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Soliton generation from randomly modulated return-to-zero pulses
Optics Communications 281 (2008) 5439–5443
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Soliton generation from randomly modulated return-to-zero pulses Stanislav A. Derevyanko a,*,1, Jaroslaw E. Prilepsky b a b
Photonics Research Group, Aston University, Aston Triangle, Birmingham B4 7ET, UK B.I. Verkin Institute for Low Temperature Physics and Engineering, NASU, 47 Lenin Avenue, Kharkov 61103, Ukraine
a r t i c l e
i n f o
Article history: Received 1 May 2008 Received in revised form 25 July 2008 Accepted 25 July 2008
Keywords: Random phase modulation Nonlinear Schrödinger equation Optical solitons
a b s t r a c t We consider return-to-zero (RZ) pulses with random phase modulation propagating in a nonlinear channel (modelled by the integrable nonlinear Schrödinger equation, NLSE). We suggest two different models for the phase fluctuations of the optical field: (i) Gaussian short-correlated fluctuations and (ii) generalized telegraph process. Using the rectangular-shaped pulse form we demonstrate that the presence of phase fluctuations of both types strongly influences the number of solitons generated in the channel. It is also shown that increasing the correlation time for the random phase fluctuations affects the coherent content of a pulse in a non-trivial way. The result obtained has potential consequences for all-optical processing and design of optical decision elements. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction Classical optical solitons have been one of the recurring topics in optical communications for more than 30 years now. The fact that they are described by a well-known integrable nonlinear Schrödinger equation (NLSE) have always made them attractive for both theoretical and numerical modelling. And although the attempts of applying classical NLSE solitons directly to the problem of data transmission have run into numerous difficulties there are still many areas where these solitons have found their niche. First of all, recent developments [1,2] demonstrated the real possibility of achieving quasi-lossless transmission over long distances with very small power variations. Such systems are well modelled by a classical NLSE which opens the possibility of observing and utilizing classical solitons without the need to refer to other related concepts (like e.g. dispersion-managed or dissipative solitons). Since the NLSE is integrable a lot of properties of the emerging signal can be obtained by analysing the input pulse form (possibly corrupted by noise). Second important aspect of classical solitons which is pertinent to the current study is the use of optical solitons for all-optical return-to-zero (RZ) pulse regeneration [3]. In particular it was demonstrated that the efficiency of carrier signal control can be enhanced by periodic conversion of an RZ signal into a classical soliton. Therefore one has to consider the problem of RZ pulse-to-soliton conversion, i.e. given the initial RZ pulse shape determine the properties of the emerging (multi)-soliton pulse
* Corresponding author. E-mail addresses: [email protected] (S.A. Derevyanko), astrayed@ yandex.ru (J.E. Prilepsky). 1 On leave from the Institute for Radiophysics and Electronics, NASU, 12 Acad. Proscura Street, Kharkov 61085, Ukraine. 0030-4018/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2008.07.079
after its propagation through a quasi-lossless span described by the classical NLSE. For different deterministic pulse shapes the problem has been studied extensively (see Refs. [4–6] and references therein). However any optical regenerator must deal not with pure deterministic waveform but rather with a pulse affected by noise (that may come from amplifier spontaneous emission, channel cross-talk and other impairments). Also any RZ pulse generator introduces random chirp which is sometimes difficult to control. Therefore when studying real-world communication systems, rather than studying deterministic pulse-to-soliton conversion problem one has to turn to its random counterpart. The aim of our study is to elucidate how the output of a integrable NLSE channel is affected by random phase modulation of the input pulse. Since asymptotically the solution of the NLSE is determined largely by the soliton part and the radiation disperses away (see e.g. [7]) we will gauge the performance of such a channel by the average number of emerging solitons. This number is highly sensitive to phase disorder as well as the correlations present in the phase of the input signal. The propagation of partially coherent pulses in the Kerr-type media has long been a subject of investigation in different contexts [8–10] including random phase modulation of temporal solitons [11,12] and nonlinear Fraunhofer diffraction of random fields [13]. Phase fluctuations are of particular importance inasmuch as the phase of a field is hardly controllable whereas the pulse energy can usually be adjusted to a desired level. As a rule, phase modulation of the incoming signal is not explicitly known and may be considered random. But such a modulation can drastically influence the coherent (soliton) component of the input pulse even in the case of deterministically chirped pulses. It is known that the energy contained in the soliton part of the pulse decreases and soliton states can eventually disappear completely in the case of the unbounded phase (e.g. for linear
5440
S.A. Derevyanko, J.E. Prilepsky / Optics Communications 281 (2008) 5439–5443
chirp) [4,6], while the opposite tendency was observed when the phase was rapidly decaying [5]. Hence even deterministic phase modulation affects the formation of solitons in a non-trivial way. The situation when the phase fluctuates randomly seems to be even more involved. In this case, as shown in [12] for the model with random frequency and linear chirp, the energy threshold for the formation of a soliton state increases and, once overcome, only some portion of the input energy is transformed into a soliton. At the same time the questions of whether we can completely destroy the coherency of the signal by the RPM and how the coherent part of the pulse ‘‘feels” the type of statistics and correlations of the phase fluctuations have so far remained unclear. The goal of our paper is to clarify these points. Here we will consider two different types of phase disorder. Firstly, weak Gaussian fluctuations of the pulse amplitude in the strongly nonlinear regime produce Gaussian statistics of the accumulated phase shift due to the self-phase modulation (GordonMollenauer effect). As an example of such a situation we may mention random phase jitter caused by the action of the additive white Gaussian noise, arising during the soliton propagation [14,15]. The Gaussian-Markovian frequency fluctuations were also considered in Refs. [16,17]. Discontinuous random jump processes, such as dichotomic and generalised telegraph process, were suggested to model the phase fluctuating field of a laser in Refs. [18,19]. In this paper we deal with both Gaussian-type and telegraph-type fluctuations of the phase and study how these fluctuations affect the coherent part of the input pulse entering the NLSE channel. As for the shape of the signal we opt for a simple RZ rectangular profile. The reason for that is twofold. Firstly, the unmodulated rectangular shape admits the complete analytical solution of the direct scattering problem (the so-called Zakharov–Shabat eigenproblem, ZSSP) associated with the NLSE [20]. Thus for the unmodulated case we know the energy thresholds for soliton generation explicitly. Secondly, such a profile always generates some amount of linear radiation in addition to the soliton part, so that the situation is more general than the case of pure multi-soliton (sech-type) pulses where radiation can be absent completely. At the same time we argue that the type of the RZ pulse shape chosen here does not restrict the generality of the conclusions made in the current study since other types of the input RZ pulse shapes would merely produce different thresholds for the soliton creation and so the picture would change only quantitatively.
2. Problem statement We define a randomly modulated RZ-pulse as
Q ðtÞ ¼ Q 0 ðtÞ exp½igðtÞ;
ð1Þ
where Q 0 ðtÞ is the pre-defined deterministic pulse shape and random process gðtÞ represents phase fluctuations. We assume that no deterministic phase shift is present initially, i.e. there is no deterministic pre-chirping. The pulse shape is characterised by its FWHM width, T, and the total power (energy) inside the bit window T b :
E¼
Z
Tb
jQ ðtÞj2 dt:
ð2Þ
0
The intensity of the RPM can be characterised by the local R.M.S. value of the phase r2g ¼ hg2 ðtÞi. Another characteristic of the RPM is the correlation time of the fluctuations, s. In this paper we will be concerned with two types of phase disorder: I. Short-correlated zero mean Gaussian process with the autocorrelation function
hgðtÞgðt 0 Þi ¼ r2g exp½jt t 0 j=s:
ð3Þ
II. The generalized telegraph process [21]. Here one first generates a Poissonian flow of time intervals ½t i ; tiþ1 where the length of the intervals, Dt ¼ tiþ1 t i , is chosen from the exponential distribution, PðDtÞ ¼ s1 exp½Dt=s, where s is a correlation time. Within each interval, the process gðtÞ takes constant value and at the end of the interval it switches to a new value which is picked independently from a given distribution P0 ðgÞ. Here we shall assume a uniform, zero mean distribution, P 0 ðgÞ with the variance r2g . The generalised telegraph process has the same type of correlation function as the coloured Gaussian (3). The inclusion of random modulation changes the number of solitons as well as the soliton creation thresholds. We will be inter~ ested here in the average number of emerging soliton states, N, in dependence on the input pulse energy, E, and the intensity of the RPM, characterised by the variance rg . To determine the quan~ one has to calculate the individual probabilities of creating tity N exactly N solitons, PN . These probabilities will depend on pulse energy, E, width T, as well as the local intensity of the disorder (characterised by rg ) and its correlation time s. Regardless of the choice of parameters the probabilities must naturally satisfy the normalization condition: 1 X
PN ðE; T; rg ; sÞ ¼ 1:
ð4Þ
N¼0
3. The method According to the inverse scattering transform technique [22], any localised initial condition for an integrable nonlinear equation evolves into a combination of solitons and quasi-linear oscillating wave packets. The parameters of formed solitons, as well as their number, can be analysed via the non self-adjoint ZSSP which in the case of focusing NLSE (i.e. for the case of anomalous chromatic dispersion) takes the following form:
iow1 =ot þ Qw2
¼ fw1 ;
iow2 =ot Q w1
¼ fw2 ;
ð5Þ
where in the case considered the initial pulse shape Q ðtÞ taken in the form (1) acts as a random potential. In ZSSP (5) f is a (generally complex) eigenvalue, and wi ðtÞ stand for the components of a vector eigenfunction. Eq. (5) are written in dimensionless soliton units, where time is normalized to the FWHM time, t ! t=T, and the power is normalized by the characteristic power P ¼ ðcLD Þ1 jb2 j=ðcT 2 Þ, where c is the nonlinear coefficient in W1/km, LD ¼ T 2 =jb2 j is the dispersion length measured in km and b2 is the (negative) group velocity dispersion (GVD) parameter measured in ps2/km. Since eigenproblem (5) is not self-adjoint complex discrete eigenvalues may arise in addition to the real continuous spectrum. Each of these discrete eigenvalues corresponds to an individual soliton. The problem of determining the average num~ thus reduces to determining the average number of solitons, N, ber of discrete eigenvalues of (5). Similarly the probabilities P N can be restored from the statistical analysis of the occurrences of N-discrete eigenvalues, that is from a simple Monte-Carlo simulation. In the case of real, unmodulated rectangular initial condition Q 0 ðtÞ the solution of (5) can be readily obtained [20,22]. The discrete eigenvalues are purely imaginary (which corresponds to solitons moving with group velocity), fn ¼ iAn =2, and the real amplitudes An of the emerging solitons are given by the solutions of the following transcendental equation (in the realworld units):
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tanðmn Þ ¼ mn = D2 m2n ;
ð6Þ
S.A. Derevyanko, J.E. Prilepsky / Optics Communications 281 (2008) 5439–5443
where
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 mn ¼ pffiffiffiffiffi E=T A2n =4 > 0; P E cET 2 D ¼ ¼ ; TP jb2 j and the total energy, E, is defined by Eq. (2). The threshold condition for the appearance of Nth soliton (initially with the amplitude An ¼ 0) is then simply
D ¼ pðN 1=2Þ;
ð7Þ
and the number of generated solitons for given D is
1 D N ¼ int þ ; 2 p
ð8Þ
where int½ means the integer part of an expression. For a fixed energy E the width threshold of the creation of Nth soliton reads as
T N ðEÞ ¼
jb2 jp2 ðN 1=2Þ2 ; cE
ð9Þ
and a similar expression holds for the energy threshold having a fixed pulsewidth
EN ðTÞ ¼
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P1
jb2 jp2 ðN 1=2Þ2 : cT
ð10Þ
Note that criterion (8) is a special case of a general criteria valid for real positive pulses Q 0 , see Ref. [23], provided that the real input profile does not bring about the appearance of the ZSSP eigenvalues with nonzero real parts (all emerging solitons have velocities different from the group velocity of the envelope). The value of N can then be found using the formula
Z 1 1 1 N ¼ int þ pffiffiffiffiffi Q 0 ðtÞ dt : 2 pT P 1
ð11Þ
~ ¼ N N¼0 NP N versus the input pulse parameters for each statistical model from Section 2. In the simulation we assumed the following values for the parameters of the nonlinear channel: T ¼ T b ¼ 20 ps, c ¼ 3 W1 km1 and jb2 j ¼ 20 ps2 =km. These values of parameters yield the dispersion length LD ¼ 20 km and the normalization power P ¼ 16 mW. The number of Monte-Carlo realisations used for averaging was 1000. The pulse energy varied from zero to the value corresponding to five-soliton regime of the unmodulated rectangular shape. Thus the number of solitons for the unmodulated pulse changed from 0 to 5. We choose to operate in the diapason where the phase gðtÞ satisfies the condition rg K p. In such a way we ensure that the bulk of the local probability density function (PDF) of the phase lies within the interval ½p; p and apart from very rare fluctuations most values of phase are sampled within this interval. This looks physically reasonable and occurs in most practical situations. In ~ as a Fig. 1 we show the average number of emerging solitons N function of rg and E for two different values of correlation time. ~ are no longer integer Obviously because of averaging the values N numbers. From this figure one can readily see that the presence of disorder significantly decreases the number of solitons imbedded into the incoming signal (and thus decreases the energy content supplied in the coherent constituent of the signal, in agreement with the results of Ref. [12]) and the increase of phase fluctuations can eventually lead to complete disintegration of the pulse for a sufficiently high strength of disorder. The transition regions in Fig. 1 where the number of soliton states essentially differs from integer values are narrow for small values of the correlation time but tend to blur as s increases. In Fig. 2 we present similar results for the RPM-model of uniformly distributed exponentially correlated telegraph process. One readily observes that these results follow the same general trend as in the Gaussian model. Again, the presence of disorder diminishes the number of solitons and the transition regions become increasingly blurred as the correlation time increases.
This formula is generally valid for any single-lobe input profile [24], which is just the case in our study. The more complicated real input shapes may produce the ZSSP eigenvalues with nonzero real parts, and in this case the criterion (11) does not hold. The proof of this fact and related details can be found in Ref. [24]. However, again, the usage of the real RZ single-lobe shape for the deterministic part of the potential, Eq. (1), does not diminish the generality of the results and conclusions made in our current paper. In the case of more complicated shapes one merely would have to resort to direct numerical simulations rather than Eqs. (8) and (11). Let us now turn to the case of randomly modulated rectangular pulse. Generally, the solution of the ZSSP for the RPM-pulse cannot be found analytically so we use direct numerical simulations for the number of emerging solitons in the channel. The algorithm is as follows. At first stage one generates a random realization of the potential with the prescribed RPM statistics. Then scattering problem (5) is solved: the goal for each run is to find the total number of discrete complex eigenvalues f of ZSSP, Eq. (5), for a single generated realization of random potential. Performing a sufficiently large number of such Monte-Carlo runs we take the algebraic average of the number of emerging soliton states. The procedure is then repeated for different values of energy E and rg while keeping the pulse width T and the correlation time s fixed.
4. Results and discussion As mentioned in the previous section the goal of the simulations is to determine the average number of emerging solitons
~ versus the RPM intensity (variance) rg and Fig. 1. The average number of solitons N pulse energy E for the Gaussian RPM with exponential correlations. The top panel (a) corresponds to correlation time s ¼ 0:05 ps while the bottom (b) is for s ¼ 0:5 ps. The other input parameters is given in the main text.
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S.A. Derevyanko, J.E. Prilepsky / Optics Communications 281 (2008) 5439–5443
Fig. 2. Same as in Fig. 1 but with the generalized telegraph process taken as a disorder model for the RPM. The top panel (a) corresponds to correlation time s ¼ 0:05 ps while the bottom (b) is for s ¼ 0:5 ps.
Let us provide a simple qualitative illustration of how RPM affects RZ pulse-to-soliton conversion. Let us take the parameters of the optical fibre as above and consider the evolution of an optical pulse with and without RPM. We choose the energy of the pulse E ¼ 22 pJ which according to criterion (11) yields a 3-soliton breather in the unmodulated case. We then introduce RPM modelled by a telegraph process with the parameters rg ¼ 2:0 and s ¼ 0:2 ps. The pulse evolution for both inputs is plotted in Fig. 3. One can immediately observe how the results obtained by analyzing ZSSP (5) are mapped into real-world evolution of a given pulse. In the unmodulated case the initial rectangular pulse quickly evolves into a 3-soliton bound state (as predicted by IST) while the randomly modulated pulse of the same energy collapses into dispersive radiation which is in accordance with our prediction. Note that in both scenarios the asymptotical form of the solution is achieved rather quickly, on scales less than one dispersion length LD . As seen from Figs. 1 and 2 the increase of the correlation time affects the average number of solitons. In order to quantify this effect in Figs. 4,5 we plot the average number of soliton states vs correlation time for both models (for fixed values of pulse width T ¼ 20 ps and energy E ¼ 80 pJ). These graphs demonstrate an interesting trend. First of all as the correlation time increases the pulse restores part of its coherence and for fully coherent phase of the pulse the same threshold criterion (8) applies whatever the strength of disorder, so the number of solitons approaches that of the unmodulated pulse (which in our case is 5). This explains the eventual growth of the soliton number with the increase of the correlation time. If we start from the regime where the average number of soliton is initially less than one, then, as one can see from the bottom curves in Figs. 3 and 4, the growth is monotonic. However if we start from the regimes where already a few solitons are present we can clearly observe a non-monotonic behavior. This demonstrates that even if the strength of phase disorder is not sufficient enough to inhibit the creation of solitons completely, the increasing correlation time can at first act in a similar fashion as an increased strength of a very
Fig. 3. The evolution of a rectangular pulse in the unmodulated case (a) and with the telegraph RPM (b).
~ versus the relative correlation time for Fig. 4. The average number of solitons N E ¼ 80 pJ and two different values of phase disorder (identified in the plot legend) for Gaussian colored disorder model of the RPM.
short-correlated disorder. Such a non-trivial result calls for further investigation which is beyond the scope of this paper.
S.A. Derevyanko, J.E. Prilepsky / Optics Communications 281 (2008) 5439–5443
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correlation time can be non-monotonic in the multi-soliton regime. For small values of phase correlation time for each model of RPM there exists an intensity threshold of disorder beyond which the probability of finding a soliton state is negligibly small. However the situation can be more involved in the case of large correlation times when the boundaries of the regions corresponding to different numbers of solitons become blurred or when considering telegraph process with the uniform RPM distribution. Acknowledgements The authors would like to thank Sergei Turitsyn for drawing our attention to the considered problem, and also Keith Blow and Leonid Pastur for valuable comments. This work was supported by the Royal Society. References ~ versus the relative correlation time for Fig. 5. The average number of solitons N E ¼ 80 pJ and two different values of phase disorder (identified in the plot legend) for the telegraph-type disorder model of the RPM.
5. Conclusions To sum up we have investigated the influence of the RPM on the soliton states in the pulse-to-soliton conversion in a NLSE channel. Since the NLSE is integrable, we can explicitly distinguish between the coherent and incoherent content of the incoming signal by means of the direct scattering problem (ZSSP), Eq. (5), associated with the NLSE. Since the non-soliton (radiative) part of the solution disperses away from the bit slot it is the soliton content which contributes most to the energy detected by a distant receiver. The effect of the RPM on the soliton content was studied within the framework of two models for the phase fluctuations: the Gaussian and telegraph-type processes. We have shown that in the presence of phase fluctuations the signal looses its coherent structure and the number of the emerging soliton states generally decreases. However the increase of the correlation time may produce a positive effect in the formation of soliton states. We have also demonstrated that the dependence of average soliton number on the
[1] J. Ania-Castañón, Opt. Express 12 (2004) 4372. [2] J.D. Ania-Castañón, T.J. Ellingham, R. Ibbotson, X. Chen, L. Zhang, S.K. Turitsyn, Phys. Rev. Lett. 96 (2006) 023902. [3] B. Dany, P. Brindel, O. Leclerc, E. Desurvire, Electron. Lett. 35 (1999) 98. [4] D. Burak, W. Nasalski, Appl. Opt. 33 (1994) 6393. [5] J.C. Bronsky, Physica D 97 (1996) 376. [6] J.E. Prilepsky, S.A. Derevyanko, S.K. Turitsyn, J. Opt. Soc. Am. B 24 (2007) 1254. [7] E.A. Kuznetsov, A.V. Mikhailov, I.A. Shimokhin, Physica D 87 (1995) 201. [8] F. Abdullaev, S. Darmanyan, P. Khabibullaev, Optical Solitons, Springer-Verlag, 1993. [9] J.N. Elgin, Phys. Lett. A 110 (1985) 441. [10] S.K. Turitsyn, S.A. Derevyanko, Phys. Rev. A, in press. [11] S.A. Akhmanov, V.A. Vysloukh, A.S. Chirkin, Sov. Phys. Uspekhi 29 (1986) 642. [12] K.J. Blow, D. Wood, Opt. Commun. 58 (1986) 349. [13] F.G. Bass, Y.S. Kivshar, V.V. Konotop, Sov. Phys. – JETP 65 (1987) 245. [14] S.A. Derevyanko, S.K. Turitsyn, D.A. Yakushev, J. Opt. Soc. Am. B 22 (2005) 743. [15] A. Mecozzi, Opt. Lett. 29 (2004) 673. [16] D.S. Elliott, M.W. Hamilton, K. Arnett, S.J. Smith, Phys. Rev. A 32 (1985) 887. [17] H. Ritch, P. Zoller, J. Cooper, Phys. Rev. A 41 (1990) 2653. [18] J.H. Eberly, K. Wódkiewicz, B.W. Shore, Phys. Rev. A 30 (1984) 2381. [19] A.G. Kofman, Phys. Rev. E 51 (1995) 4011. [20] S.V. Manakov, Sov. Phys. – JETP 38 (1974) 693. [21] V.I. Klyatskin, Stochastic Equations through the Eye of the Physicist, Elsevier, 2005. [22] S.V. Manakov, S.P. Novikov, L.P. Pitaevskii, V.E. Zakharov, Theory of Solitons, Consultants Bureau (1984). [23] Y.S. Kivshar, J. Phys. A: Math. Gen. 22 (1989) 337. [24] M. Klaus, J.K. Shaw, Phys. Rev. E 65 (2002). Art. No. 036607.
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Soliton generation from randomly modulated return-to-zero pulses Stanislav A. Derevyanko a,*,1, Jaroslaw E. Prilepsky b a b
Photonics Research Group, Aston University, Aston Triangle, Birmingham B4 7ET, UK B.I. Verkin Institute for Low Temperature Physics and Engineering, NASU, 47 Lenin Avenue, Kharkov 61103, Ukraine
a r t i c l e
i n f o
Article history: Received 1 May 2008 Received in revised form 25 July 2008 Accepted 25 July 2008
Keywords: Random phase modulation Nonlinear Schrödinger equation Optical solitons
a b s t r a c t We consider return-to-zero (RZ) pulses with random phase modulation propagating in a nonlinear channel (modelled by the integrable nonlinear Schrödinger equation, NLSE). We suggest two different models for the phase fluctuations of the optical field: (i) Gaussian short-correlated fluctuations and (ii) generalized telegraph process. Using the rectangular-shaped pulse form we demonstrate that the presence of phase fluctuations of both types strongly influences the number of solitons generated in the channel. It is also shown that increasing the correlation time for the random phase fluctuations affects the coherent content of a pulse in a non-trivial way. The result obtained has potential consequences for all-optical processing and design of optical decision elements. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction Classical optical solitons have been one of the recurring topics in optical communications for more than 30 years now. The fact that they are described by a well-known integrable nonlinear Schrödinger equation (NLSE) have always made them attractive for both theoretical and numerical modelling. And although the attempts of applying classical NLSE solitons directly to the problem of data transmission have run into numerous difficulties there are still many areas where these solitons have found their niche. First of all, recent developments [1,2] demonstrated the real possibility of achieving quasi-lossless transmission over long distances with very small power variations. Such systems are well modelled by a classical NLSE which opens the possibility of observing and utilizing classical solitons without the need to refer to other related concepts (like e.g. dispersion-managed or dissipative solitons). Since the NLSE is integrable a lot of properties of the emerging signal can be obtained by analysing the input pulse form (possibly corrupted by noise). Second important aspect of classical solitons which is pertinent to the current study is the use of optical solitons for all-optical return-to-zero (RZ) pulse regeneration [3]. In particular it was demonstrated that the efficiency of carrier signal control can be enhanced by periodic conversion of an RZ signal into a classical soliton. Therefore one has to consider the problem of RZ pulse-to-soliton conversion, i.e. given the initial RZ pulse shape determine the properties of the emerging (multi)-soliton pulse
* Corresponding author. E-mail addresses: [email protected] (S.A. Derevyanko), astrayed@ yandex.ru (J.E. Prilepsky). 1 On leave from the Institute for Radiophysics and Electronics, NASU, 12 Acad. Proscura Street, Kharkov 61085, Ukraine. 0030-4018/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2008.07.079
after its propagation through a quasi-lossless span described by the classical NLSE. For different deterministic pulse shapes the problem has been studied extensively (see Refs. [4–6] and references therein). However any optical regenerator must deal not with pure deterministic waveform but rather with a pulse affected by noise (that may come from amplifier spontaneous emission, channel cross-talk and other impairments). Also any RZ pulse generator introduces random chirp which is sometimes difficult to control. Therefore when studying real-world communication systems, rather than studying deterministic pulse-to-soliton conversion problem one has to turn to its random counterpart. The aim of our study is to elucidate how the output of a integrable NLSE channel is affected by random phase modulation of the input pulse. Since asymptotically the solution of the NLSE is determined largely by the soliton part and the radiation disperses away (see e.g. [7]) we will gauge the performance of such a channel by the average number of emerging solitons. This number is highly sensitive to phase disorder as well as the correlations present in the phase of the input signal. The propagation of partially coherent pulses in the Kerr-type media has long been a subject of investigation in different contexts [8–10] including random phase modulation of temporal solitons [11,12] and nonlinear Fraunhofer diffraction of random fields [13]. Phase fluctuations are of particular importance inasmuch as the phase of a field is hardly controllable whereas the pulse energy can usually be adjusted to a desired level. As a rule, phase modulation of the incoming signal is not explicitly known and may be considered random. But such a modulation can drastically influence the coherent (soliton) component of the input pulse even in the case of deterministically chirped pulses. It is known that the energy contained in the soliton part of the pulse decreases and soliton states can eventually disappear completely in the case of the unbounded phase (e.g. for linear
5440
S.A. Derevyanko, J.E. Prilepsky / Optics Communications 281 (2008) 5439–5443
chirp) [4,6], while the opposite tendency was observed when the phase was rapidly decaying [5]. Hence even deterministic phase modulation affects the formation of solitons in a non-trivial way. The situation when the phase fluctuates randomly seems to be even more involved. In this case, as shown in [12] for the model with random frequency and linear chirp, the energy threshold for the formation of a soliton state increases and, once overcome, only some portion of the input energy is transformed into a soliton. At the same time the questions of whether we can completely destroy the coherency of the signal by the RPM and how the coherent part of the pulse ‘‘feels” the type of statistics and correlations of the phase fluctuations have so far remained unclear. The goal of our paper is to clarify these points. Here we will consider two different types of phase disorder. Firstly, weak Gaussian fluctuations of the pulse amplitude in the strongly nonlinear regime produce Gaussian statistics of the accumulated phase shift due to the self-phase modulation (GordonMollenauer effect). As an example of such a situation we may mention random phase jitter caused by the action of the additive white Gaussian noise, arising during the soliton propagation [14,15]. The Gaussian-Markovian frequency fluctuations were also considered in Refs. [16,17]. Discontinuous random jump processes, such as dichotomic and generalised telegraph process, were suggested to model the phase fluctuating field of a laser in Refs. [18,19]. In this paper we deal with both Gaussian-type and telegraph-type fluctuations of the phase and study how these fluctuations affect the coherent part of the input pulse entering the NLSE channel. As for the shape of the signal we opt for a simple RZ rectangular profile. The reason for that is twofold. Firstly, the unmodulated rectangular shape admits the complete analytical solution of the direct scattering problem (the so-called Zakharov–Shabat eigenproblem, ZSSP) associated with the NLSE [20]. Thus for the unmodulated case we know the energy thresholds for soliton generation explicitly. Secondly, such a profile always generates some amount of linear radiation in addition to the soliton part, so that the situation is more general than the case of pure multi-soliton (sech-type) pulses where radiation can be absent completely. At the same time we argue that the type of the RZ pulse shape chosen here does not restrict the generality of the conclusions made in the current study since other types of the input RZ pulse shapes would merely produce different thresholds for the soliton creation and so the picture would change only quantitatively.
2. Problem statement We define a randomly modulated RZ-pulse as
Q ðtÞ ¼ Q 0 ðtÞ exp½igðtÞ;
ð1Þ
where Q 0 ðtÞ is the pre-defined deterministic pulse shape and random process gðtÞ represents phase fluctuations. We assume that no deterministic phase shift is present initially, i.e. there is no deterministic pre-chirping. The pulse shape is characterised by its FWHM width, T, and the total power (energy) inside the bit window T b :
E¼
Z
Tb
jQ ðtÞj2 dt:
ð2Þ
0
The intensity of the RPM can be characterised by the local R.M.S. value of the phase r2g ¼ hg2 ðtÞi. Another characteristic of the RPM is the correlation time of the fluctuations, s. In this paper we will be concerned with two types of phase disorder: I. Short-correlated zero mean Gaussian process with the autocorrelation function
hgðtÞgðt 0 Þi ¼ r2g exp½jt t 0 j=s:
ð3Þ
II. The generalized telegraph process [21]. Here one first generates a Poissonian flow of time intervals ½t i ; tiþ1 where the length of the intervals, Dt ¼ tiþ1 t i , is chosen from the exponential distribution, PðDtÞ ¼ s1 exp½Dt=s, where s is a correlation time. Within each interval, the process gðtÞ takes constant value and at the end of the interval it switches to a new value which is picked independently from a given distribution P0 ðgÞ. Here we shall assume a uniform, zero mean distribution, P 0 ðgÞ with the variance r2g . The generalised telegraph process has the same type of correlation function as the coloured Gaussian (3). The inclusion of random modulation changes the number of solitons as well as the soliton creation thresholds. We will be inter~ ested here in the average number of emerging soliton states, N, in dependence on the input pulse energy, E, and the intensity of the RPM, characterised by the variance rg . To determine the quan~ one has to calculate the individual probabilities of creating tity N exactly N solitons, PN . These probabilities will depend on pulse energy, E, width T, as well as the local intensity of the disorder (characterised by rg ) and its correlation time s. Regardless of the choice of parameters the probabilities must naturally satisfy the normalization condition: 1 X
PN ðE; T; rg ; sÞ ¼ 1:
ð4Þ
N¼0
3. The method According to the inverse scattering transform technique [22], any localised initial condition for an integrable nonlinear equation evolves into a combination of solitons and quasi-linear oscillating wave packets. The parameters of formed solitons, as well as their number, can be analysed via the non self-adjoint ZSSP which in the case of focusing NLSE (i.e. for the case of anomalous chromatic dispersion) takes the following form:
iow1 =ot þ Qw2
¼ fw1 ;
iow2 =ot Q w1
¼ fw2 ;
ð5Þ
where in the case considered the initial pulse shape Q ðtÞ taken in the form (1) acts as a random potential. In ZSSP (5) f is a (generally complex) eigenvalue, and wi ðtÞ stand for the components of a vector eigenfunction. Eq. (5) are written in dimensionless soliton units, where time is normalized to the FWHM time, t ! t=T, and the power is normalized by the characteristic power P ¼ ðcLD Þ1 jb2 j=ðcT 2 Þ, where c is the nonlinear coefficient in W1/km, LD ¼ T 2 =jb2 j is the dispersion length measured in km and b2 is the (negative) group velocity dispersion (GVD) parameter measured in ps2/km. Since eigenproblem (5) is not self-adjoint complex discrete eigenvalues may arise in addition to the real continuous spectrum. Each of these discrete eigenvalues corresponds to an individual soliton. The problem of determining the average num~ thus reduces to determining the average number of solitons, N, ber of discrete eigenvalues of (5). Similarly the probabilities P N can be restored from the statistical analysis of the occurrences of N-discrete eigenvalues, that is from a simple Monte-Carlo simulation. In the case of real, unmodulated rectangular initial condition Q 0 ðtÞ the solution of (5) can be readily obtained [20,22]. The discrete eigenvalues are purely imaginary (which corresponds to solitons moving with group velocity), fn ¼ iAn =2, and the real amplitudes An of the emerging solitons are given by the solutions of the following transcendental equation (in the realworld units):
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tanðmn Þ ¼ mn = D2 m2n ;
ð6Þ
S.A. Derevyanko, J.E. Prilepsky / Optics Communications 281 (2008) 5439–5443
where
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 mn ¼ pffiffiffiffiffi E=T A2n =4 > 0; P E cET 2 D ¼ ¼ ; TP jb2 j and the total energy, E, is defined by Eq. (2). The threshold condition for the appearance of Nth soliton (initially with the amplitude An ¼ 0) is then simply
D ¼ pðN 1=2Þ;
ð7Þ
and the number of generated solitons for given D is
1 D N ¼ int þ ; 2 p
ð8Þ
where int½ means the integer part of an expression. For a fixed energy E the width threshold of the creation of Nth soliton reads as
T N ðEÞ ¼
jb2 jp2 ðN 1=2Þ2 ; cE
ð9Þ
and a similar expression holds for the energy threshold having a fixed pulsewidth
EN ðTÞ ¼
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P1
jb2 jp2 ðN 1=2Þ2 : cT
ð10Þ
Note that criterion (8) is a special case of a general criteria valid for real positive pulses Q 0 , see Ref. [23], provided that the real input profile does not bring about the appearance of the ZSSP eigenvalues with nonzero real parts (all emerging solitons have velocities different from the group velocity of the envelope). The value of N can then be found using the formula
Z 1 1 1 N ¼ int þ pffiffiffiffiffi Q 0 ðtÞ dt : 2 pT P 1
ð11Þ
~ ¼ N N¼0 NP N versus the input pulse parameters for each statistical model from Section 2. In the simulation we assumed the following values for the parameters of the nonlinear channel: T ¼ T b ¼ 20 ps, c ¼ 3 W1 km1 and jb2 j ¼ 20 ps2 =km. These values of parameters yield the dispersion length LD ¼ 20 km and the normalization power P ¼ 16 mW. The number of Monte-Carlo realisations used for averaging was 1000. The pulse energy varied from zero to the value corresponding to five-soliton regime of the unmodulated rectangular shape. Thus the number of solitons for the unmodulated pulse changed from 0 to 5. We choose to operate in the diapason where the phase gðtÞ satisfies the condition rg K p. In such a way we ensure that the bulk of the local probability density function (PDF) of the phase lies within the interval ½p; p and apart from very rare fluctuations most values of phase are sampled within this interval. This looks physically reasonable and occurs in most practical situations. In ~ as a Fig. 1 we show the average number of emerging solitons N function of rg and E for two different values of correlation time. ~ are no longer integer Obviously because of averaging the values N numbers. From this figure one can readily see that the presence of disorder significantly decreases the number of solitons imbedded into the incoming signal (and thus decreases the energy content supplied in the coherent constituent of the signal, in agreement with the results of Ref. [12]) and the increase of phase fluctuations can eventually lead to complete disintegration of the pulse for a sufficiently high strength of disorder. The transition regions in Fig. 1 where the number of soliton states essentially differs from integer values are narrow for small values of the correlation time but tend to blur as s increases. In Fig. 2 we present similar results for the RPM-model of uniformly distributed exponentially correlated telegraph process. One readily observes that these results follow the same general trend as in the Gaussian model. Again, the presence of disorder diminishes the number of solitons and the transition regions become increasingly blurred as the correlation time increases.
This formula is generally valid for any single-lobe input profile [24], which is just the case in our study. The more complicated real input shapes may produce the ZSSP eigenvalues with nonzero real parts, and in this case the criterion (11) does not hold. The proof of this fact and related details can be found in Ref. [24]. However, again, the usage of the real RZ single-lobe shape for the deterministic part of the potential, Eq. (1), does not diminish the generality of the results and conclusions made in our current paper. In the case of more complicated shapes one merely would have to resort to direct numerical simulations rather than Eqs. (8) and (11). Let us now turn to the case of randomly modulated rectangular pulse. Generally, the solution of the ZSSP for the RPM-pulse cannot be found analytically so we use direct numerical simulations for the number of emerging solitons in the channel. The algorithm is as follows. At first stage one generates a random realization of the potential with the prescribed RPM statistics. Then scattering problem (5) is solved: the goal for each run is to find the total number of discrete complex eigenvalues f of ZSSP, Eq. (5), for a single generated realization of random potential. Performing a sufficiently large number of such Monte-Carlo runs we take the algebraic average of the number of emerging soliton states. The procedure is then repeated for different values of energy E and rg while keeping the pulse width T and the correlation time s fixed.
4. Results and discussion As mentioned in the previous section the goal of the simulations is to determine the average number of emerging solitons
~ versus the RPM intensity (variance) rg and Fig. 1. The average number of solitons N pulse energy E for the Gaussian RPM with exponential correlations. The top panel (a) corresponds to correlation time s ¼ 0:05 ps while the bottom (b) is for s ¼ 0:5 ps. The other input parameters is given in the main text.
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S.A. Derevyanko, J.E. Prilepsky / Optics Communications 281 (2008) 5439–5443
Fig. 2. Same as in Fig. 1 but with the generalized telegraph process taken as a disorder model for the RPM. The top panel (a) corresponds to correlation time s ¼ 0:05 ps while the bottom (b) is for s ¼ 0:5 ps.
Let us provide a simple qualitative illustration of how RPM affects RZ pulse-to-soliton conversion. Let us take the parameters of the optical fibre as above and consider the evolution of an optical pulse with and without RPM. We choose the energy of the pulse E ¼ 22 pJ which according to criterion (11) yields a 3-soliton breather in the unmodulated case. We then introduce RPM modelled by a telegraph process with the parameters rg ¼ 2:0 and s ¼ 0:2 ps. The pulse evolution for both inputs is plotted in Fig. 3. One can immediately observe how the results obtained by analyzing ZSSP (5) are mapped into real-world evolution of a given pulse. In the unmodulated case the initial rectangular pulse quickly evolves into a 3-soliton bound state (as predicted by IST) while the randomly modulated pulse of the same energy collapses into dispersive radiation which is in accordance with our prediction. Note that in both scenarios the asymptotical form of the solution is achieved rather quickly, on scales less than one dispersion length LD . As seen from Figs. 1 and 2 the increase of the correlation time affects the average number of solitons. In order to quantify this effect in Figs. 4,5 we plot the average number of soliton states vs correlation time for both models (for fixed values of pulse width T ¼ 20 ps and energy E ¼ 80 pJ). These graphs demonstrate an interesting trend. First of all as the correlation time increases the pulse restores part of its coherence and for fully coherent phase of the pulse the same threshold criterion (8) applies whatever the strength of disorder, so the number of solitons approaches that of the unmodulated pulse (which in our case is 5). This explains the eventual growth of the soliton number with the increase of the correlation time. If we start from the regime where the average number of soliton is initially less than one, then, as one can see from the bottom curves in Figs. 3 and 4, the growth is monotonic. However if we start from the regimes where already a few solitons are present we can clearly observe a non-monotonic behavior. This demonstrates that even if the strength of phase disorder is not sufficient enough to inhibit the creation of solitons completely, the increasing correlation time can at first act in a similar fashion as an increased strength of a very
Fig. 3. The evolution of a rectangular pulse in the unmodulated case (a) and with the telegraph RPM (b).
~ versus the relative correlation time for Fig. 4. The average number of solitons N E ¼ 80 pJ and two different values of phase disorder (identified in the plot legend) for Gaussian colored disorder model of the RPM.
short-correlated disorder. Such a non-trivial result calls for further investigation which is beyond the scope of this paper.
S.A. Derevyanko, J.E. Prilepsky / Optics Communications 281 (2008) 5439–5443
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correlation time can be non-monotonic in the multi-soliton regime. For small values of phase correlation time for each model of RPM there exists an intensity threshold of disorder beyond which the probability of finding a soliton state is negligibly small. However the situation can be more involved in the case of large correlation times when the boundaries of the regions corresponding to different numbers of solitons become blurred or when considering telegraph process with the uniform RPM distribution. Acknowledgements The authors would like to thank Sergei Turitsyn for drawing our attention to the considered problem, and also Keith Blow and Leonid Pastur for valuable comments. This work was supported by the Royal Society. References ~ versus the relative correlation time for Fig. 5. The average number of solitons N E ¼ 80 pJ and two different values of phase disorder (identified in the plot legend) for the telegraph-type disorder model of the RPM.
5. Conclusions To sum up we have investigated the influence of the RPM on the soliton states in the pulse-to-soliton conversion in a NLSE channel. Since the NLSE is integrable, we can explicitly distinguish between the coherent and incoherent content of the incoming signal by means of the direct scattering problem (ZSSP), Eq. (5), associated with the NLSE. Since the non-soliton (radiative) part of the solution disperses away from the bit slot it is the soliton content which contributes most to the energy detected by a distant receiver. The effect of the RPM on the soliton content was studied within the framework of two models for the phase fluctuations: the Gaussian and telegraph-type processes. We have shown that in the presence of phase fluctuations the signal looses its coherent structure and the number of the emerging soliton states generally decreases. However the increase of the correlation time may produce a positive effect in the formation of soliton states. We have also demonstrated that the dependence of average soliton number on the
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