About one exact solution of linear Schrӧdinger equation, describing femtosecond pulse propagation

Svetlana Stepanenko


In current paper, a model of a femtosecond laser pulse propagation in a linear medium is considered. At the study of laser physics problems, described by the Schrödinger equations, a special role is played by the problem of an exact solution existence. The present paper proposes one of the approaches for the construction of exact solution, which consists in the transform of the Schrödinger equation to the Airy equation. An exact solution is developed, using this approach. It should be stressed that even in the case of the propagation of a laser pulse in a linear medium, no exact solutions were found early for the equation under consideration in the paper, and this problem is an urgent one. The exact solution was found for the Schrödinger equation as a product of the exponential function and the Airy functions, and linearly independent solutions of the Airy equation were written, whose properties were used under the analysis of initial laser pulse distributions. Let us note that the obtained solution includes a free parameter that corresponds to a change in the amplitude of the solution oscillations on one of the boundaries of the region considered in the problem. In addition, initial distributions, corresponding to obtained solution and depending on the physical parameters of the problem, are depicted in Figures.

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N. Tzoar, M. Jain, “Self-phase modulation in long-geometry optical waveguides”, Phys. Rev. A, vol. 23, no. 3, pp. 1266–1270, 1981.

G. Agrawal, “Nonlinear Fiber Optics”, Academic, 4th ed., 2007.

T. Brabec, F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime”, Phys. Rev. Lett., vol. 78, no. 17, pp. 3282–13285, 1997.

T. Brabec, F. Krausz, “Intense few-cycle laser fields: Frontiers of nonlinear optics”, Rev. Mod. Phys., vol. 72, no. 2, pp. 545–591, 2000.

S.V. Stepanenko, A.V. Razgulin, V.A. Trofimov, “About one model, describing femtosecond pulse distribution in a medium with Kerr nonlinearity”, Applied Mathematics and Computer Science, vol. 60, М.: MAKSPress, pp. 51–61, 2019.

M. V. Berry, N. L. Balazs, “Nonspreading wave packets”, Am. J. Phys. vol. 47, no. 4, pp. 264–267, 1979.

O. Vallée, M. Soares, “Airy Functions and Applications to Physics”, World Scientific, Hackensack NJ, 2004.

A. Gil, J. Segura, and N. M. Temme, “On Non-Oscillating Integrals for Computing Inhomogeneous Airy Functions”, Math. Comput., vol.70, pp. 1183–1194, 2000.

A. Gil, J. Segura, “On the complex zeros of Airy and Bessel functions and those of their derivatives”, Anal. Appl., vol. 12, pp. 537–561, 2014.

S. Masaki, J.Segata, “Refinement of Strichartz Estimates for Airy

Equation in Nondiagonal Case and its Application”, SIAM J. Math. Anal., vol. 50, no. 3, pp. 2839–2866, 2017.

P. Caputa, Sh. Hirano, “Airy Function and 4d Quantum Gravity”, J. High En. Phys., 2018.

A. K. Ghatak, R. L. Gallawa, I. C. Goyal, “Accurate Solutions to Schrӧdinger's Equation Using Modified Airy Functions”, IEEE Journal of Quantum Electronics, vol. 28, pp. 400–403, 1992.

M. Miyagi, S. Nishida, “Pulse spreading in a single-mode fiber due to third-order dispersion”, Appl Opt., vol. 18, pp. 678–682, 1979.

A. Mahalov, S. Suslov, “An "Airy gun": Self-accelerating solutions of the time-dependent Schrӧdinger equation in vacuum”, Phys. Lett. A, vol. 377, pp. 33–38, 2012.

W. Cai, L. Wang, Sh. Wen, “Evolution of airy pulses in the present of third order dispersion”, Optik - Int. J. Light Electron Opt., vol. 124, pp. 5833–5836, 2013.

A. Mahalov, E. Suazo, S. K. Suslov, “Spiral laser beams in inhomogeneous media”, Opt. Lett., vol. 38, pp. 2763–2766, 2013.

Chr. Koutschan, E. Suazo, S. K. Suslov, “Fundamental laser modes in paraxial optics: from computer algebra and simulations to experimental observation”, Appl.Phys.B, vol. 121, pp. 315–336, 2015.

R Li., M. Imran, H. Chen, “Airy plasmons in graphene based waveguides”, IEEE MTT-S International Microwave Workshop Series on Advanced Materials and Processes for RF and THz Applications (IMWS-AMP) , pp. 1–3, 2016.

J. A. Borda-Hernández, “Propagation of time-truncated Airy-type pulses in media with quadratic and cubic dispersion”, J. Opt. Soc. Am. A, vol. 32, pp. 1791–1796, 2015.

K. K. De, H. Kaur, A. Goyal, C. N. Kumar, T. S. Raju, “Airy-Bessel modulated self-similar rogue waves in a nonlinear Schrӧdinger equation model”, Journal of Modern Optics, vol. 62, pp. 137–144, 2015.

N. K. Efremidis, “Spatiotemporal diffraction-free pulsed beams in free-space of the Airy and Bessel type”, Opt.Lett., vol. 42, no. 23, pp. 5038–5041 (2017).

R. Driben, V. V. Konotop, T. Meier, “Coupled Airy breathers”, Opt.Lett., vol. 39, pp. 5523–5526, 2014.

D.V. Karlovets, “Gaussian and Airy wave-packets of massive particles with orbital angular momentum”, Phys.Rev.A vol. 91, pp. 013847, 2015.

C. J. Zapata-Rodríguez, M. Naserpour, “Nonparaxial shape-preserving Airy beams with Bessel signature”, Opt. Lett. vol. 39, pp. 2507–2510, 2014.

M. Zamboni-Rached, K. Z. Nóbrega, C. A. Dartora, “Analytic description of Airy-type beams when truncated by finite apertures”,

Opt. Expr., vol. 20, no.18, pp. 19972–19977, 2012.

A. Banerjee, S. Roy, “Collision-mediated radiation due to Airy-soliton interaction in a nonlinear Kerr medium”, Phys. Rev. A vol. 98, pp. 033806, 2018.


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