Abstract

We demonstrate that nondiffracting beams can be generated with an arbitrary transverse shape. In particular, we show that the azimuthal complex modulation of the angular spectra of Helmholtz–Gauss wave fields constitutes a degree of freedom sufficient to tailor nondiffracting beams with an intensity pattern of choice.

© 2010 Optical Society of America

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  1. J. Durnin, J. J. Miceli, and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
    [CrossRef] [PubMed]
  2. V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, Nature 419, 145 (2002).
    [CrossRef] [PubMed]
  3. X. Tsampoula, V. Garcés-Chávez, M. Comrie, D. Stevenson, M. B. Agate, F. J. Gunn-Moore, C. T. A. Brown, and K. Dholakia, Appl. Phys. Lett. 91, 053902 (2007).
    [CrossRef]
  4. C. Yu, M. R. Wang, A. J. Varela, and B. Chen, Opt. Commun. 177, 369 (2000).
    [CrossRef]
  5. R. P. MacDonald, S. A. Boothroyd, T. Okamoto, J. Chrostowski, and B. A. Syrett, Opt. Commun. 122, 169 (1996).
    [CrossRef]
  6. Z. Ding, H. Ren, Y. Zhao, J. S. Nelson, and Z. Chen, Opt. Lett. 27, 243 (2002).
    [CrossRef]
  7. J. C. Gutiérrez-Vega and M. A. Bandres, J. Opt. Soc. Am. A Opt. Image Sci. Vis 22, 289 (2005).
    [CrossRef] [PubMed]
  8. This reduces to the Fourier–Bessel transform, also called a Hankel transform, for azimuthally symetric fields.
  9. D. M. Cottrell, J. M. Craven, and J. A. Davis, Opt. Lett. 32, 298 (2007).
    [CrossRef] [PubMed]
  10. J. E. Curtis, B. A. Koss, and D. G. Grier, Opt. Commun. 207, 169 (2002).
    [CrossRef]
  11. R. Brauer, F. Wyrowsky, and O. Byngdahl, J. Opt. Soc. Am. A 8, 572 (1991).
    [CrossRef]
  12. M. V. Berry and N. L. Balazs, Am. J. Phys. 47, 264 (1979).
    [CrossRef]
  13. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901-l (2007).
    [CrossRef]

2007 (3)

X. Tsampoula, V. Garcés-Chávez, M. Comrie, D. Stevenson, M. B. Agate, F. J. Gunn-Moore, C. T. A. Brown, and K. Dholakia, Appl. Phys. Lett. 91, 053902 (2007).
[CrossRef]

D. M. Cottrell, J. M. Craven, and J. A. Davis, Opt. Lett. 32, 298 (2007).
[CrossRef] [PubMed]

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

2005 (1)

J. C. Gutiérrez-Vega and M. A. Bandres, J. Opt. Soc. Am. A Opt. Image Sci. Vis 22, 289 (2005).
[CrossRef] [PubMed]

2002 (3)

Z. Ding, H. Ren, Y. Zhao, J. S. Nelson, and Z. Chen, Opt. Lett. 27, 243 (2002).
[CrossRef]

J. E. Curtis, B. A. Koss, and D. G. Grier, Opt. Commun. 207, 169 (2002).
[CrossRef]

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, Nature 419, 145 (2002).
[CrossRef] [PubMed]

2000 (1)

C. Yu, M. R. Wang, A. J. Varela, and B. Chen, Opt. Commun. 177, 369 (2000).
[CrossRef]

1996 (1)

R. P. MacDonald, S. A. Boothroyd, T. Okamoto, J. Chrostowski, and B. A. Syrett, Opt. Commun. 122, 169 (1996).
[CrossRef]

1991 (1)

1987 (1)

J. Durnin, J. J. Miceli, and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef] [PubMed]

1979 (1)

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

Agate, M. B.

X. Tsampoula, V. Garcés-Chávez, M. Comrie, D. Stevenson, M. B. Agate, F. J. Gunn-Moore, C. T. A. Brown, and K. Dholakia, Appl. Phys. Lett. 91, 053902 (2007).
[CrossRef]

Balazs, N. L.

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

Bandres, M. A.

J. C. Gutiérrez-Vega and M. A. Bandres, J. Opt. Soc. Am. A Opt. Image Sci. Vis 22, 289 (2005).
[CrossRef] [PubMed]

Berry, M. V.

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

Boothroyd, S. A.

R. P. MacDonald, S. A. Boothroyd, T. Okamoto, J. Chrostowski, and B. A. Syrett, Opt. Commun. 122, 169 (1996).
[CrossRef]

Brauer, R.

Broky, J.

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

Brown, C. T. A.

X. Tsampoula, V. Garcés-Chávez, M. Comrie, D. Stevenson, M. B. Agate, F. J. Gunn-Moore, C. T. A. Brown, and K. Dholakia, Appl. Phys. Lett. 91, 053902 (2007).
[CrossRef]

Byngdahl, O.

Chen, B.

C. Yu, M. R. Wang, A. J. Varela, and B. Chen, Opt. Commun. 177, 369 (2000).
[CrossRef]

Chen, Z.

Christodoulides, D. N.

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

Chrostowski, J.

R. P. MacDonald, S. A. Boothroyd, T. Okamoto, J. Chrostowski, and B. A. Syrett, Opt. Commun. 122, 169 (1996).
[CrossRef]

Comrie, M.

X. Tsampoula, V. Garcés-Chávez, M. Comrie, D. Stevenson, M. B. Agate, F. J. Gunn-Moore, C. T. A. Brown, and K. Dholakia, Appl. Phys. Lett. 91, 053902 (2007).
[CrossRef]

Cottrell, D. M.

Craven, J. M.

Curtis, J. E.

J. E. Curtis, B. A. Koss, and D. G. Grier, Opt. Commun. 207, 169 (2002).
[CrossRef]

Davis, J. A.

Dholakia, K.

X. Tsampoula, V. Garcés-Chávez, M. Comrie, D. Stevenson, M. B. Agate, F. J. Gunn-Moore, C. T. A. Brown, and K. Dholakia, Appl. Phys. Lett. 91, 053902 (2007).
[CrossRef]

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, Nature 419, 145 (2002).
[CrossRef] [PubMed]

Ding, Z.

Dogariu, A.

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

Durnin, J.

J. Durnin, J. J. Miceli, and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef] [PubMed]

Eberly, J. H.

J. Durnin, J. J. Miceli, and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef] [PubMed]

Garcés-Chávez, V.

X. Tsampoula, V. Garcés-Chávez, M. Comrie, D. Stevenson, M. B. Agate, F. J. Gunn-Moore, C. T. A. Brown, and K. Dholakia, Appl. Phys. Lett. 91, 053902 (2007).
[CrossRef]

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, Nature 419, 145 (2002).
[CrossRef] [PubMed]

Grier, D. G.

J. E. Curtis, B. A. Koss, and D. G. Grier, Opt. Commun. 207, 169 (2002).
[CrossRef]

Gunn-Moore, F. J.

X. Tsampoula, V. Garcés-Chávez, M. Comrie, D. Stevenson, M. B. Agate, F. J. Gunn-Moore, C. T. A. Brown, and K. Dholakia, Appl. Phys. Lett. 91, 053902 (2007).
[CrossRef]

Gutiérrez-Vega, J. C.

J. C. Gutiérrez-Vega and M. A. Bandres, J. Opt. Soc. Am. A Opt. Image Sci. Vis 22, 289 (2005).
[CrossRef] [PubMed]

Koss, B. A.

J. E. Curtis, B. A. Koss, and D. G. Grier, Opt. Commun. 207, 169 (2002).
[CrossRef]

MacDonald, R. P.

R. P. MacDonald, S. A. Boothroyd, T. Okamoto, J. Chrostowski, and B. A. Syrett, Opt. Commun. 122, 169 (1996).
[CrossRef]

McGloin, D.

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, Nature 419, 145 (2002).
[CrossRef] [PubMed]

Melville, H.

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, Nature 419, 145 (2002).
[CrossRef] [PubMed]

Miceli, J. J.

J. Durnin, J. J. Miceli, and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef] [PubMed]

Nelson, J. S.

Okamoto, T.

R. P. MacDonald, S. A. Boothroyd, T. Okamoto, J. Chrostowski, and B. A. Syrett, Opt. Commun. 122, 169 (1996).
[CrossRef]

Ren, H.

Sibbett, W.

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, Nature 419, 145 (2002).
[CrossRef] [PubMed]

Siviloglou, G. A.

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

Stevenson, D.

X. Tsampoula, V. Garcés-Chávez, M. Comrie, D. Stevenson, M. B. Agate, F. J. Gunn-Moore, C. T. A. Brown, and K. Dholakia, Appl. Phys. Lett. 91, 053902 (2007).
[CrossRef]

Syrett, B. A.

R. P. MacDonald, S. A. Boothroyd, T. Okamoto, J. Chrostowski, and B. A. Syrett, Opt. Commun. 122, 169 (1996).
[CrossRef]

Tsampoula, X.

X. Tsampoula, V. Garcés-Chávez, M. Comrie, D. Stevenson, M. B. Agate, F. J. Gunn-Moore, C. T. A. Brown, and K. Dholakia, Appl. Phys. Lett. 91, 053902 (2007).
[CrossRef]

Varela, A. J.

C. Yu, M. R. Wang, A. J. Varela, and B. Chen, Opt. Commun. 177, 369 (2000).
[CrossRef]

Wang, M. R.

C. Yu, M. R. Wang, A. J. Varela, and B. Chen, Opt. Commun. 177, 369 (2000).
[CrossRef]

Wyrowsky, F.

Yu, C.

C. Yu, M. R. Wang, A. J. Varela, and B. Chen, Opt. Commun. 177, 369 (2000).
[CrossRef]

Zhao, Y.

Am. J. Phys. (1)

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

Appl. Phys. Lett. (1)

X. Tsampoula, V. Garcés-Chávez, M. Comrie, D. Stevenson, M. B. Agate, F. J. Gunn-Moore, C. T. A. Brown, and K. Dholakia, Appl. Phys. Lett. 91, 053902 (2007).
[CrossRef]

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

J. Opt. Soc. Am. A Opt. Image Sci. Vis (1)

J. C. Gutiérrez-Vega and M. A. Bandres, J. Opt. Soc. Am. A Opt. Image Sci. Vis 22, 289 (2005).
[CrossRef] [PubMed]

Nature (1)

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, Nature 419, 145 (2002).
[CrossRef] [PubMed]

Opt. Commun. (3)

C. Yu, M. R. Wang, A. J. Varela, and B. Chen, Opt. Commun. 177, 369 (2000).
[CrossRef]

R. P. MacDonald, S. A. Boothroyd, T. Okamoto, J. Chrostowski, and B. A. Syrett, Opt. Commun. 122, 169 (1996).
[CrossRef]

J. E. Curtis, B. A. Koss, and D. G. Grier, Opt. Commun. 207, 169 (2002).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. Lett. (2)

J. Durnin, J. J. Miceli, and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef] [PubMed]

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

Other (1)

This reduces to the Fourier–Bessel transform, also called a Hankel transform, for azimuthally symetric fields.

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Figures (3)

Fig. 1
Fig. 1

(a) Wave-vector distribution of NDBs. In reciprocal space, the wave vectors lie on the surface of a cone with semiangle θ 0 and end in a circular ring with radius k t . (b) Typical power spectrum of an HzG beam. Notice the annular-shaped support and associated radial frequency content.

Fig. 2
Fig. 2

(a) The desired transverse intensity, I 0 ( r t ) and (b) its simulated intensity profile (log scale) from the numerical reconstruction of the CGH with a Fourier lens of focal length f = 0.4   m at z = f . (c) Normalized cross-correlation coefficient for different beams and the desired transverse intensity as a function of propagation distance. The red squares and blue circles correspond to relative space-bandwidth products α 1 = 0.34 and α 2 = 0.78 , respectively. The black line represents the case of no frequency constraints and the dashed green line corresponds to m = 0.75 for reference. Note that the ranges of z / f for which m > 0.75 in each case are roughly four ( z max = 0.26   m ) and three times ( z max = 0.18   m ) larger than for a beam without the support constraints, respectively.

Fig. 3
Fig. 3

(a) Numerical simulation of the beam profile I 0 ( r t ) at z = f and at z = z max and (b) experimental beam profile I R ( r t ) reconstructed from the CGH with f = 0.40   m . In both cases α = 0.34 . Gaussian spread takes over after the beams propagate for a distance z max z R , as expected from an HzG beam.

Equations (5)

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k t = 2 π λ sin θ 0 .
U ( r ) = exp ( i k t 2 2 k z μ ) G ( r ) W ( r t ) ,
G ( r ) = exp ( i k z ) μ exp ( r 2 μ w 0 2 ) ,
U ̃ ( u , v ; z ) = D ( z ) exp ( ω 0 2 μ 4 ρ 2 ) W ( ω 0 2 2 i u ω 0 2 2 i v ; k t ) ,
D ( z ) = ω 0 2 2 exp ( 1 4 k t 2 ω 0 2 ) exp ( i k z ) .

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