9 A PIECE WISE LINEAR SOLUION FOR NONLINEAR SRS EFFEC IN DWDM FIBER OPIC COMMUNICAION SYSEMS M. L. SINGH and I. S. HUDIARA Department of Electronics echnology Guru Nanak Dev University Amritsar-005, India E-mail: mlsingh7@yahoo.co.uk, mlsingh@sancharnet.in, hudiarais@yahoo.co.in Abstract Stimulated Raman Scattering (SRS) is one of the significant nonlinear effects in dense wavelength division multiplexed (DWDM) based fiber optic communication systems. For tackling this problem accurate evaluation of amplification and depletion of optical power at different wavelength channels due to SRS effect in DWDM fiber optic systems is required. In this paper we have presented a method for evaluating SRS effect taking into account both the pump depletion and wavelength dependence of fiber attenuation coefficient. he novelty of our method is that the nonlinear problem of SRS has been solved in a piecewise linear manner, i.e., the nonlinear problem is divided into small linear steps. In addition to this wavelength dependent fiber loss coefficient is considered, which is assumed wavelength independent in the analytical models available in the literature. Also an algorithm has been developed to increase the step size from 00 meters to km without sacrificing the required accuracy. his ten time increase in the step size has led to a significant reduction in the overall processing time, which is very useful for the simulation purpose. Keywords: Stimulated Raman Scattering, Dense Wavelength Division Multiplexing, Fiber optic communication, Fiber nonlinearities. I. INRODUCION In most of the previous studies like [-], the effect of SRS in DWDM fiber optic systems has been treated under various assumptions like undepleted pump approximation, wavelength independent fiber loss coefficient, triangular approximation of Raman gain profile, negligible loss of energy whenever a short wavelength photon is transformed into a long wavelength photon (i.e., λ i / λ k ~ ), equal power loading in all the wavelength channels and equal channel spacing. Some of the models [], [5] have taken into account the effect of pump depletion but still assume wavelength independent optical fiber loss/attenuation coefficient and hence wavelength independent effective length of fiber. In [6] it has been shown experimentally, that fiber loss coefficient varies up to 0.7 db over a 5 nm bandwidth (in.55 µm transmission window) and 00 km fiber length. In long haul optical systems with end-to-end bandwidth more than 5 nm and length of the order of several thousand
0 kilometers with in-line optical amplifiers, wavelength dependence of effective length will aggravate the SRS effect to a large extent and hence cannot be ignored. Our suggested method gives solution for evaluating SRS cross talk without any of the above mentioned assumptions. he only assumption that has been considered in our case is the triangular approximation of Raman gain profile. If the exact Raman gain profile of a fiber is known, our model can be easily modified to evaluate the SRS effect even without the triangular approximation of Raman gain profile, which makes our model capable of evaluating SRS cross talk in DWDM systems without any of the above-mentioned assumptions. However like in other models in our case also, it has been assumed that marks (binary ) are being transmitted on all the wavelength channels and also walk off effects [7-9] among different wavelength channels have been ignored. Our model is also applicable in the case of unequal channel spacing and any arbitrary input spectrum like the model mentioned in [5]. Comparison of our model with the model in [5] shows excellent agreement when, wavelength independent effective length of the fiber is considered. On the other hand when wavelength dependent fiber loss coefficient is considered, our results deviate from the results of the analytical model [5], which does not take into account the wavelength dependence of the fiber loss coefficient. his deviation indicates aggravation of SRS effect due to wavelength dependence of fiber loss coefficient. It has also been shown that this deviation is more than ust addition of linear loss coefficient especially when long haul fiber optic systems with inline amplifiers are considered. II. DEVELOPMEN OF HE MODEL In DWDM systems due to SRS effect, channels at lower wavelengths act as pump for the channels at higher wavelengths that act as stokes. his leads to power transfer among the different wavelength channels. he graphical representation of depletion and amplification of optical power at different wavelength channels due to SRS has been shown in Fig. for a four channel DWDM system. D[,] is the fractional power depleted from the channel # by the channel # D[,] D[,] D[,] Ch.# λ Ch.# λ Ch.# λ Ch.# λ D[,] D[,] Fig.. Optical power transfer among different wavelength channels due to SRS A simple case of a DWDM fiber optic system having four co-propagating channels is considered first and mathematical expressions for depletion and amplification of power at
different wavelength channels due to SRS are obtained as given in equations () (). A generalized expression for N channel system is then obtained from these equations. P M [] = P []. D[, i] i= () PM [] = P []. D[, i] + P []. D[,] i= () P P M M [] = P []. [] = P [] + { D[,] } + { P [ ]. D[,] } = { P [ ]. D[,] } = where P [i] and P M [i] are the optical power launched in the i th channel and the modified power in the i th channel due to SRS effect after propagation over a given length of optical fiber respectively for i=,,,. Channel # is the lowest wavelength channel with centre wavelength λ. D[i,] represents the fraction of power depleted form the i th channel by the th channel for >i, i.e., th channel is at a wavelength longer than that of the i th channel. he value of D[i,] is evaluated using equation (5) that is developed by modifying the expression given in []. 5 {( f f ) /.5 0 }. g.{( L ( ) 0 ) /( b. A )} D [ i, ] = ( λ / λ ). P [ ]. λ i i R max e e () () for ( f i f ).5 0 Hz. and > i D[ i, ] = 0 for ( f i f ) >.5 0 Hz. and i (5) g R max is peak Raman gain coefficient (cm/w). λ i, λ are the wavelengths (nm) of i th and th channels and f i, f are the centre frequencies (Hz) of the i th and th channels. A e is effective core area of optical fiber in cm. P [] is optical power in watts launched in the th channel and value of b varies from to depending upon the polarization state of the signals at different wavelength channels, b= is considered for scrambled polarisation []. L is actual fiber length in km and L e (λ ) is wavelength dependent effective length in km which is calculated using equation (6) given below. L e ( λ ) ( λ ) α L. = exp (6). α( λ ) where α(λ ) is wavelength dependent linear loss coefficient of optical fiber in db/km. For calculating α(λ ), equation (7) is developed by taking into account the variation in linear loss coefficient with wavelength up to 0.7 db, over 5nm bandwidth (in.55 µm transmission window) and 00 km fiber length (confirmed by the experimental results [6]).
α( λ ) = α α α max min var { α [( λ λ ) / ].( α α )} max = α + α = α α var var / / = (0.007 / 5). WDM WDM max min (7) where WDM is spectral width of DWDM signal (i.e., separation between the shortest wavelength channel and the longest wavelength channel) in nm, λ and λ are wavelengths of channel# and channel# respectively in nanometers and α var is in db/ km and α is fiber loss coefficient in db/km at centre wavelength. It is assumed that α varies linearly with wavelength. However in transmission windows other than.55 µm window the variation of fiber loss coefficient with wavelength will differ and so the factor.007/5 in equation (7) will have to be changed accordingly. his makes our model capable of taking into account the effect of wavelength dependent linear loss coefficient of fiber while calculating SRS induced spectral distortion in DWDM fiber optic systems. he equations () () are combined together to form a single equation, which is further generalized for N channel systems as given below. P M [ k] = P [ k] P [ k]. N i= k + D[ k, i] + k = P [ ]. D[, k] for k =,,.N (8) where D[k,i] = 0 for i>n and D[,k] = 0 for k= In equation (8) the term P [ k] D[ k, i] N i= k +. gives the total power depleted from the k th channel by the higher wavelength channels and P []. D[ k] k =, indicates the total power depleted by the k th channel from the lower wavelength channels. While evaluating the above equations, simultaneous power transfer among different wavelength channels is assumed. Actual optical power received in k th channel P R [k] at the receiver side is obtained using equation (9) given below. [] k P [] k. exp{ a( λ L} PR = M k ). (9) III. EFFEC OF PUMP DEPLEION As already mentioned that in DWDM systems with SRS the lower wavelength channels act as pump for the higher wavelength channels, which act as stokes. In the previous section while calculating the SRS effect over a given length of fiber, the pump is considered equally strong through out the length. But in actual case the pump is the strongest at the starting point and as we move away from the starting point the pump is depleted of its power by the stokes. So if we take into account the effect of pump depletion, the effect of SRS will be different from whatever has been predicted by the models ignoring pump depletion. For
considering the effect of pump depletion while calculating the SRS effect, the whole fiber length of La km between two successive inline amplifiers is divided into M seg = (La/Ls) small segments each of length Ls km as shown in Fig.. EDFA Optical fiber La km EDFA Segment # Ls km Ls km Segment # Segment # m Ls km Fig.. Division of inter-amplifier span into smaller segments for taking pump depletion into account while evaluating SRS effect Modified power due to SRS in all the wavelength channels, i.e., P M [k] for k=,, N is calculated at the end of the first segment. his power distribution evaluated at the end of the first segment is then treated as input spectrum for the second segment and SRS effect is evaluated for the second segment. Output spectrum of the second segment is then considered as the input spectrum for the third segment and so on. In this way the iteration process is repeated until we reach the end of the fiber. In the computer programming two different algorithms can be used for evaluating SRS effect with due consideration to the pump depletion. he flow charts for these algorithms have been explained in Fig. and Fig.. SAR Input power launched per channel P, # of channels N, channel separation, Amplifier separation La, Segment length Ls and other related parameter values Set counter m = Calculate effective length of a segment = ((-exp( α(λ).ls))/α(λ), it will be same for all the segments having length Ls km Calculate P M [k], modified power per channel due to SRS for k =,,.N m = m + m = La / Ls YES Print P M [k] for k =,, N NO P [k] = P M [k]{exp(-α(λ)ls)} for k=,,.n END Fig.. Flow chart for Algorithm #
In the algorithm #, effective length evaluated for any segment remains the same as long as the length of all the segments is same. However the attenuation in signal power over a segment length due to fiber loss coefficient is considered. In the algorithm #, the effective length for the first segment is calculated in the same way as given in algorithm #, but for the segment # to the segment #M seg, the effective length has been obtained by using the following equation (0). Also the input power distribution for the following segment is taken equal to the modified power due to SRS in the preceding segment, i.e., P [k] = P M [k] for k =,,., N, instead of taking P [k] = P M [k].exp(-α(λ).ls) as in algorithm #. L e [ m, λ] = [{ exp( a( λ). Ls. m)}/ a( λ)] [{ exp( a( λ) Ls( m ))}/ a( λ)] for m =,,, M seg. (0) SAR Input power launched per channel P, # of channels N, channel separation, Amplifier separation La, Segment length Ls and other related parameter values Set counter m = Calculate effective length of the m th segment using equation 0 Calculate P M [k], modified power per channel due to SRS for k =,,.N m = m + m = La / Ls YES Print P M [k] for k =,, N NO P [k] = P M [k] for k=,,.n END Fig.. A Flow chart for the algorithm # IV. COMPARISON OF WO ALGORIHMS he results obtained by using the algorithms # and # are compared with the results of a theoretical model given in [5]. For comparison an example given in [5] is considered, i.e., 80 channel system with 50 GHz inter-channel separation using optical fiber having loss coefficient of 0. db/km, affective core area of 55 µm and Raman gain of 7x0 - cm/w for a frequency separation of 5 Hz. Fig. 5 and Fig. 6 depicts the results of the comparison. It has been observed that algorithm # is better than the algorithm #, since it allows larger segment length of the order of km, in comparison with the algorithm #, which requires 0 times smaller segment length (00 meters) to achieve the same accuracy. Using algorithm # processing time required for calculating the SRS effect can be reduced by a factor of ten. Moreover if the exact Raman gain profile of a fiber is known, the equation (5) can be further modified very easily to make our model applicable for evaluating SRS effect even without
5 making the assumption of approximating the Raman gain profile as a triangular function. In all the remaining discussions the algorithm # is considered. While calculating Raman loss using our method we have calculated the loss in the lowest wavelength channel, which is the worst affected channel, i.e., Raman loss = 0 log 0 (P M [] / P []). Similarly the Raman gain is calculated as Raman gain = 0 log 0 (P M [80] / P [80]), since the highest wavelength channel experiences only gain due to SRS. Raman Gain/Loss (db) 6 0 - and 6 Power/channel =mw, Channel separation = 50 GHz Raman loss (Zirngibl s model) Raman Loss (Algorithm #, segment length=00 meters) Raman Loss (Algorithm #, segment length= km) Raman gain (Zirngibl s model) Raman gain (Algorithm #, segment length = km) Raman gain (Algorithm #, segment length=00 meters) 5 and 5 6-0 0 0 60 80 00 System Length (km) Fig. 5. Comparison of our model using algorithm # with Zirngibl s model [5] for evaluating Raman Gain/Loss Raman gain and Raman loss (db) 6 5 0 - - Power/ch.= mw, N=80, Channel spacing=50ghz Loss for shortest wavelength channel(algorithm #) Loss for shortest wavelength channel (Zirngibl s Method) Gain for longest wavelength channel (Algorithm #) Gain for longest wavelength channel (Zirngibl s Method) Segment length = km - 0 0 0 60 80 00 System length(km) Fig. 6. Comparison of our model using algorithm # with Zirngibl s model [5] for evaluating Raman Gain/Loss
6 V. EFFEC OF WAVELENGH DEPENDEN FIBER LOSS COEFFICIEN In Fig. 5 and Fig. 6 wavelength independent fiber loss coefficient is considered while comparing the results generated by our model with the results of Zirngibl s model in order to check the effectiveness of the algorithm # and algorithm #. he reason being that the Zirngibl s model has considered the wavelength independent fiber loss coefficient. In our model, which is capable of taking into account the wavelength dependence of fiber loss coefficient, when the wavelength dependence is considered, a definite difference in the results is observed as depicted in Fig. 7. Raman gain and Raman loss (db) 0 - Power/ch.= mw, N=80, Channel spacing=50ghz Loss for shortest wavelength channel(our Method) Loss for shortest wavelength channel (Zirngibl s Method) Gain for longest wavelength channel (our method) Gain for longest wavelength channel (Zirngibl s Method) Loss taking into account wavelength dependent fiber loss Gain taking into account wavelength dependent fiber loss - 0 0 0 60 80 00 System length(km) Fig. 7. Raman gain/loss as a function of system length, a comparison of our model with the Zirngibl s model, with and without wavelength dependent fiber loss coefficient (power/channel = mw) Although this difference of approximately 0.5 db may not be very distinctive, but in case of long haul systems of several thousand km length with in line optical amplifiers the aggravation of SRS effect due to wavelength dependence may be quite alarming. his has been shown in Fig. 8. In Fig. 8 our model is used to depict the evolution of SRS induced power tilt between the optical power levels of channel # and channel # along the length of a fiber optic system using NZDSF+ fiber (Non Zero Dispersion Shifted fiber in anomalous regime having fiber loss coefficient of 0.05 db/km @.55 µm, effective core area of 5.x0-7 cm and peak Raman gain coefficient of 0.05x0 - cm/w) with inline optical amplifiers each having gain assumed to compensate the attenuation of signal (considering fiber attenuation coefficient at centre wavelength). It has been observed in this figure that SRS induced power tilt at 000 km with inter amplifier spacing of 00 km becomes.5 db for wavelength independent fiber loss/attenuation coefficient, and it increases to. db when wavelength dependent loss coefficient is considered. his increase of 8.7 db is.7 db higher than the expected increase of 7dB due to wavelength dependence of fiber loss coefficient (gain tilt @ 0.7 db per 5 nm spectral width per 00 km fiber length). he
7 problem becomes more severe when inter amplifier spacing is reduced to 50 km keeping the same input power level (power per channel =.5mW). In this case the difference between the power tilt with and without the wavelength dependent loss coefficient becomes.65 db, which is.65 db higher than the expected increase of 7dB. 50 5 0 Power per channel =.5 mw, channel spacing = 00 GHz With wavelength dependent fiber loss, Amp. spacing=50km Without wavelength dependent fiber loss, Amp. spacing=50km With wavelength dependent fiber loss, Amp. spacing=00km Without wavelength dependent fiber loss, Amp. spacing=00km Ch# / Ch# power ratio (db) 5 0 5 0 5 0 For Amplifier Spacing = 50 km 5 For Amplifier Spacing = 00 km 0 0 00 00 00 00 500 600 700 800 900 000 System length (km) Fig. 8. Evolution of SRS-induced power tilt along a DWDM fiber optic link with inline optical amplifiers with and without the wavelength dependent fiber loss coefficient for different inter amplifier spacing VI. CONCLUSION It has been observed that in long haul fiber optic systems, the difference in the power tilt due to SRS with wavelength independent fiber loss coefficient and the power tilt due to SRS with wavelength dependent fiber loss coefficient is much more than the expected difference. his deviation is due to the fact that power depletion by the higher wavelength channels from the lower wavelength channels is actually dependent on the power levels of the concerned wavelength channels and the power levels are also varying due to wavelength dependence of fiber loss coefficient along the length of the fiber. his deviation from the expected difference will further increase with the increase in the system length, increase in power level per channel, increase in the number of channels and decrease in the amplifier spacing. So for the accurate measurement of the SRS induced power tilt, the wavelength dependence of the fiber loss coefficient should be taken into account while calculating SRS effect. aking wavelength dependent fiber loss coefficient into consideration while calculating SRS effect is possible in our proposed method. So our proposed method will be helpful in evaluation of SRS effect with better accuracy which will further be helpful in designing more effective measures to counter act the SRS effect. he use of the algorithm has made it possible to increase the step size from 00 meters to km without sacrificing the required accuracy in the calculations. his will lead to comparatively less processing time required for evaluating SRS effect and hence will result into better simulation effects.
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