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Abstrakty
Time-harmonic electromagnetic waves in vacuum are described by the Helmholtz equation Δu + ω2u = 0 for (x, y, z) ∈ ℝ3. For the evolution of such waves along the z-axis, a Schrödinger equation can be derived through a multiple scaling ansatz. It is the purpose of this paper to justify this formal approximation by proving bounds between this formal approximation and true solutions of the original system. The challenge of the presented validity analysis is the fact that the Helmholtz equation is ill-posed as an evolutionary system along the z-axis.
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Rocznik
Tom
Strony
67--72
Opis fizyczny
Bibliogr. 11 poz.
Twórcy
autor
- Institut für Analysis, Dynamik und Modellierung, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
autor
- Institut für Analysis, Dynamik und Modellierung, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
Bibliografia
- [1] J. E. Harvey, R. G. Irvin and R. N. Pfisterer, Modeling physical optics phenomena by complex ray tracing, Optical Eng. 54 (2015), no. 3, Article ID 035105.
- [2] J. D. Jackson, Classical Electrodynamics, John Wiley & Sons, New York, 1988.
- [3] L. A. Kalyakin, Asymptotic decay of a one-dimensional wave packet in a nonlinear dispersive medium, Math. USSR, Sb. 60 (1988), no. 2, 457-483.
- [4] P. Kirrmann, G. Schneider and A. Mielke, The validity of modulation equations for extended systems with cubic nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A 122 (1992), no. 1-2, 85-91.
- [5] M. Marte and S. Stenholm, Paraxial light and atom optics: The optical schrödinger equation and beyond, Phys. Rev. A. 56 (1997), 2940-2953.
- [6] D. Rafferty, U. H. Wagner, C. Rau, P. Chang, S. Alcock, R. Dockree and I. R. Robinson, Development of a computer model to simulate wavefront propagation, https://www.ucl.ac.uk/~ucapikr/projects/Dominic_report.pdf.
- [7] J. Rauch, Hyperbolic Partial Differential Equations and Geometric Optics, Grad. Stud. Math. 133, American Mathematical Society, Providence, 2012.
- [8] J. D. Schmidt, Numerical Simulation of Optical Wave Propagation with Examples in MATLAB, SPIE Press, Bellingham, 2010.
- [9] G. Schneider, Justification and failure of the nonlinear Schrödinger equation in case of non-trivial quadratic resonances, J. Differential Equations 216 (2005), no. 2, 354-386.
- [10] G. Schneider and H. Uecker, Nonlinear PDEs. A Dynamical Systems Approach, Grad. Stud. Math. 182, American Mathematical Society, Providence, 2017.
- [11] H. Yoda, P. Polynkin and M. Mansuripur, Beam quality factor of higher order modes in a step-index fiber, J. Lightwave Technol. 24 (2006), no. 3, Paper No. 1350.
Uwagi
Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023).
Typ dokumentu
Bibliografia
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bwmeta1.element.baztech-d39cb6bc-4ab0-4e71-8cd1-5ca2068daf22