The parabolic equation in the theory of wave propagation

Vlasov, S. N.; Talanov, V. I.

doi:10.1007/bf01051853

Cited by 29 publications

(28 citation statements)

References 8 publications

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“…Among other quantum mechanical analogs, the minimum-uncertainty squeezed states for atoms and photons in a cavity are reviewed in [28]. (See also [6,10,19,29] and the references therein for extensions to nonlinear geometrical optics; an optoacoustic experiment is proposed in [30].) In summary, we present multiparameter solutions to homogeneous and inhomogeneous paraxial wave equations which may be of interest in adaptive optics of (partially) coherent beams propagating through an atmospheric turbulence [17,[31][32][33] and deserve an experimental observation.…”

Section: × Expmentioning

confidence: 99%

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Spiral laser beams in inhomogeneous media

2013

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Explicit solutions of the inhomogeneous paraxial wave equation in a linear and quadratic approximation are applied to wave fields with invariant features, such as oscillating laser beams in a parabolic waveguide and spiral light beams in varying media. A similar effect of superfocusing of particle beams in a thin monocrystal film, harmonic oscillations of cold trapped atoms, and motion in magnetic field are also mentioned. Green function and generalized Fresnel integrals. In the context of quantum mechanics, a one-dimensional (1D) linear Schrödinger equation for generalized driven harmonic oscillators,(a, b, c, d, f , and g are suitable real-valued functions of time t only), can be solved by the integral superposition principle:Gx; y; tψy; 0dy;where Gx; y; t 2πμ 0 t −1∕2× expiα 0 tx 2 β 0 txy γ 0 ty 2 δ 0 tx ε 0 ty κ 0 t; (3) for certain initial data ψx; 0 φx (see [1][2][3][4] and the references therein for more details). The intrinsic connection between Hamiltonian mechanics and the process of wave propagation is anything but a new idea [5,6]. Yet, in paraxial optics, when the time variable t represents the coordinate, say s, in the direction of wave propagation, Eqs. (2) and (3) can be thought of as a generalization of the Fresnel integral [7][8][9][10].In the paraxial approximation, a 2D coherent light field in a parabolic inhomogeneous medium with coordinates r; s x; y; s is described by the following equation for the complex field amplitude: iA s −aA xx A yy bx 2 y 2 A

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Section: × Expmentioning

confidence: 99%

“…The intrinsic connection between Hamiltonian mechanics and the process of wave propagation is anything but a new idea [5,6]. Yet, in paraxial optics, when the time variable t represents the coordinate, say s, in the direction of wave propagation, Eqs.…”

Section: −1∕2mentioning

confidence: 99%

Spiral laser beams in inhomogeneous media

2013

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“…(20) are ordinary differen− tial equations which enable to describe elliptical GB self− −focusing and diffraction along arbitrary curvilinear trajec− tory in a nonlinear inhomogeneous fibres and are very useful for the analysis and numerical simulations. This gives a great advantage of the method under consideration over parabolic equation approximation [16][17][18][19][20][21][22][23]26,27] …”

Section: The General Riccati Equation For Gb Evolution In Nonlinear Smentioning

confidence: 99%

“…Moreover, the problem of GB evolution in nonlinear trans− versely and longitudinally inhomogeneous fibres is pre− sented in this paper. Firstly, this paper generalizes the prob− lem of GB diffraction in lens−like media [16][17][18][19]21,22,24,26,27] for beam evolution in nonlinear inhomogeneous media. New element is generalization of the results of pa− pers and books [28][29][30][31][32][33][34][35][36][37] for GB propagation in nonlinear sat− urable fibres with contribution of the linear refraction, the initial wave front curvature and the effect of fibre absorp− tion.…”

Section: Introductionmentioning

confidence: 99%

“…New element is generalization of the results of pa− pers and books [28][29][30][31][32][33][34][35][36][37] for GB propagation in nonlinear sat− urable fibres with contribution of the linear refraction, the initial wave front curvature and the effect of fibre absorp− tion. Next new element is generalization of the results of papers and books [25][26][27][28][29][30][31][32][33][34][35][36][37] for the case of GB evolution in nonlinear biconical tapering fibres. Moreover, we discuss in this paper the influence of nonuniformity of a nonlinear coefficient on GB evolution in under−critical and over−criti− cal regime of nonlinear fibre.…”

Section: Introductionmentioning

confidence: 99%

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Gaussian beam evolution in longitudinally and transversely inhomogeneous nonlinear fibres with absorption

Berczyński

2013

Opto-Electronics Review

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The method of complex geometrical optics (CGO) is presented, which describes Gaussian beam (GB) diffraction and self-focusing in smoothly inhomogeneous and nonlinear Kerr type and saturable fibres. CGO reduces the problem of Gaussian beam evolution in inhomogeneous and nonlinear media to the system of the first order ordinary differential equations for the complex curvature of the wave front and for GB amplitude, which can be readily solved both analytically and numerically. As a result, CGO radically simplifies the description of Gaussian beam diffraction and self-focusing effects as compared to other methods of nonlinear optics such as variational method approach, method of moments and beam propagation method. The power of CGO method is presented on the example of Gaussian beam propagation in saturable fibres with either focusing and defocusing refractive profiles. Besides, the influence of initial curvature of the wave front, phenomenon of weak absorption and effect of either transverse and longitudinal inhomogeneity on GB propagation in nonlinear fibres is discussed in this paper.

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Solving Nonlinear Parabolic Equations by a Strongly Implicit Finite Difference Scheme

Ghodgaonkar

Christov

2019

Applied Wave Mathematics II

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The parabolic equation in the theory of wave propagation

Cited by 29 publications

References 8 publications

Spiral laser beams in inhomogeneous media

Spiral laser beams in inhomogeneous media

Gaussian beam evolution in longitudinally and transversely inhomogeneous nonlinear fibres with absorption

Solving Nonlinear Parabolic Equations by a Strongly Implicit Finite Difference Scheme

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