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Propagation of chirped solitary pulses in optical transmission lines: perturbed variational approach
Optics Communications 213 (2002) 293–299 www.elsevier.com/locate/optcom
Propagation of chirped solitary pulses in optical transmission lines: perturbed variational approach Manos Manousakis, Sotiris Droulias, Panagiotis Papagiannis, Kyriakos Hizanidis * Department of Electrical and Computer Engineering, National Technical University of Athens, 9 Iroon Polytechniou, Zografou, 157 73 Athens, Greece Received 15 May 2002; received in revised form 16 September 2002; accepted 14 October 2002
Abstract The evolution of a dressed solitary pulse subjected to filtered amplification is examined. The model equation used is complex cubic Ginzburg–Landau equation (CCGLE). A system of ordinary differential equations is derived on the basis of an extended-perturbed variational method. These equations are solved numerically for a set of initial conditions in the vicinity of the fixed point (corresponding to the exact solution of CCGLE) of the dissipative system these equations model. The stability and degree of stationarity (in propagation distance) of pulses with initial (launching) parameters falling in the vicinity of the fixed point are examined in the context of this method. A fully numerical simulation of the CCGLE finally tests the results of this investigation. Detailed comparisons reveal a wide class of initial pulse profiles, which are characterized by adequate stationarity and long propagation, distances before they disintegrate. In the anomalous dispersion regime there is an adequate quantitative agreement while in the normal dispersion regime the predictability of the method is impressive. Limitations of the proposed method are also discussed. Ó 2002 Elsevier Science B.V. All rights reserved. Keywords: EDFA; Ginzburg–Landau equation; Dissipative system; Variational approach; Solitary pulse propagation
1. Introduction The modeling of nonlinear dispersive pulses propagation in optical transmission lines equipped with EDFAs has been a subject of intense research investigation. A number of modifications of the nonlinear Schr€ odinger equation (NLSE) have been
proposed in order to model the losses and amplification of the pulse during the propagation [1–5]. A very interesting relevant model (both mathematically and technically) is the complex cubic Ginzburg–Landau equation (CCGLE) [6] oU 1 oU D G2 o2 U þi þ þi i oz vg ot 2 ot2 2 þ cjU j2 U iG0 ¼ 0;
ð1Þ
*
Corresponding author. Tel.: +30-10-772-3685; fax: +30-10772-3513. E-mail address: [email protected] (K. Hizanidis).
where vg is the group velocity (in km/ps) of the pffiffiffiffiffi pulse envelope U (in W ), D is the group velocity
0030-4018/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 ( 0 2 ) 0 2 0 8 6 - 2
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M. Manousakis et al. / Optics Communications 213 (2002) 293–299
dispersion (in ps2 =km), G2 is the spectral filtering losses (in ps2 /km), c is the nonlinearity coefficient (in W1 km1 ) and G0 is the excess gain of the EDFA (in km1 ). By making the frame of reference transformation T ¼ t z=vg , i.e., moving along with the pulse envelope U, CGLE becomes oU D G2 o2 U þ þi i þ cjU j2 U iG0 ¼ 0: oz 2 2 oT 2 ð2Þ By applying the following transformations on the CCGLE, we obtain the normalization of the equation: Unnormalized dimensions Time, T Distance, z Amplitude, U
Normalization quantity pffiffiffiffiffiffiffiffiffi T0 ¼ G2 =2G0 Z0 ¼ 1=G 0 pffiffiffiffiffiffiffiffiffiffi U0 ¼ G0 =c
Normalized dimensions s ¼ T =T0 f ¼ z=Z0 u ¼ U =U0
The quantities G0 ; G2 are the reference gain and the reference spectral filtering. Hence CCGLE becomes d 2 iu1 þ ð3Þ b2 i uss þ juj u igu ¼ 0: 2 The normalized terms are expressed via the respective unnormalized, according to the equations d¼
2D ; G2
b2 ¼
G2 ; G2
g¼
G0 : G0
ð4Þ
As it can be easily observed, when the reference gain and spectral filtering are chosen as G0 ¼ G0 and G2 ¼ G2 , then the normalized CCGLE takes the following, frequently met in the scientific literature, form: d 2 iuf þ ð5Þ i uss þ juj u iu ¼ 0: 2 This equation is well known as it has been used for applications in a lot of different regions of physics. In nonlinear optics, specifically, it is useful in modeling soliton amplifying transmission lines, spatial optical solitons and passive mode-locked lasers.
Pereira–Stenflo soliton pulses can be obtained from passively mode-locked lasers [7]. This kind of laser emits chirped soliton pulses with amplitude 1þil given by us ðsÞ ¼ As ½sechðs=aÞ . The pulse parameters As , a and l are related to the soliton width Ts , the peak power Ps and frequency chirp dx as Ts ¼ aT2 , Ps ¼ jb2 jNs2 =ðcT22 Þ and dx ¼ l tanh ðs=aÞ=Ts where T2 is the dipole relaxation time (0.1 ps for fiber amplifiers) and c ¼ n2 x0 =ðcAeff Þ is the nonlinearity coefficient. In this work it serves as an averaging model of a soliton transmission line equipped with amplifiers. The advantage of this model is that it includes the spectral filtering of the amplifier, which is important because it suppresses the Gordon–Haus jitter of the soliton central frequency. The model approximates the gain as a parabolic function of frequency and this is fair as long as the pulseÕs spectral width remains small in comparison with the gain bandwidth. There is a well-known solution of the equation which is called Pereira–Stenflo type pulse uðf; sÞ ¼ Afsech½s=a g1þil expðikfÞ:
ð6Þ
This special solution was found from Bekki– Nozaki [6]. Since the equation is not integrable the method they used was a phase-plane based one and not the inverse scattering method. This chirped pulse propagates undistorted (for sufficiently large propagation distances) only as long as the chirp is given by the following relation: 3 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 l¼ dþ 32 þ 9d : ð7Þ 4 4 It is well known that Eq. (5) possesses an unstable background. However, if the noise level (numerical or externally introduced) is low, the Pereira– Stenflo solution can propagate long distances before disintegration. Therefore, from the practical point of view, this solution (as an initial condition at the launching point of an optical link) may present some advantages as pre-chirped pulse. However, the Pereira–Stenflo solution is a special solution bounded by the, obviously impractical, restriction, Eq. (7). Therefore, it becomes apparent that it is far more interesting the investigation of a much wider class of initial conditions which may
M. Manousakis et al. / Optics Communications 213 (2002) 293–299
share similar characteristics and advantages with this exact solution. The analytical tool for this investigation is the variational method. It has been used in all kinds of conservative problems in nonlinear optics. The valuable qualitative results of the method have made it very popular among the research community. The analytical (variational) treatment can be used in the problems of mechanics when the forces taken into account are not frictional, in the sense that they can be derived from a work function. The conservation of the energy is not then demanded. However, the existence of a generating work function is crucial for the validity of variational minimizing procedures. One problem that has to be encountered here is the fact that in most cases the perturbations are large. Therefore, adequate justification must be given for the applicability of a perturbation method: In the case at hand, the really large perturbations (namely, the amplifier gain and the optical power losses) have been absorbed in the process of the averaging method used to derive the CCGLE itself. The dissipative terms present in CCGLE are smaller. As far as the overall scientific literature is concerned, a lot of researchers have been using the variational method in dissipative systems, each, of course, from different perspective. The potential of this method lies on its capability of providing useful qualitative results. In [8] the variational method is based on the concept of the system and anti-system approach, in [9] on the fractional derivative formalism, while in [10], the Lagrangian of the system is split into a sum of a conservative part and a non-conservative one. In the present work we use the latter formalistic approach to find the equations that describe qualitatively and quantitatively the dynamics of a Pereira–Stenflo type pulse as it evolves under the model of CCGLE. The paper is organized as follows: In Section 2 the ordinary differential equations governing the parameters of the pulse are derived and the normalizations that had to be made in order to compare the analytical with the numerical results are discussed. In Section 3, comparisons between the direct numerical solution of the CCGLE and the analytical results of the variational method are made. Finally in Section 4 the main conclusions are summarized.
295
2. Variational analysis According to the variational method [11] the time averaged Lagrangian of the system has to be calculated Z þ1 hLi ¼ L ds: ð8Þ 1
The Lagrangian is decomposed in two parts: the conservative one (it comes from the terms of the equation that are energy conservative) and the nonconservative one Lðu; u ; 1; s; u1 ; us ; u1 ; us Þ ¼ Lcon þ Lnonc ;
ð9Þ
where i d 1 4 2 Lcon ¼ ðuu1 u u1 Þ þ jus j juj : 2 2 2
ð10Þ
The CCGLE, Eq. (5), can now be written as d ð11Þ iu1 þ uss þ juj2 u ¼ Pu ; 2 where Pu ¼ ib2 uss þ igu:
ð12Þ
The variation of the non-conservative part Lncon can be written as dLnonc ¼ Pu du þ Pu du :
ð13Þ
The ansatz that will be used is a Pereira–Stenflo type pulse uð1; sÞ ¼ f ðp1 ð1Þ; p2 ð1Þ; p3 ð1Þ; p4 ð1ÞsÞ ¼ p1 ð1Þfsech½s=p2 ð1Þ g1þip3 ð1Þ expðip4 ðfÞÞ: ð14Þ The four parameters of interest for the pulse are A p1 the amplitude, a p2 the time width, l p3 the chirp and the phase, / p4 . Taking into account the fact that ðPu Þ ¼ Pu and Eq. (13) one can find that dLnonc ¼ 2ReðPu duÞ:
ð15Þ
The dynamical equations are derived from the following: Z Z dhLcon þ Lnonc i ou dpi df ¼ 0: ð16Þ du opi Thus, the general form of the dynamical equations of the pulse parameters can now be obtained
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d ohLcon i ohLcon i ¼ 2Re d1 oðpi Þ1 opi
Z Pu
ou : opi
ð17Þ
When the relations, Eqs. (8)–(16), are applied to the set of equations, Eq. (17), the following dynamical system comes out:
d dl2 4 2 þ A a þ ð2a 2 lnð2ÞÞal0 3a 3a 3 la0 2a/0 ¼ 0;
3. Results and discussion ð18Þ
2A2 dð1 þ l2 Þ 4lb2 ð1 þ l2 Þ 2lA0 2gl þ þ þ 3a2 3a2 3 A 0 0 ð19Þ þ ð3 2 lnð2ÞÞl 2/ ¼ 0; 2 4b 4 16 4 dl 2 þ lnð2Þb2 l2 b2 þ lnð2Þl2 b2 3 3 9 3 9 2 4a ðlnð2Þ 1ÞA0 þ 4ga2 4g lnð2Þa2 þ A þ ð3 þ 2 lnð2ÞÞaa0 ¼ 0; ð20Þ 2A2 b2 ð1 þ l2 Þ 2gA2 a þ 2Aaa0 þ A2 a0 ¼ 0; 3a
developed and the results of these two approaches are discussed in Section 3. The numerical method used for the development of this application is based on the broadly used symmetrical split-step Fourier method.
ð21Þ
The procedure of comparison between the analytical and the numerical results has been divided in two main categories: the evolution of the pulse in the anomalous dispersion regime and in the normal dispersion regime. The anomalous dispersion regime is examined first. Since we are interested in the evolution of pulses that are not initially (at f ¼ 0) the exact solutions of CCGLE, the following figures refer to the evolution of pulses possessing initially parameter values different than the ones of Pereira– Stenflo type. Each one of the following figures shows the evolution when only one of the three important parameters of the pulse (amplitude, width and chirp) deviates from the respective value of the exact solution.
where the primes denote differentiation with respect to the propagation distance. The last equation of the set, Eq. (21), gives the insight into the physical mechanism that governs the problem in hand: It can be rewritten as follows: 0
ðaA2 Þ ¼
2A2 b2 ð1 þ l2 Þ þ 2gA2 a: 3a
ð22Þ
One can easily observe that Eq. (22) describes explicitly the role of the excessive gain g as an exponential enhancing factor of the energy. The term which corresponds to the filter parameter is more complicated but it surely expresses the loss of power. Another way to check the method is to set all the derivatives equal to zero. Actually the algebraic system that comes out leads to Eq. (7) as it was expected. The complexity of the dynamical system, Eqs. (18)–(22), does not allow the analytical solution and, therefore, numerical simulations are being performed. An application for the direct partial differential equation numerical solution is also
Fig. 1. The evolution of the pulse when the initial amplitude deviates from the fixed point value, variationally (a, c) and full numerical simulation (b, d) in the anomalous dispersion regime. (a, b) +5% and (c, d) +20%.
M. Manousakis et al. / Optics Communications 213 (2002) 293–299
Specifically in Fig. 1 the evolution of a pulse with amplitude that is 5% (a, b) and 20% (c, d) larger than the Pereira–Stenflo pulse is shown. It is evident that, in the case of +5% deviation, the perturbed variational method gives a quite accurate picture of the pulse evolution. However, the oscillation of the amplitude is not predicted with absolute accuracy in the case of 20% deviation. In the following, Fig. 2, the time width is now the deviating parameter and the behaviour of the pulse resembles to that of Fig. 1: The case of +5% deviation validates again the variational procedure, while the +20% divergence shows the limits of the method. In Fig. 3 the main observation is that deviation in the initial chirp value is not so crucial for the evolution of the pulse for both mild and large deviations. As far as the normal dispersion regime is concerned we observe the following: Fig. 4 is the respective of Fig. 2 in the anomalous dispersion regime. Close examination of this figure makes two things clear: firstly, the accuracy of the analytical method is really better in this case and secondly the evolution of the pulse is smoother in a general point of view. These conclusions become clearer from Figs. 5 and 6. Apart from a small shift
297
Fig. 3. The evolution of the pulse when the initial chirp deviates from the fixed point value, variationally (a, c) and full numerical simulation (b, d) in the anomalous dispersion regime. (a, b) +5% and (c, d) +20%.
Fig. 4. The evolution of the pulse when the initial amplitude deviates from the fixed point value, variationally (a, c) and full numerical simulation (b, d) in the normal dispersion regime. (a, b) +5% and (c, d) +20%.
Fig. 2. The evolution of the pulse when the initial time width deviates from the fixed point value, variationally (a, c) and full numerical simulation (b, d) in the anomalous dispersion regime. (a, b) +5% and (c, d) +20%.
in the transmission coordinate, the qualitative aspects of the two approaches are exactly the same. The features of these figures enforce the belief that the variational approach of the pulse dynamics is a useful and handy analytical tool.
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M. Manousakis et al. / Optics Communications 213 (2002) 293–299
Fig. 7. Evolution of the pulse in the anomalous dispersion regime when the initial amplitude deviates from the fixed point value by +5%; (a) variationally, (b) full numerical simulation of the partial differential equation.
Fig. 5. The evolution of the pulse when the initial time width deviates from the fixed point value, variationally (a, c) and full numerical simulation (b, d) in the normal dispersion regime. (a, b) +5% and (c, d) +20%.
Fig. 6. The evolution of the pulse when the initial chirp deviates from the fixed point value, variationally (a, c) and full numerical simulation (b, d) in the normal dispersion regime. (a, b) +5% and (c, d) +20%.
The last figure shows the limits of the perturbed variational description of the problem in terms of the length of transmission. In Fig. 7 one can observe that in the anomalous dispersion regime and beyond the normalized transmission distance value
f ¼ 3 the pulse, whose initial amplitude deviates by +5%, starts to oscillate very rapidly and finally breaks into three parts. This evolution is catastrophic from the telecommunicationsÕ applications point of view. Similar behaviour characterizes the normal dispersion regime as well. The reason for the inaccuracy of the variational technique after a certain transmission distance is that the ansatz used does not possess any radiation component. Thus, the reduction of the problem through the averaging procedure hides certain information, which is crucial. The evolution of the radiation is enhanced by the nonlinearity and results in the destabilization and the ultimate destruction of the pulse. However, from the practical point of view, one should only be interested for the maximum propagation distance possible under the circumstances: According to the normalization of the coefficients in Eq. (5), it is noteworthy that the transmission distance has been normalized by the length Zo ¼ 1=Go , where Go is the excess gain of the amplifier. Hence, if Go is properly chosen, then a normalized distance of f ¼ 1 could correspond to cases characterized by long transmission distances. For example, if Go ¼ 0:1 km1 then Zo ¼ 10 km and f ¼ 1 is actually 10 km, while if Go ¼ 0:001 km1 then Zo ¼ 1000 km and f ¼ 1 is in fact 1000 km. Thus, the results demonstrate that the proposed method can actually be applied on a plethora of realistic cases of technological interest. The stationary solutions of complex Ginzburg–Landau equation with quintic nonlinear terms have been investigated thoroughly in [12]. In our case of interest the quintic terms are
M. Manousakis et al. / Optics Communications 213 (2002) 293–299
absent and the two-photon absorption term is set to zero. So our solutions fall in the case of solitons with fixed amplitude given by the expression: uðf; sÞ ¼ pðsÞ expðið/ðsÞ kfÞÞ, where pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /ðsÞ ¼ l lnðpðsÞÞ, B ¼ g=ðdl b2 þ l2 b2 Þ, C ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lð1 þ l2 Þðd 2 þ 4b22 Þ=½2ðdl 2b2 Þ , and l given by Eq. (7), while k ¼ gðdl2 4b2 l dÞ=2ðb2 l2 þ dl b2 Þ. From the filtering parameter two-photon absorption plane obtained in [12] it is evident that our solutions lay in the domain of stable solutions with unstable background. In our work the simulations are developed in the region g=d 1 and seem to be the main reason why the instability of the background develops slowly (for the fixed point). This behaviour corresponds to the condition g 1 obtained in [12] where jdj ¼ 1. So it is fair to claim that in a more general case the propagation distance of a PS solitary pulse is large as long as g=d 1. 4. Conclusions In this work the evolution of chirped pulses in an EDFA-supported transmission line has been studied. The chirped pulse ansatz has the functional form of the exact solution of the equation under examination, namely the CCGLE. The perturbed variational method was used in order to investigate the transmission properties of a wider class of initial pulse profiles this ansatz is amenable to. Four equations for the propagation dynamics were derived for the four main parameters of the pulse. These equations have been solved numerically for a set of initial conditions in the vicinities of the fixed point of the dissipative system these evolution equations express. The main focus of this work was on the investigation of the existence of domains in the parameter space of the initial pulse profiles which correspond to pulses characterized by as much stationary (in propagation distance f) as possible evolution dynamics and long propagation distances before they disintegrate. Especially in the case of normal dispersion
299
regime divergence even up to 20% from the fixed point meet these requirements, leading to wide classes of initial pulse profiles for practical applications in optical communications. Comparison with numerical simulations of the CCGLE itself has shown that the perturbed variational approach is an effective method for systems of this nature. However, more accurate results can be obtained if one takes into account the radiation of the pulse. This will possibly lead to a dynamical system with at least one additional degree of freedom. This is a subject of current and future investigation.
References [1] B. Malomed, Resonant transmission of a chirped soliton in a long optical fiber with periodic amplification, J. Opt. Soc. Am. B 13 (1996) 677. [2] F.Kh. Abdullaev, A.A. Abdumalikov, B.B. Baizakov, Stochastic instability of chirped optical solitons in media with periodic amplification, Quant. Electron. 27 (1997) 171. [3] B. Malomed, D. Parker, N. Smyth, Resonant shape oscillations and decay of a soliton in a periodically inhomogeneous nonlinear optical fiber, Phys. Rev. E 48 (1993) 1418. [4] T. Lakoba, J. Yang, D. Kaup, B. Malomed, Conditions for stationary propagations in the strong dispersion management limit, Opt. Commun. 149 (1998) 366. [5] B. Malomed, I. Uzunov, M. Golles, F. Lederer, Two stage perturbation theory for bandwidth limited amplification of optical solitons near the zero dispersion point, Phys. Rev. E 55 (1997) 3777. [6] N. Bekki, K. Nozaki, Formation of spatial patterns and holes in the generalised GLE, Phys. Lett. A 110 (1985). [7] G.P. Agrawal, Nonlinear Fiber Optics, second ed., Academic Press, New York, 1995. [8] D. Kaup, B. Malomed, Variational principle for nonlinear waves in dissipative systems, Physica D 87 (1995) 155. [9] F. Riewe, Non-conservative Lagrangian and Hamiltonian mechanics, Phys. Rev. E. 53 (1890). [10] S. Cerda, S. Cavalcanti, J. Hickmann, A variational approach of nonlinear dissipative pulse propagation, Eur. J. Phys. D 1313 (1998). [11] D. Anderson, Variational approach to nonlinear pulse propagation in optical fibers, Phys. Rev. A 27 (1983) 3135. [12] N.N. Akhmediev, A. Ankiewicz, Solitons: Nonlinear Pulses and Beams, Chapman & Hall, London, 1997.
Propagation of chirped solitary pulses in optical transmission lines: perturbed variational approach Manos Manousakis, Sotiris Droulias, Panagiotis Papagiannis, Kyriakos Hizanidis * Department of Electrical and Computer Engineering, National Technical University of Athens, 9 Iroon Polytechniou, Zografou, 157 73 Athens, Greece Received 15 May 2002; received in revised form 16 September 2002; accepted 14 October 2002
Abstract The evolution of a dressed solitary pulse subjected to filtered amplification is examined. The model equation used is complex cubic Ginzburg–Landau equation (CCGLE). A system of ordinary differential equations is derived on the basis of an extended-perturbed variational method. These equations are solved numerically for a set of initial conditions in the vicinity of the fixed point (corresponding to the exact solution of CCGLE) of the dissipative system these equations model. The stability and degree of stationarity (in propagation distance) of pulses with initial (launching) parameters falling in the vicinity of the fixed point are examined in the context of this method. A fully numerical simulation of the CCGLE finally tests the results of this investigation. Detailed comparisons reveal a wide class of initial pulse profiles, which are characterized by adequate stationarity and long propagation, distances before they disintegrate. In the anomalous dispersion regime there is an adequate quantitative agreement while in the normal dispersion regime the predictability of the method is impressive. Limitations of the proposed method are also discussed. Ó 2002 Elsevier Science B.V. All rights reserved. Keywords: EDFA; Ginzburg–Landau equation; Dissipative system; Variational approach; Solitary pulse propagation
1. Introduction The modeling of nonlinear dispersive pulses propagation in optical transmission lines equipped with EDFAs has been a subject of intense research investigation. A number of modifications of the nonlinear Schr€ odinger equation (NLSE) have been
proposed in order to model the losses and amplification of the pulse during the propagation [1–5]. A very interesting relevant model (both mathematically and technically) is the complex cubic Ginzburg–Landau equation (CCGLE) [6] oU 1 oU D G2 o2 U þi þ þi i oz vg ot 2 ot2 2 þ cjU j2 U iG0 ¼ 0;
ð1Þ
*
Corresponding author. Tel.: +30-10-772-3685; fax: +30-10772-3513. E-mail address: [email protected] (K. Hizanidis).
where vg is the group velocity (in km/ps) of the pffiffiffiffiffi pulse envelope U (in W ), D is the group velocity
0030-4018/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 ( 0 2 ) 0 2 0 8 6 - 2
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M. Manousakis et al. / Optics Communications 213 (2002) 293–299
dispersion (in ps2 =km), G2 is the spectral filtering losses (in ps2 /km), c is the nonlinearity coefficient (in W1 km1 ) and G0 is the excess gain of the EDFA (in km1 ). By making the frame of reference transformation T ¼ t z=vg , i.e., moving along with the pulse envelope U, CGLE becomes oU D G2 o2 U þ þi i þ cjU j2 U iG0 ¼ 0: oz 2 2 oT 2 ð2Þ By applying the following transformations on the CCGLE, we obtain the normalization of the equation: Unnormalized dimensions Time, T Distance, z Amplitude, U
Normalization quantity pffiffiffiffiffiffiffiffiffi T0 ¼ G2 =2G0 Z0 ¼ 1=G 0 pffiffiffiffiffiffiffiffiffiffi U0 ¼ G0 =c
Normalized dimensions s ¼ T =T0 f ¼ z=Z0 u ¼ U =U0
The quantities G0 ; G2 are the reference gain and the reference spectral filtering. Hence CCGLE becomes d 2 iu1 þ ð3Þ b2 i uss þ juj u igu ¼ 0: 2 The normalized terms are expressed via the respective unnormalized, according to the equations d¼
2D ; G2
b2 ¼
G2 ; G2
g¼
G0 : G0
ð4Þ
As it can be easily observed, when the reference gain and spectral filtering are chosen as G0 ¼ G0 and G2 ¼ G2 , then the normalized CCGLE takes the following, frequently met in the scientific literature, form: d 2 iuf þ ð5Þ i uss þ juj u iu ¼ 0: 2 This equation is well known as it has been used for applications in a lot of different regions of physics. In nonlinear optics, specifically, it is useful in modeling soliton amplifying transmission lines, spatial optical solitons and passive mode-locked lasers.
Pereira–Stenflo soliton pulses can be obtained from passively mode-locked lasers [7]. This kind of laser emits chirped soliton pulses with amplitude 1þil given by us ðsÞ ¼ As ½sechðs=aÞ . The pulse parameters As , a and l are related to the soliton width Ts , the peak power Ps and frequency chirp dx as Ts ¼ aT2 , Ps ¼ jb2 jNs2 =ðcT22 Þ and dx ¼ l tanh ðs=aÞ=Ts where T2 is the dipole relaxation time (0.1 ps for fiber amplifiers) and c ¼ n2 x0 =ðcAeff Þ is the nonlinearity coefficient. In this work it serves as an averaging model of a soliton transmission line equipped with amplifiers. The advantage of this model is that it includes the spectral filtering of the amplifier, which is important because it suppresses the Gordon–Haus jitter of the soliton central frequency. The model approximates the gain as a parabolic function of frequency and this is fair as long as the pulseÕs spectral width remains small in comparison with the gain bandwidth. There is a well-known solution of the equation which is called Pereira–Stenflo type pulse uðf; sÞ ¼ Afsech½s=a g1þil expðikfÞ:
ð6Þ
This special solution was found from Bekki– Nozaki [6]. Since the equation is not integrable the method they used was a phase-plane based one and not the inverse scattering method. This chirped pulse propagates undistorted (for sufficiently large propagation distances) only as long as the chirp is given by the following relation: 3 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 l¼ dþ 32 þ 9d : ð7Þ 4 4 It is well known that Eq. (5) possesses an unstable background. However, if the noise level (numerical or externally introduced) is low, the Pereira– Stenflo solution can propagate long distances before disintegration. Therefore, from the practical point of view, this solution (as an initial condition at the launching point of an optical link) may present some advantages as pre-chirped pulse. However, the Pereira–Stenflo solution is a special solution bounded by the, obviously impractical, restriction, Eq. (7). Therefore, it becomes apparent that it is far more interesting the investigation of a much wider class of initial conditions which may
M. Manousakis et al. / Optics Communications 213 (2002) 293–299
share similar characteristics and advantages with this exact solution. The analytical tool for this investigation is the variational method. It has been used in all kinds of conservative problems in nonlinear optics. The valuable qualitative results of the method have made it very popular among the research community. The analytical (variational) treatment can be used in the problems of mechanics when the forces taken into account are not frictional, in the sense that they can be derived from a work function. The conservation of the energy is not then demanded. However, the existence of a generating work function is crucial for the validity of variational minimizing procedures. One problem that has to be encountered here is the fact that in most cases the perturbations are large. Therefore, adequate justification must be given for the applicability of a perturbation method: In the case at hand, the really large perturbations (namely, the amplifier gain and the optical power losses) have been absorbed in the process of the averaging method used to derive the CCGLE itself. The dissipative terms present in CCGLE are smaller. As far as the overall scientific literature is concerned, a lot of researchers have been using the variational method in dissipative systems, each, of course, from different perspective. The potential of this method lies on its capability of providing useful qualitative results. In [8] the variational method is based on the concept of the system and anti-system approach, in [9] on the fractional derivative formalism, while in [10], the Lagrangian of the system is split into a sum of a conservative part and a non-conservative one. In the present work we use the latter formalistic approach to find the equations that describe qualitatively and quantitatively the dynamics of a Pereira–Stenflo type pulse as it evolves under the model of CCGLE. The paper is organized as follows: In Section 2 the ordinary differential equations governing the parameters of the pulse are derived and the normalizations that had to be made in order to compare the analytical with the numerical results are discussed. In Section 3, comparisons between the direct numerical solution of the CCGLE and the analytical results of the variational method are made. Finally in Section 4 the main conclusions are summarized.
295
2. Variational analysis According to the variational method [11] the time averaged Lagrangian of the system has to be calculated Z þ1 hLi ¼ L ds: ð8Þ 1
The Lagrangian is decomposed in two parts: the conservative one (it comes from the terms of the equation that are energy conservative) and the nonconservative one Lðu; u ; 1; s; u1 ; us ; u1 ; us Þ ¼ Lcon þ Lnonc ;
ð9Þ
where i d 1 4 2 Lcon ¼ ðuu1 u u1 Þ þ jus j juj : 2 2 2
ð10Þ
The CCGLE, Eq. (5), can now be written as d ð11Þ iu1 þ uss þ juj2 u ¼ Pu ; 2 where Pu ¼ ib2 uss þ igu:
ð12Þ
The variation of the non-conservative part Lncon can be written as dLnonc ¼ Pu du þ Pu du :
ð13Þ
The ansatz that will be used is a Pereira–Stenflo type pulse uð1; sÞ ¼ f ðp1 ð1Þ; p2 ð1Þ; p3 ð1Þ; p4 ð1ÞsÞ ¼ p1 ð1Þfsech½s=p2 ð1Þ g1þip3 ð1Þ expðip4 ðfÞÞ: ð14Þ The four parameters of interest for the pulse are A p1 the amplitude, a p2 the time width, l p3 the chirp and the phase, / p4 . Taking into account the fact that ðPu Þ ¼ Pu and Eq. (13) one can find that dLnonc ¼ 2ReðPu duÞ:
ð15Þ
The dynamical equations are derived from the following: Z Z dhLcon þ Lnonc i ou dpi df ¼ 0: ð16Þ du opi Thus, the general form of the dynamical equations of the pulse parameters can now be obtained
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d ohLcon i ohLcon i ¼ 2Re d1 oðpi Þ1 opi
Z Pu
ou : opi
ð17Þ
When the relations, Eqs. (8)–(16), are applied to the set of equations, Eq. (17), the following dynamical system comes out:
d dl2 4 2 þ A a þ ð2a 2 lnð2ÞÞal0 3a 3a 3 la0 2a/0 ¼ 0;
3. Results and discussion ð18Þ
2A2 dð1 þ l2 Þ 4lb2 ð1 þ l2 Þ 2lA0 2gl þ þ þ 3a2 3a2 3 A 0 0 ð19Þ þ ð3 2 lnð2ÞÞl 2/ ¼ 0; 2 4b 4 16 4 dl 2 þ lnð2Þb2 l2 b2 þ lnð2Þl2 b2 3 3 9 3 9 2 4a ðlnð2Þ 1ÞA0 þ 4ga2 4g lnð2Þa2 þ A þ ð3 þ 2 lnð2ÞÞaa0 ¼ 0; ð20Þ 2A2 b2 ð1 þ l2 Þ 2gA2 a þ 2Aaa0 þ A2 a0 ¼ 0; 3a
developed and the results of these two approaches are discussed in Section 3. The numerical method used for the development of this application is based on the broadly used symmetrical split-step Fourier method.
ð21Þ
The procedure of comparison between the analytical and the numerical results has been divided in two main categories: the evolution of the pulse in the anomalous dispersion regime and in the normal dispersion regime. The anomalous dispersion regime is examined first. Since we are interested in the evolution of pulses that are not initially (at f ¼ 0) the exact solutions of CCGLE, the following figures refer to the evolution of pulses possessing initially parameter values different than the ones of Pereira– Stenflo type. Each one of the following figures shows the evolution when only one of the three important parameters of the pulse (amplitude, width and chirp) deviates from the respective value of the exact solution.
where the primes denote differentiation with respect to the propagation distance. The last equation of the set, Eq. (21), gives the insight into the physical mechanism that governs the problem in hand: It can be rewritten as follows: 0
ðaA2 Þ ¼
2A2 b2 ð1 þ l2 Þ þ 2gA2 a: 3a
ð22Þ
One can easily observe that Eq. (22) describes explicitly the role of the excessive gain g as an exponential enhancing factor of the energy. The term which corresponds to the filter parameter is more complicated but it surely expresses the loss of power. Another way to check the method is to set all the derivatives equal to zero. Actually the algebraic system that comes out leads to Eq. (7) as it was expected. The complexity of the dynamical system, Eqs. (18)–(22), does not allow the analytical solution and, therefore, numerical simulations are being performed. An application for the direct partial differential equation numerical solution is also
Fig. 1. The evolution of the pulse when the initial amplitude deviates from the fixed point value, variationally (a, c) and full numerical simulation (b, d) in the anomalous dispersion regime. (a, b) +5% and (c, d) +20%.
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Specifically in Fig. 1 the evolution of a pulse with amplitude that is 5% (a, b) and 20% (c, d) larger than the Pereira–Stenflo pulse is shown. It is evident that, in the case of +5% deviation, the perturbed variational method gives a quite accurate picture of the pulse evolution. However, the oscillation of the amplitude is not predicted with absolute accuracy in the case of 20% deviation. In the following, Fig. 2, the time width is now the deviating parameter and the behaviour of the pulse resembles to that of Fig. 1: The case of +5% deviation validates again the variational procedure, while the +20% divergence shows the limits of the method. In Fig. 3 the main observation is that deviation in the initial chirp value is not so crucial for the evolution of the pulse for both mild and large deviations. As far as the normal dispersion regime is concerned we observe the following: Fig. 4 is the respective of Fig. 2 in the anomalous dispersion regime. Close examination of this figure makes two things clear: firstly, the accuracy of the analytical method is really better in this case and secondly the evolution of the pulse is smoother in a general point of view. These conclusions become clearer from Figs. 5 and 6. Apart from a small shift
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Fig. 3. The evolution of the pulse when the initial chirp deviates from the fixed point value, variationally (a, c) and full numerical simulation (b, d) in the anomalous dispersion regime. (a, b) +5% and (c, d) +20%.
Fig. 4. The evolution of the pulse when the initial amplitude deviates from the fixed point value, variationally (a, c) and full numerical simulation (b, d) in the normal dispersion regime. (a, b) +5% and (c, d) +20%.
Fig. 2. The evolution of the pulse when the initial time width deviates from the fixed point value, variationally (a, c) and full numerical simulation (b, d) in the anomalous dispersion regime. (a, b) +5% and (c, d) +20%.
in the transmission coordinate, the qualitative aspects of the two approaches are exactly the same. The features of these figures enforce the belief that the variational approach of the pulse dynamics is a useful and handy analytical tool.
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Fig. 7. Evolution of the pulse in the anomalous dispersion regime when the initial amplitude deviates from the fixed point value by +5%; (a) variationally, (b) full numerical simulation of the partial differential equation.
Fig. 5. The evolution of the pulse when the initial time width deviates from the fixed point value, variationally (a, c) and full numerical simulation (b, d) in the normal dispersion regime. (a, b) +5% and (c, d) +20%.
Fig. 6. The evolution of the pulse when the initial chirp deviates from the fixed point value, variationally (a, c) and full numerical simulation (b, d) in the normal dispersion regime. (a, b) +5% and (c, d) +20%.
The last figure shows the limits of the perturbed variational description of the problem in terms of the length of transmission. In Fig. 7 one can observe that in the anomalous dispersion regime and beyond the normalized transmission distance value
f ¼ 3 the pulse, whose initial amplitude deviates by +5%, starts to oscillate very rapidly and finally breaks into three parts. This evolution is catastrophic from the telecommunicationsÕ applications point of view. Similar behaviour characterizes the normal dispersion regime as well. The reason for the inaccuracy of the variational technique after a certain transmission distance is that the ansatz used does not possess any radiation component. Thus, the reduction of the problem through the averaging procedure hides certain information, which is crucial. The evolution of the radiation is enhanced by the nonlinearity and results in the destabilization and the ultimate destruction of the pulse. However, from the practical point of view, one should only be interested for the maximum propagation distance possible under the circumstances: According to the normalization of the coefficients in Eq. (5), it is noteworthy that the transmission distance has been normalized by the length Zo ¼ 1=Go , where Go is the excess gain of the amplifier. Hence, if Go is properly chosen, then a normalized distance of f ¼ 1 could correspond to cases characterized by long transmission distances. For example, if Go ¼ 0:1 km1 then Zo ¼ 10 km and f ¼ 1 is actually 10 km, while if Go ¼ 0:001 km1 then Zo ¼ 1000 km and f ¼ 1 is in fact 1000 km. Thus, the results demonstrate that the proposed method can actually be applied on a plethora of realistic cases of technological interest. The stationary solutions of complex Ginzburg–Landau equation with quintic nonlinear terms have been investigated thoroughly in [12]. In our case of interest the quintic terms are
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absent and the two-photon absorption term is set to zero. So our solutions fall in the case of solitons with fixed amplitude given by the expression: uðf; sÞ ¼ pðsÞ expðið/ðsÞ kfÞÞ, where pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /ðsÞ ¼ l lnðpðsÞÞ, B ¼ g=ðdl b2 þ l2 b2 Þ, C ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lð1 þ l2 Þðd 2 þ 4b22 Þ=½2ðdl 2b2 Þ , and l given by Eq. (7), while k ¼ gðdl2 4b2 l dÞ=2ðb2 l2 þ dl b2 Þ. From the filtering parameter two-photon absorption plane obtained in [12] it is evident that our solutions lay in the domain of stable solutions with unstable background. In our work the simulations are developed in the region g=d 1 and seem to be the main reason why the instability of the background develops slowly (for the fixed point). This behaviour corresponds to the condition g 1 obtained in [12] where jdj ¼ 1. So it is fair to claim that in a more general case the propagation distance of a PS solitary pulse is large as long as g=d 1. 4. Conclusions In this work the evolution of chirped pulses in an EDFA-supported transmission line has been studied. The chirped pulse ansatz has the functional form of the exact solution of the equation under examination, namely the CCGLE. The perturbed variational method was used in order to investigate the transmission properties of a wider class of initial pulse profiles this ansatz is amenable to. Four equations for the propagation dynamics were derived for the four main parameters of the pulse. These equations have been solved numerically for a set of initial conditions in the vicinities of the fixed point of the dissipative system these evolution equations express. The main focus of this work was on the investigation of the existence of domains in the parameter space of the initial pulse profiles which correspond to pulses characterized by as much stationary (in propagation distance f) as possible evolution dynamics and long propagation distances before they disintegrate. Especially in the case of normal dispersion
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regime divergence even up to 20% from the fixed point meet these requirements, leading to wide classes of initial pulse profiles for practical applications in optical communications. Comparison with numerical simulations of the CCGLE itself has shown that the perturbed variational approach is an effective method for systems of this nature. However, more accurate results can be obtained if one takes into account the radiation of the pulse. This will possibly lead to a dynamical system with at least one additional degree of freedom. This is a subject of current and future investigation.
References [1] B. Malomed, Resonant transmission of a chirped soliton in a long optical fiber with periodic amplification, J. Opt. Soc. Am. B 13 (1996) 677. [2] F.Kh. Abdullaev, A.A. Abdumalikov, B.B. Baizakov, Stochastic instability of chirped optical solitons in media with periodic amplification, Quant. Electron. 27 (1997) 171. [3] B. Malomed, D. Parker, N. Smyth, Resonant shape oscillations and decay of a soliton in a periodically inhomogeneous nonlinear optical fiber, Phys. Rev. E 48 (1993) 1418. [4] T. Lakoba, J. Yang, D. Kaup, B. Malomed, Conditions for stationary propagations in the strong dispersion management limit, Opt. Commun. 149 (1998) 366. [5] B. Malomed, I. Uzunov, M. Golles, F. Lederer, Two stage perturbation theory for bandwidth limited amplification of optical solitons near the zero dispersion point, Phys. Rev. E 55 (1997) 3777. [6] N. Bekki, K. Nozaki, Formation of spatial patterns and holes in the generalised GLE, Phys. Lett. A 110 (1985). [7] G.P. Agrawal, Nonlinear Fiber Optics, second ed., Academic Press, New York, 1995. [8] D. Kaup, B. Malomed, Variational principle for nonlinear waves in dissipative systems, Physica D 87 (1995) 155. [9] F. Riewe, Non-conservative Lagrangian and Hamiltonian mechanics, Phys. Rev. E. 53 (1890). [10] S. Cerda, S. Cavalcanti, J. Hickmann, A variational approach of nonlinear dissipative pulse propagation, Eur. J. Phys. D 1313 (1998). [11] D. Anderson, Variational approach to nonlinear pulse propagation in optical fibers, Phys. Rev. A 27 (1983) 3135. [12] N.N. Akhmediev, A. Ankiewicz, Solitons: Nonlinear Pulses and Beams, Chapman & Hall, London, 1997.