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Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 17, Iss. 12 — Dec. 1, 2000
  • pp: 2391–2402

Uncertainty products for nonparaxial wave fields

M. A. Alonso and G. W. Forbes  »View Author Affiliations

JOSA A, Vol. 17, Issue 12, pp. 2391-2402 (2000)

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Although maximal localization is a basic notion in the consideration of phase-space representations of fields, it has not yet been pursued for general wave fields. We develop measures of spatial and directional spreads for nonparaxial waves in free space. These measures are invariant under translation and rotation and are shown to reduce to the conventional ones when applied to paraxial fields. The associated uncertainty relation sets limits to joint localization in coordinate and frequency space. This relation provides a basis for the definition of a joint localization measure that is analogous to the beam propagation factor (i.e., M2) of paraxial optics. The results are first developed for two-dimensional fields and then generalized to three dimensions.

© 2000 Optical Society of America

OCIS Codes
(030.1670) Coherence and statistical optics : Coherent optical effects
(070.2590) Fourier optics and signal processing : ABCD transforms
(260.2160) Physical optics : Energy transfer
(350.5730) Other areas of optics : Resolution
(350.7420) Other areas of optics : Waves

M. A. Alonso and G. W. Forbes, "Uncertainty products for nonparaxial wave fields," J. Opt. Soc. Am. A 17, 2391-2402 (2000)

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  1. L. Cohen, Time-Frequency Analysis (Prentice-Hall, N.J., 1995), Chap. 3.
  2. G. Folland and A. Sitaram, “The uncertainty principle: a mathematical survey,” J. Fourier Anal. Appl. 3, 207–238 (1997).
  3. See Ref. 1, p. 46.
  4. M. W. Sasnett, “Propagation of multimode laser beams—the M2 factor,” in The Physics and Technology of Laser Resonators, D. R. Hall and P. E. Jackson, eds. (Hilger, Bristol, UK, 1989), pp. 132–142.
  5. A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE 1224, 2–14 (1990).
  6. T. F. Johnston, Jr., “M2 concept characterises beam quality,” Laser Focus World 26 (5), 173–183 (1990).
  7. The usefulness of M2 as a measure of beam quality has been questioned as a result of the sensitivity to noise of the second moments. See, for example, G. N. Lawrence, “Proposed international standard for laser-beam quality falls short,” Laser Focus World 30 (7), 109–114 (1994).
  8. M. A. Porras, “The best quality optical beam beyond the paraxial approximation,” Opt. Commun. 111, 338–349 (1994).
  9. G. W. Forbes and M. A. Alonso, “Measures of spread for periodic distributions and the associated uncertainty relations,” Am. J. Phys. (to be published).
  10. Since we are considering wave fields generated by distant sources, we ignore for now any evanescent components, which correspond to complex values of θ.
  11. A. Erdélyi, Asymptotic Expansions (Dover, New York, 1956), pp. 51–57.
  12. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, New York, 1995), pp. 128–133.
  13. P. González-Casanova and K. B. Wolf, “Interpolation for solutions of the Helmholtz equation,” Numer. Methods Partial Diff. Eqns. 11, 77–91 (1995).
  14. M. A. Alonso, “Measurement of Helmholtz wave fields,” J. Opt. Soc. Am. A 17, 1256–1264 (2000).
  15. This form follows simply upon consideration of the process of wrapping a Gaussian distribution around a ring and summing the multiple values at each location. The Fourier coefficients of the result are then given as an integral over all θ of a Gaussian times exp (imθ), which is once again a Gaussian. The form of Eq. (9.4) then follows.
  16. This form is completely analogous to the standard form of the angular momentum operator used in elementary quantum mechanics. See, for example, R. L. Liboff, Quantum Mechanics (Addison-Wesley, Reading, Mass., 1980), p. 327.
  17. See the reference given in Note 7.
  18. E. Wigner, “On the correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
  19. H. W. Lee, “Theory and application of the quantum phase-space distribution functions,” Phys. Rep. 259, 147–211 (1995).
  20. A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 58, 1256–1259 (1968).
  21. M. Bastiaans, “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979).
  22. A. Lohmann, “The Wigner function and its optical production,” Opt. Commun. 42, 32–37 (1980).
  23. K. B. Wolf, M. A. Alonso, and G. W. Forbes, “Wigner functions for Helmholtz wave fields,” J. Opt. Soc. Am. A 16, 2476–2487 (1999).

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