Abstract

Spatially accelerating beams that are solutions to Maxwell equations may propagate along incomplete circular trajectories. Taking these truncated Bessel fields to the paraxial limit, some authors have sustained that it has recovered the known Airy beams (AiBs). Based on the angular spectrum representation of optical fields, we demonstrated that the paraxial approximation rigorously leads to off-axis focused beams instead of finite-energy AiBs. The latter will arise under the umbrella of a nonparaxial approach following elliptical trajectories in place of parabolas. The analytical expression of such a shape-preserving wave field under Gaussian apodization is disclosed by using third-order nonparaxial coefficients. Deviations from full-wave simulations appear more severely in beam positioning rather than its local profile.

© 2014 Optical Society of America

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References

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    [CrossRef]
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2014 (1)

V. V. Kotlyar and A. A. Kovalev, Opt. Commun. 313, 290 (2014).
[CrossRef]

2013 (1)

2012 (6)

F. Courvoisier, A. Mathis, L. Froehly, R. Giust, L. Furfaro, P. A. Lacourt, M. Jacquot, and J. M. Dudley, Opt. Lett. 37, 1736 (2012).
[CrossRef]

C. J. Zapata-Rodríguez, D. Pastor, and J. J. Miret, Opt. Express 20, 23553 (2012).
[CrossRef]

M. A. Alonso and M. A. Bandres, Opt. Lett. 37, 5175 (2012).
[CrossRef]

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, Phys. Rev. Lett. 109, 193901 (2012).
[CrossRef]

P. Aleahmad, M.-A. Miri, M. S. Mills, I. Kaminer, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 109, 203902 (2012).
[CrossRef]

I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, Phys. Rev. Lett. 108, 163901 (2012).
[CrossRef]

2011 (2)

2010 (1)

2007 (1)

1980 (1)

M. V. Berry and C. Upstill, Prog. Opt. 18, 257 (1980).
[CrossRef]

1969 (1)

M. Berry, Sci. Prog. 57, 43 (1969).

Aceves, A. B.

P. Chamorro-Posada, J. Sánchez-Curto, A. B. Aceves, and G. S. McDonald, “On the asymptotic evolution of finite energy Airy wavefunctions,” arXiv:1305.3529.

Aleahmad, P.

P. Aleahmad, M.-A. Miri, M. S. Mills, I. Kaminer, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 109, 203902 (2012).
[CrossRef]

Alonso, M. A.

Arfken, G. B.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists (Academic, New York, 2001).

Bandres, M. A.

Barwick, S.

Bekenstein, R.

I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, Phys. Rev. Lett. 108, 163901 (2012).
[CrossRef]

Berry, M.

M. Berry, Sci. Prog. 57, 43 (1969).

Berry, M. V.

M. V. Berry and C. Upstill, Prog. Opt. 18, 257 (1980).
[CrossRef]

Cannan, D.

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, Phys. Rev. Lett. 109, 193901 (2012).
[CrossRef]

Chamorro-Posada, P.

P. Chamorro-Posada, J. Sánchez-Curto, A. B. Aceves, and G. S. McDonald, “On the asymptotic evolution of finite energy Airy wavefunctions,” arXiv:1305.3529.

Chen, Z.

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, Phys. Rev. Lett. 109, 193901 (2012).
[CrossRef]

Christodoulides, D. N.

P. Aleahmad, M.-A. Miri, M. S. Mills, I. Kaminer, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 109, 203902 (2012).
[CrossRef]

G. A. Siviloglou and D. N. Christodoulides, Opt. Lett. 32, 979 (2007).
[CrossRef]

Courvoisier, F.

Dudley, J. M.

Froehly, L.

Furfaro, L.

Giust, R.

Greenfield, E.

E. Greenfield, M. Segev, W. Walasik, and O. Raz, Phys. Rev. Lett. 106, 213902 (2011).
[CrossRef]

Hu, Y.

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, Phys. Rev. Lett. 109, 193901 (2012).
[CrossRef]

Jacquot, M.

Kaminer, I.

P. Aleahmad, M.-A. Miri, M. S. Mills, I. Kaminer, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 109, 203902 (2012).
[CrossRef]

I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, Phys. Rev. Lett. 108, 163901 (2012).
[CrossRef]

Kotlyar, V. V.

V. V. Kotlyar and A. A. Kovalev, Opt. Commun. 313, 290 (2014).
[CrossRef]

Kovalev, A. A.

V. V. Kotlyar and A. A. Kovalev, Opt. Commun. 313, 290 (2014).
[CrossRef]

Lacourt, P. A.

Li, T.

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, Phys. Rev. Lett. 109, 193901 (2012).
[CrossRef]

Mathis, A.

McDonald, G. S.

P. Chamorro-Posada, J. Sánchez-Curto, A. B. Aceves, and G. S. McDonald, “On the asymptotic evolution of finite energy Airy wavefunctions,” arXiv:1305.3529.

Mills, M. S.

P. Aleahmad, M.-A. Miri, M. S. Mills, I. Kaminer, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 109, 203902 (2012).
[CrossRef]

Miret, J. J.

Miri, M.-A.

P. Aleahmad, M.-A. Miri, M. S. Mills, I. Kaminer, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 109, 203902 (2012).
[CrossRef]

Morandotti, R.

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, Phys. Rev. Lett. 109, 193901 (2012).
[CrossRef]

Nemirovsky, J.

I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, Phys. Rev. Lett. 108, 163901 (2012).
[CrossRef]

Papoulis, A.

A. Papoulis, Systems and Transformations with Applications in Optics (McGraw-Hill, 1968).

Pastor, D.

Raz, O.

E. Greenfield, M. Segev, W. Walasik, and O. Raz, Phys. Rev. Lett. 106, 213902 (2011).
[CrossRef]

Sánchez-Curto, J.

P. Chamorro-Posada, J. Sánchez-Curto, A. B. Aceves, and G. S. McDonald, “On the asymptotic evolution of finite energy Airy wavefunctions,” arXiv:1305.3529.

Segev, M.

I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, Phys. Rev. Lett. 108, 163901 (2012).
[CrossRef]

P. Aleahmad, M.-A. Miri, M. S. Mills, I. Kaminer, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 109, 203902 (2012).
[CrossRef]

E. Greenfield, M. Segev, W. Walasik, and O. Raz, Phys. Rev. Lett. 106, 213902 (2011).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, 1986).

Siviloglou, G. A.

Upstill, C.

M. V. Berry and C. Upstill, Prog. Opt. 18, 257 (1980).
[CrossRef]

Walasik, W.

E. Greenfield, M. Segev, W. Walasik, and O. Raz, Phys. Rev. Lett. 106, 213902 (2011).
[CrossRef]

Weber, H. J.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists (Academic, New York, 2001).

Yin, X.

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, Phys. Rev. Lett. 109, 193901 (2012).
[CrossRef]

Zapata-Rodríguez, C. J.

Zhang, P.

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, Phys. Rev. Lett. 109, 193901 (2012).
[CrossRef]

Zhang, X.

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, Phys. Rev. Lett. 109, 193901 (2012).
[CrossRef]

Opt. Commun. (1)

V. V. Kotlyar and A. A. Kovalev, Opt. Commun. 313, 290 (2014).
[CrossRef]

Opt. Express (2)

Opt. Lett. (5)

Phys. Rev. Lett. (4)

E. Greenfield, M. Segev, W. Walasik, and O. Raz, Phys. Rev. Lett. 106, 213902 (2011).
[CrossRef]

I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, Phys. Rev. Lett. 108, 163901 (2012).
[CrossRef]

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, Phys. Rev. Lett. 109, 193901 (2012).
[CrossRef]

P. Aleahmad, M.-A. Miri, M. S. Mills, I. Kaminer, M. Segev, and D. N. Christodoulides, Phys. Rev. Lett. 109, 203902 (2012).
[CrossRef]

Prog. Opt. (1)

M. V. Berry and C. Upstill, Prog. Opt. 18, 257 (1980).
[CrossRef]

Sci. Prog. (1)

M. Berry, Sci. Prog. 57, 43 (1969).

Other (4)

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists (Academic, New York, 2001).

A. Papoulis, Systems and Transformations with Applications in Optics (McGraw-Hill, 1968).

P. Chamorro-Posada, J. Sánchez-Curto, A. B. Aceves, and G. S. McDonald, “On the asymptotic evolution of finite energy Airy wavefunctions,” arXiv:1305.3529.

A. E. Siegman, Lasers (University Science Books, 1986).

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Figures (3)

Fig. 1.
Fig. 1.

(a) Spatial spectrum and associated caustic curve for a Gaussian-apodized Bessel field for l=50 and Ω=π/10. We also represent the modulus of the field ez at λ=500nm and different apertures: (b) complete Bessel field, (c) Ω=π/2, (d) Ω=π/10, and (e) Ω=π/25.

Fig. 2.
Fig. 2.

(a) Field |ez| at λ=500nm and evaluated from Eq. (4) for Ω=π/20 at different Bessel orders l. The solid white lines crossing at origin determine the boundaries of a sector of angle 2Ω where caustics arise in all such wave fields. The circular caustic signature of the Bessel wave field for l=50 is also represented in white. (b) Distribution of the field |ez| at y=0 and at different orders.

Fig. 3.
Fig. 3.

(a) Caustic curves for the Bessel signature (solid line), the parabolic AiB (dotted line), and the nonparaxial AiB (dashed line). Field |ez| in the plane (b) y=rlsinΩ and (c) y=0 at λ=500nm for an incomplete Bessel field of l=25×106 and Ω=π/20.

Equations (15)

Equations on this page are rendered with MathJax. Learn more.

ez(r,ϕ)=m=Amψm(r,ϕ),
ψm=ima(θ)exp(imθ)exp(ik·r)dθ,
F{ψm}=ima(θ)exp(imθ)kδ(k|k|),
ez=ila(θ)exp(ilθ+ikxcosθ+ikysinθ)dθ,
x=(l/k)sinθandy=(l/k)cosθ,
cosθθ+θ3/6,
sinθ1θ2/2,
k2y2+2klx=2l2.
(x,y)=(xl2+xl2cos2α,xl2sin2α),
ez(x,y)=exp(iky)χ(y)exp[(lkx)2Ω24χ2(y)],
ez=2πΩM(x)exp{i[ky+α(x,y)]}Ai[ζ(x,y)],
M(x)=exp(i2π/3)(2/kx)1/3,
ζ(x,y)=(lkx)M(x)γ2(x,y),
α(x,y)=ζ(x,y)γ(x,y)+γ3(x,y)/3,
γ(x,y)=(ky/2iΩ2)M2(x).

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