Abstract

We demonstrate that any two-dimensional accelerating beam can be described in a canonical form in Fourier space. In particular, we demonstrate that there is a one-to-one correspondence between complex functions in the real line (the line spectrum) and accelerating beams. An arbitrary line spectrum can be used to generate novel accelerating beams with diverse transverse shapes. The line spectra for the special cases of the families of Airy and accelerating parabolic beams are provided.

© 2009 Optical Society of America

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2009 (4)

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, Nat. Photonics 3, 395 (2009).
[CrossRef]

J. Baumgartl, G. Hannappel, D. Stevenson, D. Day, M. Gu, and K. Dholakia, Lab Chip 9, 1334 (2009).
[CrossRef] [PubMed]

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, Science 324, 229 (2009).
[CrossRef] [PubMed]

M. A. Bandres and M. Guizar-Sicairos, Opt. Lett. 34, 13 (2009).
[CrossRef]

2008 (2)

2007 (2)

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

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

1996 (1)

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

1987 (1)

1979 (1)

M. V. Berry and N. L. Balazs, Am. J. Phys. 47, 264 (1979).
[CrossRef]

Arie, A.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, Nat. Photonics 3, 395 (2009).
[CrossRef]

Balazs, N. L.

M. V. Berry and N. L. Balazs, Am. J. Phys. 47, 264 (1979).
[CrossRef]

Bandres, M. A.

Baumgartl, J.

J. Baumgartl, G. Hannappel, D. Stevenson, D. Day, M. Gu, and K. Dholakia, Lab Chip 9, 1334 (2009).
[CrossRef] [PubMed]

Berry, M. V.

M. V. Berry and N. L. Balazs, Am. J. Phys. 47, 264 (1979).
[CrossRef]

Broky, J.

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

Christodoulides, D. N.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, Science 324, 229 (2009).
[CrossRef] [PubMed]

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

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

Cottrell, D. M.

Davis, J. A.

Day, D.

J. Baumgartl, G. Hannappel, D. Stevenson, D. Day, M. Gu, and K. Dholakia, Lab Chip 9, 1334 (2009).
[CrossRef] [PubMed]

Dholakia, K.

J. Baumgartl, G. Hannappel, D. Stevenson, D. Day, M. Gu, and K. Dholakia, Lab Chip 9, 1334 (2009).
[CrossRef] [PubMed]

Dogariu, A.

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

Durnin, J.

Ellenbogen, T.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, Nat. Photonics 3, 395 (2009).
[CrossRef]

Ganany-Padowicz, A.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, Nat. Photonics 3, 395 (2009).
[CrossRef]

Gu, M.

J. Baumgartl, G. Hannappel, D. Stevenson, D. Day, M. Gu, and K. Dholakia, Lab Chip 9, 1334 (2009).
[CrossRef] [PubMed]

Guizar-Sicairos, M.

Hannappel, G.

J. Baumgartl, G. Hannappel, D. Stevenson, D. Day, M. Gu, and K. Dholakia, Lab Chip 9, 1334 (2009).
[CrossRef] [PubMed]

Kolesik, M.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, Science 324, 229 (2009).
[CrossRef] [PubMed]

Mintry, M. J.

Moloney, J. V.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, Science 324, 229 (2009).
[CrossRef] [PubMed]

Polynkin, P.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, Science 324, 229 (2009).
[CrossRef] [PubMed]

Rau, A. R. P.

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

Siviloglou, G. A.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, Science 324, 229 (2009).
[CrossRef] [PubMed]

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

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

Stevenson, D.

J. Baumgartl, G. Hannappel, D. Stevenson, D. Day, M. Gu, and K. Dholakia, Lab Chip 9, 1334 (2009).
[CrossRef] [PubMed]

Unnikrishnan, K.

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

Voloch-Bloch, N.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, Nat. Photonics 3, 395 (2009).
[CrossRef]

Am. J. Phys. (2)

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

M. V. Berry and N. L. Balazs, Am. J. Phys. 47, 264 (1979).
[CrossRef]

J. Opt. Soc. Am. A (1)

Lab Chip (1)

J. Baumgartl, G. Hannappel, D. Stevenson, D. Day, M. Gu, and K. Dholakia, Lab Chip 9, 1334 (2009).
[CrossRef] [PubMed]

Nat. Photonics (1)

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, Nat. Photonics 3, 395 (2009).
[CrossRef]

Opt. Express (1)

Opt. Lett. (3)

Phys. Rev. Lett. (1)

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

Science (1)

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, Science 324, 229 (2009).
[CrossRef] [PubMed]

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Figures (1)

Fig. 1
Fig. 1

Transverse amplitude distributions of some accelerating beams at s = 0 , 5 and their corresponding L ( k v ) line spectra.

Tables (1)

Tables Icon

Table 1 Orthogonal Families of AiB, ApB, and AipwB and Their Line Spectra

Equations (15)

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

2 ψ + i s ψ = 0 ,
| 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 ) ,
ψ ( u , v , s ) = e i s ( u λ s 2 ) e i s 3 3 F 0 ( u λ s 2 , v ) .
F 0 ( u , v ) = 1 2 π e i k ρ F ̃ 0 ( k u , k v ) d k ,
F ̃ 0 ( k u , k v ) = 1 2 π e i k ρ F 0 ( u , v ) d ρ ,
( k u 2 + k v 2 + i k u ) F ̃ 0 ( k u , k v ) = 0 .
F ̃ 0 ( k u , k v ) = L ( k v ) exp ( i k v 2 k u + i k u 3 3 ) ,
ψ ( u , v , s ) = e i s ( u λ s 2 ) e i s 3 3 × F 1 [ exp ( i k u k v 2 + i k u 3 3 ) L ( k v ) ] ( u λ s 2 , v ) ,
Ψ ( u , v , s ) = e u a ψ ( u + 2 i s a , v , s ) ,
ψ ( u + a 2 , v , s i a ) ,
F [ Ψ ] ( k u , k v ) = L ( k v ) e i k v 2 ( k u + i a ) + i ( k u + i a ) 3 3 .

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