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Pulse propagation in media with deterministic and random dispersion variations
Optics Communications 214 (2002) 353–362 www.elsevier.com/locate/optcom
Pulse propagation in media with deterministic and random dispersion variations q T. Sch€ afer a,1, R.O. Moore a,b,*,2, C.K.R.T. Jones a a
Division of Applied Mathematics, Brown University, 182 George St., P.O. Box F, Providence, RI 02912, USA Department of Mathematics, Simon Fraser University, 8888 University Dr., Burnaby, BC, Canada V5A 1S6
b
Received 8 June 2002; received in revised form 27 September 2002; accepted 25 October 2002
Abstract We obtain a widely valid description of pulse broadening in dispersion-managed fiber with a random component in the dispersion. For short distances, a quasi-linear approach in the form of a simple perturbation expansion suffices to capture the broadening for each realization of the random dispersion. For intermediate distances, over which the noise and nonlinearity interact, we demonstrate that the partial differential equation can be reduced to a relatively simple system of nonlinear stochastic ordinary differential equations. Finally, we demonstrate that, over long distances, the slow evolution of the pulse width can be obtained by applying an appropriate multiple-scales averaging procedure to yield a new, scaled noise process effecting pulse broadening. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 42.81; 05.40 Keywords: Optical fibers; Random dispersion
1. Introduction Since the discovery of exact pulse-like solutions to nonlinear wave equations, the response of these solutions to perturbations has been of both q
This material is based upon work supported by the National Science Foundation under Grant No. 0073923. * Corresponding author. Tel.: +1-604-2913760; fax: +1-6042686657. E-mail addresses: [email protected] (T. Scha¨fer), [email protected] (R.O. Moore). 1 Also corresponding author. 2 Partially supported by the Natural Sciences and Engineering Research Council of Canada.
practical and theoretical interest. In optical fiber transmission, the robustness of pulses to the random variations of a system parameter is critically important to their feasibility as carriers of information. One such parameter, the group-velocity dispersion, is particularly sensitive to material imperfections and deformations, and has been demonstrated to exhibit variations when measured directly [1,2]. In the presence of these variations, a pulse that would otherwise preserve its shape can arrive at its destination distorted or destroyed [3]. The canonical evolution equation for pulses in optical fiber is the nonlinear Schr€ odinger equa-
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 1 3 2 - 6
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tion (NLSE) [4]. Two variants of this equation, with significantly different concomitant behaviors, are of current practical interest. The first is the classical (focusing) case, in which the dispersion and nonlinear coefficients are constant and of the same sign. The NLSE is then integrable by the inverse scattering transform [5] and has soliton solutions. The influence of random dispersion on classical solitons has been studied through a variational approach [6]. The second interesting case is the dispersion-managed version of the NLSE, where the dispersion coefficient is allowed to vary (deterministically) with propagation distance. This equation is not integrable in the classical sense [7] and is not solvable by direct analytical methods. Nevertheless, a reduction to ordinary differential equations by means of a Gauss–Hermite expansion [8,9] has been used to describe the properties of a class of special solutions of this equation, referred to as dispersionmanaged solitons because of their similarity to classical solitons. If only the highest mode in this Gauss–Hermite expansion is considered, the resulting ordinary differential equations are the same as those found previously by a variational approach or by the consideration of certain moments [10]. A reduction of the NLSE to simpler equations is important for two reasons. First, to generate reliable statistics quantifying the effect of random coefficients on pulse evolution requires a large number of simulations. This is particularly true if the most relevant occurrences are exactly those that are least likely to occur in a straightforward Monte-Carlo approach, such as bit errors. It is therefore of obvious benefit to reduce the computational time for each simulation to produce meaningful results in a minimum of time. The second argument in favor of finding a reduction to the NLSE is that it is often difficult to obtain much insight from a partial differential equation. A reduction to a dynamical system of finite dimension is generally more conducive to obtaining an analytical description of the pulse behavior. The resulting ordinary differential equations can be linearized about the dispersion-managed soliton for approximations valid over shorter distances, or they can be averaged over the dispersion map
period [11] to capture the behavior over longer scales. In summary, we demonstrate a simple and widely valid characterization of pulse broadening for dispersion-managed solitons under the influence of random dispersion. We obtain this characterization by considering a quasi-linear limit of the partial differential equation for small distances, and by averaging a corresponding system of nonlinear ordinary differential equations for large distances. These simple representations can be used far more easily than the original partial differential equation to relate the magnitude of random dispersion fluctuations in a fiber-optic line to the likelihood of a bit error due to pulse broadening.
2. Pulse propagation in dispersion-managed transmission lines 2.1. The cubic nonlinear Schr€odinger equation In this section, we quickly review the basic equation for modeling pulse propagation in optical fiber lines and explain the basic idea of dispersion management. For pulses with a width larger than a few picoseconds, MaxwellÕs equations are well approximated by the NLSE [4] i
oE a 1 o2 E 2 þ i E b2 2 þ cjEj E ¼ 0: oZ 2 2 oT
ð1Þ
Here, the complex-valued function EðZ; T Þ represents the envelope of the electric field in a 2 moving frame, where jEj is the optical power measured in W. The propagation distance z has units of km and the retarded time t is measured in ps. The attenuation is modeled by a, measured in km1 . The group velocity dispersion b2 has units of ps2 km1 , and the nonlinearity c is given in W1 km1 . Typically, a and b2 depend on z, as these parameters vary along the fiber line. Standard monomode fiber has a dispersion of jb2 j ¼ 10 ps2 km1 , a loss of a ¼ 0:2 dB km1 , and a nonlinearity of c 1 W1 km1 , but these values vary greatly depending on the type of fiber that is used. Rather than focus on a particular set of parameters, we will study a dimensionless ver-
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sion of Eq. (1) where we use the full-width-halfmaximum pulse width TFWHM and the peak power P to set our scales. While the pulse width is usually between 3 and 7 ps, the peak power can, depending on the line, vary greatly. It is common to set the peak power at several milliwatts; at this power level, our assumption of weak nonlinearity is clearly valid. A dispersion-managed fiber link consists, in the most simple case, of a concatenation of two fibers, one with positive dispersion and the other with negative dispersion. The length L of the dispersion map is then the sum of the lengths of the two fibers. We use this length to set the scale of our evolution variable z. In practice, L can assume arbitrary values, as the fibers can be cut at arbitrary lengths. A standard value for the length of the dispersion map is approximately 20 km. After rescaling to dimensionless variables by applying t¼
T ; TFWHM
z¼
Z ; L
E E~ ¼ pffiffiffi ; P
ð2Þ
and renormalizing to include the attenuation by applying a ð3Þ E~ðz; tÞ ¼ Aðz; tÞ exp z ; 2 we obtain, in dimensionless units, 2
iAz þ dðzÞAtt þ cðzÞjAj A ¼ 0;
ð4Þ
where b2 ðzÞ L ; 2 TFWHM Z z aðz0 Þ dz0 : cðzÞ ¼ cPL exp dðzÞ ¼
ð5Þ
0
Eq. (4) is the basic equation for this paper. The attenuated coefficient cðzÞ is typically recovered periodically by the placement of amplifiers along the fiber line. Here, we consider the lossless or distributed-gain case, where cðzÞ is constant. This arises when the effect of loss and gain are so minor or rapid as to be insignificant to the slow evolution of the pulse, or when the fiber attenuation is compensated continuously through distributed amplification.
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2.2. Dispersion-managed solitons As mentioned before, the simplest dispersion map is formed by a concatenation of two sections of fiber with different dispersion values. This twostep map gives the following form for the dispersion coefficient: 8 0 6 z < 1=4; < dl þ dav ; ð6Þ dðzÞ ¼ dl þ dav ; 1=4 6 z < 3=4; : dl þ dav ; 3=4 6 z < 1; where dl represents the local, mean-zero variations of the dispersion and dav the residual dispersion. This relation defines the function dðzÞ in [0,1] and we extend this definition periodically for z > 1. It is well known that special solutions called solitons exist in the case of the integrable NLSE, for which d and c are both constant and positive [5]. It has also been suggested, however, that a similar class of solutions, referred to as dispersionmanaged solitons [12], exists in the case of a two-step map. A dispersion-managed soliton is a solution As of Eq. (4) with As ðz; tÞ ¼ F ðz; tÞei/ðzÞ ;
F ðz; tÞ ¼ F ðz þ 1; tÞ;
ð7Þ
0
where / ðzÞ is called the quasi-momentum. For a dispersion map given by Eq. (6), a family of dispersion-managed solitons, as well as the relation between their quasi-momentum and width, can be found analytically [10]. Due to the local variations of the dispersion, dispersion-managed solitons ‘‘breathe’’, or vary periodically in width over a map period. We might remark that, to our knowledge, no mathematical proof of the existence of dispersion-managed solitons is known. Numerical simulations [13], however, strongly support the existence of dispersion-managed solitons. Moreover, for practical purposes in a telecommunications line, the numerically known solutions are close enough to periodic to be useful for data transfer. 2.3. Random perturbations of the dispersion We are now ready to give a mathematical formulation of the problem considered in this paper: assume that we add to the dispersion map dðzÞ a
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small random term. How will a dispersion-managed soliton, that is otherwise periodic in the above sense, evolve under these random perturbations? Numerical simulations indicate that the pulse will broaden [3,14]. As we are interested in quantifying this phenomenon, we have first to define an appropriate measure of the broadening that arises from the presence of randomness. We choose the second moment for this purpose Z Z 2 2 DT t2 jAj dt t2 jAs j dt: ð8Þ In general, DT is a stochastic process depending on z. In the deterministic case, obviously A ¼ As and DT ¼ 0. Numerically, the value of DT can be found directly from simulations of the NLSE. In the following sections we will show how to compute DT approximately and compare the analytical results to direct numerical simulations.
3. Quasi-linear theory and pulse broadening 3.1. Theory Although nonlinearity is weak in most strongly dispersion-managed systems, its effect is still important as it determines the fixed point corresponding to the dispersion-managed soliton for a given pulse width. Let us therefore consider Eq. (4) with the following scaling 2 iAz þ d~ðzÞAtt þ dav Att þ jAj A ¼ 0:
ð9Þ
Here, we have explicitly separated the locally varying dispersion d~ from the small residual dispersion dav . The scaling for a dispersion-managed soliton As is then As ðz; tÞ ¼ F ðz; tÞe
ikz
:
ð10Þ
In the linear case where ¼ 0, any bounded function is a periodic solution of Eq. (9). When the nonlinearity is turned on by increasing from zero, however, very few (and in some parameter regimes, only one) solutions retain the periodicity described by Eq. (7). A dispersionmanaged soliton can therefore be characterized as a truly nonlinear object even when nonlinearity is small.
The functional form of As is mainly a Gaussian with higher order corrections [8,9]. We can therefore write As ðz; tÞ ¼ AG ðz; tÞ þ lAres ðz; tÞ;
ð11Þ
where AG denotes the Gaussian part and l 1. The Gaussian part AG is assumed to satisfy the equation iAGz þ d~ðzÞAGtt ¼ 0;
ð12Þ
which we can solve explicitly by Fourier transform. This fact that the core of a dispersionmanaged soliton exhibits primarily linear behavior has been successfully used, for example, to study pulse interactions in the time or frequency domain. As mentioned above, AG is not an arbitrary Gaussian; rather, its amplitude, width, and quasimomentum k are linked through the fiber nonlinearity. Because of the smallness of the parameters and l, we can describe the behavior of a dispersion-managed soliton under the influence of weak random dispersion, at least at the beginning of its evolution, by perturbation theory. A comparison to numerics will then show how far this approach is valid. We consider an additional random dispersion term in Eq. (9) and write iAz þ d~ðzÞAtt þ dav Att þ mdr ðzÞAtt þ jAj2 A ¼ 0: ð13Þ The noisy part of the dispersion is assumed to be low-amplitude (m 1) white noise, with a correlation function given by hdr ðzÞdr ðz0 Þi ¼ Ddðz z0 Þ:
ð14Þ
The amplitude of pffiffiffithe ffi noise term in the equation is then given by m D. Here, randomness is introduced as variations of the dispersion coefficient. Another possible source of randomness, not considered here, is random variation of the fiber spans [15]. We look for a solution to this equation of the form Aðz; tÞ ¼ As ðz; tÞ þ mA1 ðz; tÞ:
ð15Þ
The first order of this perturbation series will therefore be given by the linear equation,
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iA1z þ d~ðzÞA1tt þ dr ðzÞAGtt ¼ 0:
ð16Þ
The deviation from the zero-order solution As is driven by the term dr ðzÞAGtt , the influence of the randomness on the Gaussian part of the dispersion-managed soliton. The solution of this linear equation is given in Fourier domain by A^1 ðz; xÞ ¼ ix2 Wr ðzÞ A^G ðz; xÞ; Z z dr ðz0 Þ dz0 ; Wr ðzÞ ¼
ð17Þ
0
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deterministic system, the evolution on the slow scales can be described by the Gabitov–Turitsyn equation [17]. The equivalent PDE in the stochastic case, however, is difficult to treat analytically or numerically. We therefore look for a reduction to a finite-dimensional system in appropriate coordinates, not only to reproduce the quasi-linear result derived from the PDE, but also to extend our analysis to longer evolution scales. 4.1. The ODE model for width and chirp
where Wr is a Wiener process in z. This equation is the analytical solution of the problem, and all interesting quantities (e.g., statistical moments) can be calculated from this solution. As an example, the broadening of the pulse width due to randomness can be computed in the Fourier domain as Z 1 2 2 DT ¼ j A^x j j A^sx j dx 2p Z m A^Gx A^1x þ A^Gx A^1x dx 2p Z mWr ðzÞ 2ixð A^Gx A^G A^G A^Gx Þ dx: ð18Þ ¼ 2p This calculation shows that the evolution of DT is a scaled Wiener process, or random walk. We can proceed by evaluating the integral quantity in Eq. (18), noting that for the Gaussian core of a dispersion-managed soliton, we have ix2 R~ðzÞ
0 A^G ðz; xÞ ¼ e ð19Þ f^ðxÞ; R~ðzÞ ¼ d~ðzÞ; ^ where f is a real-valued function. [16]. This allows us to write a final closed-form expression for the pulse broadening Z ~ DT ¼ 8mWr ðzÞRðzÞ f 0 ðtÞ2 dt: ð20Þ
In Section 5 we will compare this result to numerical simulations. 4. Reduction of the NLSE to ordinary differential equations The above relation, Eq. (20), is valid for short propagation distances; for larger z, the nonlinearity will interact with the random dispersion. In a
We start again with the basic Eq. (4). The most interesting parameters of the pulse are its width and chirp, whose evolution can be described to a good approximation by ordinary differential equations. The appropriate self-similar ansatz for this variational reduction is
t MðzÞ 2 t : Aðz; tÞ ¼ aðzÞf exp ikðzÞ þ i T ðzÞ T ðzÞ ð21Þ This ansatz is inserted into the Lagrangian density of Eq. (4) and, after integrating the Lagrangian over the time t, one obtains from the Euler–Lagrange equations: dT ¼ 4dðzÞM; dz dM dðzÞC1 C2 ¼ 2: dz T3 T
ð22Þ ð23Þ
Here, C1 and C2 are integrals over the pulse shape f ðxÞ. Eqs. (22) and (23) are general and can be used for dispersion-managed or classical solitons; of course, the constants C1 and C2 will be different, as the pulse shape of a classical soliton is different from that of a dispersion-managed soliton. A system of equations identical to Eqs. (22) and (23) can be found by a momentum method or by truncation of an expansion of the solution into Gauss–Hermite eigenfunctions after applying the Lens transform [9]. In the case of a classical soliton without randomness, the dispersion d is constant and the form of the soliton is unchanged as it moves along the fiber. This is represented by a fixed point of Eqs. (22) and (23): in this case, the pulse width T is constant and the chirp M is zero.
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In dispersion-managed case, the dispersion varies periodically along the fiber line. Dispersionmanaged solitons are therefore represented by periodic solutions of Eqs. (22) and (23), and their trajectory in a phase plot of T against M is a closed curve. The associated Poincare plot (the points of the graph of T and M taken at the end of each period of the dispersion map) again consists of a single fixed point. 4.2. Influence of randomness: comparison to PDE dynamics Random dispersion introduces perturbations to these solutions. The first issue we address is to check numerically whether the reduced model of Eqs. (22) and (23) can be used to describe these perturbations. If this is true for single noise realizations, then it is clearly also true for ensemble quantities such as statistical moments. We therefore monitor the pulse width under the influence of arbitrary realizations of random dispersion. We first consider classical solitons with a hyperbolic secant trial function determining the constants C1 and C2 . Fig. 1 shows that, even for moderate values of z, the ODE solution and the PDE solution diverge significantly. While this divergence obviously depends on the noise strength,
Fig. 2. Numerical comparison of a single ODE simulation (solid line) with the same noise realization in a PDE simulation (dashed line). The amplitude of the noise is 0.02. The other parameters are dl ¼ 5, dav ¼ 0:15, C1 ¼ 1, C2 ¼ 0:1975.
it is evident that the ODE can only be taken to represent the PDE for small values of z. A similar comparison using a dispersion-managed soliton in a two-step dispersion map is depicted in Fig. 2. In this case, while there are again differences between the ODE and PDE results, they agree well for significantly longer scales than in the classical case. In particular, for moderate values of z relevant to real communication networks, the curves are almost indistinguishable. The excellent agreement demonstrates that MonteCarlo simulations using the PDE model can be replaced without significant loss of accuracy by much faster simulations using the ODE model.
5. Theory of the stochastic ordinary differential equations 5.1. Quasi-linear theory revisited
Fig. 1. Numerical comparison of a single ODE simulation (solid line) with the same noise realization in a PDE simulation (dashed line) for a classical soliton. Here, dðzÞ ¼ 1:0 and cðzÞ ¼ 2:0. The amplitude and the width of the initial pulse are one. The amplitude of the noise is 0.01.
If the ODE model in Eqs. (22) and (23) accurately represents the PDE model in Eq. (4), we should be able to rederive Eq. (20) using the ODEs. Let us rescale Eqs. (22) and (23) by taking C1 ¼ 1 and C2 ¼ N 2 . Again, we have a dispersion dðzÞ that consists of a varying part, a constant part, and a random part: dðzÞ ¼ d~ðzÞ þ dav þ mdr ðzÞ;
ð24Þ
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Furthermore, we can write T and M in expansions T ¼ TG þ lTres þ mT1 ; M ¼ MG þ lMres þ mM1 ;
ð25Þ
where TG and MG satisfy the equations TGz ¼ 4d~ðzÞMG ;
MGz ¼
d~ðzÞ : TG3
ð26Þ
The equations for T1 and M1 then become T1z ¼ 4dr ðzÞMG þ 4d~ðzÞM1 ; dr ðzÞ 3d~ðzÞT1 M1z ¼ 3 : TG4 TG
ð27Þ
This system of equations can be solved easily as it is equivalent (to the leading and first order) to sðzÞ ¼ TG þ mT1 ;
cðzÞ ¼ MG þ mM1 ;
sz ¼ 4ðd~ þ mdr Þc;
cz ¼
d~ þ mdr ; s3
ð28Þ ð29Þ
with the solution s2 ðqÞ ¼ s20 þ
4q2 ; s20
q0 ¼ d~ðzÞ þ mdr ðzÞ:
ð30Þ
As s measures the width of the pulse, we see again from this solution that, for small m, the change in width is proportional to R~ and the Wiener process Wr as in Eq. (20). We now compare the predictions of Eqs. (30) and (20) to numerical simulations. To do this, we write Eq. (20) explicitly with the system parameters. If we take the Gaussian approximation of the dispersion-managed soliton, we can write the initial condition (with T0 ¼ T ð0Þ and Mð0Þ ¼ 0) as
N t Að0; tÞ ¼ pffiffiffiffiffi B0 f0 ; T0 T0 ð31Þ f0 ðxÞ ¼ p1=4 expðx2 =2Þ; pffiffiffiffiffiffi where B20 ¼ 2 2p. This gives pffiffiffiffiffiffi 8mWr ðzÞR~ðzÞN 2 2p DT ¼ : ð32Þ T02 Fig. 3 shows that for small z, the analytical results describe very well the behavior of the ODE solution and therefore also the behavior of the full problem. For larger z, however, both the
Fig. 3. Numerical comparison of the numerical simulations (solid line) of the ODE with the PDE simulations (dots) and with the analytical model (dashed line). The amplitude of the noise is 0.05. The figure shows an average over 5000 realizations. The parameters of the considered system are dl ¼ 5:0, dav ¼ 0:15, C1 ¼ 1:0, C2 ¼ 0:117.
PDE and the ODE models predict an interaction between the noise and the nonlinear part, and this interaction renders the quasi-linear approach for finding the statistics of DT invalid. In order to find a way to describe the systemÕs properties beyond this quasi-linear regime, we have to consider the interaction of randomness and noise. 5.2. Averaging the ODEs in the presence of random dispersion We have already seen that random fluctuations in the dispersion drive the system away from its fixed point (the dispersion-managed soliton). Once the system has left this fixed point, the balance between nonlinearity and residual dispersion is affected, and this changes the dynamics. However, since nonlinearity and residual dispersion are small, these changes come into play on longer evolution scales. The systemÕs behavior can therefore be treated in the framework of a multiplescales expansion. As our equations are stochastic ordinary differential equations, we have to find a way to incorporate the noise in the multiple scale expansion. We start our analysis from Eqs. (22) and (23) written with the appropriate scaling
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dM dðzÞ N2 ¼ 3 2 dz T T
dT ¼ 4dðzÞM; dz
ð33Þ
and with the previously defined dispersion map dðzÞ ¼ d~ðzÞ þ dav þ mdr ðzÞ: It is possible to treat these equations directly by introducing an expansion at multiple scales, but the calculations are very cumbersome. We choose instead to use an appropriate coordinate transformation discussed previously [11], where the Hamiltonian of the zeroth order, given by TGz ¼ 4d~ðzÞMG ;
MGz ¼
d~ðzÞ ; TG3
is used to introduce action and angle variables, bG ¼ 4MG TG ;
XG ¼
4 þ 16MG2 ; TG2
ð34Þ
where XGz ¼ 0;
bGz ¼ d~ðzÞXG :
ð35Þ
We apply the same transformation to the full nonlinear system, Eq. (33), by introducing b ¼ 4MT ;
X¼
4 þ 16M 2 T2
ð36Þ
yielding the corresponding nonlinear equations
3=2 X Xz ¼ 8N 2 b ; 4 þ b2 ð37Þ
1=2 X 2 : bz ¼ dðzÞX 4N 4 þ b2 In order to find the slow evolution of these equations, we introduce multiple scales zn ¼ n z and the expansions XðzÞ ¼ X0 ðz0 ; z1 ; . . .Þ þ X1 ðz0 ; z1 ; . . .Þ þ bðzÞ ¼ b0 ðz0 ; z1 ; . . .Þ þ b1 ðz0 ; z1 ; . . .Þ þ
ð38Þ
In what follows, we will only consider the expansion to the order . The solution of the zeroth order of these equations is ~ ðz1 Þ; X0 ðz0 ; z1 Þ ¼ X ~ ðz1 ÞR~ðz0 Þ þ b~ðz1 Þ; b0 ðz0 ; z1 Þ ¼ X
ð39Þ
where dR~ ~ ¼ d ðz0 Þ: dz0
ð40Þ
The equations for the first order are !3=2 oX1 oX0 X 0 ¼ 8N 2 b0 ; oz0 oz1 4 þ b20 ob1 ~ ob m d X1 ¼ 0 þ dav X0 þ dr ðz0 ÞX0 oz0 oz1 !1=2 X0 4N 2 : 4 þ b20
ð41Þ
We can solve these equations by variation of constants and obtain as a solvability condition the equations for the slow variation of X and b, hence ~ ðz1 Þ and b~ðz1 Þ. The difficulty is the equations for X to see how the random term mdr ðz0 ÞX0 = can be incorporated in this formalism. If we integrate Eq. (41) over one period of d~ðz0 Þ, from z0 ¼ n to n þ 1, we obtain Z nþ1 dr ðz0 Þ dz0 ; n
a random variable whose statistical properties do not depend on n. Simple calculations show that this random term has a mean of zero and a variance equal to the strength D of the white noise, and that it is uncorrelated with values of dr ðz0 Þ for z0 > n þ 1 and z0 < n. In this way, the solvability condition produces a sequence of d-correlated random numbers for a given fiber line, where each element nn in the sequence is given by Z nþ1 nn ¼ dr ðz0 Þ dz0 : ð42Þ n
Each of these random elements contributes the ~ 1 = to the slow piecewise-constant quantity mnn X evolution of the angle variable b0 . Since this slow evolution occurs over the scale of z1 ¼ z0 , and since the random elements nn are piecewise-constant for sections of Oð1Þ in z0 and therefore of OðÞ in z1 , this piecewise-constant, d-correlated random process can be approximated by a white noise process Nðz1 Þ (or, similarly, can be thought of as a discrete representation of a continuous white noise process) with the correlation function:
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hNðz1 ÞNðz01 Þi ¼ Ddðz1 z01 Þ:
361
ð43Þ
~ ðz1 Þ and b~ðz1 Þ are The resulting equations for X then !3=2 Z 1 ~ ðz1 Þ dX X ðz ;z Þ 0 0 1 ¼ 8N 2 b0 ðz0 ;z1 Þ dz0 ; dz1 4 þ b0 ðz0 ;z1 Þ2 0 db~ðz1 Þ ~ ðz1 Þ þ 4N 2 J ðz1 Þ; ~ ðz1 Þ þ m Nðz1 ÞX ¼ dav X dz1 2 !3=2 Z 1 X0 ðz0 ;z1 Þ 4 ~ J ðz1 Þ 2Rðz0 Þb0 ðz0 ;z1 Þ 2 4 þ b0 ðz0 ;z1 Þ 0 !1=2 3 X0 ðz0 ;z1 Þ 5 dz0 : 2 4 þ b0 ðz0 ;z1 Þ ð44Þ Here, we can substitute the zero-order solution given by Eq. (39) and solve the integrals. Introducing the abbreviations ~ ðz1 Þ dl X ; 4 ð45Þ ~ ðz1 Þ dl X ~ ~s ðz1 Þ bðz1 Þ ; 4 we find 0 1 pffiffiffiffi ~ ~ dXðz1 Þ 1 1 XB C ¼ 16N 2 @ pffiffiffiffiffiffiffiffiffiffiffi2ffi pffiffiffiffiffiffiffiffiffiffiffi2ffi A; dz1 dl 4þ~sþ 4þ~s
Fig. 4. Comparison between numerical simulations using the stochastic ODEs governing the slow dynamics (Eq. (46), solid line) and simulations using the full nonlinear equations (Eq. (33), dashed line). The amplitude of the noise is 0.03; the other parameters for the full nonlinear simulations are dl ¼ 5:0, dav ¼ 0:15, C1 ¼ 1, C2 ¼ 0:1975, and Mð0Þ ¼ 0. The figure shows an average over 5000 realizations.
5.3. Comparison to numerical simulations
~sþ ðz1 Þ b~ðz1 Þ þ
2 db~ðz1 Þ ~ ðz1 Þþ 8N ~ ðz1 Þþ m Nðz1 ÞX pffiffiffiffi F ð~s ;~sþ Þ; ¼ dav X dz1 ~ dl X ! ! ~sþ ~s F ð~s ;~sþ Þ arcsinh arcsinh 2 2 0 1
1 1 B C þð~s ~sþ Þ@ pffiffiffiffiffiffiffiffiffiffiffiffi þ pffiffiffiffiffiffiffiffiffiffiffi2ffi A: 2 4þ~sþ 4þ~s
We are now in a position to compare the predictions for the evolution of DT defined in Eq. (8) as given by the slow evolution equations to the results of the full (i.e., unaveraged) nonlinear ODEs. For this purpose, we plot DT stroboscopically, taking only its values at the endpoints of each dispersion map, and compare these values to those obtained from the averaged ODEs. Fig. 4 shows reasonably good agreement between the slow evolution and the stroboscopically probed fast evolution. Most importantly, this agreement holds for much larger distances than the validity of the quasi-linear approximation given by Eq. (32).
6. Conclusion ð46Þ
It is simple to show that there exists a fixed point of this system representing the dispersionmanaged soliton, and that this fixed point has the form M ¼ 0 and T ¼ gðN Þ, where g establishes a family of pairs of N and T representing dispersionmanaged solitons.
We have verified that the key behavior of a dispersion-managed soliton under the influence of random dispersion is successfully captured through a simplified pair of stochastic ODEs. Over short distances, the leading order effect of randomness on pulse broadening can be obtained by considering just the linear evolution of the solitonÕs
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Gaussian core. In this limit, we have verified that the ODE prediction matches with the PDE prediction for pulse broadening, and that both match well with numerically obtained results over appropriately short evolution distances. Over longer scales for which no analytical results are available from the PDE, we have demonstrated that the slowly evolving stochastic ODEs obtained through a multiple-scales averaging procedure successfully reproduce the pulse broadening obtained by integrating the original ODE system. As the original ODE system has already been demonstrated to capture the evolution of a dispersion-managed soliton in the NLSE for relatively large z, this demonstrates the validity of the averaged stochastic ODEs we have derived. The importance of these results lies in their reduction of a PDE with rapidly varying coefficients to a much simpler ODE system. In both the quasilinear case and in the large-z averaged case, these ODEs are much more readily analyzed to produce outage probabilities and bit error rates for pulses in real fiber-optic lines. Similarly, the computational effort required to obtain pulse broadening statistics from the ODE system is several orders of magnitude less than would be required to integrate the original PDE system.
Acknowledgements The authors thank J. Bronski and I. Gabitov for many very valuable discussions.
References [1] L.F. Mollenauer, P.V. Mamyshev, M.J. Neubelt, Opt. Lett. 21 (21) (1996) 1724. [2] I. Brener, P.P. Mitra, D.D. Lee, D.J. Thomson, D.L. Philen, Opt. Lett. 23 (19) (1998) 1520. [3] F.Kh. Abdullaev, B.B. Baizakov, Opt. Lett. 25 (2) (2000) 93. [4] G.P. Agrawal, Nonlinear Fiber Optics, Academic Press, Boston, 1989. [5] V.E. Zakharov, A.B. Shabat, Zh. Ekcp. Teor. Fiz 61 (1971) 118. [6] F.Kh. Abdullaev, J.C. Bronski, G. Papanicolaou, Physica D 135 (2000) 369. [7] T.I. Lakoba, Phys. Lett. A 260 (1999) 68. [8] T.I. Lakoba, D.J. Kaup, Phys. Rev. E 58 (1998) 6728. [9] S.K. Turitsyn, T. Sch€afer, K.H. Spatschek, V.K. Mezentsev, Opt. Commun. 163 (1999) 122. [10] T. Sch€afer, V.K. Mezentsev, K.H. Spatschek, S.K. Turitsyn, Proc. R. Soc. A 457 (2001) 273. [11] S.K. Turitsyn, A.B. Aceves, C.K.R.T. Jones, V. Zharnitsky, Phys. Rev. E 58 (1998) R48. [12] J.H.B. Nijhof, N.J. Doran, W. Forysiak, A. Berntson, Electron. Lett. 34 (1998) 481. [13] V. Cautaerts, A. Maruta, Y. Kodama, Chaos 10 (2000) 515. [14] M. Matsumoto, H. Haus, IEEE Photon. Technol. Lett. 9 (1997) 785. [15] B.A. Malomed, A. Berntson, JOSA B 18 (2001) 1243. [16] M.J. Ablowitz, G. Biondini, Opt. Lett. 23 (1998) 1668. [17] I. Gabitov, S.K. Turitsyn, Opt. Lett. 21 (1996) 327.
Pulse propagation in media with deterministic and random dispersion variations q T. Sch€ afer a,1, R.O. Moore a,b,*,2, C.K.R.T. Jones a a
Division of Applied Mathematics, Brown University, 182 George St., P.O. Box F, Providence, RI 02912, USA Department of Mathematics, Simon Fraser University, 8888 University Dr., Burnaby, BC, Canada V5A 1S6
b
Received 8 June 2002; received in revised form 27 September 2002; accepted 25 October 2002
Abstract We obtain a widely valid description of pulse broadening in dispersion-managed fiber with a random component in the dispersion. For short distances, a quasi-linear approach in the form of a simple perturbation expansion suffices to capture the broadening for each realization of the random dispersion. For intermediate distances, over which the noise and nonlinearity interact, we demonstrate that the partial differential equation can be reduced to a relatively simple system of nonlinear stochastic ordinary differential equations. Finally, we demonstrate that, over long distances, the slow evolution of the pulse width can be obtained by applying an appropriate multiple-scales averaging procedure to yield a new, scaled noise process effecting pulse broadening. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 42.81; 05.40 Keywords: Optical fibers; Random dispersion
1. Introduction Since the discovery of exact pulse-like solutions to nonlinear wave equations, the response of these solutions to perturbations has been of both q
This material is based upon work supported by the National Science Foundation under Grant No. 0073923. * Corresponding author. Tel.: +1-604-2913760; fax: +1-6042686657. E-mail addresses: [email protected] (T. Scha¨fer), [email protected] (R.O. Moore). 1 Also corresponding author. 2 Partially supported by the Natural Sciences and Engineering Research Council of Canada.
practical and theoretical interest. In optical fiber transmission, the robustness of pulses to the random variations of a system parameter is critically important to their feasibility as carriers of information. One such parameter, the group-velocity dispersion, is particularly sensitive to material imperfections and deformations, and has been demonstrated to exhibit variations when measured directly [1,2]. In the presence of these variations, a pulse that would otherwise preserve its shape can arrive at its destination distorted or destroyed [3]. The canonical evolution equation for pulses in optical fiber is the nonlinear Schr€ odinger equa-
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 1 3 2 - 6
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tion (NLSE) [4]. Two variants of this equation, with significantly different concomitant behaviors, are of current practical interest. The first is the classical (focusing) case, in which the dispersion and nonlinear coefficients are constant and of the same sign. The NLSE is then integrable by the inverse scattering transform [5] and has soliton solutions. The influence of random dispersion on classical solitons has been studied through a variational approach [6]. The second interesting case is the dispersion-managed version of the NLSE, where the dispersion coefficient is allowed to vary (deterministically) with propagation distance. This equation is not integrable in the classical sense [7] and is not solvable by direct analytical methods. Nevertheless, a reduction to ordinary differential equations by means of a Gauss–Hermite expansion [8,9] has been used to describe the properties of a class of special solutions of this equation, referred to as dispersionmanaged solitons because of their similarity to classical solitons. If only the highest mode in this Gauss–Hermite expansion is considered, the resulting ordinary differential equations are the same as those found previously by a variational approach or by the consideration of certain moments [10]. A reduction of the NLSE to simpler equations is important for two reasons. First, to generate reliable statistics quantifying the effect of random coefficients on pulse evolution requires a large number of simulations. This is particularly true if the most relevant occurrences are exactly those that are least likely to occur in a straightforward Monte-Carlo approach, such as bit errors. It is therefore of obvious benefit to reduce the computational time for each simulation to produce meaningful results in a minimum of time. The second argument in favor of finding a reduction to the NLSE is that it is often difficult to obtain much insight from a partial differential equation. A reduction to a dynamical system of finite dimension is generally more conducive to obtaining an analytical description of the pulse behavior. The resulting ordinary differential equations can be linearized about the dispersion-managed soliton for approximations valid over shorter distances, or they can be averaged over the dispersion map
period [11] to capture the behavior over longer scales. In summary, we demonstrate a simple and widely valid characterization of pulse broadening for dispersion-managed solitons under the influence of random dispersion. We obtain this characterization by considering a quasi-linear limit of the partial differential equation for small distances, and by averaging a corresponding system of nonlinear ordinary differential equations for large distances. These simple representations can be used far more easily than the original partial differential equation to relate the magnitude of random dispersion fluctuations in a fiber-optic line to the likelihood of a bit error due to pulse broadening.
2. Pulse propagation in dispersion-managed transmission lines 2.1. The cubic nonlinear Schr€odinger equation In this section, we quickly review the basic equation for modeling pulse propagation in optical fiber lines and explain the basic idea of dispersion management. For pulses with a width larger than a few picoseconds, MaxwellÕs equations are well approximated by the NLSE [4] i
oE a 1 o2 E 2 þ i E b2 2 þ cjEj E ¼ 0: oZ 2 2 oT
ð1Þ
Here, the complex-valued function EðZ; T Þ represents the envelope of the electric field in a 2 moving frame, where jEj is the optical power measured in W. The propagation distance z has units of km and the retarded time t is measured in ps. The attenuation is modeled by a, measured in km1 . The group velocity dispersion b2 has units of ps2 km1 , and the nonlinearity c is given in W1 km1 . Typically, a and b2 depend on z, as these parameters vary along the fiber line. Standard monomode fiber has a dispersion of jb2 j ¼ 10 ps2 km1 , a loss of a ¼ 0:2 dB km1 , and a nonlinearity of c 1 W1 km1 , but these values vary greatly depending on the type of fiber that is used. Rather than focus on a particular set of parameters, we will study a dimensionless ver-
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sion of Eq. (1) where we use the full-width-halfmaximum pulse width TFWHM and the peak power P to set our scales. While the pulse width is usually between 3 and 7 ps, the peak power can, depending on the line, vary greatly. It is common to set the peak power at several milliwatts; at this power level, our assumption of weak nonlinearity is clearly valid. A dispersion-managed fiber link consists, in the most simple case, of a concatenation of two fibers, one with positive dispersion and the other with negative dispersion. The length L of the dispersion map is then the sum of the lengths of the two fibers. We use this length to set the scale of our evolution variable z. In practice, L can assume arbitrary values, as the fibers can be cut at arbitrary lengths. A standard value for the length of the dispersion map is approximately 20 km. After rescaling to dimensionless variables by applying t¼
T ; TFWHM
z¼
Z ; L
E E~ ¼ pffiffiffi ; P
ð2Þ
and renormalizing to include the attenuation by applying a ð3Þ E~ðz; tÞ ¼ Aðz; tÞ exp z ; 2 we obtain, in dimensionless units, 2
iAz þ dðzÞAtt þ cðzÞjAj A ¼ 0;
ð4Þ
where b2 ðzÞ L ; 2 TFWHM Z z aðz0 Þ dz0 : cðzÞ ¼ cPL exp dðzÞ ¼
ð5Þ
0
Eq. (4) is the basic equation for this paper. The attenuated coefficient cðzÞ is typically recovered periodically by the placement of amplifiers along the fiber line. Here, we consider the lossless or distributed-gain case, where cðzÞ is constant. This arises when the effect of loss and gain are so minor or rapid as to be insignificant to the slow evolution of the pulse, or when the fiber attenuation is compensated continuously through distributed amplification.
355
2.2. Dispersion-managed solitons As mentioned before, the simplest dispersion map is formed by a concatenation of two sections of fiber with different dispersion values. This twostep map gives the following form for the dispersion coefficient: 8 0 6 z < 1=4; < dl þ dav ; ð6Þ dðzÞ ¼ dl þ dav ; 1=4 6 z < 3=4; : dl þ dav ; 3=4 6 z < 1; where dl represents the local, mean-zero variations of the dispersion and dav the residual dispersion. This relation defines the function dðzÞ in [0,1] and we extend this definition periodically for z > 1. It is well known that special solutions called solitons exist in the case of the integrable NLSE, for which d and c are both constant and positive [5]. It has also been suggested, however, that a similar class of solutions, referred to as dispersionmanaged solitons [12], exists in the case of a two-step map. A dispersion-managed soliton is a solution As of Eq. (4) with As ðz; tÞ ¼ F ðz; tÞei/ðzÞ ;
F ðz; tÞ ¼ F ðz þ 1; tÞ;
ð7Þ
0
where / ðzÞ is called the quasi-momentum. For a dispersion map given by Eq. (6), a family of dispersion-managed solitons, as well as the relation between their quasi-momentum and width, can be found analytically [10]. Due to the local variations of the dispersion, dispersion-managed solitons ‘‘breathe’’, or vary periodically in width over a map period. We might remark that, to our knowledge, no mathematical proof of the existence of dispersion-managed solitons is known. Numerical simulations [13], however, strongly support the existence of dispersion-managed solitons. Moreover, for practical purposes in a telecommunications line, the numerically known solutions are close enough to periodic to be useful for data transfer. 2.3. Random perturbations of the dispersion We are now ready to give a mathematical formulation of the problem considered in this paper: assume that we add to the dispersion map dðzÞ a
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small random term. How will a dispersion-managed soliton, that is otherwise periodic in the above sense, evolve under these random perturbations? Numerical simulations indicate that the pulse will broaden [3,14]. As we are interested in quantifying this phenomenon, we have first to define an appropriate measure of the broadening that arises from the presence of randomness. We choose the second moment for this purpose Z Z 2 2 DT t2 jAj dt t2 jAs j dt: ð8Þ In general, DT is a stochastic process depending on z. In the deterministic case, obviously A ¼ As and DT ¼ 0. Numerically, the value of DT can be found directly from simulations of the NLSE. In the following sections we will show how to compute DT approximately and compare the analytical results to direct numerical simulations.
3. Quasi-linear theory and pulse broadening 3.1. Theory Although nonlinearity is weak in most strongly dispersion-managed systems, its effect is still important as it determines the fixed point corresponding to the dispersion-managed soliton for a given pulse width. Let us therefore consider Eq. (4) with the following scaling 2 iAz þ d~ðzÞAtt þ dav Att þ jAj A ¼ 0:
ð9Þ
Here, we have explicitly separated the locally varying dispersion d~ from the small residual dispersion dav . The scaling for a dispersion-managed soliton As is then As ðz; tÞ ¼ F ðz; tÞe
ikz
:
ð10Þ
In the linear case where ¼ 0, any bounded function is a periodic solution of Eq. (9). When the nonlinearity is turned on by increasing from zero, however, very few (and in some parameter regimes, only one) solutions retain the periodicity described by Eq. (7). A dispersionmanaged soliton can therefore be characterized as a truly nonlinear object even when nonlinearity is small.
The functional form of As is mainly a Gaussian with higher order corrections [8,9]. We can therefore write As ðz; tÞ ¼ AG ðz; tÞ þ lAres ðz; tÞ;
ð11Þ
where AG denotes the Gaussian part and l 1. The Gaussian part AG is assumed to satisfy the equation iAGz þ d~ðzÞAGtt ¼ 0;
ð12Þ
which we can solve explicitly by Fourier transform. This fact that the core of a dispersionmanaged soliton exhibits primarily linear behavior has been successfully used, for example, to study pulse interactions in the time or frequency domain. As mentioned above, AG is not an arbitrary Gaussian; rather, its amplitude, width, and quasimomentum k are linked through the fiber nonlinearity. Because of the smallness of the parameters and l, we can describe the behavior of a dispersion-managed soliton under the influence of weak random dispersion, at least at the beginning of its evolution, by perturbation theory. A comparison to numerics will then show how far this approach is valid. We consider an additional random dispersion term in Eq. (9) and write iAz þ d~ðzÞAtt þ dav Att þ mdr ðzÞAtt þ jAj2 A ¼ 0: ð13Þ The noisy part of the dispersion is assumed to be low-amplitude (m 1) white noise, with a correlation function given by hdr ðzÞdr ðz0 Þi ¼ Ddðz z0 Þ:
ð14Þ
The amplitude of pffiffiffithe ffi noise term in the equation is then given by m D. Here, randomness is introduced as variations of the dispersion coefficient. Another possible source of randomness, not considered here, is random variation of the fiber spans [15]. We look for a solution to this equation of the form Aðz; tÞ ¼ As ðz; tÞ þ mA1 ðz; tÞ:
ð15Þ
The first order of this perturbation series will therefore be given by the linear equation,
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iA1z þ d~ðzÞA1tt þ dr ðzÞAGtt ¼ 0:
ð16Þ
The deviation from the zero-order solution As is driven by the term dr ðzÞAGtt , the influence of the randomness on the Gaussian part of the dispersion-managed soliton. The solution of this linear equation is given in Fourier domain by A^1 ðz; xÞ ¼ ix2 Wr ðzÞ A^G ðz; xÞ; Z z dr ðz0 Þ dz0 ; Wr ðzÞ ¼
ð17Þ
0
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deterministic system, the evolution on the slow scales can be described by the Gabitov–Turitsyn equation [17]. The equivalent PDE in the stochastic case, however, is difficult to treat analytically or numerically. We therefore look for a reduction to a finite-dimensional system in appropriate coordinates, not only to reproduce the quasi-linear result derived from the PDE, but also to extend our analysis to longer evolution scales. 4.1. The ODE model for width and chirp
where Wr is a Wiener process in z. This equation is the analytical solution of the problem, and all interesting quantities (e.g., statistical moments) can be calculated from this solution. As an example, the broadening of the pulse width due to randomness can be computed in the Fourier domain as Z 1 2 2 DT ¼ j A^x j j A^sx j dx 2p Z m A^Gx A^1x þ A^Gx A^1x dx 2p Z mWr ðzÞ 2ixð A^Gx A^G A^G A^Gx Þ dx: ð18Þ ¼ 2p This calculation shows that the evolution of DT is a scaled Wiener process, or random walk. We can proceed by evaluating the integral quantity in Eq. (18), noting that for the Gaussian core of a dispersion-managed soliton, we have ix2 R~ðzÞ
0 A^G ðz; xÞ ¼ e ð19Þ f^ðxÞ; R~ðzÞ ¼ d~ðzÞ; ^ where f is a real-valued function. [16]. This allows us to write a final closed-form expression for the pulse broadening Z ~ DT ¼ 8mWr ðzÞRðzÞ f 0 ðtÞ2 dt: ð20Þ
In Section 5 we will compare this result to numerical simulations. 4. Reduction of the NLSE to ordinary differential equations The above relation, Eq. (20), is valid for short propagation distances; for larger z, the nonlinearity will interact with the random dispersion. In a
We start again with the basic Eq. (4). The most interesting parameters of the pulse are its width and chirp, whose evolution can be described to a good approximation by ordinary differential equations. The appropriate self-similar ansatz for this variational reduction is
t MðzÞ 2 t : Aðz; tÞ ¼ aðzÞf exp ikðzÞ þ i T ðzÞ T ðzÞ ð21Þ This ansatz is inserted into the Lagrangian density of Eq. (4) and, after integrating the Lagrangian over the time t, one obtains from the Euler–Lagrange equations: dT ¼ 4dðzÞM; dz dM dðzÞC1 C2 ¼ 2: dz T3 T
ð22Þ ð23Þ
Here, C1 and C2 are integrals over the pulse shape f ðxÞ. Eqs. (22) and (23) are general and can be used for dispersion-managed or classical solitons; of course, the constants C1 and C2 will be different, as the pulse shape of a classical soliton is different from that of a dispersion-managed soliton. A system of equations identical to Eqs. (22) and (23) can be found by a momentum method or by truncation of an expansion of the solution into Gauss–Hermite eigenfunctions after applying the Lens transform [9]. In the case of a classical soliton without randomness, the dispersion d is constant and the form of the soliton is unchanged as it moves along the fiber. This is represented by a fixed point of Eqs. (22) and (23): in this case, the pulse width T is constant and the chirp M is zero.
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In dispersion-managed case, the dispersion varies periodically along the fiber line. Dispersionmanaged solitons are therefore represented by periodic solutions of Eqs. (22) and (23), and their trajectory in a phase plot of T against M is a closed curve. The associated Poincare plot (the points of the graph of T and M taken at the end of each period of the dispersion map) again consists of a single fixed point. 4.2. Influence of randomness: comparison to PDE dynamics Random dispersion introduces perturbations to these solutions. The first issue we address is to check numerically whether the reduced model of Eqs. (22) and (23) can be used to describe these perturbations. If this is true for single noise realizations, then it is clearly also true for ensemble quantities such as statistical moments. We therefore monitor the pulse width under the influence of arbitrary realizations of random dispersion. We first consider classical solitons with a hyperbolic secant trial function determining the constants C1 and C2 . Fig. 1 shows that, even for moderate values of z, the ODE solution and the PDE solution diverge significantly. While this divergence obviously depends on the noise strength,
Fig. 2. Numerical comparison of a single ODE simulation (solid line) with the same noise realization in a PDE simulation (dashed line). The amplitude of the noise is 0.02. The other parameters are dl ¼ 5, dav ¼ 0:15, C1 ¼ 1, C2 ¼ 0:1975.
it is evident that the ODE can only be taken to represent the PDE for small values of z. A similar comparison using a dispersion-managed soliton in a two-step dispersion map is depicted in Fig. 2. In this case, while there are again differences between the ODE and PDE results, they agree well for significantly longer scales than in the classical case. In particular, for moderate values of z relevant to real communication networks, the curves are almost indistinguishable. The excellent agreement demonstrates that MonteCarlo simulations using the PDE model can be replaced without significant loss of accuracy by much faster simulations using the ODE model.
5. Theory of the stochastic ordinary differential equations 5.1. Quasi-linear theory revisited
Fig. 1. Numerical comparison of a single ODE simulation (solid line) with the same noise realization in a PDE simulation (dashed line) for a classical soliton. Here, dðzÞ ¼ 1:0 and cðzÞ ¼ 2:0. The amplitude and the width of the initial pulse are one. The amplitude of the noise is 0.01.
If the ODE model in Eqs. (22) and (23) accurately represents the PDE model in Eq. (4), we should be able to rederive Eq. (20) using the ODEs. Let us rescale Eqs. (22) and (23) by taking C1 ¼ 1 and C2 ¼ N 2 . Again, we have a dispersion dðzÞ that consists of a varying part, a constant part, and a random part: dðzÞ ¼ d~ðzÞ þ dav þ mdr ðzÞ;
ð24Þ
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359
Furthermore, we can write T and M in expansions T ¼ TG þ lTres þ mT1 ; M ¼ MG þ lMres þ mM1 ;
ð25Þ
where TG and MG satisfy the equations TGz ¼ 4d~ðzÞMG ;
MGz ¼
d~ðzÞ : TG3
ð26Þ
The equations for T1 and M1 then become T1z ¼ 4dr ðzÞMG þ 4d~ðzÞM1 ; dr ðzÞ 3d~ðzÞT1 M1z ¼ 3 : TG4 TG
ð27Þ
This system of equations can be solved easily as it is equivalent (to the leading and first order) to sðzÞ ¼ TG þ mT1 ;
cðzÞ ¼ MG þ mM1 ;
sz ¼ 4ðd~ þ mdr Þc;
cz ¼
d~ þ mdr ; s3
ð28Þ ð29Þ
with the solution s2 ðqÞ ¼ s20 þ
4q2 ; s20
q0 ¼ d~ðzÞ þ mdr ðzÞ:
ð30Þ
As s measures the width of the pulse, we see again from this solution that, for small m, the change in width is proportional to R~ and the Wiener process Wr as in Eq. (20). We now compare the predictions of Eqs. (30) and (20) to numerical simulations. To do this, we write Eq. (20) explicitly with the system parameters. If we take the Gaussian approximation of the dispersion-managed soliton, we can write the initial condition (with T0 ¼ T ð0Þ and Mð0Þ ¼ 0) as
N t Að0; tÞ ¼ pffiffiffiffiffi B0 f0 ; T0 T0 ð31Þ f0 ðxÞ ¼ p1=4 expðx2 =2Þ; pffiffiffiffiffiffi where B20 ¼ 2 2p. This gives pffiffiffiffiffiffi 8mWr ðzÞR~ðzÞN 2 2p DT ¼ : ð32Þ T02 Fig. 3 shows that for small z, the analytical results describe very well the behavior of the ODE solution and therefore also the behavior of the full problem. For larger z, however, both the
Fig. 3. Numerical comparison of the numerical simulations (solid line) of the ODE with the PDE simulations (dots) and with the analytical model (dashed line). The amplitude of the noise is 0.05. The figure shows an average over 5000 realizations. The parameters of the considered system are dl ¼ 5:0, dav ¼ 0:15, C1 ¼ 1:0, C2 ¼ 0:117.
PDE and the ODE models predict an interaction between the noise and the nonlinear part, and this interaction renders the quasi-linear approach for finding the statistics of DT invalid. In order to find a way to describe the systemÕs properties beyond this quasi-linear regime, we have to consider the interaction of randomness and noise. 5.2. Averaging the ODEs in the presence of random dispersion We have already seen that random fluctuations in the dispersion drive the system away from its fixed point (the dispersion-managed soliton). Once the system has left this fixed point, the balance between nonlinearity and residual dispersion is affected, and this changes the dynamics. However, since nonlinearity and residual dispersion are small, these changes come into play on longer evolution scales. The systemÕs behavior can therefore be treated in the framework of a multiplescales expansion. As our equations are stochastic ordinary differential equations, we have to find a way to incorporate the noise in the multiple scale expansion. We start our analysis from Eqs. (22) and (23) written with the appropriate scaling
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dM dðzÞ N2 ¼ 3 2 dz T T
dT ¼ 4dðzÞM; dz
ð33Þ
and with the previously defined dispersion map dðzÞ ¼ d~ðzÞ þ dav þ mdr ðzÞ: It is possible to treat these equations directly by introducing an expansion at multiple scales, but the calculations are very cumbersome. We choose instead to use an appropriate coordinate transformation discussed previously [11], where the Hamiltonian of the zeroth order, given by TGz ¼ 4d~ðzÞMG ;
MGz ¼
d~ðzÞ ; TG3
is used to introduce action and angle variables, bG ¼ 4MG TG ;
XG ¼
4 þ 16MG2 ; TG2
ð34Þ
where XGz ¼ 0;
bGz ¼ d~ðzÞXG :
ð35Þ
We apply the same transformation to the full nonlinear system, Eq. (33), by introducing b ¼ 4MT ;
X¼
4 þ 16M 2 T2
ð36Þ
yielding the corresponding nonlinear equations
3=2 X Xz ¼ 8N 2 b ; 4 þ b2 ð37Þ
1=2 X 2 : bz ¼ dðzÞX 4N 4 þ b2 In order to find the slow evolution of these equations, we introduce multiple scales zn ¼ n z and the expansions XðzÞ ¼ X0 ðz0 ; z1 ; . . .Þ þ X1 ðz0 ; z1 ; . . .Þ þ bðzÞ ¼ b0 ðz0 ; z1 ; . . .Þ þ b1 ðz0 ; z1 ; . . .Þ þ
ð38Þ
In what follows, we will only consider the expansion to the order . The solution of the zeroth order of these equations is ~ ðz1 Þ; X0 ðz0 ; z1 Þ ¼ X ~ ðz1 ÞR~ðz0 Þ þ b~ðz1 Þ; b0 ðz0 ; z1 Þ ¼ X
ð39Þ
where dR~ ~ ¼ d ðz0 Þ: dz0
ð40Þ
The equations for the first order are !3=2 oX1 oX0 X 0 ¼ 8N 2 b0 ; oz0 oz1 4 þ b20 ob1 ~ ob m d X1 ¼ 0 þ dav X0 þ dr ðz0 ÞX0 oz0 oz1 !1=2 X0 4N 2 : 4 þ b20
ð41Þ
We can solve these equations by variation of constants and obtain as a solvability condition the equations for the slow variation of X and b, hence ~ ðz1 Þ and b~ðz1 Þ. The difficulty is the equations for X to see how the random term mdr ðz0 ÞX0 = can be incorporated in this formalism. If we integrate Eq. (41) over one period of d~ðz0 Þ, from z0 ¼ n to n þ 1, we obtain Z nþ1 dr ðz0 Þ dz0 ; n
a random variable whose statistical properties do not depend on n. Simple calculations show that this random term has a mean of zero and a variance equal to the strength D of the white noise, and that it is uncorrelated with values of dr ðz0 Þ for z0 > n þ 1 and z0 < n. In this way, the solvability condition produces a sequence of d-correlated random numbers for a given fiber line, where each element nn in the sequence is given by Z nþ1 nn ¼ dr ðz0 Þ dz0 : ð42Þ n
Each of these random elements contributes the ~ 1 = to the slow piecewise-constant quantity mnn X evolution of the angle variable b0 . Since this slow evolution occurs over the scale of z1 ¼ z0 , and since the random elements nn are piecewise-constant for sections of Oð1Þ in z0 and therefore of OðÞ in z1 , this piecewise-constant, d-correlated random process can be approximated by a white noise process Nðz1 Þ (or, similarly, can be thought of as a discrete representation of a continuous white noise process) with the correlation function:
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hNðz1 ÞNðz01 Þi ¼ Ddðz1 z01 Þ:
361
ð43Þ
~ ðz1 Þ and b~ðz1 Þ are The resulting equations for X then !3=2 Z 1 ~ ðz1 Þ dX X ðz ;z Þ 0 0 1 ¼ 8N 2 b0 ðz0 ;z1 Þ dz0 ; dz1 4 þ b0 ðz0 ;z1 Þ2 0 db~ðz1 Þ ~ ðz1 Þ þ 4N 2 J ðz1 Þ; ~ ðz1 Þ þ m Nðz1 ÞX ¼ dav X dz1 2 !3=2 Z 1 X0 ðz0 ;z1 Þ 4 ~ J ðz1 Þ 2Rðz0 Þb0 ðz0 ;z1 Þ 2 4 þ b0 ðz0 ;z1 Þ 0 !1=2 3 X0 ðz0 ;z1 Þ 5 dz0 : 2 4 þ b0 ðz0 ;z1 Þ ð44Þ Here, we can substitute the zero-order solution given by Eq. (39) and solve the integrals. Introducing the abbreviations ~ ðz1 Þ dl X ; 4 ð45Þ ~ ðz1 Þ dl X ~ ~s ðz1 Þ bðz1 Þ ; 4 we find 0 1 pffiffiffiffi ~ ~ dXðz1 Þ 1 1 XB C ¼ 16N 2 @ pffiffiffiffiffiffiffiffiffiffiffi2ffi pffiffiffiffiffiffiffiffiffiffiffi2ffi A; dz1 dl 4þ~sþ 4þ~s
Fig. 4. Comparison between numerical simulations using the stochastic ODEs governing the slow dynamics (Eq. (46), solid line) and simulations using the full nonlinear equations (Eq. (33), dashed line). The amplitude of the noise is 0.03; the other parameters for the full nonlinear simulations are dl ¼ 5:0, dav ¼ 0:15, C1 ¼ 1, C2 ¼ 0:1975, and Mð0Þ ¼ 0. The figure shows an average over 5000 realizations.
5.3. Comparison to numerical simulations
~sþ ðz1 Þ b~ðz1 Þ þ
2 db~ðz1 Þ ~ ðz1 Þþ 8N ~ ðz1 Þþ m Nðz1 ÞX pffiffiffiffi F ð~s ;~sþ Þ; ¼ dav X dz1 ~ dl X ! ! ~sþ ~s F ð~s ;~sþ Þ arcsinh arcsinh 2 2 0 1
1 1 B C þð~s ~sþ Þ@ pffiffiffiffiffiffiffiffiffiffiffiffi þ pffiffiffiffiffiffiffiffiffiffiffi2ffi A: 2 4þ~sþ 4þ~s
We are now in a position to compare the predictions for the evolution of DT defined in Eq. (8) as given by the slow evolution equations to the results of the full (i.e., unaveraged) nonlinear ODEs. For this purpose, we plot DT stroboscopically, taking only its values at the endpoints of each dispersion map, and compare these values to those obtained from the averaged ODEs. Fig. 4 shows reasonably good agreement between the slow evolution and the stroboscopically probed fast evolution. Most importantly, this agreement holds for much larger distances than the validity of the quasi-linear approximation given by Eq. (32).
6. Conclusion ð46Þ
It is simple to show that there exists a fixed point of this system representing the dispersionmanaged soliton, and that this fixed point has the form M ¼ 0 and T ¼ gðN Þ, where g establishes a family of pairs of N and T representing dispersionmanaged solitons.
We have verified that the key behavior of a dispersion-managed soliton under the influence of random dispersion is successfully captured through a simplified pair of stochastic ODEs. Over short distances, the leading order effect of randomness on pulse broadening can be obtained by considering just the linear evolution of the solitonÕs
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Gaussian core. In this limit, we have verified that the ODE prediction matches with the PDE prediction for pulse broadening, and that both match well with numerically obtained results over appropriately short evolution distances. Over longer scales for which no analytical results are available from the PDE, we have demonstrated that the slowly evolving stochastic ODEs obtained through a multiple-scales averaging procedure successfully reproduce the pulse broadening obtained by integrating the original ODE system. As the original ODE system has already been demonstrated to capture the evolution of a dispersion-managed soliton in the NLSE for relatively large z, this demonstrates the validity of the averaged stochastic ODEs we have derived. The importance of these results lies in their reduction of a PDE with rapidly varying coefficients to a much simpler ODE system. In both the quasilinear case and in the large-z averaged case, these ODEs are much more readily analyzed to produce outage probabilities and bit error rates for pulses in real fiber-optic lines. Similarly, the computational effort required to obtain pulse broadening statistics from the ODE system is several orders of magnitude less than would be required to integrate the original PDE system.
Acknowledgements The authors thank J. Bronski and I. Gabitov for many very valuable discussions.
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