Abstract

We propose a method for generation of tunable three-dimensional (3D) helical lattices with varying helix pitch. In order to change only the lattice helix pitch, a periodically varying phase along the propagation direction is added to the central beam – one of the interference beams for lattice construction. The phase periodicity determines the helix pitch, which can be reconfigured at ease. Furthermore, a helical lattice structure with an interface (domain wall) is also achieved by changing the phase structure of the lateral beams, leading to opposite rotating direction (helicity) on different sides of the interface. When a Gaussian beam is used to probe the bulk lattice, it can evolve into a spiral beam with its helicity varying in accordance with that of the lattice. Probing along the interface with two dipole-like optical beams leads to unusual propagation dynamics, depending on the phase and size of the two beams. This approach could be further explored for studies of nonlinear interface solitons and topological interface states. In addition, the helical lattices may find applications in dynamical multi-beam optical tweezers.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]

2018 (3)

J. Noh, S. Huang, K. P. Chen, and M. C. Rechtsman, “Observation of photonic topological valley Hall edge states,” Phys. Rev. Lett. 120(6), 063902 (2018).
[Crossref] [PubMed]

X. Li, H. Ma, H. Zhang, Y. Tai, H. Li, M. Tang, J. Wang, J. Tang, and Y. Cai, “Close-packed optical vortex lattices with controllable structures,” Opt. Express 26(18), 22965–22975 (2018).
[Crossref] [PubMed]

H. Gao, Y. Li, L. Chen, J. Jin, M. Pu, X. Li, P. Gao, C. Wang, X. Luo, and M. Hong, “Quasi-Talbot effect of orbital angular momentum beams for generation of optical vortex arrays by multiplexing metasurface design,” Nanoscale 10(2), 666–671 (2018).
[Crossref] [PubMed]

2017 (5)

2016 (7)

D. Leykam and Y. D. Chong, “Edge solitons in nonlinear-photonic topological insulators,” Phys. Rev. Lett. 117(14), 143901 (2016).
[Crossref] [PubMed]

D. Leykam, M. C. Rechtsman, and Y. D. Chong, “Anomalous topological phases and unpaired Dirac cones in photonic Floquet topological insulators,” Phys. Rev. Lett. 117(1), 013902 (2016).
[Crossref] [PubMed]

T. Latychevskaia and H. W. Fink, “Inverted Gabor holography principle for tailoring arbitrary shaped three-dimensional beams,” Sci. Rep. 6(1), 26312 (2016).
[Crossref] [PubMed]

A. Zannotti, F. Diebel, M. Boguslawski, and C. Denz, “Chiral light in helically twisted photonic lattices,” Adv. Opt. Mater. 5(16), 1–7 (2016).

X. Zhang, F. Ye, Y. V. Kartashov, V. A. Vysloukh, and X. Chen, “Localized waves supported by the rotating waveguide array,” Opt. Lett. 41(17), 4106–4109 (2016).
[Crossref] [PubMed]

A. Kapoor, M. Kumar, P. Senthilkumaran, and J. Joseph, “Optical vortex array in spatially varying lattice,” Opt. Commun. 365, 99–102 (2016).
[Crossref]

S. Fu, S. Zhang, and C. Gao, “Bessel beams with spatial oscillating polarization,” Sci. Rep. 6(1), 30765 (2016).
[Crossref] [PubMed]

2015 (1)

2014 (2)

2013 (3)

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496(7444), 196–200 (2013).
[Crossref] [PubMed]

Y. Lumer, Y. Plotnik, M. C. Rechtsman, and M. Segev, “Self-localized states in photonic topological insulators,” Phys. Rev. Lett. 111(24), 243905 (2013).
[Crossref] [PubMed]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Dynamics of topological light states in spiraling structures,” Opt. Lett. 38(17), 3414–3417 (2013).
[Crossref] [PubMed]

2012 (2)

I I. L. Garanovich, S. Longhi, A. A. Sukhorukov, and Y. S. Kivshar, “Light propagation and localization in modulated photonic lattices and waveguides,” Phys. Rep. 518(1-2), 1–79 (2012).
[Crossref]

D. H. Song, C. B. Lou, L. Q. Tang, Z. Y. Ye, J. J. Xu, and Z. G. Chen, “Experiments on linear and nonlinear localization of optical vortices in optically induced photonic lattices,” Int. J. Opt. 2012, 1 (2012).
[Crossref]

2011 (3)

P. Papagiannis, Y. Kominis, and K. Hizanidis, “Power-and momentum-dependent soliton dynamics in lattices with longitudinal modulation,” Phys. Rev. A 84(1), 013820 (2011).
[Crossref]

J. Becker, P. Rose, M. Boguslawski, and C. Denz, “Systematic approach to complex periodic vortex and helix lattices,” Opt. Express 19(10), 9848–9862 (2011).
[Crossref] [PubMed]

G. M. Burrow and T. K. Gaylord, “Multi-Beam interference advances and applications: nano-electronics, photonic crystals, metamaterials, subwavelength structures, optical trapping, and biomedical structures,” Micromachines (Basel) 2(2), 221–257 (2011).
[Crossref]

2010 (2)

S. H. Lee, Y. Roichman, and D. G. Grier, “Optical solenoid beams,” Opt. Express 18(7), 6988–6993 (2010).
[Crossref] [PubMed]

G. Assanto, L. A. Cisneros, A. A. Minzoni, B. D. Skuse, N. F. Smyth, and A. L. Worthy, “Soliton steering by longitudinal modulation of the nonlinearity in waveguide arrays,” Phys. Rev. Lett. 104(5), 053903 (2010).
[Crossref] [PubMed]

2009 (1)

A. Szameit, I. L. Garanovich, M. Heinrich, A. A. Sukhorukov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, and Y. S. Kivshar, “Polychromatic dynamic localization in curved photonic lattices,” Nat. Phys. 5(4), 271–275 (2009).
[Crossref]

2008 (2)

S. Longhi and K. Staliunas, “Self-collimation and self-imaging effects in modulated waveguide arrays,” Opt. Commun. 281(17), 4343–4347 (2008).
[Crossref]

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463(1-3), 1–126 (2008).
[Crossref]

2007 (4)

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Resonant mode oscillations in modulated waveguiding structures,” Phys. Rev. Lett. 99(23), 233903 (2007).
[Crossref] [PubMed]

D. M. Jović, S. Prvanović, R. D. Jovanović, and M. S. Petrović, “Gaussian-induced rotation in periodic photonic lattices,” Opt. Lett. 32(13), 1857–1859 (2007).
[Crossref] [PubMed]

S. Longhi, “Bloch dynamics of light waves in helical optical waveguide arrays,” Phys. Rew. B 76(19), 195119 (2007).
[Crossref]

S. Vyas and P. Senthilkumaran, “Interferometric optical vortex array generator,” Appl. Opt. 46(15), 2893–2898 (2007).
[Crossref] [PubMed]

2006 (3)

2004 (1)

M. Padgett, J. Courtial, and L. Allen, “Light’s Orbital Angular Momentum,” Phys. Today 57(5), 35–40 (2004).
[Crossref]

2003 (2)

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424(6950), 817–823 (2003).
[Crossref] [PubMed]

O. V. Sinkin, R. Holzlohner, J. Zweck, and C. R. Menyuk, “Optimization of the split-step Fourier method in modeling optical-fiber communications systems,” J. Lightwave Technol. 21(1), 61–68 (2003).
[Crossref]

2001 (1)

J. Masajada and B. Dubik, “Optical vortex generation by three plane wave interference,” Opt. Commun. 198(1-3), 21–27 (2001).
[Crossref]

1992 (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Allen, L.

M. Padgett, J. Courtial, and L. Allen, “Light’s Orbital Angular Momentum,” Phys. Today 57(5), 35–40 (2004).
[Crossref]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Assanto, G.

G. Assanto, L. A. Cisneros, A. A. Minzoni, B. D. Skuse, N. F. Smyth, and A. L. Worthy, “Soliton steering by longitudinal modulation of the nonlinearity in waveguide arrays,” Phys. Rev. Lett. 104(5), 053903 (2010).
[Crossref] [PubMed]

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463(1-3), 1–126 (2008).
[Crossref]

Becker, J.

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Boguslawski, M.

A. Zannotti, F. Diebel, M. Boguslawski, and C. Denz, “Chiral light in helically twisted photonic lattices,” Adv. Opt. Mater. 5(16), 1–7 (2016).

J. Becker, P. Rose, M. Boguslawski, and C. Denz, “Systematic approach to complex periodic vortex and helix lattices,” Opt. Express 19(10), 9848–9862 (2011).
[Crossref] [PubMed]

Burrow, G. M.

G. M. Burrow and T. K. Gaylord, “Multi-Beam interference advances and applications: nano-electronics, photonic crystals, metamaterials, subwavelength structures, optical trapping, and biomedical structures,” Micromachines (Basel) 2(2), 221–257 (2011).
[Crossref]

Cai, X.

Cai, Y.

Chen, K. P.

J. Noh, S. Huang, K. P. Chen, and M. C. Rechtsman, “Observation of photonic topological valley Hall edge states,” Phys. Rev. Lett. 120(6), 063902 (2018).
[Crossref] [PubMed]

Chen, L.

H. Gao, Y. Li, L. Chen, J. Jin, M. Pu, X. Li, P. Gao, C. Wang, X. Luo, and M. Hong, “Quasi-Talbot effect of orbital angular momentum beams for generation of optical vortex arrays by multiplexing metasurface design,” Nanoscale 10(2), 666–671 (2018).
[Crossref] [PubMed]

Chen, X.

Chen, Z. G.

D. H. Song, C. B. Lou, L. Q. Tang, Z. Y. Ye, J. J. Xu, and Z. G. Chen, “Experiments on linear and nonlinear localization of optical vortices in optically induced photonic lattices,” Int. J. Opt. 2012, 1 (2012).
[Crossref]

Cheng, C.

Cheng, H.

Chong, Y. D.

D. Leykam and Y. D. Chong, “Edge solitons in nonlinear-photonic topological insulators,” Phys. Rev. Lett. 117(14), 143901 (2016).
[Crossref] [PubMed]

D. Leykam, M. C. Rechtsman, and Y. D. Chong, “Anomalous topological phases and unpaired Dirac cones in photonic Floquet topological insulators,” Phys. Rev. Lett. 117(1), 013902 (2016).
[Crossref] [PubMed]

Christodoulides, D. N.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463(1-3), 1–126 (2008).
[Crossref]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424(6950), 817–823 (2003).
[Crossref] [PubMed]

Cianci, E.

S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, “Observation of dynamic localization in periodically curved waveguide arrays,” Phys. Rev. Lett. 96(24), 243901 (2006).
[Crossref] [PubMed]

Cisneros, L. A.

G. Assanto, L. A. Cisneros, A. A. Minzoni, B. D. Skuse, N. F. Smyth, and A. L. Worthy, “Soliton steering by longitudinal modulation of the nonlinearity in waveguide arrays,” Phys. Rev. Lett. 104(5), 053903 (2010).
[Crossref] [PubMed]

Courtial, J.

M. Padgett, J. Courtial, and L. Allen, “Light’s Orbital Angular Momentum,” Phys. Today 57(5), 35–40 (2004).
[Crossref]

Denz, C.

A. Zannotti, F. Diebel, M. Boguslawski, and C. Denz, “Chiral light in helically twisted photonic lattices,” Adv. Opt. Mater. 5(16), 1–7 (2016).

J. Becker, P. Rose, M. Boguslawski, and C. Denz, “Systematic approach to complex periodic vortex and helix lattices,” Opt. Express 19(10), 9848–9862 (2011).
[Crossref] [PubMed]

Diebel, F.

A. Zannotti, F. Diebel, M. Boguslawski, and C. Denz, “Chiral light in helically twisted photonic lattices,” Adv. Opt. Mater. 5(16), 1–7 (2016).

Dreisow, F.

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496(7444), 196–200 (2013).
[Crossref] [PubMed]

A. Szameit, I. L. Garanovich, M. Heinrich, A. A. Sukhorukov, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, and Y. S. Kivshar, “Polychromatic dynamic localization in curved photonic lattices,” Nat. Phys. 5(4), 271–275 (2009).
[Crossref]

Dubik, B.

J. Masajada and B. Dubik, “Optical vortex generation by three plane wave interference,” Opt. Commun. 198(1-3), 21–27 (2001).
[Crossref]

Fan, D.

Y. Liu, Y. Ke, J. Zhou, Y. Liu, H. Luo, S. Wen, and D. Fan, “Generation of perfect vortex and vector beams based on Pancharatnam-Berry phase elements,” Sci. Rep. 7(1), 44096 (2017).
[Crossref] [PubMed]

Fink, H. W.

T. Latychevskaia and H. W. Fink, “Inverted Gabor holography principle for tailoring arbitrary shaped three-dimensional beams,” Sci. Rep. 6(1), 26312 (2016).
[Crossref] [PubMed]

Foglietti, V.

S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, “Observation of dynamic localization in periodically curved waveguide arrays,” Phys. Rev. Lett. 96(24), 243901 (2006).
[Crossref] [PubMed]

Fonseca, E. J. S.

Fu, S.

S. Fu, S. Zhang, and C. Gao, “Bessel beams with spatial oscillating polarization,” Sci. Rep. 6(1), 30765 (2016).
[Crossref] [PubMed]

Gao, C.

S. Fu, S. Zhang, and C. Gao, “Bessel beams with spatial oscillating polarization,” Sci. Rep. 6(1), 30765 (2016).
[Crossref] [PubMed]

Gao, H.

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I I. L. Garanovich, S. Longhi, A. A. Sukhorukov, and Y. S. Kivshar, “Light propagation and localization in modulated photonic lattices and waveguides,” Phys. Rep. 518(1-2), 1–79 (2012).
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I I. L. Garanovich, S. Longhi, A. A. Sukhorukov, and Y. S. Kivshar, “Light propagation and localization in modulated photonic lattices and waveguides,” Phys. Rep. 518(1-2), 1–79 (2012).
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P. Papagiannis, Y. Kominis, and K. Hizanidis, “Power-and momentum-dependent soliton dynamics in lattices with longitudinal modulation,” Phys. Rev. A 84(1), 013820 (2011).
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A. Kapoor, M. Kumar, P. Senthilkumaran, and J. Joseph, “Optical vortex array in spatially varying lattice,” Opt. Commun. 365, 99–102 (2016).
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S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, “Observation of dynamic localization in periodically curved waveguide arrays,” Phys. Rev. Lett. 96(24), 243901 (2006).
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D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424(6950), 817–823 (2003).
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D. Leykam, M. C. Rechtsman, and Y. D. Chong, “Anomalous topological phases and unpaired Dirac cones in photonic Floquet topological insulators,” Phys. Rev. Lett. 117(1), 013902 (2016).
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H. Gao, Y. Li, L. Chen, J. Jin, M. Pu, X. Li, P. Gao, C. Wang, X. Luo, and M. Hong, “Quasi-Talbot effect of orbital angular momentum beams for generation of optical vortex arrays by multiplexing metasurface design,” Nanoscale 10(2), 666–671 (2018).
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Y. Liu, Y. Ke, J. Zhou, Y. Liu, H. Luo, S. Wen, and D. Fan, “Generation of perfect vortex and vector beams based on Pancharatnam-Berry phase elements,” Sci. Rep. 7(1), 44096 (2017).
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Y. Liu, Y. Ke, J. Zhou, Y. Liu, H. Luo, S. Wen, and D. Fan, “Generation of perfect vortex and vector beams based on Pancharatnam-Berry phase elements,” Sci. Rep. 7(1), 44096 (2017).
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S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, “Observation of dynamic localization in periodically curved waveguide arrays,” Phys. Rev. Lett. 96(24), 243901 (2006).
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I I. L. Garanovich, S. Longhi, A. A. Sukhorukov, and Y. S. Kivshar, “Light propagation and localization in modulated photonic lattices and waveguides,” Phys. Rep. 518(1-2), 1–79 (2012).
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S. Longhi and K. Staliunas, “Self-collimation and self-imaging effects in modulated waveguide arrays,” Opt. Commun. 281(17), 4343–4347 (2008).
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D. H. Song, C. B. Lou, L. Q. Tang, Z. Y. Ye, J. J. Xu, and Z. G. Chen, “Experiments on linear and nonlinear localization of optical vortices in optically induced photonic lattices,” Int. J. Opt. 2012, 1 (2012).
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Y. Liu, Y. Ke, J. Zhou, Y. Liu, H. Luo, S. Wen, and D. Fan, “Generation of perfect vortex and vector beams based on Pancharatnam-Berry phase elements,” Sci. Rep. 7(1), 44096 (2017).
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H. Gao, Y. Li, L. Chen, J. Jin, M. Pu, X. Li, P. Gao, C. Wang, X. Luo, and M. Hong, “Quasi-Talbot effect of orbital angular momentum beams for generation of optical vortex arrays by multiplexing metasurface design,” Nanoscale 10(2), 666–671 (2018).
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S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, “Observation of dynamic localization in periodically curved waveguide arrays,” Phys. Rev. Lett. 96(24), 243901 (2006).
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H. Gao, Y. Li, L. Chen, J. Jin, M. Pu, X. Li, P. Gao, C. Wang, X. Luo, and M. Hong, “Quasi-Talbot effect of orbital angular momentum beams for generation of optical vortex arrays by multiplexing metasurface design,” Nanoscale 10(2), 666–671 (2018).
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S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, “Observation of dynamic localization in periodically curved waveguide arrays,” Phys. Rev. Lett. 96(24), 243901 (2006).
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J. Noh, S. Huang, K. P. Chen, and M. C. Rechtsman, “Observation of photonic topological valley Hall edge states,” Phys. Rev. Lett. 120(6), 063902 (2018).
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D. Leykam, M. C. Rechtsman, and Y. D. Chong, “Anomalous topological phases and unpaired Dirac cones in photonic Floquet topological insulators,” Phys. Rev. Lett. 117(1), 013902 (2016).
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M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496(7444), 196–200 (2013).
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Y. Lumer, Y. Plotnik, M. C. Rechtsman, and M. Segev, “Self-localized states in photonic topological insulators,” Phys. Rev. Lett. 111(24), 243905 (2013).
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M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496(7444), 196–200 (2013).
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Skuse, B. D.

G. Assanto, L. A. Cisneros, A. A. Minzoni, B. D. Skuse, N. F. Smyth, and A. L. Worthy, “Soliton steering by longitudinal modulation of the nonlinearity in waveguide arrays,” Phys. Rev. Lett. 104(5), 053903 (2010).
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G. Assanto, L. A. Cisneros, A. A. Minzoni, B. D. Skuse, N. F. Smyth, and A. L. Worthy, “Soliton steering by longitudinal modulation of the nonlinearity in waveguide arrays,” Phys. Rev. Lett. 104(5), 053903 (2010).
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D. H. Song, C. B. Lou, L. Q. Tang, Z. Y. Ye, J. J. Xu, and Z. G. Chen, “Experiments on linear and nonlinear localization of optical vortices in optically induced photonic lattices,” Int. J. Opt. 2012, 1 (2012).
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F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463(1-3), 1–126 (2008).
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I I. L. Garanovich, S. Longhi, A. A. Sukhorukov, and Y. S. Kivshar, “Light propagation and localization in modulated photonic lattices and waveguides,” Phys. Rep. 518(1-2), 1–79 (2012).
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Y. Liu, Y. Ke, J. Zhou, Y. Liu, H. Luo, S. Wen, and D. Fan, “Generation of perfect vortex and vector beams based on Pancharatnam-Berry phase elements,” Sci. Rep. 7(1), 44096 (2017).
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L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
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Adv. Opt. Mater. (1)

A. Zannotti, F. Diebel, M. Boguslawski, and C. Denz, “Chiral light in helically twisted photonic lattices,” Adv. Opt. Mater. 5(16), 1–7 (2016).

Appl. Opt. (2)

Int. J. Opt. (1)

D. H. Song, C. B. Lou, L. Q. Tang, Z. Y. Ye, J. J. Xu, and Z. G. Chen, “Experiments on linear and nonlinear localization of optical vortices in optically induced photonic lattices,” Int. J. Opt. 2012, 1 (2012).
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J. Lightwave Technol. (1)

Micromachines (Basel) (1)

G. M. Burrow and T. K. Gaylord, “Multi-Beam interference advances and applications: nano-electronics, photonic crystals, metamaterials, subwavelength structures, optical trapping, and biomedical structures,” Micromachines (Basel) 2(2), 221–257 (2011).
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Nanoscale (1)

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Figures (8)

Fig. 1
Fig. 1 Schematic of interference configuration of seven beams for optical induction of HLs in a nonlinear crystal, where k i (i=1,27) denotes seven beams, the red line arrows and the red circular arrow describe the direction of the polarization vector of these beams, and the angle θ is the incidence angle of each beam relative to z-direction.
Fig. 2
Fig. 2 (a) 3D HL with vertical periodicity (longitudinal periodicity along the propagation z-direction) T v =107μm at θ=0.02π. (b) Intensity distributions at different zplanes. The horizontal periodicity is T h =3.9μm.
Fig. 3
Fig. 3 (a) 3D hexagonal helix lattice with vertical periodicity T v =54μmat θ=0.02πby adding an additional phase term P z to the central beam at Λ=5λ. (b) Intensity distributions at different zplanes. The horizontal periodicity is kept at T h =3.9μm.
Fig. 4
Fig. 4 The formation principle of spiral beams by introducing additional radial phase
Fig. 5
Fig. 5 The phase distributions (top row) and intensity distributions (bottom row) of the HBs at θ=0.01πand m=2. The left column denotes P r =0. The middle and right columns denote A=0.5λand A=0.25λat P r 0, respectively.
Fig. 6
Fig. 6 (a) 3D HL with opposite helicities on both sides of an interface at the x=0plane at θ=0.01π. (b) Transverse intensity distributions at different zplanes. The red line denotes the plane of interface at x=0. Two black curved arrows (in the first two panels) represent the HL with opposite (anticlockwise and clockwise) helicities.
Fig. 7
Fig. 7 (a) Input transverse intensity pattern of an incident Gaussian beam with μ=0.1probing the HL. (b) and (c) are the “side-view” results of the probe beam propagating linearly along the y=0 plane through the bulk HLs (similar to Fig. 2) at θ=0.01π and θ=0.015πfor σ=0, respectively. (d) The propagation of the same probe beam through the HL interface described in Fig. 6 for σ=0. (e) and (f) are the nonlinear propagation of the same beam through the bulk HL as for (b) but with σ=0.5 and σ=1, respectively.
Fig. 8
Fig. 8 (a) Intensity pattern of an incident dipole-like beam with μ=0.15probing the interface. The propagation of the dipole-like beam with μ=0.15(b) and μ=0.2(c) in y=0 plane in the HL interface of same helicities. (d) Propagation of the probe beam in (a) in the HL interface of opposite helicities. All insets denote the transverse cross sections of the probe beam at the z=4mm plane. The other parameter θ=0.015π.

Equations (6)

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I( r )= 1 2 i=1 N E i 2 + j>i N V ij cos(( k i k j ) r +δ ϕ ij ) , i,j=1,2,3N,
[ x y z ]=[ Aωsin(ωs)(ps)+Acos(ωs) Aωcos(ωs)(ps)+Asin(ωs) p ].
[ x y z ]=[ pAωsin(ωp)+Acos(ωp) pAωcos(ωp)+Asin(ωp) 0 ].
P r = r 2 A 2 1 + tan 1 ( r 2 A 2 1 )γ,
i u z + 1 2 k 0 n ( 2 u x 2 + 2 u y 2 ) 1 2 k 0 n n e 3 γ 33 E 0 1+ I L +σ | u | 2 u=0.
u(x,y,0)=5exp( μ( (x-15× 10 -6 ) 2 + y 2 )/ Γ 2 )+5exp( μ( (x+15× 10 -6 ) 2 + y 2 )/ Γ 2 ).

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