Studia Geophysica et Geodaetica

, Volume 56, Issue 2, pp 355–372 | Cite as

A tale of two beams: an elementary overview of Gaussian beams and Bessel beams



An overview of two types of beam solutions is presented, Gaussian beams and Bessel beams. Gaussian beams are examples of non-localized or diffracting beam solutions, and Bessel beams are example of localized, non-diffracting beam solutions. Gaussian beams stay bounded over a certain propagation range after which they diverge. Bessel beams are among a class of solutions to the wave equation that are ideally diffraction-free and do not diverge when they propagate. They can be described by plane waves with normal vectors along a cone with a fixed angle from the beam propagation direction. X-waves are an example of pulsed beams that propagate in an undistorted fashion. For realizable localized beam solutions, Bessel beams must ultimately be windowed by an aperture, and for a Gaussian tapered window function this results in Bessel-Gauss beams. Bessel-Gauss beams can also be realized by a combination of Gaussian beams propagating along a cone with a fixed opening angle. Depending on the beam parameters, Bessel-Gauss beams can be used to describe a range of beams solutions with Gaussian beams and Bessel beams as end-members. Both Gaussian beams, as well as limited diffraction beams, can be used as building blocks for the modeling and synthesis of other types of wave fields. In seismology and geophysics, limited diffraction beams have the potential of providing improved controllability of the beam solutions and a large depth of focus in the subsurface for seismic imaging.


Gaussian beams Bessel beams Bessel-Gauss beams wave propagation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Aki K. and Richards P., 1980. Quantitative Seismology. W.H. Freeman, San Francisco, CA.Google Scholar
  2. Alkhalifah T., 1995. Gaussian beam depth migration for anisotropic media. Geophysics, 60, 1474–1484.Google Scholar
  3. Babich V.M. and Popov M.M., 1989. Gaussian beam summation (review). Izvestiya Vysshikh Uchebnykh Zavedenii Radiofizika, 32, 1447–1466 (in Russian, translated in Radiophysics and Quantum Electronics, 32, 1063–1081, 1990).Google Scholar
  4. Bateman H., 1915. Electrical and Optical Wave Motion. Cambridge University Press, Cambridge, U.K.Google Scholar
  5. Bouchal Z., 2003. Nondiffracting optical beams: physical properties, experiments, and applications. Czech J. Phys., 53, 537–578.CrossRefGoogle Scholar
  6. Brillouin L., 1960. Wave Propagation and Group Velocity. Academic Press, New York.Google Scholar
  7. Brittingham J.N., 1983. Focus wave modes in homogeneous Maxwell’s equations: transverse electric mode. J. Appl. Phys., 54, 1179–1189.CrossRefGoogle Scholar
  8. Burckhardt C.B., Hoffmann H. and Grandchamp P.-A., 1973. Ultrasound axicon: a device for focusing over a large depth. J. Acoust. Soc. Am., 54, 1628–1630.CrossRefGoogle Scholar
  9. Červený V., 2001. Seismic Ray Theory. Cambridge University Press, Cambridge, U.K.CrossRefGoogle Scholar
  10. Červený V., Klimeš and Pšenčík I., 2007. Seismic ray method: recent developments. Adv. Geophys, 48, 1–128.CrossRefGoogle Scholar
  11. Červený V., Popov M.M. and Pšenčík I., 1982. Computation of wavefields in inhomogeneous media — Gaussian beam approach. Geophys. J. R. Astr. Soc., 70, 109–128.CrossRefGoogle Scholar
  12. Chavez-Cerda S., 1999. A new approach to Bessel beams. J. Mod. Opt., 46, 923–930.Google Scholar
  13. Chew W.C., 1990. Waves and Fields in Inhomogeneous Media. Van Nostrand Reinhold, New York.Google Scholar
  14. Courant R. and Hilbert D., 1966. Methods of Mathematical Physics, Vol. 2. Wiley, New York.Google Scholar
  15. Deschamps G.A., 1971. Gaussian beams as a bundle of complex rays. Electron. Lett., 7, 684–685.CrossRefGoogle Scholar
  16. de Hoop M.V., Smith H., Uhlmann G. and van der Hilst R.D., 2009. Seismic imaging with the generalized Radon transform: a curvelet transform perspective. Inverse Probl., 25, 025005, DOI: 10.1088/0266-5611/25/2/025005.CrossRefGoogle Scholar
  17. Douma H. and de Hoop M.V., 2007. Leading-order seismic imaging using curvelets. Geophysics, 72, S231–S248.Google Scholar
  18. Durnin J., 1987. Exact solutions for nondiffracting beams: I. the scalar theory. J. Opt. Soc. Am. A, 4, 651–654.CrossRefGoogle Scholar
  19. Durnin J., Miceli J.J. and Eberly J.H., 1987. Diffraction-free beams. Phys. Rev. Lett., 58, 1499–1501.CrossRefGoogle Scholar
  20. Felsen L.B., 1976. Complex-source-point solutions of the field equations and their relation to the propagation and scattering of Gaussian beams. Symp. Matemat., 18. Instituto Nazionale di Alta Matematica, Academic Press, London, 40–56.Google Scholar
  21. Feng S. and Winful H.G., 2001. Physical origin of the Gouy shift. Opt. Lett., 26, 485–487.CrossRefGoogle Scholar
  22. Flammer C., 1957. Speroidal Wave Functions. Stanford Press, Stanford, CA.Google Scholar
  23. Gori F., Guattari G. and Padovani C., 1987. Bessel-Gauss beams. Optics Commun., 64, 491–495.CrossRefGoogle Scholar
  24. Gray S.H., 2005. Gaussian beam migration of common-shot records. Geophysics, 70, S71–S77.Google Scholar
  25. Gray S.H. and Bleistein N., 2009. True-amplitude Gaussian-beam migration. Geophysics, 74, S11–S23.Google Scholar
  26. Hill N.R., 1990. Gaussian beam migration. Geophysics, 55, 1416–1428.Google Scholar
  27. Hill N.R., 2001. Prestack Gaussian beam depth migration. Geophysics, 66, 1240–1250.Google Scholar
  28. ISGILW-Sanya2011, 2011. International Symposium on Geophysical Imaging with Localized waves.
  29. Keller J.B. and Streifer W. 1971. Complex rays with applications to Gaussian beams. J. Opt. Soc. Am., 61, 41–43.CrossRefGoogle Scholar
  30. Kravtsov Yu.A. and Berczynski P., 2007. Gaussian beams in inhomogeneous media: a review. Stud. Geophys. Geod., 51, 1–36.CrossRefGoogle Scholar
  31. Lu J.Y., 1997. 2D and 3D high frame rate imaging with limited diffraction beams. IEEE Trans. Ultrason. Ferroelectr. Freq. Control, 44, 839–856.CrossRefGoogle Scholar
  32. Lu J.Y., 2008. Ultrasonic imaging with limited-diffraction beams. In: Hernandez-Figueroa H.E., Zamboni-Rached M. and Recami E. (Eds.), Localized Waves. Wiley Interscience, New York, 97–128.CrossRefGoogle Scholar
  33. Lu J.Y. and Greenleaf J.F., 1992a. Nondiffracting X-waves: exact solution to free-space scalar wave equation and their finite aperture realizations. IEEE Trans. Ultrason. Ferroelectr. Freq. Control, 39, 19–31.CrossRefGoogle Scholar
  34. Lu J.Y. and Greenleaf J.F., 1992b. Experimental verification of nondiffracting X-waves. IEEE Trans. Ultrason. Ferroelectr. Freq. Control, 39, 441–446.CrossRefGoogle Scholar
  35. Lu J.Y. and Greenleaf J.F., 1994. Biomedical ultrasound beam forming. Ultrasound Med. Biol., 20, 403–428.CrossRefGoogle Scholar
  36. Lu J.Y. and Greenleaf J.F., 1995. Comparison of sidelobes of limited diffraction beams and localized waves. In: Jones J.P. (Ed.), Acoustic Imaging, Vol. 21. Plenum Press, New York, 145–152.CrossRefGoogle Scholar
  37. Lu J.Y. and Liu A., 2000. An X wave transform. IEEE Trans. Ultrason. Ferroelectr. Freq. Control, 47, 1472–1481.CrossRefGoogle Scholar
  38. Lunardi J.T., 2001. Remarks on Bessel beams, signals and superluminality. Phys. Lett. A, 291, 66–72.CrossRefGoogle Scholar
  39. McDonald K.T., 2000. Bessel beams.
  40. McDonald K.T., 2002. Gaussian laser beams via oblate spheroidal waves.
  41. McGloin D. and Dholakia K., 2005. Bessel beams: diffraction in a new light. Contemp. Phys., 46, 15–28.CrossRefGoogle Scholar
  42. Mcleod J.H., 1954. The axicon: a new type of optical element. J. Opt. Soc. Am., 44, 592–597.CrossRefGoogle Scholar
  43. Mcleod J.H., 1960. Axicons and their uses. J. Opt. Soc. Am., 50, 166–169.CrossRefGoogle Scholar
  44. Milonni P.W., 2005. Fast Light, Slow Light and Left-Handed Light. Institute of Physics Publ. Bristol, U.K.Google Scholar
  45. Mugnai D., Ranfagni A. and Ruggeri R., 2000. Observation of superluminal behaviors in wave propagation. Phys. Rev. Lett., 84, 4830–4833.CrossRefGoogle Scholar
  46. Nowack R.L., 2003. Calculation of synthetic seismograms with Gaussian beams. Pure Appl. Geophys., 160, 487–507.CrossRefGoogle Scholar
  47. Nowack R.L., 2008. Focused Gaussian beams for seismic imaging. SEG Expanded Abstracts, 27, 2376–2380.CrossRefGoogle Scholar
  48. Nowack R.L., 2011. Dynamically focused Gaussian beams for seismic imaging. Int. J. Geophys., 316581, DOI: 10.1155/2011/316581.Google Scholar
  49. Nowack R.L. and Kainkaryam S.M., 2011. The Gouy phase anomaly for harmonic and time-domain paraxial Gaussian beams. Geophys. J. Int., 184, 965–973.CrossRefGoogle Scholar
  50. Nowack R.L., Sen M.K. and Stoffa P.L., 2003. Gaussian beam migration of sparse common-shot data. SEG Expanded Abstracts, 22, 1114–1117.CrossRefGoogle Scholar
  51. Palma C., 2001. Decentered Gaussian beams, ray bundles and Bessel-Gauss beams. Appl. Optics, 36, 1116–1120.CrossRefGoogle Scholar
  52. Popov M.M., 1982. A new method of computation of wave fields using Gaussian beams. Wave Motion, 4, 85–97.CrossRefGoogle Scholar
  53. Popov M.M., 2002. Ray Theory and Gaussian Beam Method for Geophysicists. Lecture Notes, University of Bahia, Salvador, Brazil.Google Scholar
  54. Popov M.M., Semtchenok N.M., Popov P.M. and Verdel A.R., 2010. Depth migration by the Gaussian beam summation method. Geophysics, 75, S81–S93.Google Scholar
  55. Protasov M.I. and Cheverda V.A., 2006. True-amplitude seismic imaging. Dokl. Earth Sci., 407, 441–445.CrossRefGoogle Scholar
  56. Protasov M.I. and Tcheverda V.A., 2011. True amplitude imaging by inverse generalized Radon transform based on Gaussian beam decomposition of the acoustic Green’s function, Geophys.l Prospect., 59, 197–209.CrossRefGoogle Scholar
  57. Recami E., 1998. On localized “X-shaped” superluminal solutions to Maxwell equations. Physica A, 252, 586–610.CrossRefGoogle Scholar
  58. Recami E. and Zamboni-Rached M., 2009. Localized waves: a review. Adv. Imag. Electron Phys., 156, 235–353.CrossRefGoogle Scholar
  59. Recami E. and Zamboni-Rached M., 2011. Non-diffracting waves, and “frozen waves”: an introduction. International Symposium on Geophysical Imaging with Localized Waves, Sanya, China, July 24–28 2011,
  60. Recami E., Zamboni-Rached M. and Hernandez-Figuerao H.E., 2008. Localized waves: a historical and scientific introduction. In: Hernandez-Figueroa H.E., Zamboni-Rached M. and Recami E. (Eds.), Localized Waves. Wiley Interscience, New York, 1–41.CrossRefGoogle Scholar
  61. Saari P. and Reivelt K., 1997. Evidence of X-shaped propagation-invariant localized light waves. Phys. Rev. Lett., 79, 4135–4138.CrossRefGoogle Scholar
  62. Salo J. and Friberg A.T., 2008. Propagation-invariant fields: rotationally periodic and anisotropic nondiffracting waves. In: Hernandez-Figueroa H.E., Zamboni-Rached M. and Recami E. (Eds.), Localized Waves. Wiley Interscience, New York, 129–157.CrossRefGoogle Scholar
  63. Sauter T. and Paschke F., 2001. Can Bessel beams carry superluminal signals? Phys. Lett. A, 285, 1–6.CrossRefGoogle Scholar
  64. Sheppard C.J.R., 1978. Electromagnetic field in the focal region of wide-angular annular lens and mirror systems. 2, 163–166Google Scholar
  65. Sheppard C.J.R. and Wilson T., 1978. Gaussian-beam theory of lenses with annular aperture. Microwaves, Optics and Acoustics, 2(4), 105–112.CrossRefGoogle Scholar
  66. Siegman A.E., 1986. Lasers. University Science Books, Sausalito, CA.Google Scholar
  67. Stratton J.A., 1941. Electromagnetic Theory. McGraw-Hill, New York.Google Scholar
  68. Stratton J.A., 1956. Spheroidal Wave Functions. Wiley, New York.Google Scholar
  69. Turunen J. and Friberg A., 2010. Propagation-invariant optical fields. Progress in Optics, 54, 1–88.CrossRefGoogle Scholar
  70. Vertergaard Hau L., Harris S.E., Dutton Z. and Behroozi C.H., 1999. Light speed reduction to 17 metres per second in an ultracold atomic gas. Nature, 397, 594–598.CrossRefGoogle Scholar
  71. Walker S.C. and Kuperman W.A., 2007. Cherenkov-Vavilov formulation of X waves. Phys. Rev. Lett., 99, 244802.CrossRefGoogle Scholar
  72. Wang L.J., Kuzmich A. and Dogariu A., 2000. Gain-assisted superluminal light propagation. Nature, 406, 277–279.CrossRefGoogle Scholar
  73. Weber H.J. and Arfken G.B., 2004. Essential Mathematical Methods for Physicists. Elsevier, Amsterdam, The Netherlands.Google Scholar
  74. Wu R.S., 1985. Gaussian beams, complex rays, and the analytic extension of the Green’s function in smoothly inhomogeneous media. Geophys. J. R. Astr. Soc., 83, 93–110.CrossRefGoogle Scholar
  75. Wu T.T., 1985. Electromagnetic missiles. J. Appl. Phys., 57, 2370–2373.CrossRefGoogle Scholar
  76. Žáček, K., 2006. Decomposition of the wavefield into optimized Gaussian packets. Stud. Geophys. Geod., 50, 367–380.CrossRefGoogle Scholar
  77. Zamboni-Rached M. and Recami E., 2008. Subluminal wave bullets: Exact localized subluminal solutions to the wave equation. Phys. Rev. A, 77, 033824.CrossRefGoogle Scholar
  78. Zamboni-Rached M., Recami E. and Besieris I.M., 2010. Cherenkov radiation versus X-shaped localized waves. J. Opt. Soc. Am. A, 27, 928–934.CrossRefGoogle Scholar
  79. Zamboni-Rached M., Recami E. and Hernandez-Figueroa H.E., 2002. New Localized superluminal solutions to the wave equations with finite total energies and arbitrary frequencies. Eur. Phys. J. D, 21, 217–228.CrossRefGoogle Scholar
  80. Zamboni-Rached M., Recami E. and Hernandez-Figueroa H.E., 2004. Theory of frozen waves: modelling the shape of stationary wave fields. J. Opt. Soc. Am. A, 22, 2465–2475.CrossRefGoogle Scholar
  81. Zamboni-Rached M., Recami E. and Hernandez-Figuerao H.E., 2008. Structure of nondiffracting waves and some interesting applications. In: Hernandez-Figueroa H.E., Zamboni-Rached M. and Recami E. (Eds.), Localized Waves. Wiley Interscience, New York, 43–77.CrossRefGoogle Scholar
  82. Zheng Y., Geng Y. and Wu R.S., 2011. Numerical investigation of propagation of localized waves in complex media. International Symposium on Geophysical Imaging with Localized Waves, Sanya, China, July 24–28, 2011,
  83. Ziolkowski R.W., 1985. Exact solutions of the wave equation with complex source locations. J. Math. Phys., 26, 861–863.CrossRefGoogle Scholar

Copyright information

© Institute of Geophysics of the ASCR, v.v.i 2012

Authors and Affiliations

  1. 1.Department of Earth and Atmospheric SciencesPurdue UniversityWest LafayetteUSA

Personalised recommendations