Green's function for paraxial equation

Alexander G. Nerukh*, D.A. Zolotariov, D.A. Nerukh, Georgi N. Georgiev

*Corresponding author for this work

    Research output: Chapter in Book/Published conference outputConference publication


    The theory and experimental applications of optical Airy beams are in active development recently. The Airy beams are characterised by very special properties: they are non-diffractive and propagate along parabolic trajectories. Among the striking applications of the optical Airy beams are optical micro-manipulation implemented as the transport of small particles along the parabolic trajectory, Airy-Bessel linear light bullets, electron acceleration by the Airy beams, plasmonic energy routing. The detailed analysis of the mathematical aspects as well as physical interpretation of the electromagnetic Airy beams was done by considering the wave as a function of spatial coordinates only, related by the parabolic dependence between the transverse and the longitudinal coordinates. Their time dependence is assumed to be harmonic. Only a few papers consider a more general temporal dependence where such a relationship exists between the temporal and the spatial variables. This relationship is derived mostly by applying the Fourier transform to the expressions obtained for the harmonic time dependence or by a Fourier synthesis using the specific modulated spectrum near some central frequency. Spatial-temporal Airy pulses in the form of contour integrals is analysed near the caustic and the numerical solution of the nonlinear paraxial equation in time domain shows soliton shedding from the Airy pulse in Kerr medium. In this paper the explicitly time dependent solutions of the electromagnetic problem in the form of time-spatial pulses are derived in paraxial approximation through the Green's function for the paraxial equation. It is shown that a Gaussian and an Airy pulse can be obtained by applying the Green's function to a proper source current. We emphasize that the processes in time domain are directional, which leads to unexpected conclusions especially for the paraxial approximation.

    Original languageEnglish
    Title of host publicationPIERS proceedings Moscow
    Subtitle of host publicationProgress In Electromagnetics Research Symposium
    PublisherElectromagnetics Academy
    Number of pages4
    ISBN (Print)978-1-934142-22-6
    Publication statusPublished - 12 Nov 2012
    EventProgress in Electromagnetics Research Symposium - Moscow, Russian Federation
    Duration: 19 Aug 201223 Aug 2012

    Publication series

    NameProgress in electromagnetics research symposium
    PublisherElectromagnetics Academy
    ISSN (Print)1559-9450


    ConferenceProgress in Electromagnetics Research Symposium
    Abbreviated titlePIERS 2012 Moscow
    Country/TerritoryRussian Federation


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