Abstract

We present an algorithm for manipulating and controlling 3-D field patterns, with energy confined to the narrow vicinity of predefined 3-D trajectories in free-space, which are of arbitrary curvature and torsion. This is done by setting the aperture field’s phase to form smooth caustic surfaces that include the desired trajectory. The aperture amplitude distribution is constructed to manipulate both the on-axis intensity profile and the off-axis beam-width, and is updated iteratively. Once the aperture distribution is calculated, the radiation from a finite sampled aperture is computed numerically using a Fast Fourier Transform-based scheme. This allows for both verification of the design and examination of its sensitivity to parameters of realistic discrete implementation. The algorithm is demonstrated for the cases of an Airy beam of a planar trajectory, as well as for helical and conical-helical trajectory beams.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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    [Crossref]

2019 (3)

W. Liu, X. Yang, and J. Gao, “Optical transportation and accumulation of microparticles by self-accelerating cusp beams,” Phys. Rev. A 99(4), 043839 (2019).
[Crossref]

M. Zhang, Z. Ren, and P. Yu, “Improve depth of field of optical coherence tomography using finite energy airy beam,” Opt. Lett. 44(12), 3158–3161 (2019).
[Crossref]

Z. Pi, Y. Hu, Z. Chen, and J. Xu, “Large-scale sharply bending paraxial beams,” APL Photonics 4(5), 056101 (2019).
[Crossref]

2018 (2)

M. Goutsoulas and N. Efremidis, “Precise amplitude, trajectory, and beam-width control of accelerating and abruptly autofocusing beams,” Phys. Rev. A 97(6), 063831 (2018).
[Crossref]

Y. Wen, Z. Liu, S. Lin, Y. Chen, Y. Zhang, and S. Yu, “Construction, characteristics, and constraints of accelerating beams based on caustic design,” Opt. Express 26(25), 32728–32738 (2018).
[Crossref]

2017 (2)

2016 (1)

2015 (4)

D. Kuang, Y. Cao, T. Lepine, and W. Mi, “Curved surface plasmon polariton excitation with shaped beam by fifth-power phase mask,” IEEE Photonics J. 7(6), 1–5 (2015).
[Crossref]

R.-S. Penciu, V. Paltoglou, and N. K. Efremidis, “Closed-form expressions for nonparaxial accelerating beams with pre-engineered trajectories,” Opt. Lett. 40(7), 1444–1447 (2015).
[Crossref]

P. K. Shrestha, Y. T. Chun, and D. Chu, “A high-resolution optically addressed spatial light modulator based on zno nanoparticles,” Light: Sci. Appl. 4(3), e259 (2015).
[Crossref]

B. K. Singh, R. Remez, Y. Tsur, and A. Arie, “Measurement of acceleration and orbital angular momentum of airy beam and airy-vortex beam by astigmatic transformation,” Opt. Lett. 40(22), 5411–5414 (2015).
[Crossref]

2014 (5)

S. Jia and X. Zhuang, “Super-resolution imaging with airy beams,” Opt. Photonics News 25, LW4I.4 (2014).
[Crossref]

S. Jia, J. C. Vaughan, and X. Zhuang, “Isotropic 3d super-resolution imaging with a self-bending point spread function,” Nat. Photonics 8(4), 302–306 (2014).
[Crossref]

R. Schley, I. Kaminer, E. Greenfield, R. Bekenstein, Y. Lumer, and M. Segev, “Loss-proof self-accelerating beams and their use in non-paraxial manipulation of particles’ trajectories,” Nat. Commun. 5(1), 5189 (2014).
[Crossref]

A. E. Minovich, A. E. Klein, D. N. Neshev, T. Pertsch, Y. S. Kivshar, and D. N. Christodoulides, “Airy plasmons: non-diffracting optical surface waves,” Laser Photonics Rev. 8(2), 221–232 (2014).
[Crossref]

T. Vettenburg, H. I. C. Dalgarno, J. Nylk, C. Coll-Llado, D. E. K. Ferrier, T. Cizmar, F. J. Gunn-Moore, and K. Dholakia, “Light-sheet microscopy using an Airy beam,” Nat. Methods 11(5), 541–544 (2014).
[Crossref]

2013 (1)

2012 (3)

2011 (3)

2010 (2)

2009 (3)

M. Bandres, “Accelerating beams,” Opt. Lett. 34(24), 3791–3793 (2009).
[Crossref]

S. Karimkashi and A. A. Kishk, “Focused microstrip array antenna using a dolph-chebyshev near-field design,” IEEE Trans. Antennas Propag. 57(12), 3813–3820 (2009).
[Crossref]

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense airy beams,” Science 324(5924), 229–232 (2009).
[Crossref]

2008 (5)

2007 (3)

Agarwal, G. S.

Aiello, A.

Alonso, M. A.

Arie, A.

Babic, V.

V. Babič and V. Buldyrev, Short-Wavelength Diffraction Theory: Asymptotic Methods (Spring–Verlag, Berlin, 1991).

Bandres, M.

Baumgartl, J.

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008).
[Crossref]

Bekenstein, R.

R. Schley, I. Kaminer, E. Greenfield, R. Bekenstein, Y. Lumer, and M. Segev, “Loss-proof self-accelerating beams and their use in non-paraxial manipulation of particles’ trajectories,” Nat. Commun. 5(1), 5189 (2014).
[Crossref]

Besieris, I. M.

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1964).

Broky, J.

Buldyrev, V.

V. Babič and V. Buldyrev, Short-Wavelength Diffraction Theory: Asymptotic Methods (Spring–Verlag, Berlin, 1991).

Cannan, D.

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, “Nonparaxial mathieu and weber accelerating beams,” Phys. Rev. Lett. 109(19), 193901 (2012).
[Crossref]

Cao, Y.

D. Kuang, Y. Cao, T. Lepine, and W. Mi, “Curved surface plasmon polariton excitation with shaped beam by fifth-power phase mask,” IEEE Photonics J. 7(6), 1–5 (2015).
[Crossref]

Chen, H.

Chen, Y.

Chen, Z.

Z. Pi, Y. Hu, Z. Chen, and J. Xu, “Large-scale sharply bending paraxial beams,” APL Photonics 4(5), 056101 (2019).
[Crossref]

J. Zhao, P. Zhang, D. Deng, J. Liu, Y. Gao, I. D. Chremmos, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, “Observation of self-accelerating bessel-like optical beams along arbitrary trajectories,” Opt. Lett. 38(4), 498–500 (2013).
[Crossref]

I. D. Chremmos, Z. Chen, D. N. Christodoulides, and N. K. Efremidis, “Bessel-like optical beams with arbitrary trajectories,” Opt. Lett. 37(23), 5003–5005 (2012).
[Crossref]

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, “Nonparaxial mathieu and weber accelerating beams,” Phys. Rev. Lett. 109(19), 193901 (2012).
[Crossref]

Y. Hu, G. A. Siviloglou, P. Zhang, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, “Self-accelerating airy beams: Generation, control, and applications,” in Nonlinear Photonics and Novel Optical Phenomena, Z. Chen and R. Morandotti, eds. (Springer New York, New York, NY, 2012), pp. 1–46.

Chremmos, I. D.

Christodoulides, D. N.

A. E. Minovich, A. E. Klein, D. N. Neshev, T. Pertsch, Y. S. Kivshar, and D. N. Christodoulides, “Airy plasmons: non-diffracting optical surface waves,” Laser Photonics Rev. 8(2), 221–232 (2014).
[Crossref]

J. Zhao, P. Zhang, D. Deng, J. Liu, Y. Gao, I. D. Chremmos, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, “Observation of self-accelerating bessel-like optical beams along arbitrary trajectories,” Opt. Lett. 38(4), 498–500 (2013).
[Crossref]

I. D. Chremmos, Z. Chen, D. N. Christodoulides, and N. K. Efremidis, “Bessel-like optical beams with arbitrary trajectories,” Opt. Lett. 37(23), 5003–5005 (2012).
[Crossref]

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense airy beams,” Science 324(5924), 229–232 (2009).
[Crossref]

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

J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical airy beams,” Opt. Express 16(17), 12880–12891 (2008).
[Crossref]

D. N. Christodoulides, “Optical trapping: Riding along an airy beam,” Nat. Photonics 2(11), 652–653 (2008).
[Crossref]

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

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

Y. Hu, G. A. Siviloglou, P. Zhang, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, “Self-accelerating airy beams: Generation, control, and applications,” in Nonlinear Photonics and Novel Optical Phenomena, Z. Chen and R. Morandotti, eds. (Springer New York, New York, NY, 2012), pp. 1–46.

Chu, D.

P. K. Shrestha, Y. T. Chun, and D. Chu, “A high-resolution optically addressed spatial light modulator based on zno nanoparticles,” Light: Sci. Appl. 4(3), e259 (2015).
[Crossref]

Chun, Y. T.

P. K. Shrestha, Y. T. Chun, and D. Chu, “A high-resolution optically addressed spatial light modulator based on zno nanoparticles,” Light: Sci. Appl. 4(3), e259 (2015).
[Crossref]

Cizmar, T.

T. Vettenburg, H. I. C. Dalgarno, J. Nylk, C. Coll-Llado, D. E. K. Ferrier, T. Cizmar, F. J. Gunn-Moore, and K. Dholakia, “Light-sheet microscopy using an Airy beam,” Nat. Methods 11(5), 541–544 (2014).
[Crossref]

Coll-Llado, C.

T. Vettenburg, H. I. C. Dalgarno, J. Nylk, C. Coll-Llado, D. E. K. Ferrier, T. Cizmar, F. J. Gunn-Moore, and K. Dholakia, “Light-sheet microscopy using an Airy beam,” Nat. Methods 11(5), 541–544 (2014).
[Crossref]

Courvoisier, F.

Dalgarno, H. I. C.

T. Vettenburg, H. I. C. Dalgarno, J. Nylk, C. Coll-Llado, D. E. K. Ferrier, T. Cizmar, F. J. Gunn-Moore, and K. Dholakia, “Light-sheet microscopy using an Airy beam,” Nat. Methods 11(5), 541–544 (2014).
[Crossref]

de la Hoz, P.

Deng, D.

Dholakia, K.

T. Vettenburg, H. I. C. Dalgarno, J. Nylk, C. Coll-Llado, D. E. K. Ferrier, T. Cizmar, F. J. Gunn-Moore, and K. Dholakia, “Light-sheet microscopy using an Airy beam,” Nat. Methods 11(5), 541–544 (2014).
[Crossref]

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008).
[Crossref]

Ding, J.

Dogariu, A.

Dudley, J. M.

Efremidis, N.

M. Goutsoulas and N. Efremidis, “Precise amplitude, trajectory, and beam-width control of accelerating and abruptly autofocusing beams,” Phys. Rev. A 97(6), 063831 (2018).
[Crossref]

Efremidis, N. K.

Epstein, I.

Ferrier, D. E. K.

T. Vettenburg, H. I. C. Dalgarno, J. Nylk, C. Coll-Llado, D. E. K. Ferrier, T. Cizmar, F. J. Gunn-Moore, and K. Dholakia, “Light-sheet microscopy using an Airy beam,” Nat. Methods 11(5), 541–544 (2014).
[Crossref]

Froehly, L.

Fuerschbach, K.

Furfaro, L.

Gao, J.

W. Liu, X. Yang, and J. Gao, “Optical transportation and accumulation of microparticles by self-accelerating cusp beams,” Phys. Rev. A 99(4), 043839 (2019).
[Crossref]

Gao, Y.

Giust, R.

Goutsoulas, M.

M. Goutsoulas and N. Efremidis, “Precise amplitude, trajectory, and beam-width control of accelerating and abruptly autofocusing beams,” Phys. Rev. A 97(6), 063831 (2018).
[Crossref]

Greenfield, E.

R. Schley, I. Kaminer, E. Greenfield, R. Bekenstein, Y. Lumer, and M. Segev, “Loss-proof self-accelerating beams and their use in non-paraxial manipulation of particles’ trajectories,” Nat. Commun. 5(1), 5189 (2014).
[Crossref]

Gunn-Moore, F. J.

T. Vettenburg, H. I. C. Dalgarno, J. Nylk, C. Coll-Llado, D. E. K. Ferrier, T. Cizmar, F. J. Gunn-Moore, and K. Dholakia, “Light-sheet microscopy using an Airy beam,” Nat. Methods 11(5), 541–544 (2014).
[Crossref]

Heyman, E.

Hradil, Z.

Hu, Y.

Z. Pi, Y. Hu, Z. Chen, and J. Xu, “Large-scale sharply bending paraxial beams,” APL Photonics 4(5), 056101 (2019).
[Crossref]

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, “Nonparaxial mathieu and weber accelerating beams,” Phys. Rev. Lett. 109(19), 193901 (2012).
[Crossref]

Y. Hu, G. A. Siviloglou, P. Zhang, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, “Self-accelerating airy beams: Generation, control, and applications,” in Nonlinear Photonics and Novel Optical Phenomena, Z. Chen and R. Morandotti, eds. (Springer New York, New York, NY, 2012), pp. 1–46.

Jacquot, M.

Jia, S.

S. Jia, J. C. Vaughan, and X. Zhuang, “Isotropic 3d super-resolution imaging with a self-bending point spread function,” Nat. Photonics 8(4), 302–306 (2014).
[Crossref]

S. Jia and X. Zhuang, “Super-resolution imaging with airy beams,” Opt. Photonics News 25, LW4I.4 (2014).
[Crossref]

Kaganovsky, Y.

Kaminer, I.

R. Schley, I. Kaminer, E. Greenfield, R. Bekenstein, Y. Lumer, and M. Segev, “Loss-proof self-accelerating beams and their use in non-paraxial manipulation of particles’ trajectories,” Nat. Commun. 5(1), 5189 (2014).
[Crossref]

Karimkashi, S.

S. Karimkashi and A. A. Kishk, “Focused microstrip array antenna using a dolph-chebyshev near-field design,” IEEE Trans. Antennas Propag. 57(12), 3813–3820 (2009).
[Crossref]

Kishk, A. A.

S. Karimkashi and A. A. Kishk, “Focused microstrip array antenna using a dolph-chebyshev near-field design,” IEEE Trans. Antennas Propag. 57(12), 3813–3820 (2009).
[Crossref]

Kivshar, Y. S.

A. E. Minovich, A. E. Klein, D. N. Neshev, T. Pertsch, Y. S. Kivshar, and D. N. Christodoulides, “Airy plasmons: non-diffracting optical surface waves,” Laser Photonics Rev. 8(2), 221–232 (2014).
[Crossref]

Klein, A. E.

A. E. Minovich, A. E. Klein, D. N. Neshev, T. Pertsch, Y. S. Kivshar, and D. N. Christodoulides, “Airy plasmons: non-diffracting optical surface waves,” Laser Photonics Rev. 8(2), 221–232 (2014).
[Crossref]

Kolesik, M.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense airy beams,” Science 324(5924), 229–232 (2009).
[Crossref]

Kuang, D.

D. Kuang, Y. Cao, T. Lepine, and W. Mi, “Curved surface plasmon polariton excitation with shaped beam by fifth-power phase mask,” IEEE Photonics J. 7(6), 1–5 (2015).
[Crossref]

Lacourt, P. A.

Lepine, T.

D. Kuang, Y. Cao, T. Lepine, and W. Mi, “Curved surface plasmon polariton excitation with shaped beam by fifth-power phase mask,” IEEE Photonics J. 7(6), 1–5 (2015).
[Crossref]

Leuchs, G.

Li, T.

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, “Nonparaxial mathieu and weber accelerating beams,” Phys. Rev. Lett. 109(19), 193901 (2012).
[Crossref]

Lin, S.

Liu, J.

Liu, W.

W. Liu, X. Yang, and J. Gao, “Optical transportation and accumulation of microparticles by self-accelerating cusp beams,” Phys. Rev. A 99(4), 043839 (2019).
[Crossref]

Liu, Z.

Lumer, Y.

R. Schley, I. Kaminer, E. Greenfield, R. Bekenstein, Y. Lumer, and M. Segev, “Loss-proof self-accelerating beams and their use in non-paraxial manipulation of particles’ trajectories,” Nat. Commun. 5(1), 5189 (2014).
[Crossref]

Mathis, A.

Mazilu, M.

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D. Kuang, Y. Cao, T. Lepine, and W. Mi, “Curved surface plasmon polariton excitation with shaped beam by fifth-power phase mask,” IEEE Photonics J. 7(6), 1–5 (2015).
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Minovich, A. E.

A. E. Minovich, A. E. Klein, D. N. Neshev, T. Pertsch, Y. S. Kivshar, and D. N. Christodoulides, “Airy plasmons: non-diffracting optical surface waves,” Laser Photonics Rev. 8(2), 221–232 (2014).
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A. F. Peterson, S. L. Ray, and R. Mittra, Computational methods for electromagnetics, vol. 24, Chapter 4.12 (IEEE press, New York, 1998).

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P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense airy beams,” Science 324(5924), 229–232 (2009).
[Crossref]

Morandotti, R.

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, “Nonparaxial mathieu and weber accelerating beams,” Phys. Rev. Lett. 109(19), 193901 (2012).
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Neshev, D. N.

A. E. Minovich, A. E. Klein, D. N. Neshev, T. Pertsch, Y. S. Kivshar, and D. N. Christodoulides, “Airy plasmons: non-diffracting optical surface waves,” Laser Photonics Rev. 8(2), 221–232 (2014).
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Paúr, M.

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A. E. Minovich, A. E. Klein, D. N. Neshev, T. Pertsch, Y. S. Kivshar, and D. N. Christodoulides, “Airy plasmons: non-diffracting optical surface waves,” Laser Photonics Rev. 8(2), 221–232 (2014).
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A. F. Peterson, S. L. Ray, and R. Mittra, Computational methods for electromagnetics, vol. 24, Chapter 4.12 (IEEE press, New York, 1998).

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Z. Pi, Y. Hu, Z. Chen, and J. Xu, “Large-scale sharply bending paraxial beams,” APL Photonics 4(5), 056101 (2019).
[Crossref]

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P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense airy beams,” Science 324(5924), 229–232 (2009).
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A. F. Peterson, S. L. Ray, and R. Mittra, Computational methods for electromagnetics, vol. 24, Chapter 4.12 (IEEE press, New York, 1998).

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R. Schley, I. Kaminer, E. Greenfield, R. Bekenstein, Y. Lumer, and M. Segev, “Loss-proof self-accelerating beams and their use in non-paraxial manipulation of particles’ trajectories,” Nat. Commun. 5(1), 5189 (2014).
[Crossref]

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R. Schley, I. Kaminer, E. Greenfield, R. Bekenstein, Y. Lumer, and M. Segev, “Loss-proof self-accelerating beams and their use in non-paraxial manipulation of particles’ trajectories,” Nat. Commun. 5(1), 5189 (2014).
[Crossref]

Shaarawi, A. M.

Shlivinski, A.

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P. K. Shrestha, Y. T. Chun, and D. Chu, “A high-resolution optically addressed spatial light modulator based on zno nanoparticles,” Light: Sci. Appl. 4(3), e259 (2015).
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Siviloglou, G. A.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense airy beams,” Science 324(5924), 229–232 (2009).
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J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical airy beams,” Opt. Express 16(17), 12880–12891 (2008).
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G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of airy beams,” Opt. Lett. 33(3), 207–209 (2008).
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G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[Crossref]

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

Y. Hu, G. A. Siviloglou, P. Zhang, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, “Self-accelerating airy beams: Generation, control, and applications,” in Nonlinear Photonics and Novel Optical Phenomena, Z. Chen and R. Morandotti, eds. (Springer New York, New York, NY, 2012), pp. 1–46.

Stoklasa, B.

Thompson, K. P.

Tsur, Y.

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S. Jia, J. C. Vaughan, and X. Zhuang, “Isotropic 3d super-resolution imaging with a self-bending point spread function,” Nat. Photonics 8(4), 302–306 (2014).
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T. Vettenburg, H. I. C. Dalgarno, J. Nylk, C. Coll-Llado, D. E. K. Ferrier, T. Cizmar, F. J. Gunn-Moore, and K. Dholakia, “Light-sheet microscopy using an Airy beam,” Nat. Methods 11(5), 541–544 (2014).
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Z. Pi, Y. Hu, Z. Chen, and J. Xu, “Large-scale sharply bending paraxial beams,” APL Photonics 4(5), 056101 (2019).
[Crossref]

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W. Liu, X. Yang, and J. Gao, “Optical transportation and accumulation of microparticles by self-accelerating cusp beams,” Phys. Rev. A 99(4), 043839 (2019).
[Crossref]

Yin, X.

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, “Nonparaxial mathieu and weber accelerating beams,” Phys. Rev. Lett. 109(19), 193901 (2012).
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Yu, S.

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Zhang, P.

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[Crossref]

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, “Nonparaxial mathieu and weber accelerating beams,” Phys. Rev. Lett. 109(19), 193901 (2012).
[Crossref]

Y. Hu, G. A. Siviloglou, P. Zhang, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, “Self-accelerating airy beams: Generation, control, and applications,” in Nonlinear Photonics and Novel Optical Phenomena, Z. Chen and R. Morandotti, eds. (Springer New York, New York, NY, 2012), pp. 1–46.

Zhang, X.

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, “Nonparaxial mathieu and weber accelerating beams,” Phys. Rev. Lett. 109(19), 193901 (2012).
[Crossref]

Zhang, Y.

Zhao, J.

Zheng, Z.

Zhuang, X.

S. Jia, J. C. Vaughan, and X. Zhuang, “Isotropic 3d super-resolution imaging with a self-bending point spread function,” Nat. Photonics 8(4), 302–306 (2014).
[Crossref]

S. Jia and X. Zhuang, “Super-resolution imaging with airy beams,” Opt. Photonics News 25, LW4I.4 (2014).
[Crossref]

APL Photonics (1)

Z. Pi, Y. Hu, Z. Chen, and J. Xu, “Large-scale sharply bending paraxial beams,” APL Photonics 4(5), 056101 (2019).
[Crossref]

Appl. Opt. (1)

IEEE Photonics J. (1)

D. Kuang, Y. Cao, T. Lepine, and W. Mi, “Curved surface plasmon polariton excitation with shaped beam by fifth-power phase mask,” IEEE Photonics J. 7(6), 1–5 (2015).
[Crossref]

IEEE Trans. Antennas Propag. (1)

S. Karimkashi and A. A. Kishk, “Focused microstrip array antenna using a dolph-chebyshev near-field design,” IEEE Trans. Antennas Propag. 57(12), 3813–3820 (2009).
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J. Opt. Soc. Am. A (3)

Laser Photonics Rev. (1)

A. E. Minovich, A. E. Klein, D. N. Neshev, T. Pertsch, Y. S. Kivshar, and D. N. Christodoulides, “Airy plasmons: non-diffracting optical surface waves,” Laser Photonics Rev. 8(2), 221–232 (2014).
[Crossref]

Light: Sci. Appl. (1)

P. K. Shrestha, Y. T. Chun, and D. Chu, “A high-resolution optically addressed spatial light modulator based on zno nanoparticles,” Light: Sci. Appl. 4(3), e259 (2015).
[Crossref]

Nat. Commun. (1)

R. Schley, I. Kaminer, E. Greenfield, R. Bekenstein, Y. Lumer, and M. Segev, “Loss-proof self-accelerating beams and their use in non-paraxial manipulation of particles’ trajectories,” Nat. Commun. 5(1), 5189 (2014).
[Crossref]

Nat. Methods (1)

T. Vettenburg, H. I. C. Dalgarno, J. Nylk, C. Coll-Llado, D. E. K. Ferrier, T. Cizmar, F. J. Gunn-Moore, and K. Dholakia, “Light-sheet microscopy using an Airy beam,” Nat. Methods 11(5), 541–544 (2014).
[Crossref]

Nat. Photonics (3)

S. Jia, J. C. Vaughan, and X. Zhuang, “Isotropic 3d super-resolution imaging with a self-bending point spread function,” Nat. Photonics 8(4), 302–306 (2014).
[Crossref]

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008).
[Crossref]

D. N. Christodoulides, “Optical trapping: Riding along an airy beam,” Nat. Photonics 2(11), 652–653 (2008).
[Crossref]

Opt. Express (6)

Opt. Lett. (10)

M. Zhang, Z. Ren, and P. Yu, “Improve depth of field of optical coherence tomography using finite energy airy beam,” Opt. Lett. 44(12), 3158–3161 (2019).
[Crossref]

I. D. Chremmos, Z. Chen, D. N. Christodoulides, and N. K. Efremidis, “Bessel-like optical beams with arbitrary trajectories,” Opt. Lett. 37(23), 5003–5005 (2012).
[Crossref]

J. Zhao, P. Zhang, D. Deng, J. Liu, Y. Gao, I. D. Chremmos, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, “Observation of self-accelerating bessel-like optical beams along arbitrary trajectories,” Opt. Lett. 38(4), 498–500 (2013).
[Crossref]

R.-S. Penciu, V. Paltoglou, and N. K. Efremidis, “Closed-form expressions for nonparaxial accelerating beams with pre-engineered trajectories,” Opt. Lett. 40(7), 1444–1447 (2015).
[Crossref]

B. K. Singh, R. Remez, Y. Tsur, and A. Arie, “Measurement of acceleration and orbital angular momentum of airy beam and airy-vortex beam by astigmatic transformation,” Opt. Lett. 40(22), 5411–5414 (2015).
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M. Bandres, “Accelerating beams,” Opt. Lett. 34(24), 3791–3793 (2009).
[Crossref]

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

I. M. Besieris and A. M. Shaarawi, “A note on an accelerating finite energy airy beam,” Opt. Lett. 32(16), 2447–2449 (2007).
[Crossref]

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

T. Melamed and A. Shlivinski, “Practical algorithm for custom-made caustic beams,” Opt. Lett. 42(13), 2499–2502 (2017).
[Crossref]

Opt. Photonics News (1)

S. Jia and X. Zhuang, “Super-resolution imaging with airy beams,” Opt. Photonics News 25, LW4I.4 (2014).
[Crossref]

Optica (1)

Phys. Rev. A (2)

M. Goutsoulas and N. Efremidis, “Precise amplitude, trajectory, and beam-width control of accelerating and abruptly autofocusing beams,” Phys. Rev. A 97(6), 063831 (2018).
[Crossref]

W. Liu, X. Yang, and J. Gao, “Optical transportation and accumulation of microparticles by self-accelerating cusp beams,” Phys. Rev. A 99(4), 043839 (2019).
[Crossref]

Phys. Rev. Lett. (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]

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, “Nonparaxial mathieu and weber accelerating beams,” Phys. Rev. Lett. 109(19), 193901 (2012).
[Crossref]

Science (1)

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense airy beams,” Science 324(5924), 229–232 (2009).
[Crossref]

Other (4)

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1964).

V. Babič and V. Buldyrev, Short-Wavelength Diffraction Theory: Asymptotic Methods (Spring–Verlag, Berlin, 1991).

A. F. Peterson, S. L. Ray, and R. Mittra, Computational methods for electromagnetics, vol. 24, Chapter 4.12 (IEEE press, New York, 1998).

Y. Hu, G. A. Siviloglou, P. Zhang, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, “Self-accelerating airy beams: Generation, control, and applications,” in Nonlinear Photonics and Novel Optical Phenomena, Z. Chen and R. Morandotti, eds. (Springer New York, New York, NY, 2012), pp. 1–46.

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Figures (12)

Fig. 1.
Fig. 1. Mapping the beam-axis to the aperture curve. Each point on the beam-axis ${\mathbf {r}}_{\textrm {b}}(\sigma )$ is mapped to a point ${\mathbf {r}}'_{\textrm {a}}(\sigma )$ over the aperture from which the ray is emanating in the direction $[\theta _x(\sigma ),\theta _y(\sigma )]$. The mapping is carried out using (4).
Fig. 2.
Fig. 2. The local curve coordinates, $(\sigma ',n')$. For each point over the planar curve ${\mathbf {r}}'_{\textrm {a}}(\sigma ')$ we project the ray direction ${\hat {\mathbf {s}}}$ onto the unit vectors ${\hat {\boldsymbol \sigma }}'$ and ${\hat {\mathbf {n}}}'$ namely, the tangent and normal to the curve ${\mathbf {r}}'_{\textrm {a}}$. The aperture phase distribution is first constructed along ${\mathbf {r}}'_{\textrm {a}}$ via (7) and extended via (9) to other points with in the marked beam width domain $w_t$ in (12).
Fig. 3.
Fig. 3. The caustic surfaces for (a) the 3-D planar CB; (b) a CB following a helical trajectory, with both curvature and torsion. These surfaces were evaluated by setting the ray Jacobian determinant to zero for all rays emanating from the aperture.
Fig. 4.
Fig. 4. The Tukey window for various choices of $r$.
Fig. 5.
Fig. 5. The aperture field for the planar Airy-type CB. ${\mathbf {r}}'_{\textrm {a}}(\sigma )$ is marked by the black line. (a) Aperture field’s amplitude; (b) Aperture field’s phase.
Fig. 6.
Fig. 6. The planar CB example; (a) Intensity profile in the half-space $z\geq 0$ (the desired trajectory is marked by the black line). (b) The -3dB intensity widths at any $z=\textrm {const}$ plane. The desired beam-axis is marked by the black line and the dashed lines indicate the desired beam-width.
Fig. 7.
Fig. 7. Beam-width standard deviation as a function of the aperture grid spacing.
Fig. 8.
Fig. 8. Intensity peaks of the propagated field. The solid line represents the desired trajectory. $\lambda /20$: blue circles, $\lambda /2$: green squares, $3\lambda /3$: red circles.
Fig. 9.
Fig. 9. Beam-width definition in the plane containing the normal and the bi-normal local vectors at $\sigma =3.0$. The $x$ and $y$ axes are normalized with respect to the wavelength.
Fig. 10.
Fig. 10. Control of the on-axis and off-axis intensity profiles. (a,b) Iso-level surfaces of the -3dB of the CB maximum intensity in the $z>0$ half-space (desired trajectory in black lines) for the: (a) initial and (b) updated intensities. (c) Beam-width and (d) on-axis intensity profile compared to the desired ones (black line), for both the initial and the iteratively updated designs.
Fig. 11.
Fig. 11. (a): The aperture curve ${\mathbf {r}}'_{\textrm {a}}(\sigma ')$ corresponding to the beam-axis in (27). Note that the curve is forming close loops. (b): The aperture field close to the intersection point consists of the sum of terms corresponding to $r'_1$ and $r'_2$ of the form in (2).
Fig. 12.
Fig. 12. Iso-level surfaces for the conic helix beam-axis in (27).

Equations (27)

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r b ( σ ) = [ x ( σ ) , y ( σ ) , z ( σ ) ] ,
u 0 ( x , y ) = A ( x , y ) exp [ j k S ( x , y ) ] ,
[ cot θ x ( σ ) cot θ y ( σ ) ] = 1 z ˙ ( σ ) [ x ˙ ( σ ) y ˙ ( σ ) ] .
[ x ( σ ) y ( σ ) ] = [ x ( σ ) z ( σ ) cot θ x ( σ ) y ( σ ) z ( σ ) cot θ y ( σ ) ] .
s ^ ( σ ) = [ cos θ x ( σ ) , cos θ y ( σ ) , cos θ z ( σ ) ] ,
S | r a ( σ ) = [ cos θ x ( σ ) , cos θ y ( σ ) ] .
S ( σ , 0 ) = 0 σ S [ r a ( σ ) ] d σ .
S n ( σ ) = s ^ ( σ ) n ^ ( σ ) ,
S ( σ , n ) = S ( σ , 0 ) + n s ^ ( σ ) n ^ ( σ ) .
r ˙ b ( σ ) = t ^ ( σ ) , t ^ ˙ ( σ ) = K ( σ ) n ^ ( σ ) ,   n ^ ˙ ( σ ) = K ( σ ) t ^ ( σ ) + κ ( σ ) n ^ b ,   n ^ ˙ b ( σ ) = κ ( σ ) n ^ ( σ ) ,
r = r b ( σ ) + n n ^ ( σ ) + n b n ^ b ( σ ) .
A ( x , y ) = w t ( n , σ ) w l ( σ ) .
w l ( σ ) = I b ( σ ) ,   σ [ 0 , σ max ] .
F n ( σ ) = 1 β l ( n ) + β l ( n ) I b ( σ ) / I l ( n ) ( σ ) .
w t ( n , σ ) = T ( n + W t ( σ ) 2 W t ( σ ) ; r ) ,
T ( t ; r ) = { 1 2 [ 1 + cos ( 2 π ( t r 1 2 ) ) ] , 0 t < r 2 1 , r 2 t < 1 r 2 1 2 [ 1 + cos ( 2 π ( t 1 r + 1 2 ) ) ] , 1 r 2 t < 1 0 , elsewhere
e a ( r ) = 0 1 T ( t ; r ) d t = 1 r / 2.
F t ( n ) ( σ ) = 1 β t ( n ) + β t ( n ) W b ( σ ) W t ( n ) ( σ ) .
u ( x , y , z > 0 ) = 2 u 0 ( x , y ) n G ( r r ) d x d y .
G ( r , r ) = exp ( j k R ) / 4 π R ,   R = | r r | .
u ( r ) = 2 Δ x Δ y m n M , N u 0 ( m Δ x , n Δ y ) z G ( r r m n ) .
u [ m , n , z ] = 2 Δ x Δ y m n M , N u 0 [ m , n ] z G [ m m , n n , z ] ,
r b ( t ) = [ t , t , α t ] , 0 t   60 λ
S | r a ( σ ) = [ σ ¯ / 2 σ ¯ + α ¯ , σ ¯ / 2 σ ¯ + α ¯ ] ,   σ ¯ = σ / 2 ,   α ¯ = α 2 / 8.
S ( σ , 0 ) = 2 ( α ¯ ln σ ¯ + σ ¯ + α ¯ α ¯ σ ¯ ( α ¯ + σ ¯ ) ) .
r b ( t ) = [ R cos t , R sin t , P t ] ,   0.5 π t   3 π ,
r b ( t ) = 1 t max 0.8 t min [ R cos t ( t max 0.8 t ) , R sin t ( t max 0.8 t ) , P t ] ,