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Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 16, Iss. 6 — Jun. 1, 1999
  • pp: 1286–1293

Approximate description for Bessel, Bessel–Gauss, and Gaussian beams with finite aperture

Desheng Ding and Xiaojun Liu  »View Author Affiliations

JOSA A, Vol. 16, Issue 6, pp. 1286-1293 (1999)

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An approximate analysis is derived for the propagation of Bessel, Bessel–Gauss, and Gaussian beams with a finite aperture. This treatment is based on the fact that the circ function can be expanded into an approximate sum of complex Gaussian functions, so that these three beams are typically expressed as a combination of a set of infinite-aperture Bessel–Gauss beams. Correspondingly, the evaluation of the diffracted field distribution of the beams is reduced to the summation of Bessel–Gauss functions. From analytical results, the present approach provides a good description of the diffracted beams in the region far (greater than a factor of the Fresnel distance) from the aperture. A possible extension of this method to other apertured beams is also discussed.

© 1999 Optical Society of America

OCIS Codes
(000.4430) General : Numerical approximation and analysis
(050.1220) Diffraction and gratings : Apertures
(050.1960) Diffraction and gratings : Diffraction theory
(260.0260) Physical optics : Physical optics

Original Manuscript: September 29, 1998
Revised Manuscript: January 22, 1999
Manuscript Accepted: January 27, 1999
Published: June 1, 1999

Desheng Ding and Xiaojun Liu, "Approximate description for Bessel, Bessel–Gauss, and Gaussian beams with finite aperture," J. Opt. Soc. Am. A 16, 1286-1293 (1999)

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  1. Lord Rayleigh, “On the passage of electric waves through tubes, or the vibrations of dielectric cylinders,” Philos. Mag. 43, 125–132 (1897). [CrossRef]
  2. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987). [CrossRef]
  3. J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987). [CrossRef] [PubMed]
  4. R. M. Herman, T. A. Wiggins, “Production and uses of diffractionless beams,” J. Opt. Soc. Am. A 8, 932–942 (1991). [CrossRef]
  5. F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987). [CrossRef]
  6. D. K. Hsu, F. J. Margetan, D. O. Thompson, “Bessel beam ultrasonic transducer: fabrication method and experimental results,” Appl. Phys. Lett. 55, 2066–2068 (1989). [CrossRef]
  7. J. Y. Lu, J. F. Greenleaf, “Ultrasonic nondiffracting transducer for medical imaging,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 37, 438–447 (1990). [CrossRef] [PubMed]
  8. J. Y. Lu, J. F. Greenleaf, “Pulse-echo imaging using a nondiffracting beam transducer,” Ultrasound Med. Biol. 17, 265–281 (1991). [CrossRef] [PubMed]
  9. P. L. Overfelt, C. S. Kenney, “Comparison of the propagation characteristics of Bessel, Bessel–Gauss, and Gaussian beams diffracted by a circular aperture,” J. Opt. Soc. Am. A 8, 732–945 (1991). [CrossRef]
  10. A. Vasara, J. Turunen, A. T. Friberg, “Realization of general nondiffracting beams with computer-generated holograms,” J. Opt. Soc. Am. A 6, 1748–1754 (1989). [CrossRef] [PubMed]
  11. J. J. Wen, M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752–1756 (1988). [CrossRef]
  12. D. Ding, J. Lin, Y. Shui, G. Du, D. Zhang, “An analytical description of ultrasonic field produced by a circular piston transducer,” Acta Acust. (China) 18, 249–255 (1993) (in Chinese).
  13. E. Cavanagh, B. D. Cook, “Gaussian–Laguerre description of ultrasonic fields—numerical example: circular piston,” J. Acoust. Soc. Am. 67, 1136–1140 (1980). [CrossRef]
  14. In most cases, it is not desired that the aperture be placed at the waist of the Gaussian or Bessel–Gauss beam. However, in certain circumstances, that is, when a laser beam is focused through a finite-aperture lens or a pinhole, the waist size of the beam is on the order of the aperture size. See, for example, Ref. 16 and references therein; see also R. G. Schell, G. Tyras, “Irradiance from an aperture with a truncated Gaussian field distribution,” J. Opt. Soc. Am. 61, 31–35 (1971). [CrossRef]
  15. E. W. Cheney, Introduction to Approximation Theory (McGraw-Hill, New York, 1966), Chap. 6.
  16. G. Lenz, “Far-field diffraction of truncated higher-order Laguerre–Gaussian beams,” Opt. Commun. 123, 423–429 (1996). [CrossRef]
  17. I. S. Gradsthteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).

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