Dispersion-managed Solitons

Transcription

1 Proceedings of the Steklov Institute of Mathematics, Suppl., 3, pp. S98 S7. Translated from Trudy Instituta Matematiki i Mekhaniki UrO RAN, Vol. 9, No., 3. Original Russian Text Copyright c 3 by Kalisch, Pelinovsky. English Translation Copyright c 3 by MAIK Nauka/Interperiodica (Russia). Dispersion-managed Solitons in the Limit of Large Energy Henrik Kalisch and Dmitry Pelinovsky Received November, Abstract We study a dynamical system for Gaussian dispersion-managed pulses in the it of large pulse energy and map strength. We show numerically and asymptotically that the pulse energy E grows with the map strength S as E D + S, where D+ is the effective dispersion coefficient of the focusing fiber. We also inspect leading-order asymptotic solutions that describe variations of the pulse parameters along the optical fiber.. INTRODUCTION Return-to-ero pulses and dispersion-managed solitons are two elements in the endeavor to achieve higher bit rates for long-haul data transmission in dispersion-compensated optical fibers [4]. Dispersion-managed solitons can be modeled by special solutions of the periodic nonlinear Schrödinger equation [7]. An accurate approximation of dispersion-managed solitons is achieved with Gaussian pulses that are localied in time and vary periodically in distance along the optical fiber [3]. The Gaussian pulses are parametried by the pulse energy, or, equivalently, by the map strength of the dispersion variations [8]. We have recently studied Gaussian pulses that approximate dispersion-managed solitons in the it of short dispersion maps, when normal-form transformations can be applied for averaging the periodic nonlinear Schrödinger equation [6]. Here we study properties of the Gaussian pulses in the it of large pulse energy and map strength, which is the opposite it to the one in [6]. In optical communications, pulses of large energy are preferred because of a favorable signal-to-noise ratio, while strongly dispersion-managed maps reduce the four-wave mixing and improve the bit rates of communication channels in the fiber bandwidth. In this context, we show that the pulse energy E grows with larger map strength S as E D + S,whereD+ is the effective dispersion coefficient of the focusing fiber. This main result is derived by means of asymptotic analysis and is confirmed by direct numerical simulations.. MODEL Transmission of optical pulses in a dispersion-managed optical fiber is modeled by the periodic nonlinear Schrödinger (NLS) equation, This article was submitted by the authors in English Department of Mathematics, McMaster University 8 Main Street West, Hamilton, Ontario, Canada L8S 4K iu + d()u tt + γ() u u =, (.) S98

2 DISPERSION-MANAGED SOLITONS IN THE LIMIT OF LARGE ENERGY S99 where d() is a periodic dispersion coefficient and γ() is a nonlinearity coefficient, modified due to fiber loss and amplification [8]. The dispersion map period and variance can be normalied by a simple scaling transformation [6], such that the dispersion map d() takes the form d() =d + d +, n<<n+ l, d() =d + d, n+ l<<n+, n Z, (.) where l is the length of the focusing fiber such that <l<, d is the average dispersion, and d + =/l and d = /( l) are ero-mean variations of the dispersion coefficients in the focusing and defocusing fibers. In the same normaliation, the nonlinearity coefficient γ() takes the form γ() =Ke α( n ), n+ a n ++ a, n Z, (.3) where α> is the decay rate due to fiber losses, a is a position of an amplifier in the dispersion mapsuchthat a, and K is a constant that normalies the mean of the coefficient γ() to one, K = αeαa e α. Dispersion-managed solitons are approximated by the Gaussian pulses in the following form: ( u(,t) =c exp t ) a ib + iφ, (.4) where a(), b(), c(), and φ() are parameters of the Gaussian pulses varying with the distance along the fiber. The amplitude c() follows from the power balance equation as c = Ea/ (a + ib), where E is the energy of the Gaussian pulse. The width a() and the chirp b() satisfy the dynamical system da = Eγ()a5/ b d (a + b ) 3/, db = d() Eγ()a3/ (a b (.5) ) d (a + b ) 3/, and the phase shift parameter φ() is decoupled from the system with the separate equation dφ d = Eγ()a/ (3a +5b ) 8(a + b ) 3/. (.6) The map strength S of the dispersion map is defined by the minimal values of a() onaperiod : S = (.7) min a(). From (.4) and (.5) it is clear that b( )=if is a minimum of a() and that the map strength S is inverse proportional to the minimal pulse width [6]. Dispersion-managed solitons are described by the Gaussian pulses (.4), computed at a periodic solution of the system (.5) such that a(+) = a(), b(+) = b(), and φ(+) = φ()+m, where PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS Suppl. 3

3 S KALISCH, PELINOVSKY M depends on E and S [6]. The existence of periodic solutions in the system (.5) is rigorously proved in [, ]. Numerical simulations in [3, 7] confirm that the Gaussian pulses accurately approximate dispersion-managed solitons in the periodic NLS equation. It is shown in [5] that, for a positive average dispersion d >, the system (.5) has a stable elliptic fixed point in the Poincaré section of the periodic solutions (a, b), whereas for a negative average dispersion d <, the system has a stable elliptic and an unstable hyperbolic fixed point. The averaging method is developed in [6] with a rigorous proof of stability instability and bifurcations of different periodic solutions of the system (.5). Here we study asymptotic properties of the system (.5) in the it of large E and S for either sign of d. 3. ASYMPTOTIC AND NUMERICAL RESULTS We first find a numerical approximation of periodic solutions of system (.5). We use a twodimensional shooting method, where the Jacobian matrix is approximated by a first-order finitedifference operator at each time step. This algorithm does not always converge, but satisfactory convergence is found in most cases. Iterations are started with a reasonable initial approximation for a() and b(), most often stemming from the approximation obtained for slightly different parameters E and S. To integrate the system (.5) with given initial data, a four-step Runge- Kutta method is employed. Figure shows the numerical approximation of the periodic solutions a() andb(), the dispersion map d() on<<, and the phase plane (a, b) in the case when l =., d =.4, α =,ande =. a() b() d() b a Fig.. Numerical approximation of the periodic solution a() andb() on << with the parameters l =., d =.4, α =,ande =. Left: a(), b() and the dispersion map d(). Right: Phase plane (a, b). We now analye the asymptotic solutions of system (.5) in the it of large pulse energy E. We shall work independently in two separate regions: < < l and l < <, where d() =d + d + > andd() =d + d <, respectively. Let ɛ =/E be a small parameter. We define new variables ζ ± in the two regions as ɛζ ± = ± γ( )d, (3.) PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS Suppl. 3

4 DISPERSION-MANAGED SOLITONS IN THE LIMIT OF LARGE ENERGY S where the values ± are defined to be the points on the focusing and defocusing fibers, where b( ± ) =. It follows from (.5) that a( ± ) has an extremum if b( ± )hasasimpleero(cf. Figure ). The transformation (3.) is invertible since γ() >, i.e., the function ζ ± = ɛ g ± () is one-to-one with the inverse function = f ± (ɛζ ± ). It is also clear from (3.) that ζ ± =atthe points, where = ±. In new variables ζ ±, the system (.5) takes the form da a 5/ b = dζ ± (a + b ) 3/, db = ɛd(ɛζ ± ) a3/ (a b (3.) ) dζ ± (a + b ) 3/, where D(ɛζ ± )= d() γ() =f± (ɛζ ± ). The system (3.) can be reduced to the Hamiltonian form [6] with the ζ ± -dependent Hamiltonians, a / H(ζ ± )= (a + b ) / ɛd(ɛζ ±). (3.3) a We conclude from Figure that the periodic solution a() is centered and localied at the points = ±,whereζ ± =. In the asymptotic region where ζ ± O() in the it ɛ, we expand D(ɛζ ± )inapowerseriesinɛ at ɛ =. Each partial system (3.) becomes autonomous in the first order of ɛ with the following relation between a and b: b = a ( ) H ± a + ɛd ± ( ) a, (3.4) H± a + ɛd ± where H ± = H() and D ± = D() are defined separately for the two regions of ζ ±. The solution is defined in the domain a ± min a(ζ ±) a ± max,wherea± min and a± max are two roots of the quadratic equation a = ( ). H ± a + ɛd ± (3.5) We consider first the focusing fiber, where D + >. As follows from (.7), the minimal value a() is related to the map strength as S =. Under the condition that H a + + =O(ɛ p ) min a + min = min <<l with p> in the it ɛ, one can expand the smallest root of the quadratic equation (3.5) in the power series of ɛ and find the leading-order expansion as a + min = ɛ D + ( + O(ɛH + )). (3.6) We prove below that H + =O(ɛ /3 )forα =andh + =O()forα, i.e., either p =/3 or p =, and the condition of validity of the leading-order approximation (3.6) is thus satisfied. In this case, the minimum of the peak of a(ζ + ) in the focusing fiber is a + min = ɛ A min,wherea min =O() in the it ɛ. In the domain, where ζ + =O()anda(ζ + )=ɛ A min +O(ɛ 3 ), the system (3.) has the leadingorder asymptotic solution: a(ζ + )=ɛ A min + ɛ 3 A min ( ɛa min ɛ A min + D +ζ+) / + R a + (ζ +,ɛ), (3.7) PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS Suppl. 3

5 S KALISCH, PELINOVSKY and b(ζ + )=ɛd + ζ + + ɛ D ()ζ + + ɛ A min log D +ζ + + ( ɛ A min + D +ζ+ ɛa min ) / D + ζ + (ɛ A min + D +ζ+) / + R + b (ζ +,ɛ), (3.8) where the remainder terms have the order R + a (ζ +,ɛ)=o(ɛ 4 )andr + b (ζ +,ɛ)=o(ɛ 3 ) in the region, where ζ + = O(). Comparison of the leading-order asymptotic solutions (3.7) and (3.8) (dashed curve) with the numerical approximation (solid curve) is shown on Figure for the same values of parameters as on Figure a.3 b Fig.. Comparison of the numerical solution (solid curves) with the leading-order asymptotic solution (3.7) and (3.8). Left: a(). Right: b(). The function b(ζ + ) grows secularly in the its ζ + ±, where the inner asymptotic approximation (3.8) breaks. It also follows from Figure that the maximum of the function a(ζ + ) in the its ζ + ± appears beyond the inner asymptotic approximation (3.7), where a(ζ + ) = O(ɛ ). As a result, the asymptotic solutions a(ζ + ) and b(ζ + ) in (3.7) (3.8) are not uniformly valid in the it ɛ onζ + R. The inner asymptotic solutions (3.7) and (3.8) must be matched with the outer asymptotic solution in the its ζ + ± such that ɛζ + O(). The scaling of the system (3.) with b =O()andɛζ + = O() shows that the matching conditions are ζ + ± ɛζ + =Z ± a(ζ + )=ɛ /3 A(Z ± ), ζ + ± ɛζ + =Z ± b(ζ + )=B(Z ± ). (3.9) The outer asymptotic solution A(Z ± ) and B(Z ± ) in the two its satisfies the system of equations, da dz ± = A 5/ B (B + ɛ 4/3 A ) 3/, db = D(Z ± )+ A3/ (B ɛ 4/3 A ) dz ± (B + ɛ 4/3 A ) 3/. (3.) PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS Suppl. 3

6 DISPERSION-MANAGED SOLITONS IN THE LIMIT OF LARGE ENERGY S3 We do not solve here the outer asymptotic system (3.), but we exploit the scaling properties of the outer asymptotic solution (3.9). Since D(Z ± ) is independent of ɛ, it is clear from (3.3) that H(ζ + ) has the following representation at the outer asymptotic scale ɛζ + = O() in the its ζ + ± : ( H(Z ± )=ɛ /3 A / (B + ɛ 4/3 A ) / D(Z ) ±), (3.) A and therefore H(Z ± )=O(ɛ /3 )asɛ. Since H is the Hamiltonian for the system (3.), we have dh = H = ɛ D (ɛζ + ) = ɛ dζ + ζ + a(ζ + ) a(ζ + ) d()γ () γ 3 () (3.) =f+ (ɛζ + ). If γ () = (i.e., α =)andd() =d + d + > in the focusing fiber <<l,then H(Z ± ) = ζ + ± ɛζ + =Z ± H(ζ + )=H() = H +, (3.3) and H + =O(ɛ /3 )forα =, as we claimed. If γ () (i.e., α ), then H(Z ± )=H + + ɛ C ± dζ + a(ζ + ) d()γ () γ 3 () + ɛ =f+ (ɛζ + ) ζ + ζ + ± ɛζ + =Z ± C ± dζ + a(ζ + ) d()γ () γ 3 () (3.4) =f+ (ɛζ + ), where C ± =O()asɛ. The first integral in (3.4) is estimated with the inner asymptotic solution (3.7), where a(ζ + )=O(ɛ ), and the second summand in (3.4) is of order O() as ɛ. The second integral in (3.4) is estimated with the outer asymptotic solution (3.9), where a(ζ + )=O(ɛ /3 ), and the third summand in (3.4) is of order O(ɛ /3 )asɛ. As a result, H + =O()forα, as we claimed. We consider next the defocusing fiber, where D <. Under the condition that H =O(ɛ q ) for q> in the it ɛ, one can expand the largest root of the quadratic equation (3.5) in the power series of ɛ and find the leading-order expansion as: a max = H ( + O(ɛH )). (3.5) We show below that H =O(ɛ /3 ) for any value of α. In this case, the asymptotic expansion for the maximum of the peak of a(ζ ) in the defocusing fiber is a max = ɛ /3 A max,wherea max =O() in the it ɛ. The scaling of system (3.) with a =O(ɛ /3 ) shows that the inner asymptotic solution on the defocusing fiber takes the form: a(ζ )=ɛ /3 a(η,ɛ), b(ζ )=ɛ /3 b(η,ɛ), where η = ɛ /3 ζ. The leading-order inner asymptotic solution for a(ζ )andb(ζ ) takes the explicit form: 4A max a(ζ )= ɛ /3 ( 4A max + η ) + Ra (η,ɛ), (3.6) and η A 3/ max b(ζ )= ɛ /3 (4A max + η ) + R b (η,ɛ), (3.7) where the remainder terms have the order R a (η,ɛ)=o(ɛ /3 )andr b (η,ɛ)=o(ɛ /3 )inthe region, where η = O(). The leading-order asymptotic solutions (3.6) and (3.7) (dashed curve) PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS Suppl. 3

7 S4 KALISCH, PELINOVSKY Fig. 3. Comparison of the numerical solution (solid curves) with the leading-order asymptotic solution (3.6) and (3.7). Left: a(). Right: b(). are compared with the numerical approximation (solid curve) on Figure 3 for the same values of parameters as on Figure. The asymptotic solutions a(ζ )andb(ζ ) in (3.6) (3.7) are not uniformly valid in the it ɛ onζ R. The inner asymptotic solutions (3.6) and (3.7) must be matched with the outer asymptotic solution in the its ζ ±, such that ɛζ = O(). The matching conditions take the form ζ ɛζ =Z ± a(ζ )=ɛ /3 A(Z ± ), ζ ɛζ =Z ± b(ζ )=B(Z ± ), (3.8) where the functions A(Z ± )andb(z ± ) are the same as in (3.9). The functions A(Z ± )andb(z ± ) solve the same problem (3.), which is valid at the interface layer between the focusing and defocusing fiber, where D(Z ± ) changes sign. Since D(Z ± ) is independent of ɛ, the Hamiltonian H(ζ ) has the same representation (3.) with D(Z ± ) <, and therefore H(Z ± )=O(ɛ /3 )as ɛ. Similarly to the focusing case, we find from (3.) and (3.3) that H(Z ± )=H + ɛ ζ ɛζ =Z ± ζ dζ a(ζ ) d()γ () γ 3 () (3.9) =f (ɛζ ), where d() =d + d < in the defocusing fiber l<<. Since ɛ /3 A(Z ± ) a(ζ ) ɛ /3 A max, the second term in (3.9) always has the order O(ɛ /3 ), such that H = O(ɛ /3 ) in the it ɛ, as we claimed. Thus, the asymptotic solution for a() andb() onaperiod is constructed with the help of the inner asymptotic solutions (3.7) and (3.8) in the focusing fiber, the inner asymptotic solutions (3.6) and (3.7) in the defocusing fiber, and the outer asymptotic solutions A(Z ± )andb(z ± ) of the interface problem (3.) that match the inner solutions with the matching conditions (3.9). The main result of the paper is the expansion (3.6), which gives the leading-order relation between the energy E and the map strength S as E D + S. (3.) We confirm this result with numerical approximations of the periodic solutions of system (.5) for the case l =.8, d =.4, and α =. We first compute an approximate periodic solution for PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS Suppl. 3

8 DISPERSION-MANAGED SOLITONS IN THE LIMIT OF LARGE ENERGY S5 small values of E, suchase =. The calculation is then repeated with increasing values for the pulse energy E andthemapstrengths. The numerical approximation of the curve E versus S is shown in Figure 4 (dashed curve), together with the asymptotic approximation (3.) with D + = d(l/) =.54 (solid curve). 4 8 E S Fig. 4. Pulse energy E versus map strength S for l =.8, d =.4, and α =. Solid curve: numerical approximation, dashed curve: asymptotic approximation E = D + S / β.3 C Center of fit Center of fit Fig. 5. Numerical approximations of parameters of the power regression fit E = CS β. Left: β. Right: C. The center of the fit is taken at different values of S. The parameters are l =.8, d =.4, and α =,and the data are computed for <E<5. Dashed lines show asymptotic results: β =.5 andc = D +. We apply the curve fitting algorithm for the power approximation E = CS β to data points of the plot of E versus S, centered at a certain value of S. The power regression fit is shown in Figure 5 for different values of the center of the power fit in S. As the center of the power fit becomes larger, the computed value of constant C approaches D + = d(l/) =.54 and the power β tends to.5. This confirms the leading-order asymptotic result (3.). The computations are repeated with different values of the parameter l, with similar results. PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS Suppl. 3

9 S6 KALISCH, PELINOVSKY The numerical approximations of the periodic solutions of system (.5) are also obtained in the presence of loss and amplifications. Figure 6 shows the periodic solutions for the case l =., d =.4, α =.75, a =.5, and E =. Note that the minimum of a() on the focusing fiber and the maximum of a() in the defocusing fiber do not occur at the centers of the respective fibers. The locations of these points are denoted by ±. a() b() d() γ() b a Fig. 6. Numerical approximation of the periodic solution a() andb() on << in the presence of loss and amplification. The parameters are l =., d =.4, E =, a =.5, and α =.75. Left: a(), b(), dispersion map d(), and nonlinearity coefficient γ(). Right: Phase plane (a, b) β.3 C Center of fit Center of fit Fig. 7. Numerical approximations of parameters of the power regression fit E = CS β in the presence of loss and amplification. Left: β. Right:C. The center of the fit is taken at different values of S. The parameters are l =.8, d =.4, α =.75, and a =.5, and the data are computed for <E<5. Dashed lines show asymptotic results: β =.5 andc = D +. The parameters β and C of the power fit E = CS β centered at different values of S are shown on Figure 7 for the case l =.8, d =.4, α =.75, and a =.5. As the center of the power fit becomes larger, the computed value of constant C approaches D + = d( + )/γ( + ).96 and the power β tends to.5. This confirms the leading-order asymptotic result (3.) in the case PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS Suppl. 3

10 DISPERSION-MANAGED SOLITONS IN THE LIMIT OF LARGE ENERGY S7 when α. 4. SUMMARY We have studied the relation between the energy E of dispersion-managed solitons and the map strength S of the dispersion map in the it of large pulse energy. Using a matched asymptotic expansion, we have given leading-order expressions for the relation between E and S as well as for the functions a and b appearing in the dynamical system (.5). The agreement between asymptotic and numerical results confirms our main claim that the pulse energy grows like E D + S with large map strength, where D + is the effective dispersion coefficient in the focusing fiber. The leading-order asymptotic solutions describe variations of the pulse parameters along the optical fiber. The delicate problems concerning the range of applicability of the asymptotic expansions and bounds on the error terms are relegated to future work. REFERENCES. Kune, M., Periodic Solutions of a Singular Lagrangian System Related to Dispersion-Managed Fiber Communication Devices, Nonlinear Dynamics and Systems Theory,, vol., pp Kut, J.N., Holmes, P., Evangelides, S.G., and Gordon, J.P., Hamiltonian Dynamics of Dispersion-Managed Breathers, J. Opt. Soc. Am. B, 998, vol. 5, p Lakoba, T.I. and Kaup, D.J., Hermite Gaussian Expansion for Pulse Propagation in Strongly Dispersion- Managed Fibers, Phys. Rev. E, 998, vol. 58, Nakaawa, M., Kubota, H., Suuki, K., Yamada, E., and Sahara, A., Recent Progress in Soliton Transmission Technology, Chaos,, vol., Pelinovsky, D., Instabilities of Dispersion-Managed Solitons in the Normal Dispersion Regime, Phys. Rev. E,, vol. 6, Pelinovsky, D. and Zharnitsky, V., Averaging of Dispersion-Managed Pulses: Existence and Stability, SIAM J. Appl. Math., 3, in press. 7. Turitsyn, S., Fedoruk, M.P., Shapiro, E.G., Meentsev, V.K., and Turitsyna, E.G., Novel Approaches to Numerical Modeling of Periodic Dispersion-Managed Fiber Communication Systems, IEEE J. Quantum Electr.,, vol. 6, p Turitsyn, S.K. and Shapiro, E.G., Variational Approach to the Design of Optical Communication Systems with Dispersion Management, Opt. Fiber Tech., 998, vol. 4, pp PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS Suppl. 3