Nonparaxial hypergeometric modes
V.V. Kotlyar, A.A. Kovalev

Image Processing Systems Institute of the Russian Academy of Sciences,
S.P. Korolyov Samara State Aerospace University

Full text of article: Russian language.

Abstract:
We derive an analytical expression to describe the exact solution of Helmholtz equation in the cylindrical coordinates as a product of two Kummer’s functions. The solution is presented as a sum of two terms that describe the non-paraxial hypergeometric light beams propagated along the optical axis in the positive and negative directions. With the distance from the initial plane becoming much larger than the wavelength of light, the expression derived for the non-paraxial hypergeometric beam coincides with that for a paraxial hypergeometric mode.

Key words:
Schrödinger equation, Helmholtz equation, nonparaxial diffraction theory, angular spectrum of plane waves, confluent function (Kummer’s function), hypergeometric beam, hypergeometric mode.

Citation: Kotlyar VV, Kovalev AA. Nonparaxial hypergeometric modes. Computer Optics 2008; 32(3): 222-5.

References:

  1. Kotlyar VV, Skidanov RV, Khonina SN, Soifer VA. Hypergeometric modes. Optics Letters 2007; 32(7): 742-744.
  2. Karimi E. Hypergeometric-Gaussian modes. Opt. Lett. 2007; 32: 3053-3055.
  3. Kotlyar  VV, Kovalev AA. Family of hypergeometric laser beams. J. Opt. Soc. Am. A 2008; 25(1): 262-270.
  4. Bandres MA, Gutiérrez-Vega JC. Circular beams. Opt. Lett. 2008; 33: 177-179.
  5. Siegman AE. Lasers. University Science, 1986.
  6. Takenaka T, Yokota M, Fukumitsu O. Propagation of light beams beyond the paraxial approximation. J. Opt. Soc. Am. A 1985; 2: 826-829.
  7. Caron CFR, Potvliege RM. Bessel-modulated Gaussian beams with quadratic radial dependence. Opt. Commun. 1999; 164: 83-93.
  8. Rozas D, Law CT, Swartzlander GA. Propagation dynamics of optical vortices. J. Opt. Soc. Am. B 1997; 14: 3054-3065.
  9. Kotlyar VV. Generation of phase singularity through diffracting a plane or Gaussian beam by a spiral phase plate. J. Opt. Soc. Am. A 2005; 22:.849-861.
  10. Prudnikov AP, Brychkov YuA, Marichev OI. Integrals and Series. Special Functions [In Russian]. Moscow: “Nauka” (Science) Publisher; 1983.
  11. Abramovitz M, Stegun IA. Handbook of Mathematical Functions. National Bureau of Standards, Applied Math. Series, 1965.

© 2009, ИСОИ РАН
Россия, 443001, Самара, ул. Молодогвардейская, 151; электронная почта: ko@smr.ru ; тел: +7 (846) 332-56-22, факс: +7 (846 2) 332-56-20