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Modulational instability and generation of pulse trains in asymmetric dual-core nonlinear optical fibers
Physics Letters A 354 (2006) 366–372 www.elsevier.com/locate/pla
Modulational instability and generation of pulse trains in asymmetric dual-core nonlinear optical fibers R. Ganapathy a,∗,1 , Boris A. Malomed b , K. Porsezian a,∗ a Department of Physics, Pondicherry University R.V. Nagar, Pondicherry 605 014, India b Department of Interdisciplinary Studies, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel
Received 15 January 2006; accepted 1 February 2006 Available online 13 February 2006 Communicated by V.M. Agranovich
Abstract Instability of continuous-wave (CW) states is investigated in a system of two parallel-coupled fibers, with a pumped (active) nonlinear dispersive core and a lossy (passive) linear one. Modulational instability (MI) conditions are found from linearized equations for small perturbations, the results being drastically different for the normal and anomalous group-velocity dispersion (GVD) in the active core. Simulations of the full system demonstrate that the development of the MI in the former regime leads to establishment of a regular or chaotic array of pulses, if the MI saturates, or a chain of well-separated peaks with continuously growing amplitudes if the instability does not saturate. In the anomalous-GVD regime, a chain of return-to-zero (RZ) peaks, or a single RZ peak emerge, also with growing amplitudes. The latter can be used as a source of RZ pulses for optical telecommunications. © 2006 Elsevier B.V. All rights reserved. PACS: 42.81.Dp; 02.30.Ik; 42.65.Tg Keywords: Complex Ginzburg–Landau equation; Dual-core fiber; Modulational instability; Soliton; Group-velocity dispersion
1. Introduction Complex Ginzburg–Landau (CGL) equations are well known as universal models to describe pattern formation originating from the competition of loss, finite-bandwidth gain, nonlinearity, and dispersion. In particular, the cubic CGL equation is a model describing nonlinear development of instabilities in many areas, such as pipe flow and thermal convection in hydrodynamics, waves in nonequilibrium plasmas, chemical-reaction waves, etc. Dynamics generated by this equation was explored * Corresponding authors.
E-mail addresses: [email protected] (R. Ganapathy), [email protected] (B.A. Malomed), [email protected] (K. Porsezian). 1 Present address: Nonlinear Communications Research Division, Department of Electronics and Communication Engineering, Shanmugha Arts, Science, Technology and Research Academy, Deemed University, Tirumalaisamudram, Thanjavur-613 402, Tamil Nadu, India. 0375-9601/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2006.02.002
in detail, starting from the discovery of a fundamental solution in the form of a solitary pulse [1,2]. Results obtained in this and extended CGL models were summarized in reviews [3,4]. A number of models based on the CGL equations and coupled systems of such equations were developed in the context of nonlinear optics [4–6]. In particular, CGL equations describe the formation and interactions of solitary (alias return-to-zero, RZ) pulses in nonlinear optical fibers, in the presence of loss, gain, and filtering. In this context, a fundamental problem is instability of pulses, which is inevitable because the linear gain, necessary to compensate the fiber loss, makes the background around RZ pulses unstable (see, e.g., Ref. [7] and references therein). Several approaches aimed at stabilization of the solitary pulses were proposed. One of them relies on the synchronous gain modulation: a clock mechanism keeps the gain on only for a time slot allocated for the passage of the pulse through a given amplifier [8]. Another possibility is to combine amplifiers with saturable absorbers, which makes the gain effectively nonlinear; the simplest model of such a system is
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furnished by the CGL equation featuring linear and quintic loss and cubic gain. The corresponding cubic-quintic CGL equation was first introduced, as a phenomenological model, by Petviashvili and Sergeev [9]. Stabilization of pulses in the presence of linear gain and cubic nonlinearity is possible in a dual-core fiber, with the gain in one (active) core, while the other (passive) one, which actually provides for the stabilization, is lossy. The corresponding model, based on linearly coupled CGL equations, was introduced in Ref. [10] and studied by means of numerical methods in Ref. [11] (the concept of the dual-core soliton coupler, without loss and gain, was proposed in Refs. [12,13]). The equation describing the lossy core may actually be linear, making exact solutions for solitary pulses available [14], which helps to explore the pulse dynamics [15–18], including the application to multi-channel (WDM) systems [19] (in the latter case, each channel is stabilized by the linear coupling to its own linear lossy counterpart, different channels being coupled nonlinearly through the cross-phase modulation). Together with RZ pulses, the transmission of continuouswave (CW) signals in optical fibers is a subject of fundamental interest [5,6,19–21]. It is commonly known that CW states are subject to the modulational instability (MI) under the action of self-focusing nonlinearity acting in combination with anomalous group-velocity dispersion (GVD) [6,22,23]. In physically relevant situations, perturbations triggering the MI are seeded by a frequency-shifted signal wave [21]. Prior to fiber optics, the MI was predicted in hydrodynamics [24] and plasmas [25]. In optics, the MI was considered not only for uniform CW states, but also for extended (super-Gaussian) wave packets [26]. The MI in optical fibers offers a means to generate a regular array of RZ pulses, i.e., it may serve as a source of soliton trains [27]. In this capacity, it was adapted for the use in high-bit-rate optical telecommunications [28,29]. The study of CW states and their stability is also relevant to models based on equations of the CGL type, i.e., models of fiber-optic links or fiber-ring lasers, which include loss and bandwidth-limited gain (a combination of the gain and filtering). In the context of fluid mechanics, the MI in the CGL equation was studied in Ref. [30], and analyses aimed at applications to optical and other models based on CGL equations were reported in a number of later works, see, e.g., Refs. [28–33]. In this work, our first objective is to analyze the MI in the above-mentioned model of dual-core fibers with active and passive cores, the latter one obeying a linear equation. Instability conditions are obtained from linearized equations for small perturbations. Then, the development of the MI is explored in direct simulations, which demonstrates the formation of a regular or chaotic trains of pulses, with limited or growing amplitudes, depending on the sign of the GVD in the active core, and the size of the gain in it. It should be stressed that, despite many results obtained for solitons in the model of the asymmetric dual-core fiber, with gain in one core and loss in the other [10–17], CW states and their (in)stability, as well as generation of pulse trains, have not been studied in this system before. The Letter is organized as follows. In Section 2, the model is formulated, and linearized equations for small perturbations are
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derived. In Section 3, results for the MI, obtained from the linearized equations, are reported for the systems with both anomalous and normal GVD in the active core. Direct simulations of the full system are summarized in Section 4, and Section 5 concludes the Letter. 2. Formulation of the problem and linear stability analysis Light propagation in a dual-core nonlinear optical fiber with active and passive cores is governed by linearly coupled CGL equations for the respective amplitudes u and v of the electromagnetic waves [14]: 1 − iγ1 uτ τ + (σ + iγ2 )|u|2 u − iγ0 u + v = 0, iuz + (1) 2 ivz + k0 v − icvτ + iΓ0 v + u = 0, (2) where z, τ = t − z/Vgr and Vgr are, as usual, the propagation distance, reduced time, and group velocity of the carrier wave [6]. The anomalous and normal GVD in the active core corresponds, respectively, to σ = +1 and −1 (as shown in Ref. [14], the nonlinearity in the passive core may be neglected). Further, γ0 > 0 is the gain in the active core, γ1 > 0 and γ2 0 account for the dispersive and nonlinear loss, respectively (in other words, the terms with the coefficients γ0 and γ1 together account for the bandwidth-limited amplification), and Γ0 > 0 is the loss coefficient in the passive core. Finally, real parameters k0 and c measure the phase- and group-velocity mismatch between the cores, and the linear coupling constant is normalized to be 1. CW solutions to Eqs. (1) and (2) are looked for as v = P2 exp(iQz), u = P1 exp(iQz + iχ), (3) where real powers P1,2 of the two CW components, a common propagation constant of both components, Q, and a phase shift between them, χ , are determined by equations obtained by the substitution of ansatz (3) in Eqs. (1) and (2). First, P2 , χ and P1 can be eliminated: √ P1 P2 exp(−iχ) = , (4) Q − k0 − iΓ0 (Γ0 − γ0 )Q + k0 γ0 . P1 = (5) σ Γ0 + γ2 (k0 − Q) The remaining equation for the propagation constant Q is Q2 − k0 Q − (σ Q − σ k0 + Γ0 γ2 )P1 + Γ0 γ0 − 1 = 0,
(6)
where P1 should be substituted by expression (5). To analyze the MI of the CW states, we perturb them as follows: u = P1 + u exp(iQz + iχ), v = P2 + v exp(iQz), (7) where u and v are infinitesimal complex perturbations, which obey linearized equations,
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iuz
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1 − Qu + − iγ1 uτ τ + (σ + iγ2 )P1 2u + u ∗ 2
− iγ0 u + e−iχ v = 0, ivz
− v Q + k0 v
− icvτ
(8)
iχ
+ iΓ0 v + e u = 0,
(9)
the asterisk standing for the complex conjugation. To solve Eqs. (8) and (9), we define the Fourier transforms of u (z, τ ) and v (z, τ ), A(z, ω), B(z, ω) +∞ √ u (z, τ ), v (z, τ ) eiωτ dτ. = 1/ 2π −∞
The evolution of A and B is governed by equations obtained as Fourier transforms of Eqs. (8) and (9): ∂A + N11 A + N12 A† + N13 B = 0, ∂z ∂B i + N31 A + N33 B = 0, ∂z i
(10) (11)
where A† (ω) ≡ A∗ (−ω), and 1 − iγ1 ω2 + 2P1 (σ + iγ2 ) − iγ0 , N11 ≡ −Q − 2 N12 ≡ P1 (σ + iγ2 ), N13 ≡ exp(−iχ), N31 ≡ exp(iχ),
N33 ≡ −Q + k0 − cω + iΓ0 .
(12)
By adding to Eqs. (10) and (11) their complex conjugate counterparts, with ω → −ω, we arrive at a system of four linear homogeneous equations, dA + iNA = 0, dz
(13)
where A is a column of (A, A† , B, B † ) ≡ (A1 , A2 , A3 , A4 ), and N is a 4 × 4 matrix whose elements are given by Eq. (12) and N24 ≡ exp(iχ), N21 ≡ P1 (σ − iγ2 ), 1 + iγ1 ω2 + 2P1 (σ − iγ2 ) + iγ0 , N22 ≡ −Q − 2 N42 ≡ exp(−iχ), N44 ≡ −Q + k0 + cω − iΓ0
P1 (Q − k0 )2 cos(2χ) − Γ02 2 [(Q − k0 )2 + Γ0 ]2 − 2(Q − k0 )Γ0 sin(2χ) , Γ0 . χ = arctan k0 − Q P2 =
(14)
(elements of N which are not defined above are absent: N14 = N23 = N32 = N34 = N41 = N43 = 0). Stability eigenvalues can now be found from matrix system (13). 3. Continuous-wave states and their modulational instability To proceed, we set γ2 = 0, as the nonlinear loss in optical fibers is usually negligible. Then, Eqs. (6) and (5) yield the propagation constant of the CW state, Q = k0 ± (Γ0 /γ0 )(1 − γ0 Γ0 ), (15) and Eqs. (4) and (5) determine the powers of the CW components and phase shift between them,
1 − γ0 Γ0 P1 = σ k0 ± (Γ0 − γ0 ) , Γ0 γ0
(16)
Physical solutions are those with P1,2 real and positive. It follows from Eq. (16) that these conditions require, first of all, Γ0 γ0 < 1. Further, the condition P1 > 0 shows that, for k0 > 0, the solution branch corresponding to the upper signs in Eqs. (15) and (16) is physical, in the anomalous- and normalGVD regimes, σ = +1 and σ = −1, in regions, respectively, (cr) γ0 < γ0 (k0 , Γ0 ) and (cr)
γ0 (k0 , Γ0 ) < γ0 <
1 , Γ0
(17)
(cr)
is a root of the equation (cr) (cr) 1 − γ0 Γ0 = k0 . γ0 − Γ0 (cr) Γ0 γ0 where γ0
(18)
For the same case of k0 > 0, the solution branch corresponding to the lower signs in Eqs. (15) and (16) is physical, with (cr) σ = +1 and σ = −1, in the regions of γ0 < γ0 (−k0 , Γ0 ) (cr) and γ0 (−k0 , Γ0 ) < γ0 < 1/Γ0 , respectively, cf. Eq. (17). For k0 < 0, the situation is tantamount to the above one if k0 is replaced by −k0 , σ by −σ , and the two branches considered above are swapped. Proceeding to the MI analysis for CW solutions, the instability growth rate (alias instability gain) can be found, according to Eq. (13), through eigenvalues g of matrix N, as G = {Im(g)}max , where the maximum is to be taken among all eigenvalues with positive imaginary parts. In particular, an asymptotic form of the eigenvalues for ω2 → ∞ can be easily found in an explicit form. One pair of the eigenvalues is 1 1 2 2 , g1,2 = ± ω − 2σ P1 + Q + i γ0 − γ1 ω + O 2 |ω| (19) which shows that there is no instability in this asymptotic limit, provided that γ1 > 0. The second asymptotic pair does not give rise to any instability either, 1 g3,4 = cω ± Q − iΓ0 + O (20) . |ω| Note that the only actual difference of asymptotic dispersion relations (19) and (20) from their obvious counterparts for the trivial solution, u = v = 0, is the term ∓2σ P1 in the real part of pair (19). The full MI-gain spectrum was found from numerical computation of the eigenvalues. Fig. 1 shows the result for the CW solution corresponding to the upper sign in Eqs. (15) and (16) in the normal-GVD regime (σ = −1). As said above, the solution itself exists in region (17) of values of the gain in the active core. The maximum of instability growth rate G at ω = 0, which is observed in Fig. 1 and vanishes as γ0 → 0, is a consequence
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Fig. 1. The modulational-instability gain, G, vs. the perturbation frequency ω and gain coefficient γ0 in the active core, in the normal-dispersion regime, σ = −1, for CW solution (Eqs. (15) and (16)) with the upper sign. Other parameters are Γ0 = 0.12, γ1 = 0.3, k0 = 0.51, and c = 0.1. In this case, the CW (cr) solutions exist at γ0 = 0.20 < γ0 < 1/Γ0 = 8.33, as per Eqs. (17) and (18). In this figure and below, the results are displayed only in the region where the instability is present (i.e., negative-gain regions are not shown).
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Fig. 4. The same as in Fig. 3, but in the anomalous-dispersion regime (σ = +1). Other parameters are Γ0 = 0.12, k0 = 0.51, γ0 = 0.009, and c = 0.1.
Fig. 5. The modulational-instability gain, G, vs. the perturbation frequency ω and loss coefficient in the passive core, Γ0 , in the normal-dispersion regime, σ = −1, for CW solution (Eqs. (15) and (16)) with the upper sign. Other parameters are γ0 = 0.2, k0 = 0.51, γ1 = 0.1, and c = 0.05. In this case, the CW (cr) solutions exist in the interval of Γ0 ≈ 0.12 < Γ0 < 1/γ0 = 5.0, see the text. Fig. 2. The same as in Fig. 1 and for the same values of parameters, but in the anomalous-dispersion regime, σ = +1. In this case, the CW solutions exist at (cr) γ0 < γ0 ≈ 0.20.
Fig. 6. The same as in Fig. 5 and at the same values of parameters, but for the anomalous-dispersion regime, σ = +1. In this case, CW solution (15)–(16) (cr) with the upper sign exists in the interval of Γ0 > Γ0 ≈ 0.12.
Fig. 3. The modulational-instability gain, G, for CW solutions as a function of the perturbation frequency ω and the filtering coefficient in the active core, γ1 , in the normal-dispersion regime (σ = −1). Other parameters are Γ0 = 0.12, k0 = 0.51, γ0 = 0.9, and c = 0.1.
of the presence of the linear gain (γ0 ) in the active core. Naturally, the maximum value of G at ω = 0 and the width of the instability region around ω = 0 increase with γ0 . In the anomalous-GVD regime, σ = +1, the MI-gain spectrum is completely different, see a typical example in Fig. 2 (except for σ , parameters corresponding to this figure are the same as in Fig. 1, therefore the plot ends at precisely the same (cr) point, γ = γ0 ≈ 0.20, where Fig. 1 begins). In this case, G(ω = 0) = 0, while two symmetric maxima of G(ω) are found at finite values of the perturbation frequency, whose values and the width of the instability region around the maxima decrease with γ0 . Figs. 3 and 4 show the evolution of the MI-gain spectrum with the increase of the filtering coefficient, γ1 , in the normal-
and anomalous-GVD regimes, respectively. As above, in Fig. 3 the maximum value of the instability gain is at ω = 0. In the anomalous-GVD regime, G(ω = 0) = 0, and two symmetric maxima of G(ω) are again found at finite values of |ω|, as seen in Fig. 4. The dependence of the instability gain on the loss coefficient in the passive core, Γ0 , is illustrated by Figs. 5 and 6, for the normal- and anomalous-GVD regimes, respectively. In this connection, note that Eq. (18) may be regarded as one which determines a critical value of Γ0 for given γ0 ; obviously, (cr) (cr) the solution will be Γ0 (k0 , γ0 ) = γ0 (k0 → −k0 , Γ0 → γ0 ), (cr) where γ0 (k0 , Γ0 ) is defined as per Eq. (18). In particular, the CW solutions corresponding to k0 > 0 and the upper sign in Eqs. (15) and (16) exist in the normal- and anomalousGVD regimes, σ = −1 and σ = +1, in the regions of Γ0 < (cr) (cr) Γ0 (k0 , γ0 ) and Γ0 (k0 , γ0 ) < Γ0 < 1/γ0 , respectively. Finally, Figs. 7 and 8 display the dependence of the MI gain on the phase-velocity mismatch, k0 , in the normal- and anomalousdispersion regimes, respectively.
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Fig. 7. The modulational-instability gain, G, for CW solutions as a function of the perturbation frequency ω and the phase-velocity mismatch, k0 , in the normal-dispersion regime (σ = −1). Other parameters are Γ0 = 0.09, γ1 = 0.3, γ0 = 0.1, and c = 0.1.
Fig. 9. Evolution of the wave field in the active core with normal dispersion, σ = −1.0, displayed in terms of the power distribution, |u(z, τ )|2 . Parameters of the system and unperturbed CW are Γ0 = 0.12, γ0 = 0.2, γ1 = 0.3, k0 = 0.51, c = 0.1, and P1 = 0.0001634, P2 = 0.0002723. The amplitude and frequency of the initial perturbation in Eq. (21) are a0 = 0.009 and ω0 = 1.0. The eventual result is establishment of a periodic wavetrain, whose period is the same (2π ) as in the initial perturbation. Note very small values of the amplitudes of the initial CW state and established wavetrain.
Fig. 8. The same as in Fig. 7, but in the anomalous-dispersion regime (σ = +1). Other parameters are Γ0 = 0.09, γ1 = 0.3, γ0 = 0.09, and c = 0.1.
4. Direct simulations Direct numerical simulations were run to determine the outcome of the development of the MI. The CW state was initially perturbed as follows (cf. Eq. (7)):
u(z = 0, τ ) = P1 + a0 cos(ω0 τ ) eiχ ,
v(z = 0, τ ) = P2 + a0 cos(ω0 τ ) , (21) where a0 and ω0 are the amplitude and frequency of the small perturbation. Numerical simulations of Eqs. (1) and (2) with initial conditions (21) were performed (with the Mathematica software package) by means of a pseudospectral method, imposing periodic boundary conditions in τ . The simulations reveal outcomes of the instability development which depend on parameters of the system. If the amplitude of the unstable CW state is very small, a generic outcome in the model with the normal GVD is formation of periodic wavetrains in both the active and passive cores (see Figs. 9 and 10), whose period is imposed by the initial perturbation in Eq. (21), i.e., it is 2π/ω0 . If the amplitude of the unperturbed CW solution is essentially larger, the MI development leads to an irregular pattern, with a quasi-chaotic field configuration in the active core, as shown in Fig. 11. In this case, the shape of the field in the passive core also becomes irregular, see Fig. 12. In the above examples, the MI instability eventually saturates, leading to regular or irregular states with a limited amplitude. However, at larger values of gain γ0 the MI does not saturate, but rather generates a quasi-regular pattern consisting of peaks built on top a nonzero background, whose amplitudes grow indefinitely and attain extremely large values, as shown
Fig. 10. Counterpart of Fig. 9 in the passive core.
Fig. 11. The same as in Fig. 9, but for Γ0 = 0.009, γ0 = 0.01, γ1 = 0.005, k0 = 0.05, and c = 0.05. In this case, the modulational instability is seeded by initial perturbation (21) with a0 = 0.001 and ω0 = 0.01, and the instability development leads to a chaotic state.
Fig. 12. Counterpart of Fig. 11 in the passive core.
R. Ganapathy et al. / Physics Letters A 354 (2006) 366–372
Fig. 13. The same as in Fig. 9, but for Γ0 = 0.12, γ0 = 0.6, γ1 = 0.3, k0 = 0.9, c = 0.1, P1 = 1.07906, P2 = 6.47434, and a0 = 0.009, ω0 = 2.0. In this case, the CW instability generates (in the active core) a quasi-regular array of peaks with growing amplitudes.
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Fig. 15. The evolution of the field in the active core in the anomalous-dispersion regime, σ = +1. Parameters of the system, unperturbed CW, and initial perturbations are Γ0 = γ0 = 0.09, γ1 = 0.3, k0 = 1.25, c = 0.1, P1 = P2 = 1.25, and a0 = 0.0009, ω0 = 1.0. The CW instability development leads to an array of RZ peaks (in the active core).
Fig. 16. Counterpart of Fig. 15 in the passive core. Fig. 14. Counterpart of Fig. 13 in the passive core.
in Fig. 13. This outcome may be realized as a failure of the system to balance the gain with losses. In the same case, a regular pattern is also formed in the passive core, which then features something like a secondary instability, see Fig. 14. Note that, unlike the small-amplitude regular wavetrains displayed in Fig. 9, whose period was imposed by the initial perturbation, the spacing between the growing peaks is much larger than the initial perturbation period (in Fig. 13, it would be 2π/ω0 = π , while the actual spacing between the peaks is 11). The large temporal scale of the emerging pattern helps to understand why the filtering loss cannot balance the gain in this case. The outcome of the instability development is completely different in the model with the anomalous GVD. In this case, if we set γ0 = Γ0 , which yields P1 = P2 (equal amplitudes of both components), the MI triggered by the initial perturbation with a small amplitude generates a well-shaped chain of RZ peaks with growing amplitudes, as shown in Figs. 15 and 16. In this connection, we note that, according to Refs. [10,11,14], the zero solution (between the peaks) is stable at parameter values for which Figs. 15 and 16 were generated. In terms of the application to fiber-optic telecommunications, the regular array of well-separated peaks can be used as a source of RZ pulses, if the dual-core fiber is cut, and the array is released from the active core into the trunk fiber, at a value of z at which the amplitude of the pulses attains a necessary value. Keeping γ0 = Γ0 but increasing both values, along with other parameters, the nonlinear evolution leads to a single RZ pulse with a growing amplitude in the active core, see Fig. 17. This result can also be used in fiber optics, for the generation of a narrow high-power solitary pulse, by cutting the dual-core
Fig. 17. The same as in Fig. 15, but for Γ0 = 0.1, γ0 = 0.05, γ2 = 0.0, k0 = 0.2, c = 0.5, and a = 0.001, ω = 0.01.
fiber at an appropriate length. Simultaneously, the field in the passive core decays (not shown here). Finally, in the case of γ0 < Γ0 , simulations of the anomalous-GVD model demonstrate decay of the fields in both cores. 5. Conclusion We have revisited the model of the asymmetric dual-core optical fiber, with the Kerr nonlinearity, GVD (group-velocity dispersion), gain and dispersive loss in one core, and only loss in the other. The model is based on a system including a cubic complex Ginzburg–Landau equation coupled to a linear dissipative equation. While solitons in this system were investigated in detail in previous works, analysis of the modulational instability (MI) of CW states, and formation of regular or chaotic pulse arrays were not reported before. In this work, we have investigated the MI, using linearized equations for small perturbations, and simulated the full system, to determine the outcome of the development of the MI. It was found that the results are
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altogether different in the models with the normal and anomalous GVD. In the former case, depending on the parameters, the MI either saturates, leading to a pattern in the form of a periodic or chaotic pulse array, or does not saturate, generating a chain of peaks with continuously growing amplitudes. In the anomalous-GVD regime, the system may give rise to a chain of return-to-zero (RZ) peaks, or a single one, also with growing amplitudes. The latter result may be used as a source of RZ pulses for optical telecommunications.
[10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
Acknowledgements [20]
K.P. wishes to thank the DST, DAE-BRNS and UGC (Research Award), Government of India, for the financial support.
[21]
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Modulational instability and generation of pulse trains in asymmetric dual-core nonlinear optical fibers R. Ganapathy a,∗,1 , Boris A. Malomed b , K. Porsezian a,∗ a Department of Physics, Pondicherry University R.V. Nagar, Pondicherry 605 014, India b Department of Interdisciplinary Studies, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel
Received 15 January 2006; accepted 1 February 2006 Available online 13 February 2006 Communicated by V.M. Agranovich
Abstract Instability of continuous-wave (CW) states is investigated in a system of two parallel-coupled fibers, with a pumped (active) nonlinear dispersive core and a lossy (passive) linear one. Modulational instability (MI) conditions are found from linearized equations for small perturbations, the results being drastically different for the normal and anomalous group-velocity dispersion (GVD) in the active core. Simulations of the full system demonstrate that the development of the MI in the former regime leads to establishment of a regular or chaotic array of pulses, if the MI saturates, or a chain of well-separated peaks with continuously growing amplitudes if the instability does not saturate. In the anomalous-GVD regime, a chain of return-to-zero (RZ) peaks, or a single RZ peak emerge, also with growing amplitudes. The latter can be used as a source of RZ pulses for optical telecommunications. © 2006 Elsevier B.V. All rights reserved. PACS: 42.81.Dp; 02.30.Ik; 42.65.Tg Keywords: Complex Ginzburg–Landau equation; Dual-core fiber; Modulational instability; Soliton; Group-velocity dispersion
1. Introduction Complex Ginzburg–Landau (CGL) equations are well known as universal models to describe pattern formation originating from the competition of loss, finite-bandwidth gain, nonlinearity, and dispersion. In particular, the cubic CGL equation is a model describing nonlinear development of instabilities in many areas, such as pipe flow and thermal convection in hydrodynamics, waves in nonequilibrium plasmas, chemical-reaction waves, etc. Dynamics generated by this equation was explored * Corresponding authors.
E-mail addresses: [email protected] (R. Ganapathy), [email protected] (B.A. Malomed), [email protected] (K. Porsezian). 1 Present address: Nonlinear Communications Research Division, Department of Electronics and Communication Engineering, Shanmugha Arts, Science, Technology and Research Academy, Deemed University, Tirumalaisamudram, Thanjavur-613 402, Tamil Nadu, India. 0375-9601/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2006.02.002
in detail, starting from the discovery of a fundamental solution in the form of a solitary pulse [1,2]. Results obtained in this and extended CGL models were summarized in reviews [3,4]. A number of models based on the CGL equations and coupled systems of such equations were developed in the context of nonlinear optics [4–6]. In particular, CGL equations describe the formation and interactions of solitary (alias return-to-zero, RZ) pulses in nonlinear optical fibers, in the presence of loss, gain, and filtering. In this context, a fundamental problem is instability of pulses, which is inevitable because the linear gain, necessary to compensate the fiber loss, makes the background around RZ pulses unstable (see, e.g., Ref. [7] and references therein). Several approaches aimed at stabilization of the solitary pulses were proposed. One of them relies on the synchronous gain modulation: a clock mechanism keeps the gain on only for a time slot allocated for the passage of the pulse through a given amplifier [8]. Another possibility is to combine amplifiers with saturable absorbers, which makes the gain effectively nonlinear; the simplest model of such a system is
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furnished by the CGL equation featuring linear and quintic loss and cubic gain. The corresponding cubic-quintic CGL equation was first introduced, as a phenomenological model, by Petviashvili and Sergeev [9]. Stabilization of pulses in the presence of linear gain and cubic nonlinearity is possible in a dual-core fiber, with the gain in one (active) core, while the other (passive) one, which actually provides for the stabilization, is lossy. The corresponding model, based on linearly coupled CGL equations, was introduced in Ref. [10] and studied by means of numerical methods in Ref. [11] (the concept of the dual-core soliton coupler, without loss and gain, was proposed in Refs. [12,13]). The equation describing the lossy core may actually be linear, making exact solutions for solitary pulses available [14], which helps to explore the pulse dynamics [15–18], including the application to multi-channel (WDM) systems [19] (in the latter case, each channel is stabilized by the linear coupling to its own linear lossy counterpart, different channels being coupled nonlinearly through the cross-phase modulation). Together with RZ pulses, the transmission of continuouswave (CW) signals in optical fibers is a subject of fundamental interest [5,6,19–21]. It is commonly known that CW states are subject to the modulational instability (MI) under the action of self-focusing nonlinearity acting in combination with anomalous group-velocity dispersion (GVD) [6,22,23]. In physically relevant situations, perturbations triggering the MI are seeded by a frequency-shifted signal wave [21]. Prior to fiber optics, the MI was predicted in hydrodynamics [24] and plasmas [25]. In optics, the MI was considered not only for uniform CW states, but also for extended (super-Gaussian) wave packets [26]. The MI in optical fibers offers a means to generate a regular array of RZ pulses, i.e., it may serve as a source of soliton trains [27]. In this capacity, it was adapted for the use in high-bit-rate optical telecommunications [28,29]. The study of CW states and their stability is also relevant to models based on equations of the CGL type, i.e., models of fiber-optic links or fiber-ring lasers, which include loss and bandwidth-limited gain (a combination of the gain and filtering). In the context of fluid mechanics, the MI in the CGL equation was studied in Ref. [30], and analyses aimed at applications to optical and other models based on CGL equations were reported in a number of later works, see, e.g., Refs. [28–33]. In this work, our first objective is to analyze the MI in the above-mentioned model of dual-core fibers with active and passive cores, the latter one obeying a linear equation. Instability conditions are obtained from linearized equations for small perturbations. Then, the development of the MI is explored in direct simulations, which demonstrates the formation of a regular or chaotic trains of pulses, with limited or growing amplitudes, depending on the sign of the GVD in the active core, and the size of the gain in it. It should be stressed that, despite many results obtained for solitons in the model of the asymmetric dual-core fiber, with gain in one core and loss in the other [10–17], CW states and their (in)stability, as well as generation of pulse trains, have not been studied in this system before. The Letter is organized as follows. In Section 2, the model is formulated, and linearized equations for small perturbations are
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derived. In Section 3, results for the MI, obtained from the linearized equations, are reported for the systems with both anomalous and normal GVD in the active core. Direct simulations of the full system are summarized in Section 4, and Section 5 concludes the Letter. 2. Formulation of the problem and linear stability analysis Light propagation in a dual-core nonlinear optical fiber with active and passive cores is governed by linearly coupled CGL equations for the respective amplitudes u and v of the electromagnetic waves [14]: 1 − iγ1 uτ τ + (σ + iγ2 )|u|2 u − iγ0 u + v = 0, iuz + (1) 2 ivz + k0 v − icvτ + iΓ0 v + u = 0, (2) where z, τ = t − z/Vgr and Vgr are, as usual, the propagation distance, reduced time, and group velocity of the carrier wave [6]. The anomalous and normal GVD in the active core corresponds, respectively, to σ = +1 and −1 (as shown in Ref. [14], the nonlinearity in the passive core may be neglected). Further, γ0 > 0 is the gain in the active core, γ1 > 0 and γ2 0 account for the dispersive and nonlinear loss, respectively (in other words, the terms with the coefficients γ0 and γ1 together account for the bandwidth-limited amplification), and Γ0 > 0 is the loss coefficient in the passive core. Finally, real parameters k0 and c measure the phase- and group-velocity mismatch between the cores, and the linear coupling constant is normalized to be 1. CW solutions to Eqs. (1) and (2) are looked for as v = P2 exp(iQz), u = P1 exp(iQz + iχ), (3) where real powers P1,2 of the two CW components, a common propagation constant of both components, Q, and a phase shift between them, χ , are determined by equations obtained by the substitution of ansatz (3) in Eqs. (1) and (2). First, P2 , χ and P1 can be eliminated: √ P1 P2 exp(−iχ) = , (4) Q − k0 − iΓ0 (Γ0 − γ0 )Q + k0 γ0 . P1 = (5) σ Γ0 + γ2 (k0 − Q) The remaining equation for the propagation constant Q is Q2 − k0 Q − (σ Q − σ k0 + Γ0 γ2 )P1 + Γ0 γ0 − 1 = 0,
(6)
where P1 should be substituted by expression (5). To analyze the MI of the CW states, we perturb them as follows: u = P1 + u exp(iQz + iχ), v = P2 + v exp(iQz), (7) where u and v are infinitesimal complex perturbations, which obey linearized equations,
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iuz
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1 − Qu + − iγ1 uτ τ + (σ + iγ2 )P1 2u + u ∗ 2
− iγ0 u + e−iχ v = 0, ivz
− v Q + k0 v
− icvτ
(8)
iχ
+ iΓ0 v + e u = 0,
(9)
the asterisk standing for the complex conjugation. To solve Eqs. (8) and (9), we define the Fourier transforms of u (z, τ ) and v (z, τ ), A(z, ω), B(z, ω) +∞ √ u (z, τ ), v (z, τ ) eiωτ dτ. = 1/ 2π −∞
The evolution of A and B is governed by equations obtained as Fourier transforms of Eqs. (8) and (9): ∂A + N11 A + N12 A† + N13 B = 0, ∂z ∂B i + N31 A + N33 B = 0, ∂z i
(10) (11)
where A† (ω) ≡ A∗ (−ω), and 1 − iγ1 ω2 + 2P1 (σ + iγ2 ) − iγ0 , N11 ≡ −Q − 2 N12 ≡ P1 (σ + iγ2 ), N13 ≡ exp(−iχ), N31 ≡ exp(iχ),
N33 ≡ −Q + k0 − cω + iΓ0 .
(12)
By adding to Eqs. (10) and (11) their complex conjugate counterparts, with ω → −ω, we arrive at a system of four linear homogeneous equations, dA + iNA = 0, dz
(13)
where A is a column of (A, A† , B, B † ) ≡ (A1 , A2 , A3 , A4 ), and N is a 4 × 4 matrix whose elements are given by Eq. (12) and N24 ≡ exp(iχ), N21 ≡ P1 (σ − iγ2 ), 1 + iγ1 ω2 + 2P1 (σ − iγ2 ) + iγ0 , N22 ≡ −Q − 2 N42 ≡ exp(−iχ), N44 ≡ −Q + k0 + cω − iΓ0
P1 (Q − k0 )2 cos(2χ) − Γ02 2 [(Q − k0 )2 + Γ0 ]2 − 2(Q − k0 )Γ0 sin(2χ) , Γ0 . χ = arctan k0 − Q P2 =
(14)
(elements of N which are not defined above are absent: N14 = N23 = N32 = N34 = N41 = N43 = 0). Stability eigenvalues can now be found from matrix system (13). 3. Continuous-wave states and their modulational instability To proceed, we set γ2 = 0, as the nonlinear loss in optical fibers is usually negligible. Then, Eqs. (6) and (5) yield the propagation constant of the CW state, Q = k0 ± (Γ0 /γ0 )(1 − γ0 Γ0 ), (15) and Eqs. (4) and (5) determine the powers of the CW components and phase shift between them,
1 − γ0 Γ0 P1 = σ k0 ± (Γ0 − γ0 ) , Γ0 γ0
(16)
Physical solutions are those with P1,2 real and positive. It follows from Eq. (16) that these conditions require, first of all, Γ0 γ0 < 1. Further, the condition P1 > 0 shows that, for k0 > 0, the solution branch corresponding to the upper signs in Eqs. (15) and (16) is physical, in the anomalous- and normalGVD regimes, σ = +1 and σ = −1, in regions, respectively, (cr) γ0 < γ0 (k0 , Γ0 ) and (cr)
γ0 (k0 , Γ0 ) < γ0 <
1 , Γ0
(17)
(cr)
is a root of the equation (cr) (cr) 1 − γ0 Γ0 = k0 . γ0 − Γ0 (cr) Γ0 γ0 where γ0
(18)
For the same case of k0 > 0, the solution branch corresponding to the lower signs in Eqs. (15) and (16) is physical, with (cr) σ = +1 and σ = −1, in the regions of γ0 < γ0 (−k0 , Γ0 ) (cr) and γ0 (−k0 , Γ0 ) < γ0 < 1/Γ0 , respectively, cf. Eq. (17). For k0 < 0, the situation is tantamount to the above one if k0 is replaced by −k0 , σ by −σ , and the two branches considered above are swapped. Proceeding to the MI analysis for CW solutions, the instability growth rate (alias instability gain) can be found, according to Eq. (13), through eigenvalues g of matrix N, as G = {Im(g)}max , where the maximum is to be taken among all eigenvalues with positive imaginary parts. In particular, an asymptotic form of the eigenvalues for ω2 → ∞ can be easily found in an explicit form. One pair of the eigenvalues is 1 1 2 2 , g1,2 = ± ω − 2σ P1 + Q + i γ0 − γ1 ω + O 2 |ω| (19) which shows that there is no instability in this asymptotic limit, provided that γ1 > 0. The second asymptotic pair does not give rise to any instability either, 1 g3,4 = cω ± Q − iΓ0 + O (20) . |ω| Note that the only actual difference of asymptotic dispersion relations (19) and (20) from their obvious counterparts for the trivial solution, u = v = 0, is the term ∓2σ P1 in the real part of pair (19). The full MI-gain spectrum was found from numerical computation of the eigenvalues. Fig. 1 shows the result for the CW solution corresponding to the upper sign in Eqs. (15) and (16) in the normal-GVD regime (σ = −1). As said above, the solution itself exists in region (17) of values of the gain in the active core. The maximum of instability growth rate G at ω = 0, which is observed in Fig. 1 and vanishes as γ0 → 0, is a consequence
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Fig. 1. The modulational-instability gain, G, vs. the perturbation frequency ω and gain coefficient γ0 in the active core, in the normal-dispersion regime, σ = −1, for CW solution (Eqs. (15) and (16)) with the upper sign. Other parameters are Γ0 = 0.12, γ1 = 0.3, k0 = 0.51, and c = 0.1. In this case, the CW (cr) solutions exist at γ0 = 0.20 < γ0 < 1/Γ0 = 8.33, as per Eqs. (17) and (18). In this figure and below, the results are displayed only in the region where the instability is present (i.e., negative-gain regions are not shown).
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Fig. 4. The same as in Fig. 3, but in the anomalous-dispersion regime (σ = +1). Other parameters are Γ0 = 0.12, k0 = 0.51, γ0 = 0.009, and c = 0.1.
Fig. 5. The modulational-instability gain, G, vs. the perturbation frequency ω and loss coefficient in the passive core, Γ0 , in the normal-dispersion regime, σ = −1, for CW solution (Eqs. (15) and (16)) with the upper sign. Other parameters are γ0 = 0.2, k0 = 0.51, γ1 = 0.1, and c = 0.05. In this case, the CW (cr) solutions exist in the interval of Γ0 ≈ 0.12 < Γ0 < 1/γ0 = 5.0, see the text. Fig. 2. The same as in Fig. 1 and for the same values of parameters, but in the anomalous-dispersion regime, σ = +1. In this case, the CW solutions exist at (cr) γ0 < γ0 ≈ 0.20.
Fig. 6. The same as in Fig. 5 and at the same values of parameters, but for the anomalous-dispersion regime, σ = +1. In this case, CW solution (15)–(16) (cr) with the upper sign exists in the interval of Γ0 > Γ0 ≈ 0.12.
Fig. 3. The modulational-instability gain, G, for CW solutions as a function of the perturbation frequency ω and the filtering coefficient in the active core, γ1 , in the normal-dispersion regime (σ = −1). Other parameters are Γ0 = 0.12, k0 = 0.51, γ0 = 0.9, and c = 0.1.
of the presence of the linear gain (γ0 ) in the active core. Naturally, the maximum value of G at ω = 0 and the width of the instability region around ω = 0 increase with γ0 . In the anomalous-GVD regime, σ = +1, the MI-gain spectrum is completely different, see a typical example in Fig. 2 (except for σ , parameters corresponding to this figure are the same as in Fig. 1, therefore the plot ends at precisely the same (cr) point, γ = γ0 ≈ 0.20, where Fig. 1 begins). In this case, G(ω = 0) = 0, while two symmetric maxima of G(ω) are found at finite values of the perturbation frequency, whose values and the width of the instability region around the maxima decrease with γ0 . Figs. 3 and 4 show the evolution of the MI-gain spectrum with the increase of the filtering coefficient, γ1 , in the normal-
and anomalous-GVD regimes, respectively. As above, in Fig. 3 the maximum value of the instability gain is at ω = 0. In the anomalous-GVD regime, G(ω = 0) = 0, and two symmetric maxima of G(ω) are again found at finite values of |ω|, as seen in Fig. 4. The dependence of the instability gain on the loss coefficient in the passive core, Γ0 , is illustrated by Figs. 5 and 6, for the normal- and anomalous-GVD regimes, respectively. In this connection, note that Eq. (18) may be regarded as one which determines a critical value of Γ0 for given γ0 ; obviously, (cr) (cr) the solution will be Γ0 (k0 , γ0 ) = γ0 (k0 → −k0 , Γ0 → γ0 ), (cr) where γ0 (k0 , Γ0 ) is defined as per Eq. (18). In particular, the CW solutions corresponding to k0 > 0 and the upper sign in Eqs. (15) and (16) exist in the normal- and anomalousGVD regimes, σ = −1 and σ = +1, in the regions of Γ0 < (cr) (cr) Γ0 (k0 , γ0 ) and Γ0 (k0 , γ0 ) < Γ0 < 1/γ0 , respectively. Finally, Figs. 7 and 8 display the dependence of the MI gain on the phase-velocity mismatch, k0 , in the normal- and anomalousdispersion regimes, respectively.
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Fig. 7. The modulational-instability gain, G, for CW solutions as a function of the perturbation frequency ω and the phase-velocity mismatch, k0 , in the normal-dispersion regime (σ = −1). Other parameters are Γ0 = 0.09, γ1 = 0.3, γ0 = 0.1, and c = 0.1.
Fig. 9. Evolution of the wave field in the active core with normal dispersion, σ = −1.0, displayed in terms of the power distribution, |u(z, τ )|2 . Parameters of the system and unperturbed CW are Γ0 = 0.12, γ0 = 0.2, γ1 = 0.3, k0 = 0.51, c = 0.1, and P1 = 0.0001634, P2 = 0.0002723. The amplitude and frequency of the initial perturbation in Eq. (21) are a0 = 0.009 and ω0 = 1.0. The eventual result is establishment of a periodic wavetrain, whose period is the same (2π ) as in the initial perturbation. Note very small values of the amplitudes of the initial CW state and established wavetrain.
Fig. 8. The same as in Fig. 7, but in the anomalous-dispersion regime (σ = +1). Other parameters are Γ0 = 0.09, γ1 = 0.3, γ0 = 0.09, and c = 0.1.
4. Direct simulations Direct numerical simulations were run to determine the outcome of the development of the MI. The CW state was initially perturbed as follows (cf. Eq. (7)):
u(z = 0, τ ) = P1 + a0 cos(ω0 τ ) eiχ ,
v(z = 0, τ ) = P2 + a0 cos(ω0 τ ) , (21) where a0 and ω0 are the amplitude and frequency of the small perturbation. Numerical simulations of Eqs. (1) and (2) with initial conditions (21) were performed (with the Mathematica software package) by means of a pseudospectral method, imposing periodic boundary conditions in τ . The simulations reveal outcomes of the instability development which depend on parameters of the system. If the amplitude of the unstable CW state is very small, a generic outcome in the model with the normal GVD is formation of periodic wavetrains in both the active and passive cores (see Figs. 9 and 10), whose period is imposed by the initial perturbation in Eq. (21), i.e., it is 2π/ω0 . If the amplitude of the unperturbed CW solution is essentially larger, the MI development leads to an irregular pattern, with a quasi-chaotic field configuration in the active core, as shown in Fig. 11. In this case, the shape of the field in the passive core also becomes irregular, see Fig. 12. In the above examples, the MI instability eventually saturates, leading to regular or irregular states with a limited amplitude. However, at larger values of gain γ0 the MI does not saturate, but rather generates a quasi-regular pattern consisting of peaks built on top a nonzero background, whose amplitudes grow indefinitely and attain extremely large values, as shown
Fig. 10. Counterpart of Fig. 9 in the passive core.
Fig. 11. The same as in Fig. 9, but for Γ0 = 0.009, γ0 = 0.01, γ1 = 0.005, k0 = 0.05, and c = 0.05. In this case, the modulational instability is seeded by initial perturbation (21) with a0 = 0.001 and ω0 = 0.01, and the instability development leads to a chaotic state.
Fig. 12. Counterpart of Fig. 11 in the passive core.
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Fig. 13. The same as in Fig. 9, but for Γ0 = 0.12, γ0 = 0.6, γ1 = 0.3, k0 = 0.9, c = 0.1, P1 = 1.07906, P2 = 6.47434, and a0 = 0.009, ω0 = 2.0. In this case, the CW instability generates (in the active core) a quasi-regular array of peaks with growing amplitudes.
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Fig. 15. The evolution of the field in the active core in the anomalous-dispersion regime, σ = +1. Parameters of the system, unperturbed CW, and initial perturbations are Γ0 = γ0 = 0.09, γ1 = 0.3, k0 = 1.25, c = 0.1, P1 = P2 = 1.25, and a0 = 0.0009, ω0 = 1.0. The CW instability development leads to an array of RZ peaks (in the active core).
Fig. 16. Counterpart of Fig. 15 in the passive core. Fig. 14. Counterpart of Fig. 13 in the passive core.
in Fig. 13. This outcome may be realized as a failure of the system to balance the gain with losses. In the same case, a regular pattern is also formed in the passive core, which then features something like a secondary instability, see Fig. 14. Note that, unlike the small-amplitude regular wavetrains displayed in Fig. 9, whose period was imposed by the initial perturbation, the spacing between the growing peaks is much larger than the initial perturbation period (in Fig. 13, it would be 2π/ω0 = π , while the actual spacing between the peaks is 11). The large temporal scale of the emerging pattern helps to understand why the filtering loss cannot balance the gain in this case. The outcome of the instability development is completely different in the model with the anomalous GVD. In this case, if we set γ0 = Γ0 , which yields P1 = P2 (equal amplitudes of both components), the MI triggered by the initial perturbation with a small amplitude generates a well-shaped chain of RZ peaks with growing amplitudes, as shown in Figs. 15 and 16. In this connection, we note that, according to Refs. [10,11,14], the zero solution (between the peaks) is stable at parameter values for which Figs. 15 and 16 were generated. In terms of the application to fiber-optic telecommunications, the regular array of well-separated peaks can be used as a source of RZ pulses, if the dual-core fiber is cut, and the array is released from the active core into the trunk fiber, at a value of z at which the amplitude of the pulses attains a necessary value. Keeping γ0 = Γ0 but increasing both values, along with other parameters, the nonlinear evolution leads to a single RZ pulse with a growing amplitude in the active core, see Fig. 17. This result can also be used in fiber optics, for the generation of a narrow high-power solitary pulse, by cutting the dual-core
Fig. 17. The same as in Fig. 15, but for Γ0 = 0.1, γ0 = 0.05, γ2 = 0.0, k0 = 0.2, c = 0.5, and a = 0.001, ω = 0.01.
fiber at an appropriate length. Simultaneously, the field in the passive core decays (not shown here). Finally, in the case of γ0 < Γ0 , simulations of the anomalous-GVD model demonstrate decay of the fields in both cores. 5. Conclusion We have revisited the model of the asymmetric dual-core optical fiber, with the Kerr nonlinearity, GVD (group-velocity dispersion), gain and dispersive loss in one core, and only loss in the other. The model is based on a system including a cubic complex Ginzburg–Landau equation coupled to a linear dissipative equation. While solitons in this system were investigated in detail in previous works, analysis of the modulational instability (MI) of CW states, and formation of regular or chaotic pulse arrays were not reported before. In this work, we have investigated the MI, using linearized equations for small perturbations, and simulated the full system, to determine the outcome of the development of the MI. It was found that the results are
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altogether different in the models with the normal and anomalous GVD. In the former case, depending on the parameters, the MI either saturates, leading to a pattern in the form of a periodic or chaotic pulse array, or does not saturate, generating a chain of peaks with continuously growing amplitudes. In the anomalous-GVD regime, the system may give rise to a chain of return-to-zero (RZ) peaks, or a single one, also with growing amplitudes. The latter result may be used as a source of RZ pulses for optical telecommunications.
[10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
Acknowledgements [20]
K.P. wishes to thank the DST, DAE-BRNS and UGC (Research Award), Government of India, for the financial support.
[21]
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