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Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 17, Iss. 11 — Nov. 1, 2000
  • pp: 2010–2018

Propagation of Laguerre–Bessel–Gaussian beams

Anthony A. Tovar  »View Author Affiliations

JOSA A, Vol. 17, Issue 11, pp. 2010-2018 (2000)

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New exact solutions to the paraxial wave equation are obtained in the form of a product of Laguerre polynomials, Bessel functions, and Gaussian functions. In the limit of large Laguerre–Gaussian beam size, the Bessel factor dominates and the solution sets reduce to the modes of closed resonators, hollow metal waveguides, and dielectric waveguides. In the opposite limit the solutions reduce to Laguerre–Gaussian modes of open resonators and graded-index waveguides. These solutions are valid for electromagnetic waves traveling through free space, and they are valid for propagation through circularly symmetric optical systems representable by ABCD matrices as well. An interesting feature of the new solution set is the existence of three mode indices, where only two are required for an orthogonal expansion. As an example, Laguerre–Gaussian beam propagation through an optical system that contains a Bessel-like amplitude filter is discussed.

© 2000 Optical Society of America

OCIS Codes
(050.1960) Diffraction and gratings : Diffraction theory
(060.2310) Fiber optics and optical communications : Fiber optics
(140.3410) Lasers and laser optics : Laser resonators
(260.0260) Physical optics : Physical optics
(350.5500) Other areas of optics : Propagation

Original Manuscript: November 23, 1999
Revised Manuscript: June 23, 2000
Manuscript Accepted: June 23, 2000
Published: November 1, 2000

Anthony A. Tovar, "Propagation of Laguerre–Bessel–Gaussian beams," J. Opt. Soc. Am. A 17, 2010-2018 (2000)

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  1. R. H. Dicke, “Molecular amplification and generation systems and methods,” U.S. patent2,851,652 (September9, 1958).
  2. G. D. Boyd, J. P. Gordon, “Confocal multimode resona-tor for millimeter through optical wavelength masers,” Bell Syst. Tech. J. 40, 489–508 (1961). [CrossRef]
  3. G. D. Boyd, H. Kogelnik, “Generalized confocal resonator theory,” Bell Syst. Tech. J. 41, 1347–1369 (1962). [CrossRef]
  4. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), 798–804.
  5. L. W. Casperson, A. A. Tovar, “Hermite–sinusoidal–Gaussian beam in complex optical systems,” J. Opt. Soc. Am. A 15, 954–961 (1998). [CrossRef]
  6. L. W. Casperson, D. G. Hall, A. A. Tovar, “Sinusoidal–Gaussian beams in complex optical systems,” J. Opt. Soc. Am. A 14, 3341–3348 (1997). [CrossRef]
  7. J. W. Strutt ( Rayleigh), “On the passage of electric waves through tubes, or the vibrations of dielectric cylinders,” Phil. Mag. Suppl. 43, 125–132 (1897). [CrossRef]
  8. R. H. Jordan, D. G. Hall, “Free-space azimuthal paraxial wave equation: the azimuthal Bessel–Gauss beam,” Opt. Lett. 19, 427–429 (1994). [CrossRef] [PubMed]
  9. A. A. Tovar, G. H. Clark, “Concentric-circle-grating, surface-emitting laser beam propagation in complex optical systems,” J. Opt. Soc. Am. A 14, 3333–3340 (1997). [CrossRef]
  10. H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965). [CrossRef]
  11. H. Kogelnik, T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550–1567 (1966). [CrossRef] [PubMed]
  12. A. A. Tovar, L. W. Casperson, “Generalized beam matrices. II. Mode selection in lasers and periodic misaligned complex optical systems,” J. Opt. Soc. Am. A 13, 90–96 (1997). [CrossRef]
  13. I. Kimel, L. R. Elias, “Relations between Hermite and Laguerre Gaussian Modes,” IEEE J. Quantum Electron. 29, 2562–2567 (1993). [CrossRef]
  14. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), Eq. (9.1.21).
  15. See, for example, M. R. Spiegel, Schaum’s Outline Series: Mathematical Handbook (McGraw-Hill, New York, 1991), Chap. 24.
  16. L. W. Casperson, “Spatial modulation of Gaussian laser beams,” Opt. Quantum Electron. 10, 483–493 (1978). [CrossRef]

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