n (which is iposed by the energy conservation requireent) and r p,q (which guarantees that the corresponding ter is not due to SPM or XPM). The output power P of the FWM product is given by [9]: γ -al P = d Pp Pq Pr e Leffη (3) 9 where P i (i = p, q, r) represents the input peak power at the frequencies f i =ω i /π in the ark state. Assuing a perfect extinction ratio, the average input power is P av =P i /. It should be noted that equation (3) is an approxiation that holds since the power of the FWM coponents is very sall copared with each channel s power [0]. In a WDM syste it can be assued that all the peak powers at the ark state are equal (P i =P in for i=,,,n). In (3) γ is the nonlinear coefficient of the fiber [9], a is the fiber loss coefficient, L is the total fiber length, L -al ( - e ) a = is the effective length of the fiber, d is the degeneracy factor (d =3 when eff / p=q, d =6 when p q) and η is the ixing efficiency given by: al a 4e sin ( ΔβL / ) η = + (4a) a + ( Δβ ) al [ ] e In (4a), Δβ represents the phase isatch and ay be expressed in ters of the channel frequencies f i : πλ Δβ = c πλ = Δf c ( f f )( f f ) D( λ ) p r q ( p r)( q r) D( λ ) r o ο ( λ ) dd o + dλ ( λ ) dd o + dλ λ c λ Δ f c ( f f ) + ( f f ) p o (( p o) + ( q o) ) q o (4b) or approxiately, 5

6 πλ D Δβ Δf ( p r)( q r) (4c) c In (4b), D is the fiber chroatic dispersion coefficient, λ is the wavelength of the signal and c is the speed of light in vacuu. Equation (4c) is derived using the fact that for typical values of D, dd/dλ and Δf, the second ter in the brackets is uch saller than the first one [4]. Contrary to previous studies, the statistics of the space state is also considered. The aplitude of the optical fields E () and E (s), in the ark and the space state respectively, at a given channel n is written as [3]: E ( ) = P e al n exp[ jθ ] + n P ( ) F ( IM ) ( exp[ jθ ) F ( IM ) ] ( ark) (5a) E ( s) ( s) ( s) = P exp[ jθ ] ( space) (5b) F ( IM ) F ( IM ) where P n and θ n are the input peak power and the phase in the ark state, respectively, of the given channel n and ( ) ( ) jθ jθ pqn jθ ppr PF ( IM ) exp[ jθ F ( IM )] = B BqBr P e + B Bq Ppqn e + BpBr Pppr e (6a) p p q r n p p q r n p p q r= n p= q r p= q r ( s) ( s) P ( ) exp[ jθ ( )] = B B B P e + B B P e (6b) F IM F IM while P and θ =θ p +θ q -θ r are the peak power and the phase of the FWM noise generated fro a channel cobination (p, q, r). Furtherore, B i =0 or B i = is the bit value of channel i. These expressions for the electric field will be used in the next section in order to derive an expression for the photocurrent at the receiving photodiode. q r jθ p r ppr jθ B) Calculation of the Photocurrent At the receiver, the photocurrent is proportional to the optical power and hence to E where E=E () or E=E (s) []. In practical applications, it can be assued that Δβ>>a which generally 6

7 holds for D ps/n/k and channel spacing Δf 0GHz. For large L one can also use the fact that exp(-al)<<. Assuing a single fiber span without optical aplification, all other noises at the receiver except FWM can be ignored. This is especially true for high input powers and in this case the photocurrent at the detector is written as: S ( ) ( ) al = k E kpn e + kδ P e I al n (7a) S ( s) ( s) = k E kδ I (7b) s where k is the receiver responsivity and γc = πλ DΔf 3 δ P al / in (8a) I = 3 B B B p q r d cos θ p n q n ( θ ) n (8b) I s = d cos + d BpBqBr θ BpBqBr sinθ (8c) 3 p n q n 3 p n q n r n r n Equations (7a) and (7b) provide an expression for the photocurrent in the ark and space state in ters of two new variables I and I s given by (8b) and (8c) respectively. It is interesting to note that for a given nuber of channels, these new variables depend only on the bits and the phases of the optical signals. III. STATISTICAL BEHAVIOR OF THE PHOTOCURRENTS S () AND S (s) Coputation of the pdf by using Monte Carlo (MC) Siulation In this section the pdf of I and I s will be coputed using Monte Carlo (MC) siulations. The optical phases of all channels are assued to be uniforly distributed within [0, π], due to phase noise [], and the data bits are assued to be in the ark and space state with equal 7

8 probability, P(B i =0)=P(B i =)=/. Hence, the statistics of I and I s will depend only on the total nuber of channels N and the channel nuber n, which is assued to be the central channel n= N /. The variables I and I s are related to the optical phases and the values of the optical bits in a rather coplicated anner and consequently, their pdf can not be derived in closed for. In order to obtain the pdf of I and I s, through which the pdf of S () and S (s) will be deterined, a series of v Monte Carlo (MC) experients were perfored. v=0 for the cases N=8, 6 channels and v=0 0 for the case N=3 channels (due to the increased coputation tie required). The obtained results, both for the space and ark states, are plotted in Figures (a) and (b). It is easily deduced that in all cases the pdf exhibits an alost exponential behavior of the for y = Ae bx (solid lines in Figures (a)-(b)). Specifically, the pdf in the space state follows an exponential decay while in the ark state the pdf can be approxiated by an asyetrical double-sided exponential distribution. Hence, the exponential approxiation can be used away fro I s, =0, in the region of the tails in order to provide an accurate estiation for the pdf. In order to evaluate the paraeters A and b with respect to N, a series of Monte Carlo (MC) experients were perfored for various values of N<3. The values of the paraeters A and b calculated for each N are shown in Figures (a)-(d). To reduce the nuber of diagras, the paraeter A is taken equal to the peak of the pdf of I and is the sae for I >0 and I <0. Given this value of A, the value of b for I >0 is calculated so that the exponential Ae bx approxiates the right tails of the pdf. The sae is done for I <0. In the case of the pdf of I s the paraeters A and b are calculated so that the tail of the pdf of I s is accurately approxiated. As shown in the figures (a)-(d), A and b approxiately exhibit a y o +y e -N/t dependence. The values of y 0, y and t in each case are given in the figures. Given the nuber of channels N, Figures (a)-(d) are useful 8

9 in deterining A and b for the ark and the space states, and consequently the shape of the pdfs of I and I s without having to perfor MC siulations. For N>3, the paraeters A and b do not vary significantly and hence their values obtained for N=3 can be used. Once the pdf of I and I s is deterined, the pdf of S () and S (s) can also be deterined, using the theore of transforation of rando variables [5]. Applying this theore and using (7a) and (7b): ( ) al ( ) S kpn e f = ( ) ( S ) f S al I (9a) al kδ P ne kδ Pn e f ( S ( s) ( s) S s ) ( S ) = f I (9b) s kδ kδ where f X is the pdf of the variables X=I, I s, S () and S (s) respectively. Hence, given the nuber of channels N in the syste the pdf of I and I s can be deterined using Figures (a)-(d). The coputation of the pdf of S () and S (s) can then be carried out using equations (7a)-(7b). These pdfs will be used in the following sections in order to estiate the perforance of the syste. Fro figure (b), it can be seen that the pdf for the ark state is not syetric around x=0. In order to justify the asyetry of the pdf, the odd order oents I u+ + u+ x f I ( x) = dx, of I can be exained. If one of the odd order oents is nonzero, then the pdf I (x) is not syetric. Indeed if I (x) were syetric around x=0, then the f f product of the odd function x u+ with f I would be an odd function of x for every integer u, and hence, the integral of x u+ f I fro to, which gives I would be equal to zero. u + Therefore in order to show that f I is not syetric, it is sufficient to show that there exists one u + u for which I 0. 9

10 As it can be seen fro the Appendix, it is easy to expand the ters of the odd order oents into a positive linear superposition of ters of type cosθ, whose ean value is either 3 if θ 0 or 0 if θ 0. Following the above ethod, one can obtain I = 0 for u=0 and I > 0 for u=. Hence, at least one odd oent of I is non-zero and it can be concluded that the pdf of Ι is asyetric. Since both the syetric exponential and the Gaussian pdf are even functions, they ay not provide an accurate description for the statistical behavior of I. This will be further illustrated in the next sections. IV. COMPARISON WITH OTHER MODELS A) The Syetrical Exponential pdf Approxiation The difference between the syetrical exponential approxiation and the nuerically coputed pdf can be deonstrated by calculating the probability P e that an error occurs in the ark state, which is written as: Q P = f ( ) (ξ ) dξ (0) e S where Q is the receiver decision threshold. The pdf f ) of the photocurrent S () in the ark state can be evaluated fro (9a) using the syetrical exponential approxiation [4] for the pdf: ( S f exp I σ ( I ) = e () σ where σ is the noise variance of I given in [4]. We have chosen to copute P e and not the average BER because in [4] no pdf was given for the space state that allows the direct coputation of the error probability P e0 in this state. 0

11 In the reainder of the paper the paraeters used in the calculations are as follows: λ=.55μ, c=3 0 8 /s, L=80K, a=0.db/k, γ=.4 (K W) - and k=.8 A/W. Figure 3(a) shows the pdf of the photocurrent for N=6, D=0ps/n/K, P in = 5dB, Δf=5GHz and gives a first glance of the effectiveness of the syetrical exponential approxiation in describing the statistics of the photocurrent. It should be noticed that only the left part of the pdf, is shown in the figure. This happens because for practical values of the BER, only this part of the pdf is needed to copute P e using (0). In Figs. 3(b) and 3(c), P e is plotted as a function of the receiver threshold for N=6, D=0ps/n/K, P in =5dB and Δf=5GHz and N=8, D=5ps/n/K, P in =8dB and Δf=00GHz respectively. For the nuerically calculated pdfs, P e was coputed using nuerical integration of (0). In figures 3(b) and 3(c) there is an obvious deviation between the two odels, which is several orders of agnitude in the exained cases, and iplies that the average BER will be uch higher for the exponential approxiation. For exaple as shown in Figure 3(b), for a threshold Q=60μA the values of P e are 0-5 and in the case of syetrical exponential pdf and the nuerically coputed pdf respectively. Hence, it is evident that the syetrical exponential odel ay overestiate the syste bit error rate by about two orders of agnitude for error probabilities of the order 0-9. However, it can be used to easily estiate an upper bound of the BER. To further illustrate the difference between the syetrical exponential and the nuerically coputed pdf, the values of σ obtained in the case of the syetrical exponential pdf using the closed for forula of [4] is plotted (with solid lines) in Fig. 3(d). Also plotted are the values of σ obtained by fitting the nuerical pdf with /σ / / exp( / I /σ ) for I <0. As observed by the figure there is soe difference between the values of σ and σ and this is a further indication that the syetrical exponential pdf fails to describe the statistical behavior of

12 I. It should also be noted that Fig 3(d) provides an alternative approxiation for the nuerically coputed pdf of I for I <0. Once the value of σ is obtained fro Fig 3(d) then the pdf of [4] (i.e. equation ()) can be used to approxiate the pdf for I <0 by replacing σ with σ. B) The Gaussian pdf Approxiation The ost coon approach in the calculation of the BER in the presence of FWM is to assue that the FWM noise is Gaussian. According to the Gaussian approxiation [3] the error probability is written as: = = g Q t e Q erfc dt e P g π () where Q g is the Q-factor given by = = = = r q p ppr n r q p pqn n r q p al n r q p ppr n r q p al n s s g P P P P e k P P k kp e S S Q ) ( ) ( ) ( ) ( σ σ (3) To copute the value of the BER obtained by the actual pdfs of S () and S (s) the following forula is used. + = Q S Q S e d f d f P s ξ ξ ξ ξ ) ( ) ( ) ( ) ( (4) where the first and second ters are the probabilities that an error occurs in the space and ark states, respectively, while the decision level Q is chosen so as to iniize the error probability P e. This can be done by solving the equation dp e /dq=0 using well-known nuerical ethods.

13 The accuracy of the Gaussian approxiation in predicting P e and the BER is exained in Figure 4(a) for N=6, D=0ps/n/K and Δf=5GHz and in Figure 4(b) for the above case and for N=3, D=5ps/n/K and Δf=0GHz. Fro Fig. 4 it is evident that the Gaussian odel cannot be used to estiate the BER accurately. The error in the estiation of the Bit Error rate (BER) is ore pronounced for sall error rates (<0-9 ). For exaple for P in =db, P e =0-8 for the Gaussian approxiation and P e =0-5 in the nuerically calculated case as shown in Figure 4(b). The results of Fig 4 are obtained in the case where D ps/n/k. If the WDM syste operates near the zero dispersion wavelength the forulation of equations (7) and (8) is not applicable. In order to investigate the validity of the Gaussian approxiation in such systes, the pdf of the photocurrents S () and S (s) ust be estiated through MC siulations applying S () =k E () and S (s) =k E (s) where E () and E (s) are given by (5) and (6) and P is obtained using (3), (4a) and (4b) setting D=0 (Note that (4c) does not apply in this case). The results obtained by the MC siulations (dots) and the Gaussian approxiation (lines) in the case of D=0 (for the central channel) are illustrated in Figure 5 for N=6 channels, P in =0dB, dd/dλ=0.07ps/n /K and Δf=00GHz. Inspecting Figure 5, it is understood that even though the difference between the pdfs of the present and the Gaussian odel is reduced in the ark state, it reains observable since the FWM products are correlated and the central liit theore is not valid in this case as well. V. ESTIMATION OF POWER LIMITS DUE TO FWM In practice, in order to evaluate the perforance of the syste, it is useful to have a relation between the BER and the input power. With the ethod presented here, it is quite easy to 3

14 calculate the BER, using nuerical integration, given the characteristics of the syste. Once the BER is deterined one ay calculate other useful syste perforance easures such as the Packet Error Rate in IP over WDM systes []. Figures 6(a) to 6(c) depict the variations of the BER with respect to the variations of the channel spacing, signal power and fiber chroatic dispersion in a DWDM syste. The error probability was calculated using nuerical integration of (4). For the calculations of the BER, the optiu decision threshold was chosen, i.e the threshold, which iniized the error probability P e. The diagras show that P e tends to diinish rapidly below a certain power value. This happens due to the fact that since cosθ, the rando variables I and I s cannot exceed a certain value I,ax and I s,ax obtained by setting θ i =0 and B i = in (8b) and (8c) respectively. Hence, for very low input powers the distributions of the ark and space state ay not overlap at all which iplies an error free transission [3]. However, in practical systes the existence of other noises (e.g. theral noise) will force the two distributions to overlap, preventing the BER fro becoing zero. When the dispersion or the channel spacing is increased, lower values of P e are achieved at a given input power. This can be explained by equations (4a) and (4b) which give the ixing efficiency η and phase isatch Δβ. Fro these equations it is easy to see that the ixing efficiency increases when the dispersion D and/or the channel spacing Δf is reduced. Since the power P of the FWM product given by (3), is proportional to the ixing efficiency η, it is expected that the noise power will be increased and that the perforance of the syste will degrade. A siilar behavior is observed when the nuber of channels is reduced. This takes place because the nuber of the neighboring channels on each side of the central channel is reduced and therefore, the nuber of channel cobinations (p,q,r) that satisfy the condition 4

15 p+q-r=n is also reduced. Hence, the power of the FWM noise will decrease when the nuber of channels is decreased. It is interesting to note that an increase in the nuber of channels above a certain value does not significantly change the syste perforance. For exaple for D=5ps/n/K, Δf=5GHz and P in =4.5dB a BER=0-9 is achieved for N=8 channels, a BER=7 0-7 is achieved for N=6 channels and a BER=0-6 is achieved for N=3 channels. This happens because the channels, which are far away fro the central one, do not play a significant role in the production of the FWM noise. Fro figures 6(a)-6(c), it is confired that a siple way to eliinate the ipact of FWM is to use nonzero dispersion fibers and WDM systes with large channel spacing or with fewer nuber of channels. For exaple for N=6 channels, Δf=5GHz and Pin=8dB, an increase of D fro 5 to 0ps/n/K causes a reduction of BER fro 0-3 to So graphs like 6(a) to 6(c) are useful in deterining the axiu input power allowed in a WDM link, given its characteristics, i.e. the channel spacing Δf, the fiber dispersion coefficient D and the total nuber of channels. For exaple for N=3, D=ps/n/K and Δf=50GHz, the input power of each channel ust not exceed 4.9dB if the BER is not to exceed 0-9. Note that this result is obtained for a fiber length of L=80K. Since the FWM influence is not uch different in systes with a fiber length longer than the effective length defined as (-exp(-al))/a, these results are also valid when the total fiber length is longer than the effective length. Note, that the analysis carried out so far, assues a single span syste. If the syste consists of ultiple spans and optical aplifiers are used to copensate the optical losses, then additional FWM noise products are generated in each span. The phase of these products depends on the phase of the channels at the input of the span. It can be assued [4] that the dispersion of each fiber span differs slightly and that the length of the fibers used in the span is different. 5

16 Hence, it is possible to assue that the phases of the channels at the beginning of each span are independent of each other. This iplies that the phases of the products in different spans are independent as well. Also, the net dispersion in each span causes a walk-off of neighbouring channels by at least one bit period. Hence, the FWM noise products in each span can be assued independent and the pdf of the total FWM noise can be approxiated by the convolution of the individual FWM noise pdfs. These assuptions ay not be valid in all cases. If for exaple, the dispersion is copletely copensated in each span, then the phases of the channels can not be considered independent. In such cases, the pdf of S and S s can be coputed by altering the FWM efficiency η in equation (4a) in order to take account the nuber of spans as in [4]. This however prohibits the use of the auxiliary variables I and I s whose applicability assues the FWM efficiency given by (4a). Consequently, the MC siulations ust be perfored again, this tie for S and S s directly. VI. INCORPORATION OF OTHER NOISES In actual systes, receivers can also suffer fro other noises such as the theral and the Aplified Spontaneous Eission (ASE) noise. In order to include these noises, a technique based on the Moent Generating Function (MGF) of the decision variable can be used [], [5] while the error probability can be estiated using the saddle-point approxiation through the MGF. To illustrate the above technique, a siple optically prealified receiver with a rectangular optical filter and an integrate-and-dup electrical filter will be assued. The Moent Generating Function (MGF) M Z (s) of the decision variable Z at the receiver is defined as the expected value of e sz. Under the assuption that the quantu efficiency of the 6

17 photodetector equals to unity, the MGF M Z (s) of Z is related to the MGF M X (s) of the energy X of the incident optical field at the input of the aplifier through [5]: M s) = E Z ( { M } = Gs Z X ( s) M X N os N os μ (5) In (5), N o =n sp (G-) is the power spectral density of the aplified spontaneous eission (ASE) noise, while G and n sp are the gain and the spontaneous eission paraeter of the optical aplifier respectively. μ+ is equal to the product BT b of the bandwidth B of the optical filter and the bit duration T b (μ is assued to be an integer). Hence, the MGF of Z can be directly coputed fro the MGF of X by a siple change of variables and ultiplication by (-N 0 s) -μ. Assuing rectangular pulses in all channels for the ark state it is easy to show that X=ST b, where S=S () and S=S (s) for the ark and space states of the central channel respectively. To estiate the MGF of X=ST b one can use the approxiate exponential forulas for the pdf of S. For exaple, in the space state where X=kδ Τ b I s, M X (s) is approxiately written as: A M ( ) ( exp( ( X s b + s ) I s, ax ) ) (6) b + s where s =kδ Τ b and the exponential approxiation Ae bx for the pdf of I s is used (see section III). Note that a ore accurate estiation of the MGF of X could be perfored by directly using the pdf coputed fro the Monte-Carlo siulations and nuerical integration. Using (5) and (6) the MGF of the decision variable at the receiver including the influence of the FWM and ASE noises can be coputed. A siilar expression as in (6) can be derived for the ark state. In order to take into account the electrical theral noise, the MGF of the decision variable Z is ultiplied by exp(σ th s / ) which is the MGF of the theral noise. The power of the theral noise is equal to σ th =K B T K T b / (q R L ), where K B is Boltzann s constant, R L the load resistor of the photodetector, T K the teperature (in Kelvin), while q is the charge of the electron. Using the 7

18 above rearks the MGF of the decision variable can be approxiately coputed and the saddle point approxiation can be used to estiate the error probabilities fro the MGF []. Equation (5) also provides an indication of the validity of the Gaussian odel for the description of the statistics of the decision variable in the presence of FWM and ASE noises. The shape of the MGF of Z is related to the respective one of X, which as indicated by (6) can not be assued Gaussian. Consequently, it is expected that the MGF of Z will also not be Gaussian, especially if the FWM noise (aplified by the optical aplifier) is doinant. This iplies that the Gaussian odel ay not give accurate values for the error probability. However, if the receiver is doinated by the Gaussian distributed theral noise (e.g. in the absence of optical preaplification), the pdf of the decision variable can be approxiated by a Gaussian distribution, accurately enough. VII. CONCLUSIONS In this paper the statistical behaviour of the FWM noise was investigated and its iplications in the perforance of a WDM syste were analyzed as well. Using nuerical MC siulations, the pdf of the FWM noise in the space state was calculated for the first tie. It was also shown both by MC siulations and theoretical considerations that the pdf in the ark state is not syetric as previously assued in the literature. Hence, the proposed odel is considered ore accurate copared to the odels used so far. By fitting the data obtained fro nuerical siulations, approxiate expressions of both pdfs are derived and diagras are given that allow the coputation of these expressions given the nuber of channels. Using these diagras the pdfs can be estiated without having to perfor any MC siulations. The odel is then used to estiate the perforance of a WDM syste for various values of its paraeters (fiber 8

19 dispersion, channel spacing and input power). The obtained results can be very useful in the design of practical systes. A coparison between the present odel and two odels previously proposed was carried out and it was shown that these odels ay not provide an accurate value for the BER. Finally, soe guidelines for the incorporation of other noises (theral and Aplified Spontaneous Eission ASE) are presented. APPENDIX CALCULATION OF THE ODD ORDER MOMENTS In this appendix the first two odd order oents <I u+ > of the photocurrent I in the ark state will be calculated. For u=0, I = u + I is given by: I = = 3 3 e = 3 d p n q n d BpBqBr cos p n q n B B B p q r ( θ θ ) cos( θ θn ) = 0 n = (A) θ since cos ( ) = cos( + ) = 0 is: θ n θ p θ q θ r θ n. However for u=, the third oent 3 I of I I 3 = = 3 7 d BpBqBr cos p n q n e ( θ θ ) [ eep' q' r' ] + [ eep' q' r' ep'' q'' r'' ] p' q' r' n 3 9 = 3 e 3 p' q' r' p'' q'' r'' (A) where e = d B B B cos( θ )/ p n / q n p q r θ. n Using siple atheatical operations one can show that the ters in the sus of (A) have ean value either or 0. Indeed, it is easy to show that every ter of the sus of (A) can be expanded into a linear superposition of ters of the type cosθ=cos(a p θ p +a q θ q +a r θ r - 9

20 a n θ n + +a p θ p +a q θ q +a r θ r +a n θ n ) whose ean value will be either if θ 0 or 0 if θ 0. The coefficient of every ter of the above type, in the superposition is positive. For exaple, working with the ters of the third su of (A) one can write: θ=θ -n +θ p q r -n -θ p q r -n =θ p +θ q -θ r -θ n +θ p +θ q -θ r -θ n -θ p -θ q +θ r +θ n (A3) For the ters for which r =n, p=r, p =r, q=p and q =q one obtains θ 0. Since θ 0, one has cos θ = and the ean value of the corresponding ters will be. For the ters for which all the phases θ i in θ do not cancel out (θ 0), θ will be given by a linear cobination of the independent rando variables θ k each of which is uniforly distributed in [0,π] and hence <cosθ>=0. Therefore, for these ters the ean value vanishes. Fro the above rearks one can 3 easily conclude that I > 0. ACKNOWLEDGEMENTS The authors would like to thank Mr. George Kakaletris for his valuable prograing assistance. REFERENCES. H. J. Thiele, R. I. Killey and P. Bayvel, Investigation of XPM Distortion in Transission over Installed Fiber, IEEE Photonics Technology Letters, (000).. M. Shtaif and M. Eiselt, Analysis of intensity interference caused by cross-phase-odulation in dispersive optical fibers, IEEE Photon. Technol. Lett. 0, (998). 3. K. Inoue, K. Nakanishi, K. Oda and H. Toba, Crosstalk and Power Penalty Due to Fiber Four-Wave Mixing in Multichannel Transissions, J. Lightwave Technol., (994). 0

21 4. M. Eiselt, Liits on WDM Systes Due to Four-Wave Mixing: A Statistical Approach, J. Lightwave Technol. 7, 6-67 (999). 5. J. G. Proakis, Digital Counications (4 th ed. McGraw-Hill, New York, 000). 6. K. Inoue, Polarization Effect on Four-Wave Mixing Efficiency in a Single-Mode Fiber, IEEE J. Quantu Electron. 8, (99). 7. N. Shibata, K. Nosu, K. Iwashita and Y. Azua, Transission Liitations Due to Fiber Nonlinearities in Optical FDM Systes, IEEE J. Select. Area Coun. 8, (990). 8. G. P. Agrawal, Nonlinear Fiber Optics ( nd ed., Acadeic, New York, 995). 9. S. Song, C. T. Allen, K. R. Dearest and R. Hui, Intensity-Dependent Phase-Matching Effects on Four-Wave Mixing in Optical Fibers, J. Lightwave Technol. 7, (999). 0. K. O. Hill, D. C. Johnson, B. S. Kawasaki and R. I. MacDonald, CW three-wave ixing in single-ode optical fibers, J. Appl. Phys. 49, (978).. G. H. Einarsson, Principles of Lightwave Counications (John Wiley & Sons, Chichester, 996).. J. Tang, C. K. Siew and L. Zhang, Optical nonlinear effects on the perforance of IP traffic over GMPLS-based DWDM networks, Coputer Counications 6, (003). 3. P.E. Green, Fiber Optic Networks (Prentice-Hall, New Jersey, 993). 4. K. Inoue, Phase-isatching characteristic of four-wave ixing in fiber lines with ultistage optical aplifiers, Optics Letters 7, (99). 5. T. Kaalakis and T. Sphicopoulos, Asyptotic Behavior of In-Band Crosstalk Noise in WDM Networks, IEEE Photon. Technol. Lett. 5, (003).

22 Figures N=8 channels N=6 channels N=3 channels pdf I s (a) Figure a

23 pdf N=3Ch N=8Ch N=6Ch I (b) Figure b 3

24 A for the central channel 6x0-3 5x0-3 4x0-3 3x0-3 x0-3 x0-3 A=A(N) A=3,9x0-4 -N /, ,984e N (a) Figure a 4

25 b for the central channel -0,05-0,06-0,07-0,08-0,09-0,0-0, -0, b=b(n) -N /,879 b=-0, ,7867e -0, N (b) Figure b 5

26 A for the central channel 0,6 0,60 0,58 0,56 0,54 0,5 0,50 0,48 A=A(N) -N / 6,8535 A=0, ,33e 0, N (c) Figure c 6

27 b for the central channel,5,0,5,0 0,5 0,0-0,5 -,0 b=b(n) for I <0 b=b(n) for I >0 -N / 3,9303 b= -,0309-0,90364e -N /,689 b=,4907+9,9385e -, N (d) Figure d 7

28 Syetrical exponential Nuerically coputed pdf (Ap - ) 0,0 E-4 E-6 E-8 0,0 4,0x0-5 8,0x0-5,x0-4 Photocurrent (Ap) (a) Figure 3a 8

29 P e 0, 0,0 E-3 E-4 E-5 E-6 E-7 E-8 E-9 Syetrical exponential E-0 3,0x0-5 6,0x0-5 9,0x0-5,x0-4 Threshold (Ap) Nuerically coputed (b) Figure 3b 9

30 Syetrical exponential Nuerically coputed P e ,0x0-4,0x0-3,5x0-3,0x0-3 Threshold (Ap) (c) Figure 3c 30

31 σ for central channel,4,,0 0,8 0,6 0,4 0, Std (Nuerically Coputed) Std (Syetrical Exponential) 0, N (d) Figure 3d 3

32 P e Nuerically coputed Gaussian distribution ,0x0-5 6,0x0-5 9,0x0-5,x0-4 Threshold (Ap) (a) Figure 4a 3

33 Nuerically coputed Gaussian distribution P e 0 - N=3 D=5ps/n/K Δf=0GHz N=6 D=0ps/n/K Δf=5GHz P in (db) (b) Figure 4b 33

34 pdf (Ap - ) MC (space) MC (ark) Gaussian (ark) Gaussian (space) x x0-5 8x0-5 x0-4 Photocurrent (Ap) Figure 5 34

35 Δf=0GHz Δf=5GHz Δf=50GHz Δf=00GHz 0 - D D D 3 D D D D 3 D D D 3 D D P e (a) P in (db) Figure 6a 35

36 Δf=0GHz Δf=5GHz Δf=50GHz Δf=00GHz 0 - D D D 3 D D D D 3 D D D 3 D D P e (b) P in (db) Figure 6b 36

37 P e Δf=0GHz Δf=5GHz Δf=50GHz Δf=00GHz D D D 3 D D D D 3 D D D 3 D D (c) P in (db) Figure 6c 37

38 Figure captions Fig.. Pdf of a) the space and b) the ark state. The squares represent the case where N=8, the circles the case where N=6 and the triangles correspond to N=3. The exponential fittings for each pdf are also shown with solid lines. Fig.. (a)-(b) The paraeters A and b in the space state, for the central channel, with respect to the nuber of channels N and (c)-(d) the paraeters A and b in the ark state, for the central channel, with respect to the nuber of channels N. Fig. 3. a) The pdf of the photocurrent for N=6, D=0ps/n/K, P in =5dB and Δf=5GHz, b) The error probability P e for the ark state, with respect to the receiver threshold for the sae paraeters, c) P e as a function of receiver threshold for N=8, D=5ps/n/K, P in =8dB and Δf=00GHz and d) Standard deviation of the su I for the central channel with respect to the nuber of channels N for the syetrical distribution (solid line) and the nuerically coputed (Dashed line). Fig. 4. a) Error probability in the ark state with respect to the receiver threshold in the case N=6, D=0ps/n/K, P in =5dB and Δf=5GHz for the Gaussian and nuerical odels, b) BER dependence with respect to the input power for the Gaussian and the nuerical odels. Fig. 5. The pdf of the photocurrent in the ark and the space state for N=6, D=0ps/n/K, dd/dλ=0.07ps/n /K, P in =0dB and Δf=00GHz. The solid lines represent the Gaussian distribution. 38

39 Fig. 6. BER as a function of the input peak power P in in the ark state, for a) N=8, b) N=6 and c) N=3. The values of the chroatic dispersion used are D = ps / n / K, D = 5 ps / n / K and D = 0 ps / n / 3 K. 39