Abstract

There is great interest in Airy beams because they appear to propagate in a curved path. These beams are usually generated by inserting a cubic phase mask onto the input plane of a Fourier transform system. Here, we utilize a fast Fresnel diffraction algorithm to easily derive both the propagation dynamics and the Gouy phase shift for these beams. The trajectories of these beams can be modified by adding additional linear and quadratic phase terms onto the cubic phase mask. Finally, we have rewritten the equations regarding the propagating Airy beams completely in laboratory coordinates for use by experimentalists. Experimental results are included. We expect that these results will be of great importance in applications of Airy beams.

© 2014 Optical Society of America

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References

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    [CrossRef]
  2. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901 (2007).
    [CrossRef]
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    [CrossRef]
  4. M. A. Bandres, “Accelerating parabolic beams,” Opt. Lett. 33, 1678–1680 (2008).
    [CrossRef]
  5. J. A. Davis, M. J. Mitry, M. A. Bandres, and D. M. Cottrell, “Observation of accelerating parabolic beams,” Opt. Express 16, 12866–12871 (2008).
    [CrossRef]
  6. J. A. Davis, M. J. Mitry, M. A. Bandres, I. Ruiz, K. P. McAuley, and D. M. Cottrell, “Generation of accelerating Airy and accelerating parabolic beams using phase-only patterns,” Appl. Opt. 48, 3171–3177 (2009).
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    [CrossRef]
  14. J. A. Davis and D. M. Cottrell, “Ray matrix analysis of the fast Fresnel transform with applications towards liquid crystal displays,” Appl. Opt. 51, 644–650 (2012).
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  15. M. Sypek, “Light propagation in the Fresnel region. New numerical approach,” Opt. Commun. 116, 43–48 (1995).
    [CrossRef]
  16. D. Mendlovic, A. Zalevsky, and N. Konforti, “Computation considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. 44, 407–414 (1997).
    [CrossRef]
  17. D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
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  18. X. Pang, G. Gbur, and T. D. Visser, “The Gouy phase of Airy beams,” Opt. Lett. 36, 2492–2494 (2011).
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    [CrossRef]
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2012 (4)

2011 (4)

2010 (2)

2009 (1)

J. A. Davis, M. J. Mitry, M. A. Bandres, I. Ruiz, K. P. McAuley, and D. M. Cottrell, “Generation of accelerating Airy and accelerating parabolic beams using phase-only patterns,” Appl. Opt. 48, 3171–3177 (2009).

2008 (4)

2007 (2)

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

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32, 979–981 (2007).
[CrossRef]

1999 (1)

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

1997 (1)

D. Mendlovic, A. Zalevsky, and N. Konforti, “Computation considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. 44, 407–414 (1997).
[CrossRef]

1995 (1)

M. Sypek, “Light propagation in the Fresnel region. New numerical approach,” Opt. Commun. 116, 43–48 (1995).
[CrossRef]

1979 (1)

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47, 264–267 (1979).
[CrossRef]

Balazs, N. L.

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47, 264–267 (1979).
[CrossRef]

Bandres, M. A.

J. A. Davis, M. J. Mitry, M. A. Bandres, I. Ruiz, K. P. McAuley, and D. M. Cottrell, “Generation of accelerating Airy and accelerating parabolic beams using phase-only patterns,” Appl. Opt. 48, 3171–3177 (2009).

M. A. Bandres, “Accelerating parabolic beams,” Opt. Lett. 33, 1678–1680 (2008).
[CrossRef]

J. A. Davis, M. J. Mitry, M. A. Bandres, and D. M. Cottrell, “Observation of accelerating parabolic beams,” Opt. Express 16, 12866–12871 (2008).
[CrossRef]

Bernardo, L. M.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

Berry, M. V.

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47, 264–267 (1979).
[CrossRef]

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]

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

Chen, Z.

Chremmos, I.

Christodoulides, D. N.

Cottrell, D. M.

Davis, J. A.

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]

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

Efremidis, N. K.

Ferreira, C.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

Garcia, J.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

Gbur, G.

Hu, Y.

Huang, S.

Konforti, N.

D. Mendlovic, A. Zalevsky, and N. Konforti, “Computation considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. 44, 407–414 (1997).
[CrossRef]

Li, Q.

Lou, C.

Marinho, F.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

Mas, D.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

McAuley, K. P.

J. A. Davis, M. J. Mitry, M. A. Bandres, I. Ruiz, K. P. McAuley, and D. M. Cottrell, “Generation of accelerating Airy and accelerating parabolic beams using phase-only patterns,” Appl. Opt. 48, 3171–3177 (2009).

Mendlovic, D.

D. Mendlovic, A. Zalevsky, and N. Konforti, “Computation considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. 44, 407–414 (1997).
[CrossRef]

Miret, J. J.

Mitry, M. J.

J. A. Davis, M. J. Mitry, M. A. Bandres, I. Ruiz, K. P. McAuley, and D. M. Cottrell, “Generation of accelerating Airy and accelerating parabolic beams using phase-only patterns,” Appl. Opt. 48, 3171–3177 (2009).

J. A. Davis, M. J. Mitry, M. A. Bandres, and D. M. Cottrell, “Observation of accelerating parabolic beams,” Opt. Express 16, 12866–12871 (2008).
[CrossRef]

Pang, X.

Papazoglou, D.

Pastor, D.

Prakash, J.

Ruiz, I.

J. A. Davis, M. J. Mitry, M. A. Bandres, I. Ruiz, K. P. McAuley, and D. M. Cottrell, “Generation of accelerating Airy and accelerating parabolic beams using phase-only patterns,” Appl. Opt. 48, 3171–3177 (2009).

Sand, D.

Siviloglou, G. A.

Slovick, B. A.

Sypek, M.

M. Sypek, “Light propagation in the Fresnel region. New numerical approach,” Opt. Commun. 116, 43–48 (1995).
[CrossRef]

Tuvey, C. S.

Tzortzakis, S.

Visser, T. D.

Wang, Q.

Wang, X. Z.

Xu, J.

Yariv, A.

A. Yariv, Optical Electronics in Modern Communications, 5th ed. (Oxford University, 1997), Chap. 4.6.

Zalevsky, A.

D. Mendlovic, A. Zalevsky, and N. Konforti, “Computation considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. 44, 407–414 (1997).
[CrossRef]

Zapata-Rodriguez, C. J.

Zhang, P.

Am. J. Phys. (1)

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47, 264–267 (1979).
[CrossRef]

Appl. Opt. (4)

J. Mod. Opt. (1)

D. Mendlovic, A. Zalevsky, and N. Konforti, “Computation considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. 44, 407–414 (1997).
[CrossRef]

Opt. Commun. (2)

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

M. Sypek, “Light propagation in the Fresnel region. New numerical approach,” Opt. Commun. 116, 43–48 (1995).
[CrossRef]

Opt. Express (3)

Opt. Lett. (9)

Phys. Rev. Lett. (1)

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

Other (1)

A. Yariv, Optical Electronics in Modern Communications, 5th ed. (Oxford University, 1997), Chap. 4.6.

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Figures (2)

Fig. 1.
Fig. 1.

Comparison of theory and experiment for cases in which the Fourier transform lens has a focal length of 52.65 cm, β=300rad, and data are shown for no external lens, and for cases in which the focal length of the external lens is F=160cm and F=240cm.

Fig. 2.
Fig. 2.

Computed values of the Gouy phase along the path of the accelerating Airy beam using the parameters pertinent to our experimental system. Again, the Fourier transform lens has a focal length of 52.65 cm and β=300rad. Values are shown for no external lens and for the cases where the focal length of the external lens is F=160cm and F=240cm.

Equations (18)

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g^F(u,z)=I1[Z*(ξ,z)G(ξ)]=I1[Z*(ξ,z)I{g(u)}].
Ai(u)=exp[iaξ3]exp[i2πuξ]dξ.
A^iF(u,z)=exp[i(aξ3bξ2)]exp[i2πuξ]dξ.
A^iF(u,z)=exp[ia(b3a)3]exp[ia{ξ(b3a)3}]exp[i2πξu]exp[i3aξ(b3a)2]dξ.
A^iF(u,z)=expi(2πbu3a)i2a(b3a)3exp[iaη3]expi2πη(ub26πa)dη.
A^iF(u,z)=expi(2πbu3a)i2a(b3a)3Ai(ub26πa).
u=b26πa.
ΦG=b327a2.
A^iF(u,z)=exp[i(aξ3bξ2cξ2)]exp[i{2π(u+u0)ξ}dξ.
A^iF(u,z)=exp[i(aξ3Bξ2)]exp[i2πUξ]dξ.
A^iF(u,z)=expi(2πBu3a)i2a(B3a)3Ai(UB26πa).
U=uu0=B26πa.
ΦG=B327a2.
u=sinθ/λ=x/λfFT.
FZ=fFT2/z.
a=β/(NΔ/2)3=8β/(NΔ)3.
x=π(NΔ)348λfFT3β(z+fFT2F)2.
ΦG=B327a2=[πz/λfFT2+π/λF]327[8β/(NΔ)3]2.

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