Abstract

In this paper it is shown how one can use Bessel beams to obtain a stationary localized wave field with high transverse localization, and whose longitudinal intensity pattern can assume any desired shape within a chosen interval 0≤zL of the propagation axis. This intensity envelope remains static, i.e., with velocity υ=0; and because of this we call “Frozen Waves” the new solutions to the wave equations (and, in particular, to the Maxwell equations). These solutions can be used in many different and interesting applications, such as optical tweezers, atom guides, optical or acoustic bistouries, various important medical purposes, etc.

© 2004 Optical Society of America

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References

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  1. For a review, see: E.Recami, M.Zamboni-Rached, K.Z.Nóbrega, C.A.Dartora, and H.E.Hernández-Figueroa, �??On the localized superluminal solutions to the Maxwell equations,�?? IEEE Journal of Selected Topics in Quantum Electronics 9, 59-73 (2003); and references therein.
    [CrossRef]
  2. Z. Bouchal and J. Wagner, �??Self-reconstruction effect in free propagation wavefield,�?? Opt. Commun. 176, 299-307 (2000).
    [CrossRef]
  3. Z. Bouchal, �??Controlled spatial shaping of nondiffracting patterns and arrays,�?? Opt. Lett. 27, 1376-1378 (2002).
    [CrossRef]
  4. J. Rosen and A. Yariv, �??Synthesis of an arbitrary axial field profile by computer-generated holograms,�?? Opt. Lett. 19, 843-845 (1994).
    [CrossRef] [PubMed]
  5. R. Piestun, B. Spektor and J. Shamir, �??Unconventional light distributions in three-dimensional domains,�?? J. Mod. Opt. 43, 1495-1507 (1996).
    [CrossRef]
  6. J. Durnin, J. J. Miceli and J. H. Eberly, �??Diffraction-free beams,�?? Phys. Rev. Lett. 58, 1499-1501 (1987).
    [CrossRef] [PubMed]

IEEE J. Sel. Topics in Quantum Electron.

For a review, see: E.Recami, M.Zamboni-Rached, K.Z.Nóbrega, C.A.Dartora, and H.E.Hernández-Figueroa, �??On the localized superluminal solutions to the Maxwell equations,�?? IEEE Journal of Selected Topics in Quantum Electronics 9, 59-73 (2003); and references therein.
[CrossRef]

J. Mod. Opt.

R. Piestun, B. Spektor and J. Shamir, �??Unconventional light distributions in three-dimensional domains,�?? J. Mod. Opt. 43, 1495-1507 (1996).
[CrossRef]

Opt. Commun.

Z. Bouchal and J. Wagner, �??Self-reconstruction effect in free propagation wavefield,�?? Opt. Commun. 176, 299-307 (2000).
[CrossRef]

Opt. Lett.

Phys. Rev. Lett.

J. Durnin, J. J. Miceli and J. H. Eberly, �??Diffraction-free beams,�?? Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

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Figures (2)

Fig. 1.
Fig. 1.

(a) Comparison between the intensity of the desired longitudinal function F(z) and that of our Frozen Wave (FW), Ψ(ρ=0, z, t), obtained from Eq. (9). The solid line represents the function F(z), and the dotted one our FW. (b) 3D-plot of the field intensity of the FW chosen in this case by us.

Fig. 2.
Fig. 2.

(a) The Frozen Wave with Q=0.99996ω 0/c and N=30, approximately reproducing the chosen longitudinal pattern represented by Eq. (14). (b) A different Frozen wave, now with Q=0.99980ω0/c (but still with N=30) forwarding the same longitudinal pattern. We can observe that in this case (with a lower value for Q) a higher transverse localization is obtained.

Equations (16)

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ψ ( ρ , z , t ) = J 0 ( k ρ ρ ) e i β z e i ω t
k ρ 2 = ω 2 c 2 β 2 ,
ω β > 0 and k ρ 2 0
Ψ ( ρ , z , t ) = e i ω 0 t n = N N A n J 0 ( K ρ n ρ ) e i β n z ,
0 β n ω 0 c
n = N N A n e i β n z F ( z ) with 0 z L
β n = Q + 2 π L n ,
0 Q ± 2 π L N ω 0 c
Ψ ( ρ = 0 , z , t ) = e i ω 0 t e i Q z n = N N A n e i 2 π L n z ,
A n = 1 L 0 L F ( z ) e i 2 π L n z d z
Ψ ( ρ , z , t ) = e i ω 0 t e i Q z n = N N A n J 0 ( K ρ n ρ ) e i 2 π L n z ,
k ρ n 2 = ω 0 2 ( Q + 2 π n L ) 2
F ( z ) = { 4 ( z l 1 ) ( z l 2 ) ( l 2 l 1 ) 2 for l 1 z l 2 1 for l 3 z l 4 4 ( z l 5 ) ( z l 6 ) ( l 6 l 5 ) 2 for l 5 z l 6 0 elsewhere ,
F ( z ) = { 4 ( z l 1 ) ( z l 2 ) ( l 2 l 1 ) 2 for l 1 z l 2 0 elsewhere ,
Z = R tan θ ,
R L ω 0 2 c 2 β n = N 2 1

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