Abstract

We report the first observation of accelerating parabolic beams. These accelerating parabolic beams are similar to the Airy beams because they exhibit the unusual ability to remain diffraction-free while having a quadratic transverse shift during propagation. The amplitude and phase masks required to generate these beams are encoded onto a single liquid crystal display. Experimental results agree well with theory.

© 2008 Optical Society of America

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References

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  1. G. A. Siviloglou and D. N. Christodoulides, "Accelerating finite energy Airy beams," Opt. Lett. 32, 979-981 (2007).
    [CrossRef] [PubMed]
  2. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, "Observation of accelerating Airy beams," Phys. Rev. Lett. 99, 213901 (2007).
    [CrossRef]
  3. I. M. Besieris and A. M. Shaarawi, "A note on an accelerating finite energy Airy beam," Opt. Lett. 32, 2447-2449 (2007).
    [CrossRef] [PubMed]
  4. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, "Ballistic dynamics of Airy beams," Opt. Lett. 33, 207-209 (2008).
    [CrossRef] [PubMed]
  5. M. V. Berry and N. L. Balazs, "Nonspreading wave packets," Am. J. Phys. 47, 264-267 (1979).
    [CrossRef]
  6. M. A. Bandres, "Accelerating parabolic beams," Opt. Lett., doc. ID 96139 (posted 20 June 2008, in press).
  7. M. A. Bandres and J. C. Gutierrez-Vega, "Airy-Gauss beams and their transformation by paraxial optical systems," Opt. Express 15, 16719-16728 (2007).
    [CrossRef] [PubMed]
  8. K. Banerjee, S. P. Bhatnagar, V. Choudhry, and S. S. Kanwal, "The anharmonic oscillator," Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 360, 575-586 (1978).
    [CrossRef]
  9. J. A. Davis, D. M. Cottrell, J. Campos, M. J. Yzuel, and I. Moreno, "Encoding amplitude information onto phase-only filters," Appl. Opt. 38, 5004-5013 (1999).
    [CrossRef]
  10. J. B. Bentley, J. A. Davis, M. A. Bandres, and J. C. Gutierrez-Vega, "Generation of helical Ince-Gaussian beams with a liquid-crystal display," Opt. Lett. 31, 649-651 (2006).
    [CrossRef] [PubMed]
  11. J. A. Davis, C. S. Tuvey, O. Lopez-Coronado, J. Campos, M. J. Yzuel, and C. Iemmi, "Tailoring the depth of focus for optical imaging systems using a Fourier transform approach," Opt. Lett. 32, 844-846 (2007).
    [CrossRef] [PubMed]
  12. J. A. Davis, P. Tsai, D. M. Cottrell, T. Sonehara, and J. Amako, "Transmission variations in liquid crystal spatial light modulators caused by interference and diffraction effects," Opt. Eng. 38, 1051-1057 (1999).
    [CrossRef]

2008 (1)

2007 (5)

2006 (1)

1999 (2)

J. A. Davis, D. M. Cottrell, J. Campos, M. J. Yzuel, and I. Moreno, "Encoding amplitude information onto phase-only filters," Appl. Opt. 38, 5004-5013 (1999).
[CrossRef]

J. A. Davis, P. Tsai, D. M. Cottrell, T. Sonehara, and J. Amako, "Transmission variations in liquid crystal spatial light modulators caused by interference and diffraction effects," Opt. Eng. 38, 1051-1057 (1999).
[CrossRef]

1979 (1)

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

1978 (1)

K. Banerjee, S. P. Bhatnagar, V. Choudhry, and S. S. Kanwal, "The anharmonic oscillator," Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 360, 575-586 (1978).
[CrossRef]

Amako, J.

J. A. Davis, P. Tsai, D. M. Cottrell, T. Sonehara, and J. Amako, "Transmission variations in liquid crystal spatial light modulators caused by interference and diffraction effects," Opt. Eng. 38, 1051-1057 (1999).
[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.

Banerjee, K.

K. Banerjee, S. P. Bhatnagar, V. Choudhry, and S. S. Kanwal, "The anharmonic oscillator," Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 360, 575-586 (1978).
[CrossRef]

Bentley, J. B.

Berry, M. V.

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

Besieris, I. M.

Bhatnagar, S. P.

K. Banerjee, S. P. Bhatnagar, V. Choudhry, and S. S. Kanwal, "The anharmonic oscillator," Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 360, 575-586 (1978).
[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] [PubMed]

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

Campos, J.

Choudhry, V.

K. Banerjee, S. P. Bhatnagar, V. Choudhry, and S. S. Kanwal, "The anharmonic oscillator," Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 360, 575-586 (1978).
[CrossRef]

Christodoulides, D. N.

Cottrell, D. M.

J. A. Davis, P. Tsai, D. M. Cottrell, T. Sonehara, and J. Amako, "Transmission variations in liquid crystal spatial light modulators caused by interference and diffraction effects," Opt. Eng. 38, 1051-1057 (1999).
[CrossRef]

J. A. Davis, D. M. Cottrell, J. Campos, M. J. Yzuel, and I. Moreno, "Encoding amplitude information onto phase-only filters," Appl. Opt. 38, 5004-5013 (1999).
[CrossRef]

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] [PubMed]

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

Gutierrez-Vega, J. C.

Kanwal, S. S.

K. Banerjee, S. P. Bhatnagar, V. Choudhry, and S. S. Kanwal, "The anharmonic oscillator," Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 360, 575-586 (1978).
[CrossRef]

Moreno, I.

Shaarawi, A. M.

Siviloglou, G. A.

Sonehara, T.

J. A. Davis, P. Tsai, D. M. Cottrell, T. Sonehara, and J. Amako, "Transmission variations in liquid crystal spatial light modulators caused by interference and diffraction effects," Opt. Eng. 38, 1051-1057 (1999).
[CrossRef]

Tsai, P.

J. A. Davis, P. Tsai, D. M. Cottrell, T. Sonehara, and J. Amako, "Transmission variations in liquid crystal spatial light modulators caused by interference and diffraction effects," Opt. Eng. 38, 1051-1057 (1999).
[CrossRef]

Tuvey, C. S.

Yzuel, M. J.

American Journal of Physics (1)

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

Appl. Opt. (1)

Opt. Eng. (1)

J. A. Davis, P. Tsai, D. M. Cottrell, T. Sonehara, and J. Amako, "Transmission variations in liquid crystal spatial light modulators caused by interference and diffraction effects," Opt. Eng. 38, 1051-1057 (1999).
[CrossRef]

Opt. Express (1)

Opt. Lett. (5)

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]

Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences (1)

K. Banerjee, S. P. Bhatnagar, V. Choudhry, and S. S. Kanwal, "The anharmonic oscillator," Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 360, 575-586 (1978).
[CrossRef]

Other (1)

M. A. Bandres, "Accelerating parabolic beams," Opt. Lett., doc. ID 96139 (posted 20 June 2008, in press).

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Figures (5)

Fig. 1.
Fig. 1.

(a) Theoretical intensity (top row) and phase (bottom row) of the Fourier spectrum of several accelerating parabolic beams with n={0,3,6} and a=0.02. (b) Few quartic oscillator eigenfunctions.

Fig. 2.
Fig. 2.

Experimental results comparing the Airy and parabolic beams with n=0, a=0.02 and κ=68.5 µm.

Fig. 3.
Fig. 3.

Experimental results for accelerating parabolic beams with n={0,3,6}, a=0.02 and κ=68.5 µm at several transverse z-planes.

Fig. 4.
Fig. 4.

Experimental results for acelerating parabolic beams with n=4, a={0.01,0.05,0.1} and κ=68.5 µm at several transverse z-planes.

Fig. 5.
Fig. 5.

Transverse deflection of an accelerating parabolic beamwith n=0, a=0.01 and κ=68.5 µm as a function of the propagation distance for launch angles of θ=0,+2.1,-4.2 mrad.

Equations (6)

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ϕ n ( η , ξ , z ) = e i ( z 2 k κ 2 ia ) ( η 2 ξ 2 ) 2 e i ( z 2 k κ 2 i a ) 3 3 Θ n ( η ) Θ n ( i ξ ) ,
( x κ ( z 2 k κ 2 ) 2 + iaz k κ 2 , y κ ) = ( η 2 2 ξ 2 2 , η ξ ) .
x s = 1 κ 3 ( z 2 k ) 2 .
𝓕 [ ϕ n ] ( k x , k y ) exp [ i κ 3 ( k x 3 3 a 2 k x κ 2 + k x k y 2 ) ]
× [ exp ( a κ 2 ( k x 2 + k y 2 ) ) Θ n ( 2 k y κ ) ] .
x s = θ z + 1 κ 3 ( z 2 k ) 2 .

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