Abstract

We study numerically the interference resulting from the superposition of two Bessel beams propagating in free space. We discuss how to obtain such beams and show the existence of the self-imaging effect during propagation. The evolution of the superimposed Bessel beams is analyzed on the basis of the evolution of the individual beams. Our exact numerical predictions contradict previous approximated analytical treatments, showing that they can lead to quantitatively wrong results and misleading conclusions.

© 1998 Optical Society of America

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References

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  1. J. Durnin, J. Opt. Soc. Am. A 4, 2383 (1987).
    [CrossRef]
  2. J. Durnin, J. J. Miceli, and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
    [CrossRef] [PubMed]
  3. R. Borghi and M. Santarsiero, Opt. Lett. 22, 262 (1997).
    [CrossRef] [PubMed]
  4. D. Ding and Z. Lu, Appl. Phys. Lett. 68, 608 (1996).
    [CrossRef]
  5. X. Liu, Appl. Phys. Lett. 71, 722 (1997); D. Ding and Z. Lu, Appl. Phys. Lett. 71, 723 (1997).
    [CrossRef]
  6. V. E. Peet and R. V. Tsubin, Phys. Rev. A 56, 1613 (1997).
    [CrossRef]
  7. S. Klewitz, P. Leiderer, S. Herminghaus, and S. Sogomonian, Opt. Lett. 21, 248 (1996).
    [CrossRef] [PubMed]
  8. H. Sõnajalg and P. Saari, Opt. Lett. 21, 1162 (1996).
    [CrossRef]
  9. H. Sõnajalg, M. Rätsep, and P. Saari, Opt. Lett. 22, 310 (1997).
    [CrossRef]
  10. S. Chávez-Cerda, G. S. McDonald, and G. H. C. New, Opt. Commun. 123, 225 (1996).
    [CrossRef]
  11. Z. Jaroszewicz, A. Kolodziejczyk, A. Kujawski, and C. Gomez-Reino, Opt. Lett. 21, 839 (1996).
    [CrossRef] [PubMed]
  12. K. Patorsky, in Progress in Optics XXVII, E. Wolf, ed. (Elsevier, Amsterdam, 1990), p. 3, and references therein.
  13. E. Tepichin, P. Andrés, and J. Ibarra, Opt. Commun. 125, 27 (1996).
    [CrossRef]

1997

X. Liu, Appl. Phys. Lett. 71, 722 (1997); D. Ding and Z. Lu, Appl. Phys. Lett. 71, 723 (1997).
[CrossRef]

V. E. Peet and R. V. Tsubin, Phys. Rev. A 56, 1613 (1997).
[CrossRef]

R. Borghi and M. Santarsiero, Opt. Lett. 22, 262 (1997).
[CrossRef] [PubMed]

H. Sõnajalg, M. Rätsep, and P. Saari, Opt. Lett. 22, 310 (1997).
[CrossRef]

1996

1987

J. Durnin, J. Opt. Soc. Am. A 4, 2383 (1987).
[CrossRef]

J. Durnin, J. J. Miceli, and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef] [PubMed]

Andrés, P.

E. Tepichin, P. Andrés, and J. Ibarra, Opt. Commun. 125, 27 (1996).
[CrossRef]

Borghi, R.

Chávez-Cerda, S.

S. Chávez-Cerda, G. S. McDonald, and G. H. C. New, Opt. Commun. 123, 225 (1996).
[CrossRef]

Ding, D.

D. Ding and Z. Lu, Appl. Phys. Lett. 68, 608 (1996).
[CrossRef]

Durnin, J.

J. Durnin, J. J. Miceli, and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef] [PubMed]

J. Durnin, J. Opt. Soc. Am. A 4, 2383 (1987).
[CrossRef]

Eberly, J. H.

J. Durnin, J. J. Miceli, and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef] [PubMed]

Gomez-Reino, C.

Herminghaus, S.

Ibarra, J.

E. Tepichin, P. Andrés, and J. Ibarra, Opt. Commun. 125, 27 (1996).
[CrossRef]

Jaroszewicz, Z.

Klewitz, S.

Kolodziejczyk, A.

Kujawski, A.

Leiderer, P.

Liu, X.

X. Liu, Appl. Phys. Lett. 71, 722 (1997); D. Ding and Z. Lu, Appl. Phys. Lett. 71, 723 (1997).
[CrossRef]

Lu, Z.

D. Ding and Z. Lu, Appl. Phys. Lett. 68, 608 (1996).
[CrossRef]

McDonald, G. S.

S. Chávez-Cerda, G. S. McDonald, and G. H. C. New, Opt. Commun. 123, 225 (1996).
[CrossRef]

Miceli, J. J.

J. Durnin, J. J. Miceli, and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef] [PubMed]

New, G. H. C.

S. Chávez-Cerda, G. S. McDonald, and G. H. C. New, Opt. Commun. 123, 225 (1996).
[CrossRef]

Patorsky, K.

K. Patorsky, in Progress in Optics XXVII, E. Wolf, ed. (Elsevier, Amsterdam, 1990), p. 3, and references therein.

Peet, V. E.

V. E. Peet and R. V. Tsubin, Phys. Rev. A 56, 1613 (1997).
[CrossRef]

Rätsep, M.

Saari, P.

Santarsiero, M.

Sogomonian, S.

Sõnajalg, H.

Tepichin, E.

E. Tepichin, P. Andrés, and J. Ibarra, Opt. Commun. 125, 27 (1996).
[CrossRef]

Tsubin, R. V.

V. E. Peet and R. V. Tsubin, Phys. Rev. A 56, 1613 (1997).
[CrossRef]

Appl. Phys. Lett.

D. Ding and Z. Lu, Appl. Phys. Lett. 68, 608 (1996).
[CrossRef]

X. Liu, Appl. Phys. Lett. 71, 722 (1997); D. Ding and Z. Lu, Appl. Phys. Lett. 71, 723 (1997).
[CrossRef]

J. Opt. Soc. Am. A

J. Durnin, J. Opt. Soc. Am. A 4, 2383 (1987).
[CrossRef]

Opt. Commun.

E. Tepichin, P. Andrés, and J. Ibarra, Opt. Commun. 125, 27 (1996).
[CrossRef]

S. Chávez-Cerda, G. S. McDonald, and G. H. C. New, Opt. Commun. 123, 225 (1996).
[CrossRef]

Opt. Lett.

Phys. Rev. A

V. E. Peet and R. V. Tsubin, Phys. Rev. A 56, 1613 (1997).
[CrossRef]

Phys. Rev. Lett.

J. Durnin, J. J. Miceli, and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef] [PubMed]

Other

K. Patorsky, in Progress in Optics XXVII, E. Wolf, ed. (Elsevier, Amsterdam, 1990), p. 3, and references therein.

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Figures (4)

Fig. 1
Fig. 1

On-axis intensity for two interfering Bessel beams with the same normalized amplitude and spatial frequencies k1=4 and k2=k1/5 versus propagation distance normalized to the Rayleigh length.

Fig. 2
Fig. 2

(a) Detail of the initial evolution of the beams. (b) On-axis intensity for two propagating Bessel beams with the same amplitude and spatial frequencies k1=4 (dotted curve) and k2=k1/5 (dashed curve) and the SBB (solid curve) versus the normalized propagation distance. Observe that the SBB decays slowly to the Bessel beam with frequency k2 once the beam with frequency k1 has completely decayed.

Fig. 3
Fig. 3

Width of the central peak of the interacting Bessel beams with the same normalized amplitude and out of phase with k1=4 and k2=k1/5. The widths of the Bessel beams propagating alone are also plotted versus normalized propagation distance.

Fig. 4
Fig. 4

Evolution of the SBB: (a) tridimensional plot, (b) contours of the transverse profiles, showing the Talbot effect along the propagation. In this simulation we use out-of-phase Bessel beams with the same normalized amplitude and spatial frequencies k1=4 and k2=k1/5.

Equations (6)

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2E+k2E=0,
2x2+2y2Hx, y+k2-kz2Hx, y=0.
h0kr=expiΦ0δkr-kr0+a expiΦ1δkr-kr1,
Hjinr, z=J0krjr-iN0krjrexpikzjz+iϕj,
Hjoutr, z=J0krjr+iN0krjrexpikzjz+iϕj,
Ir, z=J02kr0r+J02kr1r+2J02kr0rJ02kr1r×coskz0-kz1z+θ,

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