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/Report/Conference proceedingConference contribution

Abstract

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
Pages460-463
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

Conference

ConferenceProgress in Electromagnetics Research Symposium
Abbreviated titlePIERS 2012 Moscow
CountryRussian Federation
CityMoscow
Period19/08/1223/08/12

Fingerprint

Green's function
Trajectories
Caustics
Solitons
Nonlinear equations
Fourier transforms
Electrons

Cite this

Nerukh, A. G., Zolotariov, D. A., Nerukh, D. A., & Georgiev, G. N. (2012). Green's function for paraxial equation. In PIERS proceedings Moscow: Progress In Electromagnetics Research Symposium (pp. 460-463). (Progress in electromagnetics research symposium). Electromagnetics Academy.
Nerukh, Alexander G. ; Zolotariov, D.A. ; Nerukh, D.A. ; Georgiev, Georgi N. / Green's function for paraxial equation. PIERS proceedings Moscow: Progress In Electromagnetics Research Symposium. Electromagnetics Academy, 2012. pp. 460-463 (Progress in electromagnetics research symposium).
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title = "Green's function for paraxial equation",
abstract = "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.",
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Nerukh, AG, Zolotariov, DA, Nerukh, DA & Georgiev, GN 2012, Green's function for paraxial equation. in PIERS proceedings Moscow: Progress In Electromagnetics Research Symposium. Progress in electromagnetics research symposium, Electromagnetics Academy, pp. 460-463, Progress in Electromagnetics Research Symposium, Moscow, Russian Federation, 19/08/12.

Green's function for paraxial equation. / Nerukh, Alexander G.; Zolotariov, D.A.; Nerukh, D.A.; Georgiev, Georgi N.

PIERS proceedings Moscow: Progress In Electromagnetics Research Symposium. Electromagnetics Academy, 2012. p. 460-463 (Progress in electromagnetics research symposium).

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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AU - Nerukh, D.A.

AU - Georgiev, Georgi N.

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Y1 - 2012/11/12

N2 - 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.

AB - 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.

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M3 - Conference contribution

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SN - 978-1-934142-22-6

T3 - Progress in electromagnetics research symposium

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EP - 463

BT - PIERS proceedings Moscow

PB - Electromagnetics Academy

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Nerukh AG, Zolotariov DA, Nerukh DA, Georgiev GN. Green's function for paraxial equation. In PIERS proceedings Moscow: Progress In Electromagnetics Research Symposium. Electromagnetics Academy. 2012. p. 460-463. (Progress in electromagnetics research symposium).