Abstract

In this Letter a new class of light beam arisen from the symmetrization of the spectral cubic phase of an Airy beam is presented. The symmetric Airy beam exhibits peculiar features. It propagates at initial stages with a single central lobe that autofocuses and then collapses immediately behind the autofocus. Then, the beam splits into two specular off-axis parabolic lobes like those corresponding to two Airy beams accelerating in opposite directions. Its features are analyzed and compared to other kinds of autofocusing beams; the superposition of two conventional Airy beams having opposite accelerations (in rectangular coordinates) and also to the recently demonstrated circular Airy beam (in cylindrical coordinates). The generation of a symmetric Airy beam is experimentally demonstrated as well. Besides, based on its main features, some possible applications are also discussed.

© 2014 Optical Society of America

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References

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[CrossRef]

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[CrossRef]

I. D. Chremmos, Z. Chen, D. N. Christodoulides, and N. K. Efremidis, Phys. Rev. A 85, 023828 (2012).
[CrossRef]

2011 (5)

2010 (3)

2009 (1)

J. Kasparian and J.-P. Wolf, J. Eur. Opt. Soc. (Rapid Publications) 4, 09039 (2009).
[CrossRef]

2007 (3)

1999 (1)

Alieva, T.

Bandres, M. A.

M. A. Bandres, I. Kaminer, M. S. Mills, B. M. Rodrguez-Lara, E. Greenfield, M. Segev, and D. N. Christodoulides, Opt. Phot. News 24(6), 30 (2013).
[CrossRef]

Besieris, I. M.

Broky, J.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

Calvo, M. L.

Cámara, A.

Campos, J.

Cheben, P.

Chen, Z.

I. D. Chremmos, Z. Chen, D. N. Christodoulides, and N. K. Efremidis, Phys. Rev. A 85, 023828 (2012).
[CrossRef]

I. D. Chremmos, P. Zhang, J. Prakash, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, Opt. Lett. 36, 3675 (2011).
[CrossRef]

Y. Hu, G. A. Siviloglou, P. Zhang, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, in Nonlinear Photonics and Novel Optical Phenomena, Z. Chen and R. Morandotti, eds., Vol. 170 of Springer Series in Optical Sciences (Springer, 2013), pp. 1–46.

Choi, D.

Chremmos, I. D.

Christodoulides, D. N.

M. A. Bandres, I. Kaminer, M. S. Mills, B. M. Rodrguez-Lara, E. Greenfield, M. Segev, and D. N. Christodoulides, Opt. Phot. News 24(6), 30 (2013).
[CrossRef]

I. D. Chremmos, Z. Chen, D. N. Christodoulides, and N. K. Efremidis, Phys. Rev. A 85, 023828 (2012).
[CrossRef]

I. D. Chremmos, N. K. Efremidis, and D. N. Christodoulides, Opt. Lett. 36, 1890 (2011).
[CrossRef]

D. G. Papazoglou, N. K. Efremidis, D. N. Christodoulides, and S. Tzortzakis, Opt. Lett. 36, 1842 (2011).
[CrossRef]

I. D. Chremmos, P. Zhang, J. Prakash, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, Opt. Lett. 36, 3675 (2011).
[CrossRef]

A. Salandrino and D. N. Christodoulides, Physics 4, 69 (2011).

N. K. Efremidis and D. N. Christodoulides, Opt. Lett. 35, 4045 (2010).
[CrossRef]

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

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

Y. Hu, G. A. Siviloglou, P. Zhang, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, in Nonlinear Photonics and Novel Optical Phenomena, Z. Chen and R. Morandotti, eds., Vol. 170 of Springer Series in Optical Sciences (Springer, 2013), pp. 1–46.

Cottrell, D. M.

Couairon, A.

P. Panagiotopoulos, D. G. Papazoglou, A. Couairon, and S. Tzortzakis, Nat. Commun. 4, 2622 (2013).
[CrossRef]

Davis, J. A.

Dogariu, A.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

Efremidis, N. K.

I. D. Chremmos, Z. Chen, D. N. Christodoulides, and N. K. Efremidis, Phys. Rev. A 85, 023828 (2012).
[CrossRef]

D. G. Papazoglou, N. K. Efremidis, D. N. Christodoulides, and S. Tzortzakis, Opt. Lett. 36, 1842 (2011).
[CrossRef]

I. D. Chremmos, N. K. Efremidis, and D. N. Christodoulides, Opt. Lett. 36, 1890 (2011).
[CrossRef]

I. D. Chremmos, P. Zhang, J. Prakash, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, Opt. Lett. 36, 3675 (2011).
[CrossRef]

N. K. Efremidis and D. N. Christodoulides, Opt. Lett. 35, 4045 (2010).
[CrossRef]

Y. Hu, G. A. Siviloglou, P. Zhang, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, in Nonlinear Photonics and Novel Optical Phenomena, Z. Chen and R. Morandotti, eds., Vol. 170 of Springer Series in Optical Sciences (Springer, 2013), pp. 1–46.

Greenfield, E.

M. A. Bandres, I. Kaminer, M. S. Mills, B. M. Rodrguez-Lara, E. Greenfield, M. Segev, and D. N. Christodoulides, Opt. Phot. News 24(6), 30 (2013).
[CrossRef]

Heyman, E.

Hu, Y.

Y. Hu, G. A. Siviloglou, P. Zhang, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, in Nonlinear Photonics and Novel Optical Phenomena, Z. Chen and R. Morandotti, eds., Vol. 170 of Springer Series in Optical Sciences (Springer, 2013), pp. 1–46.

Hwang, Ch.-Y.

Kaganovsky, Y.

Kaminer, I.

M. A. Bandres, I. Kaminer, M. S. Mills, B. M. Rodrguez-Lara, E. Greenfield, M. Segev, and D. N. Christodoulides, Opt. Phot. News 24(6), 30 (2013).
[CrossRef]

Kasparian, J.

J. Kasparian and J.-P. Wolf, J. Eur. Opt. Soc. (Rapid Publications) 4, 09039 (2009).
[CrossRef]

Kim, K.-Y.

Lee, B.

Martinez Matos, O.

Martínez-Matos, O.

Mills, M. S.

M. A. Bandres, I. Kaminer, M. S. Mills, B. M. Rodrguez-Lara, E. Greenfield, M. Segev, and D. N. Christodoulides, Opt. Phot. News 24(6), 30 (2013).
[CrossRef]

Moreno, I.

Panagiotopoulos, P.

P. Panagiotopoulos, D. G. Papazoglou, A. Couairon, and S. Tzortzakis, Nat. Commun. 4, 2622 (2013).
[CrossRef]

Papazoglou, D. G.

P. Panagiotopoulos, D. G. Papazoglou, A. Couairon, and S. Tzortzakis, Nat. Commun. 4, 2622 (2013).
[CrossRef]

D. G. Papazoglou, N. K. Efremidis, D. N. Christodoulides, and S. Tzortzakis, Opt. Lett. 36, 1842 (2011).
[CrossRef]

Prakash, J.

Rodrguez-Lara, B. M.

M. A. Bandres, I. Kaminer, M. S. Mills, B. M. Rodrguez-Lara, E. Greenfield, M. Segev, and D. N. Christodoulides, Opt. Phot. News 24(6), 30 (2013).
[CrossRef]

Rodrigo, J. A.

Salandrino, A.

A. Salandrino and D. N. Christodoulides, Physics 4, 69 (2011).

Segev, M.

M. A. Bandres, I. Kaminer, M. S. Mills, B. M. Rodrguez-Lara, E. Greenfield, M. Segev, and D. N. Christodoulides, Opt. Phot. News 24(6), 30 (2013).
[CrossRef]

Shaarawi, A. M.

Siviloglou, G. A.

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

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

Y. Hu, G. A. Siviloglou, P. Zhang, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, in Nonlinear Photonics and Novel Optical Phenomena, Z. Chen and R. Morandotti, eds., Vol. 170 of Springer Series in Optical Sciences (Springer, 2013), pp. 1–46.

Tzortzakis, S.

P. Panagiotopoulos, D. G. Papazoglou, A. Couairon, and S. Tzortzakis, Nat. Commun. 4, 2622 (2013).
[CrossRef]

D. G. Papazoglou, N. K. Efremidis, D. N. Christodoulides, and S. Tzortzakis, Opt. Lett. 36, 1842 (2011).
[CrossRef]

Vaveliuk, P.

Wolf, J.-P.

J. Kasparian and J.-P. Wolf, J. Eur. Opt. Soc. (Rapid Publications) 4, 09039 (2009).
[CrossRef]

Yzuel, M. J.

Zhang, P.

I. D. Chremmos, P. Zhang, J. Prakash, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, Opt. Lett. 36, 3675 (2011).
[CrossRef]

Y. Hu, G. A. Siviloglou, P. Zhang, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, in Nonlinear Photonics and Novel Optical Phenomena, Z. Chen and R. Morandotti, eds., Vol. 170 of Springer Series in Optical Sciences (Springer, 2013), pp. 1–46.

Appl. Opt. (1)

J. Eur. Opt. Soc. (1)

J. Kasparian and J.-P. Wolf, J. Eur. Opt. Soc. (Rapid Publications) 4, 09039 (2009).
[CrossRef]

Nat. Commun. (1)

P. Panagiotopoulos, D. G. Papazoglou, A. Couairon, and S. Tzortzakis, Nat. Commun. 4, 2622 (2013).
[CrossRef]

Opt. Express (4)

Opt. Lett. (6)

Opt. Phot. News (1)

M. A. Bandres, I. Kaminer, M. S. Mills, B. M. Rodrguez-Lara, E. Greenfield, M. Segev, and D. N. Christodoulides, Opt. Phot. News 24(6), 30 (2013).
[CrossRef]

Phys. Rev. A (1)

I. D. Chremmos, Z. Chen, D. N. Christodoulides, and N. K. Efremidis, Phys. Rev. A 85, 023828 (2012).
[CrossRef]

Phys. Rev. Lett. (1)

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

Physics (1)

A. Salandrino and D. N. Christodoulides, Physics 4, 69 (2011).

Other (2)

All the equations are given for one transverse dimension. However, the extension to two transverse dimensions is straightforward because of the rectangular symmetry.

Y. Hu, G. A. Siviloglou, P. Zhang, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, in Nonlinear Photonics and Novel Optical Phenomena, Z. Chen and R. Morandotti, eds., Vol. 170 of Springer Series in Optical Sciences (Springer, 2013), pp. 1–46.

Supplementary Material (1)

» Media 1: MOV (944 KB)     

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Figures (5)

Fig. 1.
Fig. 1.

1D and 2D wrapped spectral phase masks for a conventional Airy beam and for the resulting SAB when the phase parity is changed. The unwrapped phase ranges corresponds to [20π,20π].

Fig. 2.
Fig. 2.

(a) Intensity distribution for a 1D symmetric Airy beam versus (s,ξ). The intensity scale [0,1] corresponds to [0,Ivf]. (b)–(f) Intensity patterns for a 2D SAB as a function of (sx,sy) for several planes ξi=0, ξf, 1.5ξf, 2ξf, 2.5ξf, ξf being the distance between the initial plane and the autofocus plane. The intensity scale [0,1] for each 2D pattern corresponds to [0,Ivmax(ξi)]. All these intensity distributions were obtained by numerical integration of Eq. (2).

Fig. 3.
Fig. 3.

(a) Intensity distribution for a 1D dual Airy beam (DAB) versus (s,ξ). The intensity scale [0,1] corresponds to [0,Ivf]. (b)–(f) Intensity patterns for a 2D DAB versus (sx,sy) at the planes ξi=0, ξf, 1.5ξf, 2ξf, 2.5ξf. The intensity scale [0,1] corresponds to [0,Ivmax(ξi)]. All these intensity distributions were obtained from the superposition of well-known analytical expressions for the Airy beam with opposite accelerations.

Fig. 4.
Fig. 4.

Normalized peak intensity for SAB, Ivmax/Ivf, DAB Idmax/Ivf, and conventional Airy beam Iumax/Ivf as a function of ξ.

Fig. 5.
Fig. 5.

(a) Experimental intensity profile on the xz plane for a 2D symmetric Airy beam versus (x,z). The intensity scale [0,1] corresponds to [0,Ivf]. (b)–(f) Experimental intensity patterns for a 2D SAB as a function of (x,y) for different planes zi. The density scale [0,1] for each 2D pattern corresponds to [0,Ivmax(zi)]. The SAB propagation from z=0 to 20 mm is shown in Media 1.

Equations (3)

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u0=eaxx0Ai(x/x0)ipU0=x0ea33eaK2x02GaeiK3x033cpeiKa2x0,
v(x˜,z˜)=(12π)+V0ei2πz˜[1K˜2/(8π2)]V(K˜,z˜)eiK˜x˜dK˜,
V0=V(K˜,0)=x˜0ea3/3eaK˜2x˜02ei|K˜|3x˜03/3F{v0}.

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