Generation and stability of optical bullets in quadratic nonlinear media

Transcription

1 1 March 1998 Optics Communications Generation and stability of optical bullets in quadratic nonlinear media Dmitry V. Skryabin 1, William J. Firth Department of Physics and Applied Physics, John Anderson Building, UniÕersity of Strathclyde, 107 Rottenrow, Glasgow G4 0NG, UK Received 1 September 1997; accepted 6 November 1997 Abstract We show numerically that modulational instability of two-dimensional solitary waves in quadratic media leads to the formation of a train of quasi-stable three-dimensional light bullets of mutually trapped fundamental and second harmonic pulses. The dynamics and stability of these bullets are studied both numerically and variationally. Small deviations from the stationary bullet profile generally lead to long lived pulsations. Regions of unstable behaviour Ž decay. are found for both positive and negative mismatches. q 1998 Elsevier Science B.V. PACS: Tg; Kf Keywords: Optical bullets; Modulational instability; Quadratic nonlinearity Stable solutions to nonlinear wave problems can often develop modulational instability Ž MI. when additional dimensions are added to the problem, e.g. an intense continuous-wave field in an optical fibre is unstable due to temporal modulation wx 1. MI may lead to break-up of the initial wave profile into a number of fragments, which can form solitons during the subsequent evolution. These phew2 5x and quadratic w6 8x nonlinear media. In the latter case, nomena have been extensively studied for both cubic which is the topic of the present work, the solitons are two-colour, composed of mutually trapped fundamental and second harmonic fields. Some years ago, Kanashev and Rubenchik made an analytical study of MI of two-dimensional Ž 2D. spatial solitons in quadratic nonlinear media wx 6. The main results of their analytic work are perturbative, valid only for unstable modes with temporal frequencies close to zero. 1 dmitry@phys.strath.ac.uk However, the frequency with maximal growth rate is usually outside the range of this approximation. This frequency is important because it defines the period of the modulation arising in the development of the MI. Numerical analysis is thus the best way currently available to investigate the full range of relevant perturbation frequenwx 5 undertook such stud- cies. Akhmediev and co-workers ies for the case of anomalous group-velocity dispersion Ž GVD. in saturable nonlinear media. They found that a 2D solitary stripe breaks up into 3D solitons optical bullets wx 9. ŽFor convenience we use the term soliton in the physical sense rather than the restrictive mathematical one.. Recent experiments on self-chanelling of optical pulses in air w11x are encouraging for the practical relevance of the optical bullet concept. Considering now media with quadratic nonlinearity, parametric gap-solitons have recently been predicted in Bragg gratings, allowing band gaps at both first and secw12 x. Because of their large dispersion ond harmonics Bragg structures may provide better conditions for experimental realisation of optical bullets due to quadratic non r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S

2 80 D.V. Skryabin, W.J. FirthrOptics Communications linearity than bulk nonlinear crystals. In such crystals GVD is significantly weaker than diffraction, respectively in relation to typical pulse durations and beam widths. Regarding the stability of optical bullets, the global approach to the problem of 3D solitons in quadratic media w6,10x shows that they can be stable. However no numerical studies of MI of 2D solitary stripes or of the internal stability of the resulting bullets have so far been reported. In the present work we describe numerical investigations of MI of 2D spatial solitons in quadratic media and show that it leads to the formation of a train of 3D optical bullets. We then apply numerical and variational approaches to analyse the stability of the bullets with respect to internal perturbations. We consider interaction of first and second harmonic optical fields propagating in a dielectric medium with x Ž2. nonlinear susceptibility, under the conditions of type I phase matching. Both GVDs are supposed to be anomalous and spatial and temporal walk-off are neglected. Then evolution of the field envelopes E and E in the refer- 1 2 ence frame moving with group velocity can be described through the following dimensionless system of equations, 2 ˆ ) iezqet ql E1qE1 E2s0, 2 ˆ 2 2iEzq2aEt ql E2qE1s2bE 2, 1 Ž. where z and t correspond respectively to the normalised propagation and time coordinates, and cylindrical symme- ˆ 2 try is assumed, so that the transverse Laplacian LsE r q Ž 1rr. Er depends only on the scaled radial variable r. b is the normalised phase mismatch, a is the ratio of the absolute GVD values. For a discussion of these equations and the physical meaning of dimensionless units used see Refs. w8,13 x. The following stability analysis of 2D spatial solitary stripes and our computer code for simulation of the 3D dynamics are independent of the particular value of a. However, for simplicity in getting stationary 3D solitary solutions and studying their stability we assume as0.5 henceforth, and t in Ž. 1 then becomes mathematically equivalent to a third transverse coordinate. Now we can reduce the partial differential equations Ž. 1 to a set of ordinary differential equations for the soliton profile. The imk z substitution E sa r e Ž ms1,2. m m results in the following system, Here D is the dimension of the problem: for Ds3 ' 2 2 optical bullet rs r qt, while Ds2, with rsr, corresponds to the cylindrical soliton filament whose MI is to be studied. The parameter k is the effective propagation constant, which must obey k) maxž 0,y br2. to ensure exponential decay of the tails of both harmonics forming the soliton. For given D and b there is a family of solitons with different energies, each with different k, which is thus a free parameter. To find these families we solve Eqs. Ž. 2 using finite differences. Our first objective is to study MI of spatial soliton filaments, so we solve Eqs. Ž. 2 to find stationary 2D soliton profiles A Ž r. 1,2. To them we add time dependent perturbations of the general form Ž m m. e Žq. Ž z,r. e iv t qe Žy.) Ž z,r. e yiv t e imk z. Ž. Linearisation of Eqs. 1 then leads to the set of equations Žq. 2 ˆ Žq. Žq. Žy. ieze1 s kqv yl e1 ya1e2 ya2e 1, Žy. 2 ˆ Žy. Žy. Žq. ieze1 sy kqv yl e1 qa1e2 qa2e 1, Ž ˆ. Žq. 2 1 ieze2 s 2kqbqaV y L e Žq. ya e Žq , Ž ˆ. Žy. 2 1 ieze2 sy 2kqbqaV y L e Žy. qa e Žy Ž. To investigate the growth rates of such perturbations we solve Eqs. Ž. 3 using a Crank-Nicholson scheme with initial conditions localised around rs0. This method was successfully applied earlier to study modulational instabilwx 5. After transients ity in saturable nonlinear media have d 2 A1 Dy1 da1 q sk A1yA1A 2, d r 2 r d r d 2 A2 Dy1 da2 2 q s22kqb A2yA 1. Ž 2. d r 2 r d r Fig. 1. Maximal perturbation growth rate versus perturbation frequency, k s2.

3 D.V. Skryabin, W.J. FirthrOptics Communications Ž. Ž. Ž. Fig D spatial soliton filament break up, ks2, bs0. Field at a input, zs0, b zs3, c zs4. died out we calculate the average growth rate over a sufficiently large propagation range in z. Note that the 1D analog of system Ž. 3 was recently studied for the case of normal GVD in Ref. w13 x. Perturbation growth rates versus V for different values of phase mismatch b are presented in Fig. 1. Our method also yields the maximal-growth eigenfunctions, which are localised in the same region as the soliton, with a maximum at rs0, so maintaining the cylindrical symmetry. This is the so-called neck-type wx 3 of instability and it is common for MI due to anomalous dispersion in both quadratic and cubic media w3,6 x. To check our stability analysis we performed direct numerical simulations of Eqs. Ž. 1 using a split-step method Žfast Fourier transform for the dispersion part, Crank- Nicholson for the diffraction part and second order Runge-Kutta for nonlinear terms.. In the time dimension we used a 128 grid-point computational window and choose the integration step to fit into the window six periods of modulation at the maximally unstable frequency V max. Initial conditions were a stationary 2D soliton plus small Ž 1%. noise. Typical simulation results are shown in Fig. 2. The initial soliton profile breaks up into six humps as predicted by stability analysis. They form into a train of quasi-solitary waves which oscillate with propagation. Thus, while the MI leads to the formation of opticalbullet-like pulses, stationary 3D optical bullets are not found in our simulations, at least not within the propagation distances attainable on our computer system. The internal stability of the optical bullets Ži.e. against perturbations depending only on r. is governed by the standard criterion w14,15x that solitary solutions are stable if Ek Q)0. Here Q is the energy invariant, H 1 2 Qs dv < E < 2 q2< E < 2.

4 82 D.V. Skryabin, W.J. FirthrOptics Communications Fig. 3. Comparison between numerical Ž solid lines. and variational Ž dotted lines. data for 3D optical bullets: Ž a. Energy Q versus k; Ž. ba versus r. 1,2 Plots of Q versus k for positive, negative and zero phase mismatches are presented in Fig. 3Ž. a. Note that 3D solitons can be unstable for values of k close to their existence boundary for either sign of mismatch. In contrast, it is known that 1D and 2D quadratic solitons can be unstaw15,16 x. As an independent check we applied a variational ble only for negative phase mismatches approach, using Gaussian trial functions A sa e yg m r 2 m m. Minimisation of the system Lagrangian was done in general form for arbitrary D, but we present results for Ds3 only, because results for the cases Ds1,2 have been previously published w16 x. / 2 5r2 4g1 g2 2 2 Ž 1. kqg ž 1 2g1 3r2 a2 2g Ž 2. ' 2 ž g 2 g s, a s kqg 1q, / a s 2kqbq3g r2 1q. Ž. 4 Here g 1 is a real positive root of the cubic equation: 18g 1 3 qbg 1 2 yk 2 Ž 2kqb. s0. Ž 5. It is interesting to note that for zero mismatch Eqs. Ž.Ž. 4, 5 give bullet parameters in explicit form. The results of the variational and numerical approaches are compared in Fig. 3. The maximal error of the variational-gaussian approximation is about 12% for the total energy. These results confirm the tendency for variational-gaussian approximaw16 x, presumably because the Lagrangian integral weights large tions to give poorer results with increasing dimension radii more heavily as D increases. To study instability scenarios we performed a series of numerical simulations of the evolution of slightly-perturbed 3D optical bullets. Starting simulation of Eqs. Ž. 1 with a slightly low total energy, we observed spreading and decay of 3D solitons. A high total energy led to long-lived in-phase pulsations of the two harmonics. These scenarios are typical for both negative and positive mismatches, see Fig. 4Ž. a, 4Ž. b. Both spreading and oscillations are generally faster for negative mismatch. Small but arbitrary deviations from the stationary soliton profile for Ek Q) 0 lead to long-lived pulsations of solitary shape, see Fig. 4Ž. c. In the 1D version of Eqs. Ž. 1 pulsations were attributed to the existence of a perturbation eigenmode of the discrete spectrum with a pure imaginary eigenvalue w17 x. We expect that a similar phenomenon may occur here, in three dimensions. An important practical question is whether it is possible to generate bullets from an initial Gaussian pump pulse. For this reason we carried out a series of numerical simulations of Eqs. Ž. 1 with initial conditions in a form of Gaussian function in space and time and for various moderate values of b, a and temporal walk-off. We found that providing the input energy and Hamiltonian are suffi- ciently close w18x to the values for a bullet, pulsating mutually trapped 3D solitary waves can be generated. In conclusion, we numerically calculated perturbation growth rates for 2D spatial solitons in x Ž2. media in the presence of anomalous GVD. We demonstrated the possibility of generation of a train of quasi-stable 3D optical bullets due to fragmentation of these 2D solitons and observed generation of quasi-bullets from spatio-temporal Gaussian input pulses. Analysis of the internal stability of 3D solitons reveals that instability is possible for both positive and negative phase mismatches. Inside the stability region our numerics show that small deviations from

5 D.V. Skryabin, W.J. FirthrOptics Communications Fig. 4. Ž. a Optical bullet instability scenarios for bs1, ks0.04. Ž. b The same but for bsy1, ks0.52. Ž. c Long-lived pulsation, bs0, ks1. Solid Ž dotted. lines correspond to fundamental Ž second. harmonic. Deviations from stationary transverse profiles are 2% for Ž. a Ž. c. the bullet profile do not decay, but lead to long lived pulsation of both fundamental and second harmonic fields. Note: After the acceptance of the present work for publication the paper by Malomed et al. w19x related to the existence and stability of optical bullets due to quadratic nonlinearity was published. Acknowledgements D.V.S. acknowledges support from an University of Strathclyde John Anderson award and the ORS Awards Scheme. This work was supported in part by EPSRC Grant GRrL References wx 1 K. Tai, A. Hasegawa, A. Tomita, Phys. Rev. Lett. 56 Ž wx 2 V.I. Bespalov, V.I. Talanov, JETP Lett. 3 Ž wx 3 V.E. Zakharov, A.M. Rubenchik, Sov. Phys. JETP 38 Ž wx 4 G.S. McDonald, K.S. Syed, W.J. Firth, Optics Comm. 96 Ž

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