Abstract

We analyze and describe the evolution of the Poynting vector and angular momentum of the Airy beam as it propagates through space. A numerical approach is used to show the Poynting vector follows the tangent line of the direction of propagation. A similar approach is used to show that while the total angular momentum of the Airy beam is zero, the angular momentum of the main intensity peak and the Airy “tail” are non-zero.

©2008 Optical Society of America

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References

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  1. L. Allen, M. J. Padgett, and M. Babiker, Progress in Optics XXXIX, pp. 291–372 (1999).
    [Crossref]
  2. J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [Crossref] [PubMed]
  3. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651 (1987).
    [Crossref]
  4. M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47, 264 (1979).
    [Crossref]
  5. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of Accelerating Airy Beams,” Phys. Rev. Lett. 99, 213901 (2007).
    [Crossref]
  6. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32, 979 (2007).
    [Crossref] [PubMed]
  7. I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, “Nondispersive accelerating wave packets,” Am. J. Phys. 62, 519–521 (1994).
    [Crossref]
  8. D. M. Greenberger, “Comment on ‘Non-Spreading Wave Packets’,” Am. J. Phys. 48, 256 (1980).
    [Crossref]
  9. L. Torner, J.P. Torres, and S. Carrasco, “Digital spiral imaging,” Opt. Express 13, 873 (2005).
    [Crossref] [PubMed]
  10. J. D. Jackson, Classical Electrodynamics, (Wiley, New York, 1962).
  11. M. Born and E. Wolf, Principles of Optics, 7th Edition, (Cambridge University Press, Cambridge, 1999).
  12. L. Allen and M. J. Padgett, “The Poynting vector in Laguerre-Gaussian beams and the interpretation of their angular momentum desnity,” Opt. Commun. 184, 67–71 (2000).
    [Crossref]
  13. H. I. Sztul and R. R. Alfano, “Double-slit interference with Laguerre-Gaussian beams,” Opt. Lett. 31, 999–1001 (2006).
    [Crossref] [PubMed]
  14. M. V. Vasnetsov, J. P. Torres, D. V. Petrov, and L. Torner, “Observation of the orbital angular momentum spectrum of a light beam,” Opt. Lett. 28, 2285–2287 (2003).
    [Crossref] [PubMed]
  15. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett. 33, 207–209 (2008).
    [Crossref] [PubMed]

2008 (1)

2007 (2)

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32, 979 (2007).
[Crossref] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of Accelerating Airy Beams,” Phys. Rev. Lett. 99, 213901 (2007).
[Crossref]

2006 (1)

2005 (1)

2003 (1)

2000 (1)

L. Allen and M. J. Padgett, “The Poynting vector in Laguerre-Gaussian beams and the interpretation of their angular momentum desnity,” Opt. Commun. 184, 67–71 (2000).
[Crossref]

1994 (1)

I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, “Nondispersive accelerating wave packets,” Am. J. Phys. 62, 519–521 (1994).
[Crossref]

1987 (2)

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651 (1987).
[Crossref]

1980 (1)

D. M. Greenberger, “Comment on ‘Non-Spreading Wave Packets’,” Am. J. Phys. 48, 256 (1980).
[Crossref]

1979 (1)

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47, 264 (1979).
[Crossref]

Alfano, R. R.

Allen, L.

L. Allen and M. J. Padgett, “The Poynting vector in Laguerre-Gaussian beams and the interpretation of their angular momentum desnity,” Opt. Commun. 184, 67–71 (2000).
[Crossref]

L. Allen, M. J. Padgett, and M. Babiker, Progress in Optics XXXIX, pp. 291–372 (1999).
[Crossref]

Babiker, M.

L. Allen, M. J. Padgett, and M. Babiker, Progress in Optics XXXIX, pp. 291–372 (1999).
[Crossref]

Balazs, N. L.

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47, 264 (1979).
[Crossref]

Berry, M. V.

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47, 264 (1979).
[Crossref]

Besieris, I. M.

I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, “Nondispersive accelerating wave packets,” Am. J. Phys. 62, 519–521 (1994).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th Edition, (Cambridge University Press, Cambridge, 1999).

Broky, J.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett. 33, 207–209 (2008).
[Crossref] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of Accelerating Airy Beams,” Phys. Rev. Lett. 99, 213901 (2007).
[Crossref]

Carrasco, S.

Christodoulides, D. N.

Dogariu, A.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett. 33, 207–209 (2008).
[Crossref] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of Accelerating Airy Beams,” Phys. Rev. Lett. 99, 213901 (2007).
[Crossref]

Durnin, J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651 (1987).
[Crossref]

Eberly, J. H.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

Greenberger, D. M.

D. M. Greenberger, “Comment on ‘Non-Spreading Wave Packets’,” Am. J. Phys. 48, 256 (1980).
[Crossref]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, (Wiley, New York, 1962).

Miceli, J. J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

Padgett, M. J.

L. Allen and M. J. Padgett, “The Poynting vector in Laguerre-Gaussian beams and the interpretation of their angular momentum desnity,” Opt. Commun. 184, 67–71 (2000).
[Crossref]

L. Allen, M. J. Padgett, and M. Babiker, Progress in Optics XXXIX, pp. 291–372 (1999).
[Crossref]

Petrov, D. V.

Shaarawi, A. M.

I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, “Nondispersive accelerating wave packets,” Am. J. Phys. 62, 519–521 (1994).
[Crossref]

Siviloglou, G. A.

Sztul, H. I.

Torner, L.

Torres, J. P.

Torres, J.P.

Vasnetsov, M. V.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th Edition, (Cambridge University Press, Cambridge, 1999).

Ziolkowski, R. W.

I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, “Nondispersive accelerating wave packets,” Am. J. Phys. 62, 519–521 (1994).
[Crossref]

Am. J. Phys. (3)

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47, 264 (1979).
[Crossref]

I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, “Nondispersive accelerating wave packets,” Am. J. Phys. 62, 519–521 (1994).
[Crossref]

D. M. Greenberger, “Comment on ‘Non-Spreading Wave Packets’,” Am. J. Phys. 48, 256 (1980).
[Crossref]

J. Opt. Soc. Am. A (1)

Opt. Commun. (1)

L. Allen and M. J. Padgett, “The Poynting vector in Laguerre-Gaussian beams and the interpretation of their angular momentum desnity,” Opt. Commun. 184, 67–71 (2000).
[Crossref]

Opt. Express (1)

Opt. Lett. (4)

Phys. Rev. Lett. (2)

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of Accelerating Airy Beams,” Phys. Rev. Lett. 99, 213901 (2007).
[Crossref]

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

Other (3)

L. Allen, M. J. Padgett, and M. Babiker, Progress in Optics XXXIX, pp. 291–372 (1999).
[Crossref]

J. D. Jackson, Classical Electrodynamics, (Wiley, New York, 1962).

M. Born and E. Wolf, Principles of Optics, 7th Edition, (Cambridge University Press, Cambridge, 1999).

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Figures (4)

Fig. 1.
Fig. 1. The intensity of the finite-energy Airy wave in the sx -sy plane, given by Eq. (3) with a=0.15 at ξ=0.025 (a), ξ=0.05 (b), ξ=0.075 (c), ξ=0.1 (d). The inset shows the behavior of Ai(x).

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Fig. 2.
Fig. 2. The numerically calculated Poynting vector in the sx -sy plane, of the finite-energy Airy wave given by Eq. (6) with a=0.15 at ξ=0 (a), ξ=0.025 (b), ξ=0.050 (c), ξ=0.075 (d). The intensity of the Airy field is shown in the background of each frame.

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Fig. 3.
Fig. 3. The numerically calculated angular momentum in the ξ ^ -direction of the finite-energy Airy wave given by Eq. (7) with a=0.15 at ξ=0.025 (a), ξ=0.050 (b), ξ=0.075 (c), ξ=0.100 (d). Reds are positive values, blues are negative values, and green is zero.

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Fig. 4.
Fig. 4. The angular momentum spectrum of the finite-energy Airy wave given by eqs. (8, 9) with a=0.15 at ξ=0 (a), ξ=0.025 (b), ξ=0.05 (c), ξ=0.075 (d), ξ=0.100 (e), and ξ=0.125 (f).

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Equations (12)

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2 i ∂ u ( s x , s y , ξ ) ∂ ξ + ∇ τ 2 u ( s x , s y , ξ ) = 0
u ( s x , s y , ξ ) = Ai ( s x − ( ξ 2 ) 2 ) Ai ( s y − ( ξ 2 ) 2 ) exp ⁡ [ i ( s x ξ 2 + s y ξ 2 ) − i ( ξ 3 6 ) ] ,
u ( s x , s y , ξ ) = Ai ( s x − ( ξ 2 ) 2 + i a ξ ) Ai ( s y − ( ξ 2 ) 2 + i a ξ ) ×
exp ⁡ [ a s x + a s y − ( a ξ 2 2 ) − i ( ξ 3 6 ) − i ( a 2 ξ 2 ) + i ξ ( s x + s y ) 2 ] .
x m = λ o 2 ξ 2 16 π 2 x o 2 and y m = λ o 2 ξ 2 16 π 2 y o 2 .
S → = ( c 4 π ) E → × B → ,
c 4 π < E → × B → > = c 8 π ( E → × B * → + E * → × B → )
= c 8 π [ i ω ( u ∇ τ u * − u * ∇ τ u ) + 2 ω k ∣ u ∣ 2 ξ ̂ ] .
j ξ ( s x , s y , ξ ) = r → × 〈 E → × B → 〉 ξ
= s x · S s y + s y · S s x .
u ( r , ϕ ; z ) = 1 2 π ∑ m a m ( r , z ) exp ⁡ ( im ϕ ) ,
P m = C m ∑ − ∞ ∞ C n .

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