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Applied Optics

Applied Optics


  • Vol. 37, Iss. 1 — Jan. 1, 1998
  • pp: 9–21

Improved beam propagation method equations

Enrico Nichelatti and Giulio Pozzi  »View Author Affiliations

Applied Optics, Vol. 37, Issue 1, pp. 9-21 (1998)

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Improved beam propagation method (BPM) equations are derived for the general case of arbitrary refractive-index spatial distributions. It is shown that in the paraxial approximation the discrete equations admit an analytical solution for the propagation of a paraxial spherical wave, which converges to the analytical solution of the paraxial Helmholtz equation. The generalized Kirchhoff–Fresnel diffraction integral between the object and the image planes can be derived, with its coefficients expressed in terms of the standard ABCD matrix. This result allows the substitution, in the case of an unaberrated system, of the many numerical steps with a single analytical step. We compared the predictions of the standard and improved BPM equations by considering the cases of a Maxwell fish-eye and of a Luneburg lens.

© 1998 Optical Society of America

OCIS Codes
(050.1940) Diffraction and gratings : Diffraction
(350.5500) Other areas of optics : Propagation

Original Manuscript: March 28, 1997
Revised Manuscript: July 28, 1997
Published: January 1, 1998

Enrico Nichelatti and Giulio Pozzi, "Improved beam propagation method equations," Appl. Opt. 37, 9-21 (1998)

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  1. M. D. Feit, J. A. Fleck, “Light propagation in graded-index optical fibers,” Appl. Opt. 17, 3990–3998 (1978). [CrossRef] [PubMed]
  2. J. van der Donk, P. E. Lagasse, “Analysis of geodesic lenses by beam propagation method,” Electron. Lett. 16, 292–294 (1980). [CrossRef]
  3. J. Van Roey, J. van der Donk, P. E. Lagasse, “Beam-propagation method: analysis and assessment,” J. Opt. Soc. Am. 71, 803–810 (1981). [CrossRef]
  4. J. J. Gribble, J. M. Arnold, “Beam-propagation method ray equation,” Opt. Lett. 13, 611–613 (1988). [CrossRef] [PubMed]
  5. G. N. Lawrence, S.-H. Hwang, “Beam propagation in gradient refractive index media,” Appl. Opt. 31, 5201–5210 (1992). [CrossRef] [PubMed]
  6. A. Di Sebastiano, G. Pozzi, “Improved beam-propagation method equations for the analysis of integrated optics lenses,” Opt. Lett. 17, 472–474 (1992). [CrossRef] [PubMed]
  7. C. E. Pearson, Numerical Methods in Engineering and Science (Van Nostrand Reinhold, New York, 1986), Chap. 6.
  8. G. Pozzi, “Multislice approach to lens analysis,” Adv. Imaging Electron Phys. 93, 173–217 (1995). [CrossRef]
  9. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1987), Chap. 4.
  10. A. R. Mickelson, Physical Optics (Van Nostrand Reinhold, New York, 1992), Chap. 7. [CrossRef]
  11. L. W. Casperson, “Beam propagation in tapered quadratic index waveguides: numerical solutions,” J. Lightwave Technol. LT-3, 256–263 (1985). [CrossRef]
  12. L. W. Casperson, “Beam propagation in periodic quadratic-index waveguides,” Appl. Opt. 24, 4395–4403 (1985). [CrossRef] [PubMed]
  13. A. A. Tovar, L. W. Casperson, “Beam propagation in parabolically tapered graded-index waveguides,” Appl. Opt. 33, 7733–7739 (1994). [CrossRef] [PubMed]
  14. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), Chap. 9.
  15. Ref. 14, Chap. 10.
  16. G. Dattoli, J. C. Gallardo, A. Torre, “An algebraic view to the operatorial ordering and its application to optics,” Riv. Nuovo Cimento 11, 1–79 (1988). [CrossRef]
  17. S. A. Collins, “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60, 1168–1177 (1970). [CrossRef]
  18. A. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986), Chap. 20.
  19. K. Tanaka, “Paraxial theory of rotationally distributed-index media by means of Gaussian constants,” Appl. Opt. 23, 1700–1706 (1984). [CrossRef] [PubMed]
  20. J. N. McMullin, “The ABCD matrix in arbitrarily tapered quadratic-index waveguides,” Appl. Opt. 25, 2184–2187 (1986). [CrossRef] [PubMed]
  21. mathematica V. 2.2 (Wolfram Research, Champaign, Ill., 1994).
  22. Ref. 7, Chap. 7.

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