All-optical-switching and pulse amplification and steering in nonlinear fiber arrays

Aceves, Alejandro B.; Angelis, C. De; Luther, G. G.; Rubenchik, Alexander M.; Turitsyn, Sergei K.

doi:10.1016/0167-2789(95)00123-l

Cited by 36 publications

(40 citation statements)

References 25 publications

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“…Indeed, from the asymptotic form of CNLS self-focusing solution 3.1, we see that the magnitudes of ? and jj 2 are OL , 3 , and that of the nonparaxial term is O L , 5 . This suggests that the paraxial approximation breaks down when L = O p .…”

Section: Davey-stewartson Equations the Davey-stewartson Equations Dsmentioning

confidence: 99%

See 1 more Smart Citation

Self-Focusing in the Perturbed and Unperturbed Nonlinear Schrödinger Equation in Critical Dimension

Fibich¹,

Papanicolaou²

1999

SIAM J. Appl. Math.

245

307

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Abstract.The formation of singularities of self-focusing solutions of the nonlinear Schr odinger equation NLS in critical dimension is characterized by a delicate balance between the focusing nonlinearity and di raction Laplacian, and is thus very sensitive to small perturbations. In this paper we i n troduce a systematic perturbation theory for analyzing the e ect of additional small terms on self focusing, in which the perturbed critical NLS is reduced to a simpler system of modulation equations that do not depend on the spatial variables transverse to the beam axis. The modulation equations can be further simpli ed, depending on whether the perturbed NLS is power conserving or not. We review previous applications of modulation theory and present several new ones that include: Dispersive saturating nonlinearities, self-focusing with Debye relaxation, the Davey Stewartson equations, self-focusing in optical ber arrays and the e ect of randomness. An important and somewhat surprising result is that various small defocusing perturbations lead to a generic form of the modulation equations, whose solutions have slowly decaying focusing-defocusing oscillations. In the special case of the unperturbed critical NLS, modulation theory leads to a new adiabatic law for the rate of blowup which is accurate from the early stages of self-focusing and remains valid up to the singularity point. This adiabatic law preserves the lens transformation property of critical NLS and it leads to an analytic formula for the location of the singularity as a function of the initial pulse power, radial distribution and focusing angle. The asymptotic limit of this law agrees with the known loglog blowup behavior. However, the loglog behavior is reached only after huge ampli cations of the initial amplitude, at which point the physical basis of NLS is in doubt. We also include in this paper a new condition for blowup of solutions in critical NLS and an improved version of the Dawes-Marburger formula for the blowup location of Gaussian pulses.

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Section: Davey-stewartson Equations the Davey-stewartson Equations Dsmentioning

confidence: 99%

“…We nally note that when this approach is applied to the perturbed CNLS with the wrong ansatz typically a Gaussian or a sech, the reduced equation fails to capture the delicate balance of critical self-focusing and can lead to erroneous predictions. 5. Applications of modulation theory.…”

mentioning

confidence: 99%

Self-Focusing in the Perturbed and Unperturbed Nonlinear Schrödinger Equation in Critical Dimension

Fibich¹,

Papanicolaou²

1999

SIAM J. Appl. Math.

245

307

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“…As a result, discretization effects can be avoided by simply taking a very fine grid in the radial coordinate. In the last few years, however, there has been a growing interest in anisotropic effects in the NLS, either in the initial conditions (e.g., amalgamation [4], astigmatism [6], random noise [8,19]) or in the equation (e.g., polarization effects [8,5,7], fiber arrays [9,1]). In such cases one cannot use a radially-symmetric code and the NLS has to be solved using a (D + 1)-dimensional code.…”

Section: Dimensionality and Anisotropymentioning

confidence: 99%

“…It often happens, however, that these additional terms prevent the singularity formation and limit the growth of the gradients. 1 In such cases there is no real need for these specialized methods and one can use standard finite-difference methods. To do that reliably, however, requires better understanding of discretization effects in simulations of blowup solutions.…”

Section: Introductionmentioning

confidence: 99%

Discretization effects in the nonlinear Schrödinger equation

Fibich¹,

Ilan²

2003

Applied Numerical Mathematics

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We show that discretization effects in finite-difference simulations of blowup solutions of the nonlinear Schrödinger equation (NLS) initially accelerate self focusing but later arrest the collapse, resulting instead in focusing-defocusing oscillations. The modified equation of the semi-discrete NLS, which is the NLS with highorder anisotropic dispersion, captures the arrest of collapse but not the subsequent oscillations. Discretization effects in perturbed NLS equations are also discussed.

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“…In particular, we will investigate numerically the dynamics in the critical case σ = 2 N when letting the positive parameter α approach zero. In this paper, we will study the question of local and global existence of a unique solution for system (4). Specifically, we will prove the short time existence of the unique solution, when 1 σ < The proof will follow the ideas of [23] and [48] and use the important fact of the conservation of the corresponding energy and the Hamiltonian of (4).…”

Section: Introductionmentioning

confidence: 99%

Nonlinear Schrödinger–Helmholtz equation as numerical regularization of the nonlinear Schrödinger equation

2008

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A regularized α-system of the nonlinear Schrödinger (NLS) equation with 2σ nonlinear power in dimension N is studied. We prove short time existence and uniqueness of solution in the case 1 σ <. And in the case 1 σ < 3 (when N = 1) or in the case 1 σ < 4 N (when N > 1) we show global in time existence of solutions. When α → 0 + , the solutions of this regularized system will converge to the solutions of the classical NLS in the appropriate range when the latter exists. Consequently, we propose this regularized system as a numerical regularization to shed light on the profile of the blow-up solutions of the original NLS equation in the range 2 N σ < 4 N , and in particular for the classical critical case N = 2, σ = 1. Following the modulation theory, we derive the reduced system of ordinary differential equations for the Schrödinger-Helmholtz (SH) system. We observe that the reduced equations for this SH system are more complicated than the equations of some other perturbation regularizations of the classical NLS equation. The detailed analysis of the reduced system on how the regularization prevents singularity formation will be presented in a forthcoming paper.

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All-optical-switching and pulse amplification and steering in nonlinear fiber arrays

Cited by 36 publications

References 25 publications

Self-Focusing in the Perturbed and Unperturbed Nonlinear Schrödinger Equation in Critical Dimension

Self-Focusing in the Perturbed and Unperturbed Nonlinear Schrödinger Equation in Critical Dimension

Discretization effects in the nonlinear Schrödinger equation

Nonlinear Schrödinger–Helmholtz equation as numerical regularization of the nonlinear Schrödinger equation

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