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Journal of the Optical Society of America A

Journal of the Optical Society of America A

| OPTICS, IMAGE SCIENCE, AND VISION

  • Vol. 17, Iss. 6 — Jun. 1, 2000
  • pp: 1043–1047

Waveletlike basis function approach to the propagation of paraxial beams

Robert M. Potvliege  »View Author Affiliations


JOSA A, Vol. 17, Issue 6, pp. 1043-1047 (2000)
http://dx.doi.org/10.1364/JOSAA.17.001043


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Abstract

A semianalytical method is described for calculating the diffraction integral for paraxial propagation through an optical system. The field at the input plane is represented by a linear superposition of nearly Gaussian basis functions that keep a simple analytical form under ABCD propagation. The coefficients of the superposition are obtained by a numerical fit. The flexibility of the basis functions makes the method well suited to dealing with sharp local variations of the input field.

© 2000 Optical Society of America

OCIS Codes
(000.4430) General : Numerical approximation and analysis
(050.1970) Diffraction and gratings : Diffractive optics
(260.1960) Physical optics : Diffraction theory

History
Original Manuscript: September 14, 1999
Revised Manuscript: March 7, 2000
Manuscript Accepted: March 7, 2000
Published: June 1, 2000

Citation
Robert M. Potvliege, "Waveletlike basis function approach to the propagation of paraxial beams," J. Opt. Soc. Am. A 17, 1043-1047 (2000)
http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-17-6-1043


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References

  1. P. Baues, “Huygens principle in inhomogeneous isotropic media and a general integral equation applicable to optical resonators,” Opto-Electronics 1, 37–44 (1969). [CrossRef]
  2. P. Baues, “The connection of geometrical optics with the propagation of Gaussian beams and the theory of optical resonators,” Opto-Electronics 1, 103–118 (1969). [CrossRef]
  3. S. A. Collins, “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60, 1168–1177 (1970). [CrossRef]
  4. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).
  5. The definition of the transfer matrix adopted here follows that given in Ref. 4.
  6. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, San Diego, Calif., 1980).
  7. Since the coefficients of the expansion are not obtained by projection of the input field on the basis functions, the only practical constraint on their normalization is that their maximum value should not be so large, or so small, as to hinder the numerical computations.
  8. A mathematically rigorous discussion of the completeness of pseudowavelet bases is outside the scope of this work. It is clear, however, that any smoothly varying field distribution can be approached at any desirable level of accuracy by a linear superposition of such functions, as is the case with spline bases. A basis containing only pseudowavelets with the same parameter a and with n varying from 0 to N by a step of one is equivalent to a basis of N Laguerre–Gaussian modes. Bases containing pseudowavelets with different values of a, on the other hand, are subsets of overcomplete bases formed by the union of several Laguerre–Gaussian bases.
  9. Apart for the width of the slit, this configuration is identical to the setup used by Durnin et al. in their experimental realization of a diffraction-free Bessel beam [J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987)]. [CrossRef] [PubMed]
  10. F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994). [CrossRef]
  11. V. Bagini, R. Borghi, F. Gori, A. M. Pacileo, M. Santarsiero, D. Ambrosini, G. Schirripa Spagnolo, “Propagation of axially symmetric flattened Gaussian beams,” J. Opt. Soc. Am. A 13, 1385–1394 (1996). [CrossRef]
  12. D. Aiello, R. Borghi, M. Santarsiero, S. Vicalvi, “The integrated density of focused flattened Gaussian beams,” Optik 109, 97–103 (1998).
  13. B. Lü, S. Luo, B. Zhang, “Propagation of flattened Gaussian beams with rectangular symmetry passing through a paraxial optical ABCD system with and without aperture,” Opt. Commun. 164, 1–6 (1999). [CrossRef]
  14. B. Lü, B. Zhang, S. R. Luo, “Far-field intensity distribution, M2 factor, and propagation of flattened Gaussian beams,” Appl. Opt. 38, 4581–4584 (1999). [CrossRef]
  15. The calculations reported in Refs. 13 and 14 are based on Eq. (B5).

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