Theory of SHG in a medium with combined nonlinear response

Trofimov, Vyacheslav A.; Kharitonov, Dmitry M.; Fedotov, Mikhail V.

doi:10.1364/josab.35.003069

Cited by 4 publications

(4 citation statements)

References 70 publications

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“…At the end of Appendix A, we note that the proposed approach for developing the analytical solution on the basis of the problem invariant allows us not only to derive an evolution of the pulse shape but also the phase evolution. This was demonstrated in [42] under the analysis of the SHG problem.…”

Section: Institutional Review Board Statement: Not Applicablementioning

confidence: 84%

Conservative Finite-Difference Schemes for Two Nonlinear Schrödinger Equations Describing Frequency Tripling in a Medium with Cubic Nonlinearity: Competition of Invariants

Trofimov

Loginova

2021

Mathematics

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Two 1D nonlinear coupled Schrödinger equations are often used for describing optical frequency conversion possessing a few conservation laws (invariants), for example, the energy’s invariant and the Hamiltonian. Their influence on the properties of the finite-difference schemes (FDSs) may be different. The influence of each of both invariants on the computer simulation result accuracy is analyzed while solving the problem describing the third optical harmonic generation process. Two implicit conservative FDSs are developed for a numerical solution of this problem. One of them preserves a difference analog of the energy invariant (or the Hamiltonian) accurately, while the Hamiltonian (or the energy’s invariant) is preserved with the second order of accuracy. Both FDSs possess the second order of approximation at a smooth enough solution of the differential problem. Computer simulations demonstrate advantages of the implicit FDS preserving the Hamiltonian. To illustrate the advantages of the developed FDSs, a comparison of the computer simulation results with those obtained applying the Strang method, based on either an implicit scheme or the Runge–Kutta method, is made. The corresponding theorems, which claim the second order of approximation for preserving invariants for the FDSs under consideration, are stated.

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Section: Institutional Review Board Statement: Not Applicablementioning

confidence: 84%

Conservative Finite-Difference Schemes for Two Nonlinear Schrödinger Equations Describing Frequency Tripling in a Medium with Cubic Nonlinearity: Competition of Invariants

Trofimov

Loginova

2021

Mathematics

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“…In this case, all complex amplitudes A 1 , A 2 , A 3 , U, V, w 1 depend only on the spatial coordinate z. Therefore, the following initial condition: Below we use also the analytical solution of the problem ( 14) developed in papers [ 34,35 ], as well as numerical solutions of the problems ( 1) and ( 12), respectively. For some values of the problem parameters, the difference between the solutions of problems ( 1) and ( 14) can be can be enough big even at large phase mismatching ∆ 31 k and, consequently, low TH intensity, which is less than 1.2 • 10 −4 ).…”

Section: Long Pulse Durationmentioning

confidence: 99%

“…Our theoretical studies of this process [ 34 ] showed that a SHG can possess a bistable mode under such conditions. Further, we have derived expressions for interacting waves phases evolution [ 35 ] without using the pump-wave energy non-depletion approximation. However, in all these papers (and many others), the cascaded processes, connected with weak tripled frequency wave, and its influence on SHG efficiency, for example, are not taken into account.…”

Section: Introductionmentioning

confidence: 99%

High efficiency frequency conversion based on cascaded processes under large phase mismatching between some interacting pulses

Zheng,

Trofimov,

Kharitonov

et al. 2024

Nonlinear Frequency Generation and Conversion: Materials and Devices XXIII

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We study the frequency up-and down-conversion based on the cascaded process which accompanies large phase mismatching between fundamental and intermediate waves. So, for example, we propose using the cascaded second-harmonic generation (SHG) to implement the frequency conversion process, which is similar to that occurring in a medium with cubic susceptibility. At phase matching between the fundamental wave and the third-harmonic wave, the third harmonic generation (THG) occurs with high efficiency (94.5%).We demonstrate that the cascaded process may also influence negatively on the frequency conversion processes. SHG in a medium with combined quadratic and cubic nonlinear response accounting for weak THG at large phase mismatching demonstrates that in some cases , the SH intensity evolution is different if we take into account of weak THG process or not. So, then the intensity at doubled frequency may be much lower in certain sections of a medium in comparison with corresponding values without accounting for THG. We study such an influence of weak THG both for pulses with long duration by developing analytical approach and for short pulses based on computer simulation.

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“…An influence of the self-modulation as well as cross-modulation of the laser pulse on the frequency doubling efficiency was investigated experimentally and theoretically (see, for example [2, 19–28]). It should be stressed that the most general analytical theory of this process has been developed recently in [28], taking into account the time-dependent pulse shape and without using the basic wave energy non-depletion approximation on the base of using the integrals of motion (conservation laws) of the problem. We notice that the authors of [22] have obtained the most (but not all) solutions of this problem in the framework of long pulse duration approximation.…”

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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Theory of SHG in a medium with combined nonlinear response

Cited by 4 publications

References 70 publications

Conservative Finite-Difference Schemes for Two Nonlinear Schrödinger Equations Describing Frequency Tripling in a Medium with Cubic Nonlinearity: Competition of Invariants

Conservative Finite-Difference Schemes for Two Nonlinear Schrödinger Equations Describing Frequency Tripling in a Medium with Cubic Nonlinearity: Competition of Invariants

High efficiency frequency conversion based on cascaded processes under large phase mismatching between some interacting pulses

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

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