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Journal of the Optical Society of America

Journal of the Optical Society of America

  • Vol. 63, Iss. 6 — Jun. 1, 1973
  • pp: 689–698

Smith—Purcell radiation from a line charge moving parallel to a reflection grating

P. M. van den Berg  »View Author Affiliations

JOSA, Vol. 63, Issue 6, pp. 689-698 (1973)

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A rigorous solution is obtained for the problem of radiation from an electric line charge that moves, at a constant speed, parallel to an electrically perfectly conducting grating. The relevant vectorial electromagnetic problem is reduced to a two-dimensional scalar one. With the aid of a Green’s-function formulation of the problem, an integral equation of the second kind for the surface current density on a single period of the grating surface is derived. This integral equation is solved numerically by a method of moments. Some numerical results pertaining to the radiation from a moving line charge above a sinusoidal grating are presented.

P. M. van den Berg, "Smith—Purcell radiation from a line charge moving parallel to a reflection grating," J. Opt. Soc. Am. 63, 689-698 (1973)

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  1. S. J. Smith and E. M. Purcell, Phys. Rev. 92, 1069 (1953).
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  12. For extensive literature about the Rayleigh hypothesis in scattering by a periodic surface we refer to B. A. Lippmann, J. Opt. Soc. Am. 43, 408 (1953); R. Petit and M. Cadilhac, C.R. Acad. Sci. (Paris) 262, 468 (1966); R. F. Millar, Proc. Camb. Philos. Soc. 65, 773 (1969); M. Neviere and M. Cadilhac, Opt. Commun. 2, 235 (1970); R. F. Millar, Proc. Camb. Philos. Soc. 69, 217 (1971).
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  15. A. referee has drawn the author's attention to an unpublished NDA Report (18-8) by B. A. Lippmann, in which the same Green's function has been derived in connection with his "variational formulation of a grating problem."
  16. The application of the surface impedance (Leontovich) boundary condition would introduce an extra parameter, viz., the frequency-dependent surface impedance. Since the surface-impedance boundary condition holds only in a limited frequency domain, we cannot represent the field quantities as Fourier integrals.
  17. Note that this Green's function is the same as the one used by B. A. Lippmann in NDA Report No. 18-8.
  18. This phenomenon is just the "lateral wave" case of the "Wood anomalies" (Rayleigh wavelengths) observed with diffraction gratings [see J. E. Stewart and W. S. Gallaway, Appl. Opt. 1, 421 (1962)].
  19. P. M. van den Berg, Thesis, Delft University of Technology, The Netherlands (1971).
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