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Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 17, Iss. 2 — Feb. 1, 2000
  • pp: 294–303

Propagation-invariant wave fields with finite energy

Rafael Piestun, Yoav Y. Schechner, and Joseph Shamir  »View Author Affiliations

JOSA A, Vol. 17, Issue 2, pp. 294-303 (2000)

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Propagation invariance is extended in the paraxial regime, leading to a generalized self-imaging effect. These wave fields are characterized by a finite number of transverse self-images that appear, in general, at different orientations and scales. They possess finite energy and thus can be accurately generated. Necessary and sufficient conditions are derived, and they are appropriately represented in the Gauss–Laguerre modal plane. Relations with the following phenomena are investigated: classical self-imaging, rotating beams, eigen-Fourier functions, and the recently introduced generalized propagation-invariant wave fields. In the paraxial regime they are all included within the generalized self-imaging effect that is presented. In this context we show an important relation between paraxial Bessel beams and Gauss–Laguerre beams.

© 2000 Optical Society of America

OCIS Codes
(030.4070) Coherence and statistical optics : Modes
(050.1960) Diffraction and gratings : Diffraction theory
(110.6760) Imaging systems : Talbot and self-imaging effects
(140.3300) Lasers and laser optics : Laser beam shaping
(350.5030) Other areas of optics : Phase
(350.5500) Other areas of optics : Propagation
(350.7420) Other areas of optics : Waves

Original Manuscript: May 6, 1999
Revised Manuscript: October 5, 1999
Manuscript Accepted: October 8, 1999
Published: February 1, 2000

Rafael Piestun, Yoav Y. Schechner, and Joseph Shamir, "Propagation-invariant wave fields with finite energy," J. Opt. Soc. Am. A 17, 294-303 (2000)

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  1. Part of this study appeared in R. Piestun, Y. Y. Schechner, J. Shamir, “Generalized self-imaging in free space,” presented at the DO’97, EOS Topical Meeting on Diffractive Optics, Savonlinna, Finland, July 7–9, 1997.
  2. K. Patorski, “Self-imaging phenomenon and its applications,” in Progress in Optics, E. Wolf, ed. (Pergamon, Oxford, UK, 1989), Vol. XXVII, pp. 3–108.
  3. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987). [CrossRef]
  4. G. Indebetouw, “Polychromatic self-imaging,” J. Mod. Opt. 35, 243–252 (1988). [CrossRef]
  5. G. Indebetouw, “Nondiffracting optical fields: some remarks on their analysis and synthesis,” J. Opt. Soc. Am. A 6, 150–152 (1989). [CrossRef]
  6. P. Szwaykowski, J. Ojeda-Castaneda, “Nondiffracting beams and the self-imaging phenomenon,” Opt. Commun. 83, 1–4 (1991). [CrossRef]
  7. J. Turunen, A. Vasara, A. T. Friberg, “Propagation invariance and self-imaging in variable-coherence optics,” J. Opt. Soc. Am. A 8, 282–289 (1991). [CrossRef]
  8. G. Indebetouw, “Quasi-self-imaging using aperiodic sequences,” J. Opt. Soc. Am. A 9, 549–558 (1992). [CrossRef]
  9. V. Yu. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992). [CrossRef]
  10. G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993). [CrossRef]
  11. E. Abramochkin, V. Volostnikov, “Spiral-type beams,” Opt. Commun. 102, 336–350 (1993). [CrossRef]
  12. Y. Y. Schechner, R. Piestun, J. Shamir, “Wave propagation with rotating intensity distributions,” Phys. Rev. E 54, R50–R53 (1996). [CrossRef]
  13. Z. Bouchal, R. Horak, J. Wagner, “Propagation-invariant electromagnetic fields: theory and experiment,” J. Mod. Opt. 43, 1905–1920 (1996). [CrossRef]
  14. R. Piestun, Y. Y. Schechner, J. Shamir, “Self-imaging with finite energy,” Opt. Lett. 22, 200–202 (1997). [CrossRef] [PubMed]
  15. V. V. Kotlyar, V. A. Soifer, S. N. Khonina, “An algorithm for the generation of laser beams with longitudinal periodicity: rotating images,” J. Mod. Opt. 44, 1409–1416 (1997). [CrossRef]
  16. R. Piestun, J. Shamir, “Generalized propagation-invariant wave fields,” J. Opt. Soc. Am. A 15, 3039–3044 (1998). [CrossRef]
  17. S. N. Khonina, V. V. Kotlyar, V. A. Soifer, M. Honkanen, J. Lautanen, J. Turunen, “Generation of rotating Gauss–Laguerre modes with binary-phase diffractive optics,” J. Mod. Opt. 46, 227–238 (1999).
  18. R. Simon, N. Mukunda, “Iwasawa decomposition in first-order optics: universal treatment of shape-invariant propagation for coherent and partially coherent beams,” J. Opt. Soc. Am. A 15, 2146–2155 (1998). [CrossRef]
  19. W. D. Montgomery, “Self-imaging objects of infinite aperture,” J. Opt. Soc. Am. 57, 772–778 (1967). [CrossRef]
  20. W. D. Montgomery, “Algebraic formulation of diffraction applied to self-imaging,” J. Opt. Soc. Am. 58, 1112–1124 (1968). [CrossRef]
  21. J. M. Cowley, A. F. Moodie, “Fourier images. III. Finite sources,” Proc. Phys. Soc. London Sect. B 70, 505–513 (1957). [CrossRef]
  22. A. Kalestynski, B. Smolinska, “Self-restoration of the autoidolon of defective periodic objects,” Opt. Acta 25, 125–134 (1978). [CrossRef]
  23. A. P. Smirnov, “Fresnel images of periodic transparencies of finite dimensions,” Opt. Spectrosc. (USSR) 44, 208–212 (1978).
  24. R. Moignard, J. L. de Bougrenet de laTocnaye, “3D self-imaging condition for finite aperture objects,” Opt. Commun. 132, 41–47 (1996). [CrossRef]
  25. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996).
  26. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).
  27. H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966). [CrossRef]
  28. Y. Y. Schechner, “Rotation phenomena in waves,” MSc. Thesis (Technion–Israel Institute of Technology, Haifa, Israel, 1996).
  29. J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974). [CrossRef]
  30. Y. Y. Schechner, J. Shamir, “Parameterization and orbital angular momentum of anisotropic dislocations,” J. Opt. Soc. Am. A 13, 967–973 (1996). [CrossRef]
  31. Note that there is a change of sign in Eq. (33) compared with that equation in Ref. 16. This is simply due to a different convention for the direction of positive rotations (γ).
  32. Note that in this case the condition n→∞ that appears in expression (29) is redundant, since it is implicit in Eq. (24) together with the rest of the conditions of expression (29). In effect, in the limit (Δz/z0)→0, we obtain a uniform periodicity in z and, according to Eq. (24), for γ=0 we have nj-n1≈zˆ→02πNj(z0/Δz)→zˆ→0∞ for all j. Thus nj→∞.
  33. G. Nienhuis, L. Allen, “Paraxial wave optics and harmonic-oscillators,” Phys. Rev. A 48, 656–665 (1993). [CrossRef] [PubMed]
  34. A. W. Lohmann, D. Mendlovic, “Self-Fourier objects and other self-transform objects,” J. Opt. Soc. Am. A 9, 2009–2012 (1992). [CrossRef]
  35. M. J. Caola, “Self-Fourier functions,” J. Phys. A 24, L1143–L1144 (1991). [CrossRef]
  36. S. G. Lipson, “‘Self-Fourier objects and other self-transform objects’: comment,” J. Opt. Soc. Am. A 10, 2088–2089 (1993). [CrossRef]
  37. M. W. Coffey, “Self-reciprocal Fourier functions,” J. Opt. Soc. Am. A 9, 2453–2455 (1994). [CrossRef]
  38. G. Cincotti, F. Gori, M. Santarsiero, “Generalized self-Fourier functions,” J. Phys. A 25, L1191–L1194 (1992). [CrossRef]
  39. M. Nazarathy, J. Shamir, “First-order optics—a canonical operator representation: lossless systems,” J. Opt. Soc. Am. 72, 356–364 (1982). [CrossRef]
  40. G. Slekys, K. Staliunas, C. O. Weiss, “Motion of phase singularities in a class-B laser,” Opt. Commun. 119, 433–446 (1995). [CrossRef]
  41. M. Abramowitz, I. A. Stegun, eds. Handbook of Mathematical Functions (Dover, New York, 1965).

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