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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 25 — Dec. 3, 2012
  • pp: 27253–27262

A diffractive mechanism of focusing

W. B. Case, E. Sadurni, and W. P. Schleich  »View Author Affiliations

Optics Express, Vol. 20, Issue 25, pp. 27253-27262 (2012)

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We examine the free time evolution of a rectangular one dimensional Schrödinger wave packet of constant phase during the early stage which in the paraxial wave approximation is identical to the diffraction of a scalar field from a single slit. Our analysis, based on numerics and the Cornu spiral reveals considerable intricate detail behavior in the density and phase of the wave. We also point out a concentration of the intensity that occurs on axis and propose a new measure of width that expresses this concentration.

© 2012 OSA

OCIS Codes
(050.1940) Diffraction and gratings : Diffraction
(260.1960) Physical optics : Diffraction theory

ToC Category:
Diffraction and Gratings

Original Manuscript: September 5, 2012
Revised Manuscript: October 22, 2012
Manuscript Accepted: October 24, 2012
Published: November 19, 2012

W. B. Case, E. Sadurni, and W. P. Schleich, "A diffractive mechanism of focusing," Opt. Express 20, 27253-27262 (2012)

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  1. M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon Press, London, 1964).
  2. G. Möllenstedt and C. Jönsson, “Elektronenmehrfachinterferenz an regelmäßig hergestellen Feinspalten,” Z. f. Phys.155, 427–474 (1959). [CrossRef]
  3. C. Jönsson, “Electron diffraction at multiple slits,” Am. J. Phys.42, 4–11 (1974). [CrossRef]
  4. J. A. Leavitt and F. A. Bills, “Single-slit diffraction pattern of a thermal atomic potassium beam,” Am. J. Phys.37, 905–912 (1969). [CrossRef]
  5. A. Zeilinger, R. Gähler, C. G. Shull, W. Treimer, and W. Mampe “Single- and double-slit diffraction of neutrons,” Rev. Mod. Phys.60, 1067–1073 (1988). [CrossRef]
  6. O. Nairz, M. Arndt, and A. Zeilinger, “Quantum interference experiments with large molecules,” Am. J. Phys.71, 319–325 (2003). [CrossRef]
  7. In a seminal paper, Marcos Moshinsky studied the propagation of a matter wave suddenly released from a shutter. For this reason these functions are sometimes called Moshinsky functions. See M. Moshinsky, “Diffraction in time,” Phys. Rev.88, 625–631 (1952). [CrossRef]
  8. J. Goldemberg and H. M. Nussenzveig, “On the possibility of the experimental observation of diffraction in time effects,” Rev. Mex. Fis.VI.3, 105–115 (1957).
  9. M. Bozic, D. Arsenovic, and L. Vuskovic, “Transverse momentum distribution of atoms in an interferometer,” Z. Naturforsch.56a, 173–177 (2001).
  10. M. Oberthaler and T. Pfau, “One-, two-and three-dimensional nanostructures with atom lithography,” J. Phys.: Condens. Matter15, R233–R255 (2003). [CrossRef]
  11. W. Schnitzler, N. M. Linke, R. Fickler, J. Meijer, F. Schmidt-Kaler, and K. Singer, “Deterministic ultracold ion source targeting the Heisenberg limit,” Phys. Rev. Lett.102, 070501 (2009). [CrossRef] [PubMed]
  12. T. Sleator, T. Pfau, V. Balykin, and J. Mlynek, “Imaging and focusing of an atomic beam with a large period standing light wave,” Appl. Phys. B.54, 375–379 (1992). [CrossRef]
  13. W. B. Case, M. Tomandl, S. Deachapunya, and M. Arndt, “Realization of optical carpets in the Talbot and Talbot-Lau configurations,” Opt. Express17(23), 20966–20974 (2009). [CrossRef] [PubMed]
  14. A. Turlapov, A. Tonyushkin, and T. Sleator, “Talbot-Lau effect for atomic de Broglie waves manipulated with light,” Phys. Rev. A71, 043612 (2005). [CrossRef]
  15. See for example: M. Mützel, S. Tandler, D. Haubrich, D. Meschede, K. Peithmann, M. Flaspöhler, and K. Buse, “Atom lithography with a holographic light mask,” Phys. Rev. Lett.88, 083601 (2002). [CrossRef] [PubMed]
  16. A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett.85, 2733–2736 (2000). [CrossRef] [PubMed]
  17. Z. Liao, M. Al-Amri, and M.S. Zubairy, “Quantum lithography beyond the diffraction limit via Rabi oscillations,” Phys. Rev. Lett.105, 183601 (2010). [CrossRef]
  18. I. Bialynicki-Birula, M. A. Cirone, J. P. Dahl, M. Fedorov, and W. P. Schleich, “In- and outbound spreading of a free-particle s-wave,” Phys. Rev. Lett.89, 060404 (2002). [CrossRef] [PubMed]
  19. M. Andreata and D. Dodonov, “On shrinking and expansion of radial wave packets,” J. Phys. A: Math. Gen.36, 7113–7128 (2003). [CrossRef]
  20. K. Vogel, F. Gleisberg, N. L. Harshman, P. Kazemi, R. Mack, L. Plimak, and W. P. Schleich, “Optimally focusing wave packets,” Chemical Physics375, 133–143 (2010). [CrossRef]
  21. R. Mack, V. P. Yakovlev, and W. P. Schleich, “Correlations in phase space and the creation of focusing wave packets,” J. Mod. Opt.57, 1437–1444 (2010). [CrossRef]
  22. T. Reisinger, A. Patel, H. Reingruber, K. Fladischer, W. E. Ernst, G. Bracco, H. I. Smith, and B. Holst, “Poisson’s spot with molecules,” Phys. Rev. A79, 053823 (2009). [CrossRef]
  23. L. Novotny, “The history of near-field optics” in Progress in Optics vol. 50, E. Wolf, ed. (Elsevier, Amsterdam, 2007) pp. 137–184. [CrossRef]
  24. D. Courjon, Near-field Microscopy and Near-Field Optics (World Scientific Publishing, Singapore, 2003). [CrossRef]
  25. T. A. Klar, E. Engel, and S. W. Hell, “Breaking Abbe’s diffraction resolution limit in fluorescence microscopy with stimulated emission depletion beams of various shapes,” Phys. Rev. E64, 066613 (2001). [CrossRef]
  26. J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett.58, 1499–1501 (1987). [CrossRef] [PubMed]
  27. The paraxial approximation is expected to hold as long as the wavelength is much less than the slit width.
  28. R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965).
  29. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).
  30. R. W. Wood, Physical Optics (Optical Society of America, Washington, 1988).
  31. M. J. W. Hall, “Incompleteness of trajectory-based interpretations of quantum mechanics,” J. Phys. A: Math. Gen.37, 9549–9556 (2004). [CrossRef]
  32. E. Sadurní, W. B. Case, and W. P. Schleich, in preparation.
  33. M. Gonçalves (personal communication, 2011).
  34. A. Jaouadi, N. Gaaloul, B. Viaris de Lesegno, M. Telmini, L. Pruvost, and E. Charron, “Bose-Einstein condensation in dark power-law laser traps,” Phys. Rev. A82, 023613 (2010). [CrossRef]
  35. T. Kraemer, J. Herbig, M. Mark, T. Weber, C. Chin, H. C. Nägerl, and R. Grimm, “Optimized production of a cesium Bose-Einstein condensate,” App. Phys. B79, 1013–1019 (2004). [CrossRef]

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