OSA's Digital Library

Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 18, Iss. 1 — Jan. 1, 2001
  • pp: 177–184

Relationship between elegant Laguerre–Gauss and Bessel–Gauss beams

Miguel A. Porras, Riccardo Borghi, and Massimo Santarsiero  »View Author Affiliations

JOSA A, Vol. 18, Issue 1, pp. 177-184 (2001)

View Full Text Article

Enhanced HTML    Acrobat PDF (192 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



We show that the elegant Laguerre–Gauss light beams of high radial order n are asymptotically equal to Bessel–Gauss light beams. The Bessel–Gauss beam equivalent to each elegant Laguerre–Gauss beam is found and shown to have almost identical propagation factors M2. In the limit n, elegant Laguerre–Gauss beams can be identified with Durnin’s Bessel beam. Our results suggest a new experimental procedure for generating light beams with nondiffractinglike properties directly from the output of a stable resonator.

© 2001 Optical Society of America

OCIS Codes
(050.1940) Diffraction and gratings : Diffraction
(140.3300) Lasers and laser optics : Laser beam shaping
(260.1960) Physical optics : Diffraction theory
(350.5500) Other areas of optics : Propagation

Original Manuscript: May 16, 2000
Revised Manuscript: July 28, 2000
Manuscript Accepted: July 28, 2000
Published: January 1, 2001

Miguel A. Porras, Riccardo Borghi, and Massimo Santarsiero, "Relationship between elegant Laguerre–Gauss and Bessel–Gauss beams," J. Opt. Soc. Am. A 18, 177-184 (2001)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. N. G. Vakhimov, “Open resonators with mirrors having variable reflection coefficients,” Radio Eng. Electron. Phys. (USSR) 10, 1439–1446 (1965).
  2. H. Zucker, “Optical resonators with variable reflectivity mirrors,” Bell Syst. Tech. J. 49, 2349–2376 (1970). [CrossRef]
  3. J. A. Arnaud, “Mode coupling in first-order optics,” J. Opt. Soc. Am. 61, 751–758 (1971). [CrossRef]
  4. L. W. Casperson, “Beam modes in complex lenslike media and resonators,” J. Opt. Soc. Am. 66, 1373–1379 (1976). [CrossRef]
  5. A. E. Siegman, “Hermite–Gaussian functions of complex argument as optical-beam eigenfunctions,” J. Opt. Soc. Am. 63, 1093–1094 (1973). [CrossRef]
  6. J. A. Arnaud, “Hamiltonian theory of beam mode propagation,” Prog. Opt. 11, 247–304 (1973). [CrossRef]
  7. R. Pratesi, L. Ronchi, “Generalized Gaussian beams in free space,” J. Opt. Soc. Am. 67, 1274–1276 (1977). [CrossRef]
  8. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).
  9. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).
  10. T. Takenaka, M. Yokota, O. Fukumitsu, “Propagation of light beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 2, 826–829 (1985). [CrossRef]
  11. E. Zauderer, “Complex argument Hermite–Gaussian and Laguerre–Gaussian beams,” J. Opt. Soc. Am. A 3, 465–469 (1986). [CrossRef]
  12. E. Heyman, I. Beracha, “Complex multipole pulsed beams and Hermite pulsed beams: a novel expansion scheme for transient radiation from well-collimated apertures,” J. Opt. Soc. Am. A 9, 1779–1793 (1992). [CrossRef]
  13. M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975). [CrossRef]
  14. M. Yokota, T. Takenaka, O. Fukumitsu, “Relations between conventional and complex beams,” Trans. Inst. Electron. Commun. Eng. Jpn. 68-C, 1130–1131 (1985).
  15. S. Saghafi, C. J. R. Sheppard, “Near field and far field of elegant Hermite–Gaussian and Laguerre–Gaussian modes,” J. Mod. Opt. 45, 1999–2009 (1998). [CrossRef]
  16. C. J. R. Sheppard, T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” Microwaves Opt. Acoust. 2, 105–112 (1978). [CrossRef]
  17. F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987). [CrossRef]
  18. J. Durnin, J. J. Miceli, J. H. Eberly, “Nondiffracting beams,” Phys. Rev. Lett. 58, 1499–1502 (1987). [CrossRef] [PubMed]
  19. J. Rosen, B. Salik, A. Yariv, H. K. Liu, “Pseudonondiffracting slitlike beam and its analogy to the pseudonondispersing pulse,” Opt. Lett. 20, 423–425 (1995). [CrossRef] [PubMed]
  20. Q. Cao, S. Chi, “Axially symmetric on-axis flat-top beam,” J. Opt. Soc. Am. A 17, 447–455 (2000). [CrossRef]
  21. V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, G. Schirripa Spagnolo, “Generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).
  22. M. Santarsiero, “Propagation of generalized Bessel–Gauss beams through ABCD optical systems,” Opt. Commun. 132, 1–7 (1996). [CrossRef]
  23. R. Borghi, M. Santarsiero, “M2 factor of Bessel–Gauss beams,” Opt. Lett. 22, 262–264 (1997). [CrossRef] [PubMed]
  24. F. Gori, G. Guattari, C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987). [CrossRef]
  25. C. Palma, R. Borghi, G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125, 113–121 (1996). [CrossRef]
  26. R. Borghi, “Superposition scheme for J0-correlated partially coherent sources,” IEEE J. Quantum Electron. 35, 849–856 (1999). [CrossRef]
  27. A. V. Shchegrov, E. Wolf, “Partially coherent conical beams,” Opt. Lett. 25, 141–143 (2000). [CrossRef]
  28. R. H. Jordan, D. G. Hall, “Free-space azimuthal paraxial wave equation: the azimuthal Bessel–Gauss solution,” Opt. Lett. 19, 427–429 (1994). [CrossRef] [PubMed]
  29. D. G. Hall, “Vector-beam solution of Maxwell’s wave equation,” Opt. Lett. 21, 9–12 (1996). [CrossRef] [PubMed]
  30. C. Olson, P. L. Greene, G. W. Wicks, D. G. Hall, S. Rishton, “High-order azimuthal spatial modes of concentric-circle-grating surface-emitting semiconductor lasers,” Appl. Phys. Lett. 72, 1284–1286 (1998). [CrossRef]
  31. P. L. Greene, D. G. Hall, “Properties and diffraction of vector Bessel–Gauss beams,” J. Opt. Soc. Am. A 15, 3020–3027 (1998). [CrossRef]
  32. B. Glushko, B. Kryzhanovsky, D. Sarkisyan, “Self-phase-matching mechanism for efficient harmonic generation processes in a ring pump beam geometry,” Phys. Rev. Lett. 71, 243–246 (1993). [CrossRef] [PubMed]
  33. S. P. Tewari, H. Huang, R. W. Boyd, “Theory of third-harmonic generation using Bessel beams, and self-phase matching,” Phys. Rev. A 54, 2314–2325 (1996). [CrossRef] [PubMed]
  34. C. F. R. Caron, R. M. Potvliege, “Phase matching and harmonic generation in Bessel–Gauss beams,” J. Opt. Soc. Am. B 15, 1096–1106 (1998). [CrossRef]
  35. J. Arlt, K. Dholakia, L. Allen, M. J. Padgett, “Efficiency of second-harmonic generation with Bessel beams,” Phys. Rev. A 60, 2438–2441 (1999). [CrossRef]
  36. C. Altucci, R. Bruzzese, D. D’Antuoni, C. de Lisio, S. Solimeno, “Harmonic generation in gases by use of Bessel–Gauss laser beams,” J. Opt. Soc. Am. B 17, 34–42 (2000). [CrossRef]
  37. P. Pääkkönen, J. Turunen, “Resonators with Bessel–Gauss modes,” Opt. Commun. 156, 359–366 (1998). [CrossRef]
  38. A. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 2–14 (1990). [CrossRef]
  39. J. Durnin, J. J. Miceli, J. H. Eberly, “Comparison of Bessel and Gaussian beams,” Opt. Lett. 13, 79–80 (1988). [CrossRef] [PubMed]
  40. B. Hafizi, P. Sprangle, “Diffracted effects in directed radiation beams,” J. Opt. Soc. Am. A 8, 705–717 (1991). [CrossRef]
  41. I. S. Gradshtein, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, New York, 1980).
  42. P. M. Mejı́as, H. Weber, R. Martı́nez-Herrero, A. González-Ureña, eds., Proceedings of the Workshop on Laser Beam Characterization (Sociedad Espanola de Optica, Madrid, 1993); H. Weber, N. Reng, J. Lüdtke, P. M. Mejı́as, eds., Laser Beam Characterization (Festkörper-Laser-Institute, Berlin, 1994); M. Morin, A. Giesen, eds., Third International Workshop on Laser Beam and Op-tics Characterization, Proc. SPIE2870 (1996); A. Giesen, M. Morin, eds., Proceedings of the Fourth International Workshop on Laser Beam and Optics Characterization (Institut für Strahlwerkzeuge, Munich, 1997).
  43. R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD law,” Opt. Commun. 65, 322–328 (1988). [CrossRef]
  44. P. A. Bélanger, “Beam propagation and the ABCD ray matrices,” Opt. Lett. 16, 196–198 (1991). [CrossRef]
  45. A. Siegman, “Defining the effective radius of curvature of a nonideal optical beam,” IEEE J. Quantum Electron. 27, 1146–1148 (1991). [CrossRef]
  46. S. Saghafi, C. J. R. Sheppard, “The beam propagation factor for higher-order Gaussian beams,” Opt. Commun. 153, 207–210 (1998). [CrossRef]
  47. M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993). [CrossRef]
  48. See, for example, M. J. Padgett, L. Allen, “The angular momentum of light: optical spanners and the rotational frequency shift,” Opt. Quantum Electron. 31, 1–12 (1999), and references therein. [CrossRef]
  49. R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1978).

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


Fig. 1 Fig. 2 Fig. 3

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited