Abstract

We generate abruptly autofocusing beams that produce vortices at the focus. We give explicit equations for the phase-only Fourier masks that generate these beams including explanations for controlling the focal distance and numerical aperture. We show experimental results for the focal distance, the vortex pattern and show that the diameter of the focused beam can be made smaller than the size of a comparable Airy beam from a lens. Finally we show how to move the focus spot in three dimensions by encoding additional optical elements onto the phase pattern.

© 2012 OSA

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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(8), 979–981 (2007).
    [CrossRef] [PubMed]
  2. M. A. Bandres, “Accelerating parabolic beams,” Opt. Lett. 33(15), 1678–1680 (2008).
    [CrossRef] [PubMed]
  3. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
    [CrossRef] [PubMed]
  4. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett. 33(3), 207–209 (2008).
    [CrossRef] [PubMed]
  5. J. A. Davis, M. J. Mintry, M. A. Bandres, and D. M. Cottrell, “Observation of accelerating parabolic beams,” Opt. Express 16(17), 12866–12871 (2008).
    [CrossRef] [PubMed]
  6. J. A. Davis, M. J. Mitry, M. A. Bandres, I. Ruiz, K. P. McAuley, and D. M. Cottrell, “Generation of accelerating Airy and accelerating parabolic beams using phase-only patterns,” Appl. Opt. 48(17), 3170–3176 (2009).
    [CrossRef] [PubMed]
  7. H. T. Dai, Y. J. Liu, D. Luo, and X. W. Sun, “Propagation dynamics of an optical vortex imposed on an Airy beam,” Opt. Lett. 35(23), 4075–4077 (2010).
    [CrossRef] [PubMed]
  8. H. T. Dai, Y. J. Liu, D. Luo, and X. W. Sun, “Propagation dynamics of an optical vortex imposed on an Airy beam; experimental implementation,” Opt. Lett. 36(9), 1617–1619 (2011).
    [CrossRef] [PubMed]
  9. N. K. Efremidis and D. N. Christodoulides, “Abruptly autofocusing waves,” Opt. Lett. 35(23), 4045–4047 (2010).
    [CrossRef] [PubMed]
  10. D. G. Papazoglou, N. K. Efremidis, D. N. Christodoulides, and S. Tzortzakis, “Observation of abruptly autofocusing waves,” Opt. Lett. 36(10), 1842–1844 (2011).
    [CrossRef] [PubMed]
  11. I. Chremmos, N. K. Efremidis, and D. N. Christodoulides, “Pre-engineered abruptly autofocusing beams,” Opt. Lett. 36(10), 1890–1892 (2011).
    [CrossRef] [PubMed]
  12. I. Chremmos, P. Zhang, J. Prakash, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, “Fourier-space generation of abruptly autofocusing beams and optical bottle beams,” Opt. Lett. 36(18), 3675–3677 (2011).
    [CrossRef] [PubMed]
  13. 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(6), 1051–1057 (1999).
    [CrossRef]
  14. J. Wang, J. Bu, M. Wang, Y. Yang, and X. Yuan, “Generation of high quality Airy beams with blazed micron-optical cubic phase masks,” Appl. Opt. 50, 6626–6631 (2011).
  15. J. A. Davis and D. M. Cottrell, “Ray matrix analysis of the fast Fresnel transform with applications towards liquid crystal displays,” Appl. Opt. 51(5), 644–650 (2012).
    [CrossRef] [PubMed]

2012 (1)

2011 (5)

2010 (2)

2009 (1)

2008 (3)

2007 (2)

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[CrossRef] [PubMed]

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007).
[CrossRef] [PubMed]

1999 (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(6), 1051–1057 (1999).
[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(6), 1051–1057 (1999).
[CrossRef]

Bandres, M. A.

Broky, J.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett. 33(3), 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(21), 213901 (2007).
[CrossRef] [PubMed]

Bu, J.

J. Wang, J. Bu, M. Wang, Y. Yang, and X. Yuan, “Generation of high quality Airy beams with blazed micron-optical cubic phase masks,” Appl. Opt. 50, 6626–6631 (2011).

Chen, Z.

Chremmos, I.

Christodoulides, D. N.

Cottrell, D. M.

Dai, H. T.

Davis, J. A.

Dogariu, A.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett. 33(3), 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(21), 213901 (2007).
[CrossRef] [PubMed]

Efremidis, N. K.

Liu, Y. J.

Luo, D.

McAuley, K. P.

Mintry, M. J.

Mitry, M. J.

Papazoglou, D. G.

Prakash, J.

Ruiz, I.

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(6), 1051–1057 (1999).
[CrossRef]

Sun, X. W.

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(6), 1051–1057 (1999).
[CrossRef]

Tzortzakis, S.

Wang, J.

J. Wang, J. Bu, M. Wang, Y. Yang, and X. Yuan, “Generation of high quality Airy beams with blazed micron-optical cubic phase masks,” Appl. Opt. 50, 6626–6631 (2011).

Wang, M.

J. Wang, J. Bu, M. Wang, Y. Yang, and X. Yuan, “Generation of high quality Airy beams with blazed micron-optical cubic phase masks,” Appl. Opt. 50, 6626–6631 (2011).

Yang, Y.

J. Wang, J. Bu, M. Wang, Y. Yang, and X. Yuan, “Generation of high quality Airy beams with blazed micron-optical cubic phase masks,” Appl. Opt. 50, 6626–6631 (2011).

Yuan, X.

J. Wang, J. Bu, M. Wang, Y. Yang, and X. Yuan, “Generation of high quality Airy beams with blazed micron-optical cubic phase masks,” Appl. Opt. 50, 6626–6631 (2011).

Zhang, P.

Appl. Opt. (3)

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(6), 1051–1057 (1999).
[CrossRef]

Opt. Express (1)

Opt. Lett. (9)

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett. 33(3), 207–209 (2008).
[CrossRef] [PubMed]

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007).
[CrossRef] [PubMed]

M. A. Bandres, “Accelerating parabolic beams,” Opt. Lett. 33(15), 1678–1680 (2008).
[CrossRef] [PubMed]

H. T. Dai, Y. J. Liu, D. Luo, and X. W. Sun, “Propagation dynamics of an optical vortex imposed on an Airy beam,” Opt. Lett. 35(23), 4075–4077 (2010).
[CrossRef] [PubMed]

H. T. Dai, Y. J. Liu, D. Luo, and X. W. Sun, “Propagation dynamics of an optical vortex imposed on an Airy beam; experimental implementation,” Opt. Lett. 36(9), 1617–1619 (2011).
[CrossRef] [PubMed]

N. K. Efremidis and D. N. Christodoulides, “Abruptly autofocusing waves,” Opt. Lett. 35(23), 4045–4047 (2010).
[CrossRef] [PubMed]

D. G. Papazoglou, N. K. Efremidis, D. N. Christodoulides, and S. Tzortzakis, “Observation of abruptly autofocusing waves,” Opt. Lett. 36(10), 1842–1844 (2011).
[CrossRef] [PubMed]

I. Chremmos, N. K. Efremidis, and D. N. Christodoulides, “Pre-engineered abruptly autofocusing beams,” Opt. Lett. 36(10), 1890–1892 (2011).
[CrossRef] [PubMed]

I. Chremmos, P. Zhang, J. Prakash, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, “Fourier-space generation of abruptly autofocusing beams and optical bottle beams,” Opt. Lett. 36(18), 3675–3677 (2011).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[CrossRef] [PubMed]

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Figures (7)

Fig. 1
Fig. 1

Steps in building the mask for a radially autofocusing vortex beam. (a) cubic phase for β = 50, (b) linear radial phase for period of n = 20 pixels, (c) sum of the two, (d) sum with azimuthal phase with charge of l = 10.

Fig. 2
Fig. 2

Images taken at distances from the focal plane of (columns) z=0,10,20,30,40cm for (top row) n=12,β=40rad , (middle row) n=6,β=40rad , (bottom row) n=6,β=95rad .

Fig. 3
Fig. 3

Effective focal length as a function of 1/n for values of β=50rad (squares), β=100rad (diamonds) and β=200rad (circles).

Fig. 4
Fig. 4

Images taken at z= z f =100cm for circular harmonic values of =0,10,20,30 using n=2.28,β=195rad .

Fig. 5
Fig. 5

Images of the focused spot for (left) a glass lens having a focal length of 100 cm and (right) for the abruptly focusing Airy beam at a focus distance z= z f =100cm .

Fig. 6
Fig. 6

Movement of the axial position of the focused spot by encoding an additional lens function having a focal length of F onto the cubic phase mask.

Fig. 7
Fig. 7

Movement of the transverse position of the focused spot by moving the center of the additional lens function by +5pixels,0,5pixels . The focused spot moves by +220μm,0,220μm .

Equations (7)

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ϕ=β r 1 3 +2π r 1 /nΔ+2πϕ
ρ= ρ 0 α z 2 =fλ/nΔ[π W 3 /48λ f 3 β] z 2
z f = ρ 0 /α = 48 f 4 λ 2 β πΔ W 3 n = 48 f 4 λ 2 β π N 3 Δ 4 n
N.A.=2 ρ 0 α = π W 3 12Δ f 2 nβ
N.A . Radial N.A . LENS = 2 ρ 0 α W/2 f 1 = 2 ρ 0 α W/2 z f = 2 ρ 0 α W/( 2 ρ 0 /α ) = 4 ρ 0 W = 4fλ WnΔ
( 0 f 1/f 0 )
( 1 z 0 1 )( 0 f 1/f 0 )( 1 0 1/F 1 )=( z/ff/F f 1/f 0 )

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