Nonlinear distortions of hyper-Gaussian beams

Karamzin, Yu. N.; Sukhorukov, A. P.; Trofimov, Vyacheslav A.

doi:10.1007/bf01039455

Cited by 4 publications

(5 citation statements)

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“…Two types of solitary waves that were both predicted in the early 1970's are being studied. On one hand, one finds solitary waves that result from a balance between nonlinearity and diffraction (or dispersion) [1]. This type of solitary wave has been intensively investigated experimentally over the past few years [2].…”

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confidence: 99%

Velocity-Locked Solitary Waves in Quadratic Media

et al. 2010

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confidence: 99%

Velocity-Locked Solitary Waves in Quadratic Media

et al. 2010

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“…Another approach for computer simulation is based on developing of the conservative finite-difference schemes [107]. They allow preserving the difference analogues of the corresponding conservation laws.…”

Section: Introductionmentioning

confidence: 99%

Generalized nonlinear Schrödinger equations describing the Second Harmonic Generation of femtosecond pulse, containing a few cycles, and their integrals of motion

2019

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An interaction of laser pulse, containing a few cycles, with substance is a modern problem, attracting attention of many researches. The frequency conversion is a key problem for a generation of such pulses in various ranges of frequencies. Adequate description of such pulse interaction with a medium is based on a slowly evolving wave approximation (SEWA), which has been proposed earlier for a description of propagation of the laser pulse, containing a few cycles, in a medium with cubic nonlinear response. Despite widely applicability of the frequency conversion for various nonlinear optics problems solutions, SEWA has not been applied and developed for a theoretical investigation of the frequency doubling process until present time. In this study the set of generalized nonlinear Schrödinger equations describing a second harmonic generation of the super-short femtosecond pulse is derived. The equations set contains terms, describing the pulses self-steepening, and the second order dispersion (SOD) of the pulse, a diffraction of the beam as well as mixed derivatives. We propose the transform of the equations set to a type, which does not contain both the mixed derivatives and time derivatives of the nonlinear terms. This transform allows us to derive the integrals of motion of the problem: energy, spectral invariants and Hamiltonian. We show the existence of two specific frequencies (singularities in the Fourier space) inherent to the problem. They may cause an appearance of non-physical absolute instability of the problem solution if the spectral invariants are not taken into account. Moreover, we claim that the energy preservation at the laser pulses propagation may not occur if these invariants do not preserve. Developed conservation laws, in particular, have to be used for developing of the conservative finite-difference schemes, preserving the conservation laws difference analogues, and for developing of adequate theory of the modulation instability of the laser pulses, containing a few cycles.

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“…First, we should emphasize the conservative finite-difference schemes (CFDS) for the nonlinear Schrödinger equation. They were first proposed in [ 1 ], and then investigated, in detail, in papers [ 2 – 5 ] and other papers of these authors. These finite-difference schemes are nonlinear ones and, therefore, require an iterative process for their implementation, which leads to additional computation time costs.…”

Section: Introductionmentioning

confidence: 99%

Implicit finite-difference schemes, based on the Rosenbrock method, for nonlinear Schrödinger equation with artificial boundary conditions

Trofimov

Trykin

2018

PLoS ONE

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We investigate the effectiveness of using the Rosenbrock method for numerical solution of 1D nonlinear Schrödinger equation (or the set of equations) with artificial boundary conditions (ABCs). We compare the computer simulation results obtained during long time interval at using the finite-difference scheme based on the Rosenbrock method and at using the conservative finite-difference scheme. We show, that the finite-difference scheme based on the Rosenbrock method is conditionally conservative one. To combine the advantages of both numerical methods, we propose new implicit and conditionally conservative combined method based on using both the conservative finite-difference scheme and conditionally conservative Rosenbrock method and investigate its effectiveness. The combined method allows decreasing the computer simulation time in comparison with the corresponding computer simulation time at using the Rosenbrock method. In practice, the combined method is effective at computation during short time interval, which does not require an asymptotic stability property for the finite-difference scheme. We generalize also the combined method with ABCs for 2D case.

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Nonlinear distortions of hyper-Gaussian beams

Cited by 4 publications

References 0 publications

Velocity-Locked Solitary Waves in Quadratic Media

Velocity-Locked Solitary Waves in Quadratic Media

Generalized nonlinear Schrödinger equations describing the Second Harmonic Generation of femtosecond pulse, containing a few cycles, and their integrals of motion

Implicit finite-difference schemes, based on the Rosenbrock method, for nonlinear Schrödinger equation with artificial boundary conditions

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