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Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 21, Iss. 2 — Feb. 1, 2004
  • pp: 193–198

Parametric characterization of rotationally symmetric hard-edged diffracted beams

Baida Lü and Shirong Luo  »View Author Affiliations

JOSA A, Vol. 21, Issue 2, pp. 193-198 (2004)

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The truncated second-order moments and generalized M<sup>2</sup> factor (M<sub>G</sub><sup>2</sup> factor) of two-dimensional beams in the Cartesian coordinate system are extended to the case of three-dimensional rotationally symmetric hard-edged diffracted beams in the cylindrical coordinate system. It is shown that the propagation equations of truncated second-order moments and the M<sub>G</sub><sup>2</sup> factor take forms similar to those for the nontruncated case. The closed-form expression for the M<sub>G</sub><sup>2</sup> factor of rotationally symmetric hard-edged diffracted flattened Gaussian beams is derived that depends on the truncation parameter β and beam order <i>N</i>. For N → ∞, the M<sub>G</sub><sup>2</sup> factor equals 4/∛ corresponding to the value of truncated plane waves, which guarantees consistency of the formalism.

© 2004 Optical Society of America

OCIS Codes
(050.1940) Diffraction and gratings : Diffraction
(110.1220) Imaging systems : Apertures
(350.5500) Other areas of optics : Propagation

Baida Lü and Shirong Luo, "Parametric characterization of rotationally symmetric hard-edged diffracted beams," J. Opt. Soc. Am. A 21, 193-198 (2004)

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  1. A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE 1224, 2–14 (1990).
  2. International standards organization (ISO) Document, “Lasers and laser-related equipment-test methods for laser beam parameters-beam widths, divergence angle and beam propagation factor,” ISO 11146 (ISO, Geneva, Switzerland, 1999).
  3. A. E. Sigeman, “How to (maybe) measure laser beam quality,” in DPSS Lasers: Application and Issues, M. W. Dowley, ed., Vol. 17 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1998), pp. 184–199.
  4. J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752–1756 (1988).
  5. D. Ding and X. Liu, “Approximate description for Bessel, Bessel–Gauss, and Gaussian beams with finite aperture,” J. Opt. Soc. Am. A 16, 1286–1293 (1999).
  6. A. Belafhal and M. Ibnchaikh, “Propagation properties of Hermite–Cosh–Gaussian laser beams,” Opt. Commun. 186, 269–276 (2000).
  7. B. Lü and S. Luo, “Approximate propagation equations of flattened Gaussian beams passing through a paraxial ABCD system with hard-edge aperture,” J. Mod. Opt. 48, 2169–2178 (2001).
  8. R. Martínez-Herrero and P. M. Mejías, “Second-order spatial characterization of hard-edge diffracted beams,” Opt. Lett. 18, 1669–1671 (1993).
  9. R. Martínez-Herrero, P. M. Mejías, and M. Arias, “Parametric characterization of coherent, lowest-order Gaussian beams propagating through hard-edged apertures,” Opt. Lett. 20, 124–126 (1995).
  10. R. Martínez-Herrero, P. M. Mejías, S. Bosch, and A. Carnicer, “Spatial width and power-content ratio of hard-edge diffracted beams,” J. Opt. Soc. Am. A 20, 388–391 (2003).
  11. C. Pare and P.-A. Belanger, “Propagation law and quasi-invariance properties of the truncated second-order moment of a diffracted laser beam,” Opt. Commun. 123, 679–693 (1996).
  12. S. Amarande, A. Giesen, and H. Hügel, “Propagation analysis of self-convergent beam width and characterization of hard-edge diffracted beams,” Appl. Opt. 39, 3914–3924 (2000).
  13. B. Lü and S. Luo, “Beam propagation factor of hard-edge diffracted cosh–Gaussian beams,” Opt. Commun. 178, 275–281 (2000).
  14. B. Lü and S. Luo, “Generalized M2 factor of hard-edged diffracted flattened Gaussian beams,” J. Opt. Soc. Am. A 18, 2098–2101 (2001).
  15. S. A. Collins, “Lens-system diffraction integral written terms of matrix optics,” J. Opt. Soc. Am. 60, 1168–1177 (1970).
  16. A. Erdelyi, Tables of Integral Transforms (McGraw-Hill, New York, 1954), Vol. 2, pp. 51, 431; Vol. 1, p. 137.
  17. F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
  18. V. Bagini, R. Borghi, F. Gori, A. M. Pacileo, M. Santarsiero, D. Ambrosini, and G. Schirripa Spagnolo, “Propagation of axially symmetric flattened Gaussian beams,” J. Opt. Soc. Am. A 13, 1385–1394 (1996).

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