Project Number: IST Project Title: ATLAS Deliverable Type: PU*

Transcription

1 Project Number: IST Project Title: ATLAS Deliverable Type: PU* Deliverable Number: D11 Contractual Date of Delivery to the CEC: June 30, 001 Actual Date of Delivery to the CEC: July 4, 001 Title of Deliverable: Description of numerical code and analytical tools for optimising the design of a Tbit/s dense-. Workpackage contributing to the Deliverable: WP1 Nature of the Deliverable: R** Editor Organisation: FUB Author(s): F. Matera (FUB), A. Pinto, P. Almeida (AVE), A. Schiffini (PIR), A. Pizzinat, A. Galtarossa (CPADOR), M. Guglielmucci (ISCTI) Abstract: We describe the routines of a code able to simulate the performance of optical transmission systems, with particular details for the wavelength division multiplexed (WDM) systems with capacity of the order of 1 Tb/s. We introduce two novel routines for the simulation of wavelength converters based on the four waves mixing (FWM) in semiconductor optical amplifiers and of the Raman effect. We also report some simple analytical tools necessary to prepare a first draft of a design for WDM systems with capacity of 1 Tb/s. Keyword list: ATLAS, simulations, split-step, Kerr, Raman, all optical wavelength conversion * Type: PU-public, PP-limited, RE-restricted, CO-internal ** Nature: P-Prototype, R-Report, D-Demonstrator, O-Other ATLAS-FUB page 1/66

2 Executive summary In this deliverable we describe the routines of a code able to simulate the performance of optical transmission systems, with particular attention for the wavelength division multiplexed (WDM) systems with capacity of the order of 1 Tb/s. We describe in details the routines for the transmitter, for the fibre, for the amplifier and for the evaluation of the Q factor and of the bit error rate. We also report a novel routine for the simulation of wavelength converters based on the FWM in semiconductor optical amplifiers and for the first time we introduce a novel split-step method to simulate the fibre propagation also in the presence of Raman effect that permits to evaluate the relative crosstalk in the case of modulated signals. In Sect. 3 we report some simple analytical tools necessary to prepare a first draft of a design for WDM systems with capacity of 1 Tb/s and we show the main characteristics of a 5x40 Gb/s system operating over 500 km. In the appendix A, B and C we respectively report several details of the split-step method in the presence of polarization mode dispersion (PMD), an overview on the soliton transmission and the details of a routine to simulate a polarization independent wavelength converter based on the FWM in semiconductor optical amplifiers. ATLAS-FUB page /66

3 1. INTRODUCTION The telecommunication field has shown a deep and fast evolution in the last years, mainly due to the recent impressive progresses in the development of optical fibers, optical amplifiers, receivers and transmitters [1-]. The feasibility of high capacity transmission systems has been demonstrated by means of several experiments [3-4]. Numerical simulations of optical transmission systems have been a fundamental tool for investigating optical communication systems, since they are constituted by many components and many effects degrade the signal along the propagation. As a result, analytical approaches are useful only for a limited number of cases. Reliability and effectiveness in terms of time consuming of numerical simulation tools have become a more and more important issue. Several methods can be used to simulate the behaviour of an optical transmission system, but we will focalize our attention on the results obtained by means of the split step method which results very interesting in terms of time effectiveness thanks to the use of the fast Fourier transform algorithm [5]. This method is very friendly in describing the evolution of a signal under the action of deterministic effects as chromatic dispersion and Kerr effects, but in the presence of stochastic processes as Amplified Spontaneous Emission (ASE) noise and random mode coupling some carefulness should be used. In this deliverable we report all the techniques that can be used in the simulation of optical transmission systems operating with capacity of the order of 1 Tb/s with particular attention to the tools necessary to study the cases of interest of the ATLAS project. We remember that the main theoretical target of this project is the transmission of 5x40 Gb/s system over km with wavelength conversion of some channels along the line. The structure of the report is essentially divided in two parts: in the first part (Section ) we discuss some general features about the modeling of an optical system and the evaluation of ATLAS-FUB page 3/66

4 its performances, while in the last (Section 3) we report some analytical tools to have a first design of a Tb/s system. In particular, in Section, we describe the main routines to simulate an optical transmission system; some of these routines are well known but we report them to have a complete document in which the reader can find all the details to study the numerical simulations on optical telecommunication transmission systems. We also introduce some novel routines: a new split-step method that permits to study the signal evolution along the fibre in presence of Raman effect, and a routine able to reproduce the signal behaviour in a wavelength converter based on the Four Wave Mixing (FWM) in Semiconductor Optical Amplifier (SOA) that is one of the devices used in the ATLAS project and one of the most interesting to achieve a wavelength conversion process with high performance. At our knowledge it is the first time that a simulation tool able to evaluate the behaviour of a modulated WDM signal in the presence of Raman effect is reported. Details on the split-step method in the presence of Polarization Mode Dispersion (PMD) are reported in Appendix A. As far as section 3 is concerned we report the main analytical tools that are necessary to design a Tb/s WDM system based on dispersion management [-3] with RZ channels at 40 Gb/s that is one of the main topics of ATLAS. The Tb/s transmission could be obtained also with other techniques and one of the well known possibilities is the soliton propagation. At the moment such a technique seems overcome by the return to zero (RZ) transmission in links requiring strong dispersion management, as those based on G.65 fibres, where the concept of soliton could not be rigorously used since the pulses show strong variations in time and shape. However due to the strong nonlinear behaviour of the pulses, many Authors continue to use the term of soliton pulses also for very strong dispersion management where the pulses loose their initial shape after some kilometers. It has to be pointed out that the techniques proposed to increase the performance of soliton systems could be used also for other nonlinear RZ ATLAS-FUB page 4/66

5 pulses as for instance the filtering process []. Due to this reason in Appendix B we report a short review on soliton transmission. From a system point of view, the method to design a WDM system consists of two main steps: a first draft of the system can be obtained by means of the analytical tools (reported in sect. 3) and after the simulation tools permit to define the details and to do an overall performance test.. MODELING OF OPTICAL SYSTEMS Schematically, an optical communication system is composed by a transmitter, a transmission line and a receiver [1-]. The transmission line is supposed to be composed by optical fibres and optical amplifiers that replace the electrical regenerators. Sophisticated routines have been developed for the transmitter [6] and the receiver [7] and take into account their not-ideal behaviour. However since in most of the high capacity systems the role of the signal propagation is much more important, transmitters and receivers are frequently considered as ideal. For high capacity systems erbium Doped Fibre Amplifiers (EDFA) are used and, for distances shorter than 000 km, the equalisation of the gain is not a problem; as a consequence an amplifier model with a flat gain can be assumed. However a routine that takes into account the gain behaviour in the EDFA is available in the FUB tools; it was obtained during a cooperation with FRANCE TELECOM in the framework of the COST 39. The details of such a routine can be found in ref. [8]. One of the most important tools of the simulation code, is the routine of the optical fibre that mainly consists in the use of the numerical split-step method to solve the nonlinear Schroedinger equation that describes the signal propagation in optical fibre. ATLAS-FUB page 5/66

6 In the following we report the main routines of the simulation code..1 Transmitter An optical transmission can be obtained either by externally modulating a CW laser or directly modulating the current of a semiconductor laser. The first method is more expensive but the quality of the transmitted signal is very good and it permits to reach the highest capacity. In the case of a transmitter composed by a CW laser with a negligible linewidth and a Mach- Zehnder modulator the signal can be described in terms of power P and phase φ as [6] π Vt () π () Pt () = P0 cos, φ() t = K Vt Vπ Vπ (.1) where V(t) is the drive voltage, V π the switch voltage, K V A V B = VA + VB is the chirp parameter [] and VA and VB are the peak-to-peak voltages applied to the electrodes A and B of the modulator. In the case of small modulation depth the field at the transmitter output, in the case of Gaussian shape can be written as ik t Et ( ) = E exp T 0 (.) in this case the constant K, coincides with the chirp constant C defined in ref. [5]. For K=0 we have the ideal behaviour of a transmitter with Gaussian pulses. In the case of the non return to zero (NRZ) transmission, the signal is amplitude modulated by a train of risen cosine pulses with a rising time T r =T/4, being T the inverse of the bit rate R. In the RZ case, the transmitted signal is modulated by either a Gaussian or a hyperbolic sechant (soliton) electrical signal. ATLAS-FUB page 6/66

7 To give some examples, in our simulations, in the case of single-channel systems the transmitted bit stream consists of several repetitions of the following sequence of 3 bit: Such a sequence is sampled so to obtain 048 samples. In the case of WDM signals, the transmitted bit stream for the generic channel is obtained by randomly translating the sequence adopted in the single channel case. In the case of WDM systems, the overall optical bandwidth of the simulated signal depends on the sampling time. Moreover a simple routine can be used to generate pseudo random sequences of arbitrary length with the same method used in the Pulse Pattern Generators.. Fibre The simulation of the signal propagation is based on the solution of the nonlinear Schrödinger equation by the split-step method [5]. In this Section we describe the scalar split-step method. We also introduce a novel routine inside the split-step method to include the Raman effect. To accurately simulate signal propagation, polarization effects should be taken into account; as a consequence the solution of the coupled nonlinear Schrödinger equations is required, as reported in the Appendix A. Unfortunately, this solution is computationally complex and the program implementing it requires a long machine time to be executed. This is also caused by the presence of random mode coupling, which requires the choice of a smaller step in the propagation direction. However if some propagation conditions are verified, the signal propagation can be simulated by means of the scalar nonlinear Schrödinger equation. These conditions are often verified in conventional optical systems and therefore the performance of a large variety of optical systems can be evaluated numerically solving the nonlinear Schroedinger equation. ATLAS-FUB page 7/66

8 In presence of loss, chromatic dispersion or Group Velocity Dispersion (GVD), and Kerr nonlinearity, the nonlinear Schrödinger equation can be written as [5] i U ξ + α L D U = 1 sign(β ) U + i τ 6 3 L D U L' - D τ 3 L D L NL U U (.3) Here ξ=z/ld, LD = T0 / β, L'D = T0 3 / β3, T 0 is an arbitrary constant, β is the GVD, β3 is the third order dispersion. U=E/P where E is the electric field and P the input power. The nonlinear coefficient γ is included in the nonlinear length L NL = (γp) -1. Sign() is 1 in the normal dispersion region while -1 in the anomalous one. The constant T 0 is generally the time duration of a RZ pulse, while in the case of NRZ signal is more convenient to assume T 0 equal to the pulse rise time, T r. Several methods have been proposed to solve eq. (.3), most of them can be classified in two categories: finite difference- methods and pseudospectral methods. Among the pseudospectral methods, one of the most important is the split-step Fourier method (SSFM), that results very interesting under the point of view of the calculation time thanks to the use of the Fast-Fourier Transform (FFT). The SSFM is based on the division of the fibre into small steps in the propagation length. The fiber propagation is simulated by approximately solving the propagation equation in each step starting from the field at the output of the previous step. The approximate solution of equation (.3) in a single step can be more easily expressed by rewriting this equation as ATLAS-FUB page 8/66

9 + idβ + Da + idγ + Dβ3 ζτ ζ U(, ) = 0 (.4) where the real operators appearing in equation (.4) are defined as 1 D β = sign( β ) τ ; Da = αld L D 1 L Dγ = U ; D D β 3 = LNL 6 L' D τ 3 3 (.5) The formal solution of equation (.4) can be written as idγ Ls [ i Dβ + Da + Dβ3] LS U( ζτ, ) = e e U( 0, τ) (.6) where L s is the length of the step. The second member of equation (.6) shows that the field at the step output can be determined by applying two operators to the field at the step input: first a linear operator taking into account dispersion and attenuation then a nonlinear operator accounting for the Kerr effect. It is to be noted that equation (.6) is not an exact solution of (1) since the nonlinear operator depends on the local value of the field and not only on the field at the step input. However, if the step is sufficiently small, the nonlinear operator can be approximately evaluated starting from the field at the step input. Thus the dispersive (linear) and the nonlinear contribution in the step can be evaluated separately. In particular, in the dispersive step the signal is Fourier transformed by means of the FFT and hence multiplied by the transfer function of the fiber piece. In agreement with equation (.3) such transfer function can be written as α Ls i ( β ω Ls / + β ω 3 3 Ls / 6) F( ω) = e e (.7) ATLAS-FUB page 9/66

10 The resulting signal is anti-fourier transformed and is multiplied by the operator representing the Kerr effect, that in agreement with equation (.6), is e ^ idγ Ls α L s e = exp 1 i γ Po( 0, τ) L s (.8) α Ls where P o (0,τ) is the optical power at the step input. It is to be noted that, if the nonlinear operator would be evaluated using the power P o (0,τ), the nonlinear effect would be overestimated since the optical power attenuates during the propagation along the step. Thus α Ls the term e 1 Ls ( α ) is inserted to evaluate the nonlinear operator using the average optical power in L s. The accuracy of the method depends on the length L s. A good rule is that Ls has to be much shorter than any characteristic length of the fiber (L D, L NL, L' D,..). In the case of WDM systems a larger number of steps is required to take into account the effects of pulse collisions []. Novel split-step method including the Raman effect The Raman effect is a fibre nonlinearity due to the interaction between a photon and an optical phonon that produces a power exchange among the signals located at different frequencies according to a gain curve reported in fig..1 [5]. As well known, the study of the Raman effect for WDM system is very difficult and at the moment it has been deeply theoretically investigated only in the case of CW channels. At our knowledge a simulation tool able to simulate the behaviour of a general WDM signal has not been presented yet. The main difficulty, for a simulation activity, is due to the impossibility to ATLAS-FUB page 10/66

11 ATLAS-FUB page 11/66 define a routine, similar to the split-step method. In the split-step method the Kerr nonlinearity can be treated quite easily since, for short steps, the output signal can be calculated as shown by the equations (.6) and (.8). Unfortunately for N channels we cannot define an equivalent function for the Raman contribution as reported in eq. (.8). We show a method to simulate the Raman effect by introducing a particular step in which the signal behaviour is obtained by numerically solving a system of differential equations. We call this procedure modified split-step method. It has to be pointed out that the effect of the signal modulation in the Raman crosstalk evaluation has been studied, from a statistical point of view, by taking into account the overlap time of the pulses located at different frequencies [9-13]. This way also the contribution of the GVD has been taken into account. However such an approach gives very approximate results and the real evolution of signal in the presence of Raman effect is not reproduced. Before describing our modified split-step method we give some general information on the Raman effect. The Raman interaction between two signals, p and s, can be described by the following coupled equations [5] p s s s p s s s s s s s p s p p s p p p p p p p A A g A A A i A t A i t A z A A A g A A A i A t A i t A z A ' ' + + = = γ α β β γ α β β (.9)

12 where g s and g p are the Raman gains of s and p. We have neglected the contribution of the FWM and taken into account the contribution of the Kerr effect in terms of self phase modulation (SPM) and cross phase modulation (XPM). In such equations we should introduce also a term due to the Self Frequency Raman Shift (SFRS) described by Ai TR t Ai (on each channel), where T R is related to the slope of the Raman gain. However in ref. [9] it was verified that the SFRS can be neglected for pulses longer than 3 ps and this is our case. On the other side, in the same reference it was shown that such a contribution could be simply included in the nonlinear part of the split-step method. Fig..1: Raman gain curve In the case of N channels we have to consider N equations instead of as reported in eqs (.9). In the case of CW case the system of eq. (.9) transforms in a set of N equations for the intensity di j dz ν ( ν ) i i ν j IiI j gr ( νi νk ) I jik = gr (.10) i k νk ATLAS-FUB page 1/66

13 Such system of equations can be solved by means of a Runge-Kutta method. In particular MATLAB has a routine, named ODE45, that uses the 4 th order Runge-Kutta method. It has to be pointed out that the analysis on the CW case can give a first hint on the contribution of the Raman effect in a WDM system, especially in cases in which the signal shows a temporal broadening so large that can be assumed as CW. This is the case of the 40 Gb/s transmission in G.65 fibres in which the pulses show a complete overlap after few kilometres. Eq. (.10) permits only to evaluate the power variations among the channels but neither the BER nor the Q factor can be extrapolated. We underline that from the system performance point of view the power exchange among the channel is not an effect so detrimental, since in principle equalisers could compensate for it. On the contrary, the most detrimental contributions are the power fluctuations that increase the variance of the decision variable. From several analysis we have verified that the Raman effect induces strong fluctuations on the ones, while is quite negligible for the zeroes. However it is clear that if a system shows power variations among the channels smaller than 1 db, it is very likely that the Raman effect will induce a negligible crosstalk. According to this consideration the solution of eq. (.10) is a first tool to verify the conditions in which the Raman effect can be neglected. Approximating the Raman gain curve with a triangular shape and considering the channels with equal input power, a simple expression can be derived for the Raman power threshold in WDM systems. According to ref. [], the following relationship corresponds to a power variation among the channels lower than 0.5 db Pch < (.11) NcBtotLeff ATLAS-FUB page 13/66

14 where P ch is the channel power, N c the channel number, B tot the total optical bandwidth occupied by the channels and L eff the effective length Leff αz e = 1. α We remember that the CW behaviour can be extended to the modulation case by taking into account the statistics on the overlapping of the pulses as reported in ref. [1], however it is a very strong approximation. Now we describe our modified split-step method to include Raman effect. First of all we assume that at a position z of the link the N WDM signals can be described in the time domain by N vectors, each one composed by Np*Nb elements, where Nb are the number of bits and Np the number of points for each bit. Each element of the vector represents a time interval t, and at each time position i t (1=1,.. Np*Nb), each channel has a particular value of power. As a consequence at each i t, we could define a set of N equations as reported in eq. (.10), and by starting from the distance z we can see the behaviour of the signal at z+ z. Obviously in the z interval we have other effects that modify the signal behaviour as the GVD, the Kerr effect and so on. But such effects can be separately evaluated in the z length by the split-step method. In particular the presence of the GVD produces the time shift among the pulse located at different frequencies. First of all we start from the consideration that the Raman length, L R =1/g MAX, is much longer than the dispersion length and the nonlinear length as reported in fig. (.). As a consequence in the Raman step we may consider many conventional steps (dispersive+nonlinear). Practically in a Nx40 Gb/s system operating in G.65 fibres we can have steps with a length shorter than 1 m (case of N=5), while the Raman length is generally longer than 1 km. Supposing to evaluate the Raman contribution in a step Lr< L R /10, the signal at the position z+lr is obtained in the following way: ATLAS-FUB page 14/66

15 first the signal at z+lr is obtained by the signal at z by using the conventional split-step method that includes Kerr and GVD, at the position z+lr we calculate the intensity I(t) of each channel and for each t, we solve the system on N equations reported in eq. () for the length Lr, from the intensity of the N channels we obtain the field by a simple square root operation since the the phase in the Raman process is maintained. We are ready to start with the next step. The Raman analysis slows the split-step procedure due to the presence of the Runge-Kutta routines, however such a delay is not so much relevant since the Raman step is used after many conventional steps. Such a numerical code has been tested by considering several data that were available in literature [9-13]. In particular in fig. (.3) we report a comparison among our code (ramanbit), some experimental data (sperimentale) reported in ref. [1] and the statistical theory based on the pulse overlapping (raman). The main system parameters are: input power for channel equal to 0 dbm, wavelength of the first channel 1530 nm, fibre length 100 km, GVD equal to 16 ps/nm/km. As shown by the figure the comparison shows a good agreement. ATLAS-FUB page 15/66

16 Fig..:scheme of the modified split-step penalty [db] ramanbit raman sperimentale numero segnali Fig..3: comparison among the results obtained by the esperiments [1] the theory on the Raman crosstalk based on the pulse overlapping (Raman) and our simulation code (Ramanbit) In fig..4 we report the behaviour of signal in the presence of Raman effect for our system of 5x 40 Gb/s after a link 100 km long with a peak power of 40 mw per channel and with zero GVD and a loss of 0.5 db/km. Detailed results on the Raman impairment for the Nx40 Gb/s will be reported in another paper. ATLAS-FUB page 16/66

17 Fig..4a, 40 Gb/s output signal without Raman effect (Kerr nonlinearity is not included) Fig..4b: as fig..4a but in the presence of Raman effect..3 Amplifiers At the moment the main optical amplifiers for optical communications are EDFA, SOA and Raman. The model of the SOA has been deeply investigated in the framework of the UPGRADE project and details can be found in ref. [14]. In the ATLAS project SOA are investigated only to be used as wavelength converters and details will be given in Sect..5 and in Appendix C. Raman amplifiers are getting more and more important but their study was not foreseen in the first two years of the project. However the techniques reported in Sect.., for Raman simulation, can be used for the model of the Raman amplifier. ATLAS-FUB page 17/66

18 The simulation of the EDFA should take into consideration all the emission an absorption processes of the amplifier and this complete study results much complex. In ref. [8] we have proposed a simulation code based on the subdivision of erbium fibre in sections and on the balance of the photon flux in each section. However when the distances are not too long a very simple model for the EDFA can be used, supposing to use equalising process in order to have a flat gain in all the WDM bandwidth. In this case the EDFA amplification process is simply obtained by multiplying the electrical field for the total gain G T and by adding to each spectral component of the signal an independent noise term. The real and imaginary parts of the noise spectral components are independent Gaussian variables with variance [] σ =F h ν(g T -1) ν/, where F accounts for incomplete population inversion (compare Section 3.1), h is the Plank constant, ν is the signal carrier frequency and ν is the bandwidth occupied by each Fourier component of the discrete Fourier spectrum..4. Optical Filters Different types of optical filters can be used in optical communication systems, for example Fabry-Perot filters, grating filters, interference filters and acusto-optic filters. We do not report many details on these devices, since they can be simulated according to their characteristics. Generally such devices can be simulated by means of a transfer function H(ν). For instance the transfer function of a Fabry-Perot filter can be written as [] H FP( ν) 1 R = πi ( ν ν0) 1 Rexp ν (.1) ATLAS-FUB page 18/66

19 where R is the reflectivity of the interferometer mirrors and ν is the spacing between adjacent resonant frequencies. To obtain a good quality filter, the interferometer loss has to be small, this implies 1- R <<1. In this approximation, the 3-dB bandwidth B o of the Fabry-Perot filter is given by B o = 1 R π R ν (.13) Generally, the factor I= ( 1 R)/( π R), that depends only on the mirrors reflectivity, is called filter finesse, so that the filter is characterized by means of the free spectral range ν and the finesse I. Around a given resonance peak the transfer function of a Fabry-Perot filter can be approximated by a Lorentzian function, writing H FP( ν) 1 1 i ( ν ν0) B 0 (.14).5 Wavelength converters Wavelength converters will be fundamental devices in the future optical networks. At the moment several devices have been proposed for such function, and the most important are based on the nonlinear processes in SOA. In particular SOA permit to have three different wavelength converters based on different processes: cross gain, nonlinear phase variation and FWM. Among these the most interesting, especially for the high capacity, is the one based on the FWM. In principle, the generated signal is a replica of the input one in a wavelength converter based on FWM in SOA, but with a shift in frequency and an inverted spectrum. The efficiency is generally low (15-5 db lower than the input signal) but it can be amplified by means of an EDFA amplifier. Therefore a simple routine for a wavelength converter can be ATLAS-FUB page 19/66

20 obtained by considering: a frequency shift, an attenuation, a spectral inversion and the presence of a quantity of ASE noise due to the amplification of the SOA. Finally an EDFA can be located at the output of the device. It has to be pointed out that, by using particular configurations of pumps, also wavelength converters that do not perform the spectral inversion can be obtained as reported in the Appendix C. More complex models for the wavelength converters based on the FWM in SOA have been presented, for example introducing the phase noise of the SOA as reported in ref. []. However, to take into account all the effects that are present in a SOA, the model of the wavelength converters gets very complex since many equations must be solved. However, as reported in ref. [15] after analytical calculations the behaviour of the gain of a SOA can be obtained by solving the following equation: dg t, z) m = dt g ( t, z) ( m 0 1 S Sτ S g + ε ( exp[ g ( t, z) ] 1) m 1+ ε exp[ g ( t, z) ] S m ( exp[ g ( t, z) ] 1) S( t,0) m + ε exp[ ( t, z) ] S( t,0) 1 g m 1 τ S ( t,0) d dt S( t,0) + (.15) where: g 0 (z)=lng 0 (z) with G 0 (z) the unsatured gain; τ s the lifetime of the carrier; S s the saturation density of the photons; S(t,0) the input photon density; ε is the sum of the contribution characterizing the ultrafast phenomena ε shb and ε ch (the contributions due to the holes are neglected). ATLAS-FUB page 0/66

21 Eq. (.15) can be solved with the Runge-Kutta method. The phase behaviour at the output of the SOA results: 1 ( ( t, z) ( z) ) + 1 Φ( t, z) = α α ε N g g m N αt ε ch 0 c α T ε ch v ( exp[ ( t, z) ] 1) S( t,0) + Φ( t,0) g m (.16) where Φ(t,0) is the initial phase. From the knowledge of the gain and of the phase we can have the output signal 1 E( t, z) = E( t,0) exp g ( t, z) + iφ( t, z) (.17) m where the input is given by the probe and by the pump m ) B (d n sit à te in frequenza (GHz) m ) B (d à n sit te in frequenza (GHz) Fig..5: behaviour of a SOA wavelength convert: (a) input, (b) output In fig. (.5) we report the behaviour of a SOA wavelength converter for a 40 Gb/s signal with an average power of dbm and a pump of 10 dbm. The pulses have a duration of 5 ps and the frequency distance between the pump and the signal is equal to 400 GHz. ATLAS-FUB page 1/66

22 At the output we can see the replica of the signal but also other two secondary effects, the presence of a replica of pump due to a second order FWM (beating between the pump and the converted signal) and the presence of a modulation of the pump. This effect has been experimentally observed in the measurements performed on the SOA by the OPTOSPEED Italia. In Appendix C we report several details on the simulation of a polarization independent WC that has been obtained in the ATLAS project..6 Receiver At the link output the optical signal is transformed in an electrical current by a PIN photodiode, and then it is electrically filtered. The signal timing is recovered by a baseband phase locked loop (PLL) and a threshold decision device decides the received bit [1-]. In the simulation we assume an ideal PLL behaviour. In the program, the PIN photodiode and the electrical front end are simulated by a square law device and a Gaussian noise source for the receiver noise. After detection the signal is filtered by the electrical filter (simulated with a digital filter) and time synchronization is carried out. The delay between the output signal and the transmitted one is estimated finding the maximum of the correlation between the received and the transmitted signal. Once known the overall delay the sampling instants are easily determined. Different electrical filters can be used according to the signal characteristics and many details can be found in the normative G.957 (Bessel-Thomson). The bandwidth of an electrical filter generally varies between 0.6 and 0.8 R. After synchronization, the eye opening of the received optical signal is evaluated and compared with the eye opening of the transmitted signal to obtain the eye penalty. The Q factor is also evaluated to take into account the noise effects. Since the jitter can be an ATLAS-FUB page /66

23 important phenomenon for some optical systems the average jitter and its variance can be evaluated too. In particular the jitter variance is evaluated by detecting the shift of the pulse peak at the fiber output. Details on the Q factor and on the time jitter are reported in the section of the system performance..7 System performance Nowadays optical systems are required to operate at very low Bit Error Rate (BER). To correctly evaluate the BER a too long computational time is required if Monte Carlo Methods are used. For example, if the BER is required to be lower than 10-9, to correctly estimate its value by simulations more than bits should be considered. This makes impractical any direct measurement of BER and an important role is then played by indirect measurements: one of the most important indirect technique is the Q factor [16-17]. Before describing the methods to evaluate the system performance we have to report some information on the kinds of signal degradation and in particular on their statistical behaviour in order to correctly take into account all the average processes. Signal degradation The signal degradation along the link can be described in terms of three different contributions: deterministic effects, random effects and non-stationary processes [18]. The impact of each contribution in a simulative code is very different and it is not very easy to find a code that can efficiently manage them. As it will be better shown in the following paragraphs it might be useful to consider more than one performance evaluation parameter to describe the impact of these different contributions. Here let us briefly summarize some typical features of these effects. Deterministic processes are responsible of a distortion of signal shape that is always the same as the simulation of the signal propagation is repeated. ATLAS-FUB page 3/66

24 Typical examples of deterministic effects are those due to fiber chromatic dispersion and Kerr nonlinearity. When they act separately on a pulse, they are responsible of temporal and spectral broadening, respectively [5]. Beyond dispersion and nonlinearity there are many others effects that lead to deterministic signal distortions, i.e. filtering or saturation of optical amplifiers. The signal shape distortion depends on the number of interacting bits or, in other words, from the memory of the effect under consideration [18]. In fact, generally, signal distortion depends not only on the characteristics of the single pulse and of the fiber parameters but in principle on all of the transmitted message. Practically the dependence of the signal in a particular bit on distant bits becomes weaker and weaker and the channel can be considered with a finite memory so that the signal received in the time interval (t 0, t 0 +T) depends only on the signal transmitted in the time interval (t 0 - nt, t 0 +T+nT), T being the bit time and n an integer defining the memory of the channel. A typical example of random effect is given by ASE noise due to optical amplifiers. It perturbs the signal on a time scale faster than the bit time. The ASE noise has several detrimental effects: first it is responsible of energy fluctuations which induce a lower limit to the signal energy to distinguish a one from a zero level, thus reducing the signal to noise ratio at the receiver. Second, it accumulates along the line and eventually can compete with the signal in saturating the amplifiers located further down the link, third it can nonlinearly interact with the signal []. We refer to a nonstationary process if the temporal scale of the random process is much longer than the bit time. A typical example is the PMD. In principle an optical fiber with a constant and time-independent PMD behaves as a couple of unbalanced delay lines, so a deterministic pulse broadening can be observed due to the time delay between the two orthogonally polarized components of the signal [1-][5][19]. In a real fiber the random fluctuations in the fiber structure induce a local birefringence and a random mode coupling ATLAS-FUB page 4/66

25 that lead to a random power exchange between the polarized modes and a pulse broadening depending on the input state of polarization of the signal [0]. Moreover local birefringence and random coupling are very sensitive to environmental conditions, like temperature, pressure and tension. Generally, PMD is measured by the relative group delay between two particular polarization states, the principal states of polarization [19], that substantially behave as the two polarization modes of an ideal birefringent fiber. The principal states and hence their relative group delay, depends also on signal wavelength. So, in a real fiber PMD is a statistical process that fluctuates in time and strongly depends on signal wavelength. The temporal scale of its fluctuation is typically longer than 1 minute [1] and as a consequence a signal operating with a bit rate higher than 6 Mbit/s sees the same PMD during the time interval necessary for the measurement of the BER at Conversely BER measurement obtained in different times can give different values because of PMD fluctuations. System performance evaluation When the decision variable can be assumed as Gaussian the Q factor is related to the BER by the formula [1]: ( BER) 1 Q = erfc (.18) and in the hypothesis of optimum decision variable the Q factor is defined as m m σ + σ 1 0 Q = (.19) 1 0 where m 1 and m 0 are the mean values of the decision variable I k, obtained by sampling the current I(t) at the electrical filter output at the instant t k, for the bit 1 and for the bit 0 while σ 1 and σ 0 are the corresponding standard deviations. Experimentally the Q factor is obtained ATLAS-FUB page 5/66

26 by varying the decision threshold until reaching the minimum value for the BER and evaluating the corresponding experimental Q factor, Q exp, by the formula ( BER) 1 Q = erfc [16]. exp The BER can be numerically found with a relatively short bit sequence, but the problem is that unfortunately, the decision variable can be rigorously considered as Gaussian in a very limited number of cases, often not useful in the field of the optical communications. However, some approximations can be made to give a useful evaluation of the performance. It can be assumed that the degradation of the signal at the receiver is affected by two sources of noise: amplitude noise and time position noise (jitter); the latter effect being relevant mainly for RZ systems. This classification permits to evaluate the system performance considering the two effects separately and in particular the amplitude fluctuations are taken into account by means of the Q factor defined as a sort of signal-to-noise ratio [17-18], while the time jitter is taken into account simply evaluating its standard deviation []. Rigorously speaking, the Gaussian approximation is not easy to be satisfied even for simple propagation conditions. As an example, let us consider the ideal propagation condition of a signal in an amplified link, in which all the degrading effects are neglected, a part from the ASE noise. The field ASE noise has a Gaussian statistics but the photodiode performs a square operation and thus the Probability Density Function (PDF) of the decision variable is not Gaussian any more. Fortunately, along with the filtering process at the receiver, the resulting PDF turns slightly towards Gaussian, as can be understood by applying the central limit theorem [3-4]. This occurs when the optical filter in front of the receiver has a bandwidth, B, much larger than the bit rate, R, or in other words, if the number of degrees of freedom of the system, m=b/r is larger than 10, so that the decision variable can be assumed as the sum of m independent random variables. If the Gaussian approximation is not ATLAS-FUB page 6/66

27 verified the BER obtained on the basis of the Q factor can be used only as a qualitative value useful in the optimization of the system design, but not for a quantitative estimation of BER. In these conditions, for a better characterization of system performance it can be useful to evaluate from the simulations both the Q factor and standard deviation of time jitter, assuming that signal degradation is due to independent contribution of amplitude fluctuations and time jitter. In absence of jitter and by assuming a Gaussian distribution for the decision variable, eq. (.18) is valid and an error probability lower than 10-9 corresponds to Q>6. Conversely in the presence of only time jitter and assuming a Gaussian statistics for it, the BER can be analytically evaluated according to the theory reported in ref. []. The results depend on the duty cycle of the signal, but a simple rule can be extracted from [] that permits to say that to have an error probability lower than 10-9 the standard deviation of the jitter, σ t, must satisfy the following condition: σ t R<0.06 (.0) So, signal transmission can be obtained if both requirements on Q factor and standard deviation of time jitter are satisfied at the same time[18]. The impact of pattern effects on the evaluation of the error probability can be taken into account by modifying the basic formulation of Q-factor and time jitter. In fact, that approach gives over-estimated standard deviations due to the intersymbol interference, and hence also a pessimistic evaluation of the BER. In principle in any specific bit sequence the total probability error should be calculated as a sum of the probability of error of each elementary bit, instead of calculating only two error probability distributions for the zero and the one bit, respectively. The two procedures would lead to the same result in absence of pattern effect, but they will differ more and more as the number of interacting bits involved in the ATLAS-FUB page 7/66

28 pattern effect increases [5]. In order to save computational time and at the same time without loosing too much in accuracy, it is important to estimate the number of interacting bits. If the interacting bits are n+1, there are p possible different configurations of n+1 bits, where n 1 p = +. For each of the p pattern a correspondent Q i factor can be evaluated on the central bit of the pattern according to the following formula: Q < I > I i th i = (.1) σ i where I th is the decision threshold and <I i >, σ i are the mean and the standard deviation of the sampled current of the i-th pattern. The average are supposed on l different realizations of the random processes. Then the overall Q factor is obtained as where min{ ( I )} th ( [ min{ BER( )}]) 1 Q = erfc (.) BER means the minimum value of the function BER I ) that is given by I th ( th p 1 Qi BER( I th ) = erfc. (.3) p i= 1 In the most of practical cases it is sufficient to assume that pattern effect is mainly due to the interaction with the next neighbors of each bit, or it can be said that the channel has a three bit memory (n=1, p=8), but in some particular cases, i.e. the saturation of SOA amplifiers, the number of interacting bits should be increased at least up to five bits ( n=, p=3) to obtain an accurate result [5]. ATLAS-FUB page 8/66

29 In analogy to the Q-factor, the impact of pattern effect on jitter evaluation can be calculated as reported in ref. []. Another important parameter both for the reliability and time effectiveness of a simulation code is the length of the bit sequence. For the estimation of the σ i in each of the p equations (.1), l independent realizations of the noise processes are simulated by changing the seed of the random generators. In this way the evaluation of the Q i can be obtained by propagating n+1 bits for each p pattern for l different realizations of the noise process, that means by propagating l times a sequence of p(n+1) bits. However due to the independence of the patterns, the same statistical results can be obtained by considering sequences with a number of bits N greater than p(n+1), reducing the number of different realizations from l to l. It can be demonstrated that the Q factor can be mainly affected by some critical bit patterns and as a consequence to evaluate the system performance is not necessary to take into consideration an enormous quantity of bits, but it is sufficient a bit sequence that contains all the most critical patterns reducing the computational time [18]. The number of required runs and the length of the bit sequences depend on the considered propagation case and preliminary simulations must be done to find stable values of the system performance parameters. If nonstationary processes are present a further statistical investigation is necessary. In fact the evaluation of the BER can be considered valid only for a particular realization of the nonstationary process and the physical fluctuation of the BER in time will correspond to different realizations of nonstationary process in the simulations. An example is given by an optical system strongly affected by PMD where the nonstationarity of the process is simulated by changing the seeds of random generator for the parameters representing the local birefringence and axes of the fiber slices [0][6]. At this aim it should be defined the ATLAS-FUB page 9/66

30 maximum time of out service (outage probability) of a system, i.e. the time interval in which a system can operate with a BER higher than Such a condition depends on the quality of service and it is not the subject of this deliverable. A simple performance evaluation parameter is the eye degradation penalty (EDP). The EDP is generally defined as [] I1min I 0 max EDP = (.4) [ I1min I 0 max ] ideal that represents the ratio between the eye closure at the link output and the eye closure in the ideal propagation condition or back-to-back. It can give a qualitative behaviour of the signal at the link output, but generally there is no relationship with the BER, except for some particular cases. For an instance it is a common knowledge that a good performance is achieved for a EDP lower than 1 db [], but, as it has been shown in ref. [0], also in the presence of a EDP equal to 3 db a BER lower than 10-9 can be measured..8 Check of the validity of the numerical model In the previous years several tests have been performed to check the validity of the simulation code. The codes used in ATLAS project, are from FUB, PIR, CPADOR and AVE and at the beginning of the project, several tests were made among these partners to check their validity. The comparison showed a good agreement and the main results were reported in the milestone M11. Before the ATLAS project the same partners cooperated to make a comparison among simulation and experimental results obtained in the framework of other two European projects: ESTHER and UPGRADE. Many details can be found in the deliverables and in refs. [18][5]. ATLAS-FUB page 30/66

31 3. THEORETICAL TOOLS In this section we will describe the main tools to design very high capacity (Tb/s) systems. First we report a general description of this topic, and after this we will refer to the cases that are more interesting for the ATLAS project. 3.1 High capacity optical transmission systems As well known the main techniques to achieve high capacity transmission is the soliton propagation and the dispersion management method. In the first technique the impairment of the GVD is limited by the appropriate use of the Kerr effect, while in the latter the GVD is periodically compensated with suitable devices (dispersion compensating fibre or fibre gratings). Today the concept of soliton is a bit obsolete since it has been demonstrated that dispersion management is the key to achieve the maximum transmission?distance. However the soliton concept in the dispersion management vanishes since, especially in links with high GVD, the pulse shows strong fluctuations in time and shape and as a consequence the fundamental concept of pulse that maintains its shape along the propagation falls. By the way, such pulses show a very strong nonlinear behaviour that can be suitably studied by using some of the analytical techniques used for the soliton propagation. For this reason we continue to speak of soliton propagation also in links with G.65 fibres operating at 40 Gb/s, where the pulse shape is completely destroyed after some kilometres. It is well known that the maximum capacity (nx40 Gb/s) can be achieved by using the following techniques: a) RZ pulses (higher performance with respect to NRZ), ATLAS-FUB page 31/66

32 b) fibres with a GVD having an absolute values higher than ps /km (to reduce FWM), c) periodically compensation of the GVD. Furthermore other characteristics are required as the use of optical amplifiers with flat gain and compensators for the third order chromatic dispersion. In the future another strong increasing of capacity will be obtained by the introduction of the in-line synchronous modulation and by the use of the optical 3-R regenerators. Before listing some technical rules to design a very high capacity WDM system in this section we report some informations on the dispersion management technique. Conversely the details on the soliton techniques are reported in the Appendix B. It has to be pointed out that some of the techniques specifically studied in soliton regime could be extended to the dispersion management to further increase the maximum capacity: for example the polarization multiplexing and the in-line filtering. 3. Dispersion management the rule In links with dispersion management the length of the fiber pieces is chosen according L link β (link) + L comp β (comp) = L per <β > (3.1) where L link and β (link) are the length and the GVD of the compensated fiber, L comp and β (comp) are the length and the GVD of the compensating fiber, while L per = L link + L comp and <β (link) > the average GVD required for the link. ATLAS-FUB page 3/66