Abstract

We demonstrate the existence of accelerating parabolic beams that constitute, together with the Airy beams, the only orthogonal and complete families of solutions of the two-dimensional paraxial wave equation that exhibit the unusual ability to remain diffraction-free and freely accelerate during propagation. Since the accelerating parabolic beams, like the Airy beams, carry infinite energy, we present exact finite-energy accelerating parabolic beams that still retain their unusual features over several diffraction lengths.

© 2008 Optical Society of America

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  1. G. A. Siviloglou and D. N. Christodoulides, Opt. Lett. 32, 979 (2007).
    [CrossRef] [PubMed]
  2. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007).
    [CrossRef]
  3. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Opt. Lett. 33, 207 (2008).
    [CrossRef] [PubMed]
  4. K. Unnikrishnan and A. R. P. Rau, Am. J. Phys. 64, 1034 (1996).
    [CrossRef]
  5. C. P. Boyer, E. G. Kalnins, and J. W. Miller, J. Math. Phys. 16, 499 (1975).
    [CrossRef]
  6. I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, Am. J. Phys. 62, 519-521 (1994).
    [CrossRef]
  7. K. Banerjee, S. P. Bhatnagar, V. Choudhry, and S. S. Kanwal, Proc. R. Soc. London 360, 575 (1978).
    [CrossRef]
  8. I. M. Besieris and A. M. Shaarawi, Opt. Lett. 32, 2447 (2007).
    [CrossRef] [PubMed]
  9. M. A. Bandres and J. C. Gutiérrez-Vega, Opt. Express 15, 16719 (2007).
    [CrossRef] [PubMed]
  10. J. A. Davis, M. J. Mitry, M. A. Bandres, and D. M. Cottrell, “Observation of accelerating parabolic beams,” Opt. Express (to be published).
    [PubMed]

2008

2007

1996

K. Unnikrishnan and A. R. P. Rau, Am. J. Phys. 64, 1034 (1996).
[CrossRef]

1994

I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, Am. J. Phys. 62, 519-521 (1994).
[CrossRef]

1978

K. Banerjee, S. P. Bhatnagar, V. Choudhry, and S. S. Kanwal, Proc. R. Soc. London 360, 575 (1978).
[CrossRef]

1975

C. P. Boyer, E. G. Kalnins, and J. W. Miller, J. Math. Phys. 16, 499 (1975).
[CrossRef]

Bandres, M. A.

M. A. Bandres and J. C. Gutiérrez-Vega, Opt. Express 15, 16719 (2007).
[CrossRef] [PubMed]

J. A. Davis, M. J. Mitry, M. A. Bandres, and D. M. Cottrell, “Observation of accelerating parabolic beams,” Opt. Express (to be published).
[PubMed]

Banerjee, K.

K. Banerjee, S. P. Bhatnagar, V. Choudhry, and S. S. Kanwal, Proc. R. Soc. London 360, 575 (1978).
[CrossRef]

Besieris, I. M.

I. M. Besieris and A. M. Shaarawi, Opt. Lett. 32, 2447 (2007).
[CrossRef] [PubMed]

I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, Am. J. Phys. 62, 519-521 (1994).
[CrossRef]

Bhatnagar, S. P.

K. Banerjee, S. P. Bhatnagar, V. Choudhry, and S. S. Kanwal, Proc. R. Soc. London 360, 575 (1978).
[CrossRef]

Boyer, C. P.

C. P. Boyer, E. G. Kalnins, and J. W. Miller, J. Math. Phys. 16, 499 (1975).
[CrossRef]

Broky, J.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Opt. Lett. 33, 207 (2008).
[CrossRef] [PubMed]

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

Choudhry, V.

K. Banerjee, S. P. Bhatnagar, V. Choudhry, and S. S. Kanwal, Proc. R. Soc. London 360, 575 (1978).
[CrossRef]

Christodoulides, D. N.

Cottrell, D. M.

J. A. Davis, M. J. Mitry, M. A. Bandres, and D. M. Cottrell, “Observation of accelerating parabolic beams,” Opt. Express (to be published).
[PubMed]

Davis, J. A.

J. A. Davis, M. J. Mitry, M. A. Bandres, and D. M. Cottrell, “Observation of accelerating parabolic beams,” Opt. Express (to be published).
[PubMed]

Dogariu, A.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Opt. Lett. 33, 207 (2008).
[CrossRef] [PubMed]

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

Gutiérrez-Vega, J. C.

Kalnins, E. G.

C. P. Boyer, E. G. Kalnins, and J. W. Miller, J. Math. Phys. 16, 499 (1975).
[CrossRef]

Kanwal, S. S.

K. Banerjee, S. P. Bhatnagar, V. Choudhry, and S. S. Kanwal, Proc. R. Soc. London 360, 575 (1978).
[CrossRef]

Miller, J. W.

C. P. Boyer, E. G. Kalnins, and J. W. Miller, J. Math. Phys. 16, 499 (1975).
[CrossRef]

Mitry, M. J.

J. A. Davis, M. J. Mitry, M. A. Bandres, and D. M. Cottrell, “Observation of accelerating parabolic beams,” Opt. Express (to be published).
[PubMed]

Rau, A. R. P.

K. Unnikrishnan and A. R. P. Rau, Am. J. Phys. 64, 1034 (1996).
[CrossRef]

Shaarawi, A. M.

I. M. Besieris and A. M. Shaarawi, Opt. Lett. 32, 2447 (2007).
[CrossRef] [PubMed]

I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, Am. J. Phys. 62, 519-521 (1994).
[CrossRef]

Siviloglou, G. A.

Unnikrishnan, K.

K. Unnikrishnan and A. R. P. Rau, Am. J. Phys. 64, 1034 (1996).
[CrossRef]

Ziolkowski, R. W.

I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, Am. J. Phys. 62, 519-521 (1994).
[CrossRef]

Am. J. Phys.

K. Unnikrishnan and A. R. P. Rau, Am. J. Phys. 64, 1034 (1996).
[CrossRef]

I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, Am. J. Phys. 62, 519-521 (1994).
[CrossRef]

J. Math. Phys.

C. P. Boyer, E. G. Kalnins, and J. W. Miller, J. Math. Phys. 16, 499 (1975).
[CrossRef]

Opt. Express

J. A. Davis, M. J. Mitry, M. A. Bandres, and D. M. Cottrell, “Observation of accelerating parabolic beams,” Opt. Express (to be published).
[PubMed]

M. A. Bandres and J. C. Gutiérrez-Vega, Opt. Express 15, 16719 (2007).
[CrossRef] [PubMed]

Opt. Lett.

Phys. Rev. Lett.

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

Proc. R. Soc. London

K. Banerjee, S. P. Bhatnagar, V. Choudhry, and S. S. Kanwal, Proc. R. Soc. London 360, 575 (1978).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Transverse amplitude distributions of some accelerating parabolic beams with λ = 0 at s = 0 .

Fig. 2
Fig. 2

(a), (b) Propagation dynamics and (c)–(e), (f)–(h) transverse intensity distributions at different s planes of the zero-order diffraction-free accelerating parabolic beam and zero-order finite-energy accelerating parabolic beam ( a = 0.05 ) , respectively.

Fig. 3
Fig. 3

Amplitude (upper row) and phase (bottom row) of the Fourier spectrum of several accelerating parabolic beams with λ = 0 .

Equations (14)

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ψ ( u , v , s ) = ψ ( u + Δ ( s ) , v , 0 ) ,
ψ ( u , v , s ) = e i s p 2 ψ ( u , v , 0 ) ,
e i s p 2 = e i 2 s 3 3 e i s u e i s 2 p u e i s H ̂ ,
H ̂ = ( u 2 + v 2 ) + u ,
H ̂ F ( u , v ) = λ F ( u , v ) ,
( η η 2 + η 4 4 ) N ( η ) = E N ( η ) ,
( ξ ξ 2 + ξ 4 4 ) X ( ξ ) = E X ( ξ ) .
ψ n ( u , v , s , λ ) = e i s ( η 2 ξ 2 ) 2 + i s 3 3 ϴ n ( η ) ϴ n ( i ξ ) ,
+ ψ n ( u , v , s , λ ) ψ m ( u , v , s , γ ) d u d v δ n m δ ( λ γ ) .
ϕ n ( u , v , 0 , λ ) e a u ϴ n ( η ) ϴ n ( i ξ ) ,
ϕ n ( u , v , s , λ ) = e i ( s i a ) ( η 2 ξ 2 ) 2 e i ( s i a ) 3 3 ϴ n ( η ) ϴ n ( i ξ ) ,
( u ( λ + a 2 ) ( s i a ) 2 , v ) = ( η 2 2 ξ 2 2 , η ξ ) .
u ( s ) = u ( 0 ) + 2 Im ( a ) s ,
F [ ϕ n ( λ ) ] ( k u , k v ) i n e i ( k u + i a ) ( k v 2 λ ) e i ( k u + i a ) 3 3 Θ n ( 2 k v ) ,

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