Abstract

The topological effects accompanied by phase structuring during the interaction between optical fields and nonlinear crystals are presented and demonstrated. The topological phase transition in the optical field is determined during the quasi-phase matched second harmonic frequency conversion process. The mapping relationship between the corresponding topological invariant and the phase parameters is derived, and two critical transition points are obtained. The transition of the total orbital angular momentum (OAM) in the propagation direction is verified to be the physical origin of this topological regulation through OAM spectrum analysis. This work provides a new perspective for examining nonlinear light-matter interaction, which can inspire promising applications in structured light generation and optical information processing.

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

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    [Crossref]
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    [Crossref]
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2019 (5)

P. Chen, L.-L. Ma, W. Hu, Z.-X. Shen, H. K. Bisoyi, S.-B. Wu, S.-J. Ge, Q. Li, and Y.-Q. Lu, “Chirality invertible superstructure mediated active planar optics,” Nat. Commun. 10(1), 2518 (2019).
[Crossref]

W. Zhang, J. Tang, P. Chen, G. Cui, Y. Ming, W. Hu, and Y.-Q. Lu, “Evolution of orbital angular momentum in a soft quasi-periodic structure with topological defects,” Opt. Express 27(15), 21667–21676 (2019).
[Crossref]

E. Cohen, H. Larocque, F. Bouchard, F. Nejadsattari, Y. Gefen, and E. Karimi, “Geometric phase from Aharonov–Bohm to Pancharatnam–Berry and beyond,” Nat. Rev. Phys. 1(7), 437–449 (2019).
[Crossref]

F. Tang, H. C. Po, A. Vishwanath, and X. Wan, “Comprehensive search for topological materials using symmetry indicators,” Nature 566(7745), 486–489 (2019).
[Crossref]

T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” Rev. Mod. Phys. 91(1), 015006 (2019).
[Crossref]

2018 (6)

V.-C. Su, C. H. Chu, G. Sun, and D. P. Tsai, “Advances in optical metasurfaces: fabrication and applications [Invited],” Opt. Express 26(10), 13148–13182 (2018).
[Crossref]

Y. Xu, J. Sun, J. Frantz, M. I. Shalaev, W. Walasik, A. Pandey, J. D. Myers, R. Y. Bekele, A. Tsukernik, J. S. Sanghera, and N. M. Litchinitser, “Reconfiguring structured light beams using nonlinear metasurfaces,” Opt. Express 26(23), 30930–30943 (2018).
[Crossref]

M.-Y. Wang, J. Tang, H.-J. Wang, Y. Ming, Y. Zhang, G.-X. Cui, and Y.-Q. Lu, “Generation of second-harmonic Ince-Gaussian beams,” Appl. Phys. Lett. 113(8), 081105 (2018).
[Crossref]

D. Wei, C. Wang, H. Wang, X. Hu, D. Wei, X. Fang, Y. Zhang, D. Wu, Y. Hu, J. Li, S. Zhu, and M. Xiao, “Experimental demonstration of a three-dimensional lithium niobate nonlinear photonic crystal,” Nat. Photonics 12(10), 596–600 (2018).
[Crossref]

T. Xu, K. Switkowski, X. Chen, S. Liu, K. Koynov, H. Yu, H. Zhang, J. Wang, Y. Sheng, and W. Krolikowski, “Three-dimensional nonlinear photonic crystal in ferroelectric barium calcium titanate,” Nat. Photonics 12(10), 591–595 (2018).
[Crossref]

A. Karnieli and A. Arie, “All-optical Stern-Gerlach effect,” Phys. Rev. Lett. 120(5), 053901 (2018).
[Crossref]

2017 (4)

S. N. Alperin and M. E. Siemens, “Angular momentum of topologically structured darkness,” Phys. Rev. Lett. 119(20), 203902 (2017).
[Crossref]

Y. Ming, P. Chen, W. Ji, B.-Y. Wei, C.-H. Lee, T.-H. Lin, W. Hu, and Y.-Q. Lu, “Tailoring the photon spin via light–matter interaction in liquid-crystal-based twisting structures,” npj Quantum Mater. 2(1), 6 (2017).
[Crossref]

S. Trajtenebrg-Mills and A. Arie, “Shaping light beams in nonlinear processes using structured light and patterned crystals,” Opt. Mater. Express 7(8), 2928–2942 (2017).
[Crossref]

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19(1), 013001 (2017).
[Crossref]

2016 (5)

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological states in photonic systems,” Nat. Phys. 12(7), 626–629 (2016).
[Crossref]

M. J. Escuti, J. Kim, and M. W. Kudenov, “Controlling light with geometric-phase holograms,” Opt. Photonics News 27(2), 22–29 (2016).
[Crossref]

J. Zhu, P. Zhang, D. Fu, D. Chen, R. Liu, Y. Zhou, H. Gao, and F. Li, “Probing the fractional topological charge of a vortex light beam by using dynamic angular double slits,” Photonics Res. 4(5), 187–190 (2016).
[Crossref]

G. Gbur, “Fractional vortex Hilbert's hotel,” Optica 3(3), 222–225 (2016).
[Crossref]

B. Yang, X.-H. Hong, R.-E. Lu, Y.-Y. Yue, C. Zhang, Y.-Q. Qin, and Y.-Y. Zhu, “2D wave-front shaping in optical superlattices using nonlinear volume holography,” Opt. Lett. 41(13), 2927–2929 (2016).
[Crossref]

2015 (1)

Y. Ming, J. Tang, Z.-X. Chen, F. Xu, L.-J. Zhang, and Y.-Q. Lu, “Generation of N00N state with orbital angular momentum in a twisted nonlinear photonic crystal,” IEEE J. Sel. Top. Quantum Electron. 21(3), 225–230 (2015).
[Crossref]

2014 (3)

X.-H. Hong, B. Yang, C. Zhang, Y.-Q. Qin, and Y.-Y. Zhu, “Nonlinear volume holography for wave-front engineering,” Phys. Rev. Lett. 113(16), 163902 (2014).
[Crossref]

N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nat. Mater. 13(2), 139–150 (2014).
[Crossref]

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics 8(11), 821–829 (2014).
[Crossref]

2012 (2)

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett. 100(1), 013101 (2012).
[Crossref]

N. V. Bloch, K. Shemer, A. Shapira, R. Shiloh, I. Juwiler, and A. Arie, “Twisting light by nonlinear photonic crystals,” Phys. Rev. Lett. 108(23), 233902 (2012).
[Crossref]

2011 (1)

L. Novotny and N. van Hulst, “Antennas for light,” Nat. Photonics 5(2), 83–90 (2011).
[Crossref]

2010 (1)

M. Z. Hasan and C. L. Kane, “Colloquium: Topological insulators,” Rev. Mod. Phys. 82(4), 3045–3067 (2010).
[Crossref]

2009 (1)

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3(7), 395–398 (2009).
[Crossref]

2006 (2)

R. Zambrini and S. M. Barnett, “Quasi-intrinsic angular momentum and the measurement of its spectrum,” Phys. Rev. Lett. 96(11), 113901 (2006).
[Crossref]

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006).
[Crossref]

2002 (1)

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88(5), 053601 (2002).
[Crossref]

1997 (1)

J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, “Second-harmonic generation and the conservation of orbital angular momentum with high-order Laguerre-Gaussian modes,” Phys. Rev. A 56(5), 4193–4196 (1997).
[Crossref]

1996 (1)

K. Dholakia, N. B. Simpson, and M. J. Padgett, “Second-harmonic generation and the orbital angular momentum of light,” Phys. Rev. A 54(5), R3742–R3745 (1996).
[Crossref]

1993 (1)

M. Yamada, N. Nada, M. Saitoh, and K. Watanabe, “First-order quasi-phase matched LiNbO3waveguide periodically poled by applying an external field for efficient blue second-harmonic generation,” Appl. Phys. Lett. 62(5), 435–436 (1993).
[Crossref]

1992 (2)

N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17(3), 221–223 (1992).
[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]

1979 (1)

1974 (1)

J. F. Nye, M. V. Berry, and C. Frank Frederick, “Dislocations in wave trains,” Proc. R. Soc. Lond. A 336(1605), 165–190 (1974).

1962 (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127(6), 1918–1939 (1962).
[Crossref]

Aieta, F.

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett. 100(1), 013101 (2012).
[Crossref]

Allen, L.

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88(5), 053601 (2002).
[Crossref]

J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, “Second-harmonic generation and the conservation of orbital angular momentum with high-order Laguerre-Gaussian modes,” Phys. Rev. A 56(5), 4193–4196 (1997).
[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]

Alperin, S. N.

S. N. Alperin and M. E. Siemens, “Angular momentum of topologically structured darkness,” Phys. Rev. Lett. 119(20), 203902 (2017).
[Crossref]

Alpmann, C.

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19(1), 013001 (2017).
[Crossref]

Amo, A.

T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” Rev. Mod. Phys. 91(1), 015006 (2019).
[Crossref]

Andrews, D. L.

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19(1), 013001 (2017).
[Crossref]

Arie, A.

A. Karnieli and A. Arie, “All-optical Stern-Gerlach effect,” Phys. Rev. Lett. 120(5), 053901 (2018).
[Crossref]

S. Trajtenebrg-Mills and A. Arie, “Shaping light beams in nonlinear processes using structured light and patterned crystals,” Opt. Mater. Express 7(8), 2928–2942 (2017).
[Crossref]

N. V. Bloch, K. Shemer, A. Shapira, R. Shiloh, I. Juwiler, and A. Arie, “Twisting light by nonlinear photonic crystals,” Phys. Rev. Lett. 108(23), 233902 (2012).
[Crossref]

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3(7), 395–398 (2009).
[Crossref]

Armstrong, J. A.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127(6), 1918–1939 (1962).
[Crossref]

Baker, M.

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19(1), 013001 (2017).
[Crossref]

Banzer, P.

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19(1), 013001 (2017).
[Crossref]

Barnett, S. M.

R. Zambrini and S. M. Barnett, “Quasi-intrinsic angular momentum and the measurement of its spectrum,” Phys. Rev. Lett. 96(11), 113901 (2006).
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Bauer, T.

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Figures (5)

Fig. 1.
Fig. 1. Schematic illustration of the topological phase transition of the optical field during nonlinear frequency conversion. As an example, a pump optical field with a topological invariant ${t_\omega } = 0$ is converted into an SH optical field with a topological invariant ${t_{2\omega }} = \textrm{7}$after the quasi-phase matched SHG process. The wave-vector matching is indicated in the center of the illustration.
Fig. 2.
Fig. 2. Diagram of the topological phase transition in SHG. The red and blue lines correspond to the topological invariant values of the pump and SH optical fields, respectively. The critical positions of the topological phase transition are marked by yellow stars. The topological phase transition regions are marked by Roman numerals.
Fig. 3.
Fig. 3. Topological modulation of the fractional optical vortices in SHG: (a)–(d) and (i)–(l) Phase distributions with various values of α for pump optical fields; (e)–(h) and (m)–(p) Structured phase distributions of the corresponding SH optical fields. The phase singularities are marked by dotted circles with directional arrows.
Fig. 4.
Fig. 4. Optical fields topological modulation for M = 7 in SHG: (a)–(d) and (i)–(l) Phase distributions with various values of α for pump optical fields; (e)–(h) and (m)–(p) Structured phase distributions of the corresponding SH optical fields. The phase singularities are marked by dotted circles with directional arrows.
Fig. 5.
Fig. 5. OAM spectrums of optical fields for M = 7 before and after SHG. The red and the blue histograms represent OAM spectrums for pump and SH optical fields, respectively. The potential topological phase transition is indicated with an illustration in each subgraph; the yellow solid circles represent the case for no topological phase transition, the yellow solid stars represent the critical transition points where the topological phase transition just about to happen, and the yellow solid triangles represent the case for topological phase transition.

Equations (8)

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

d A ω d z = 2 i d e f f ω 1 2 k 1 c 2 A 2 ω A ω e i Δ k z d A 2 ω d z = i d e f f ω 2 2 k 2 c 2 A ω 2 e i Δ k z ,
A 2 ω ( z 0 ) = 2 i d 33 ω 2 2 k 2 c 2 | A ω | 2 exp [ i 2 φ ω ( x , y ) ] 0 z 0 s i g n [ cos ( 2 π Λ z ) ] e i Δ k z d z   2 i d 33 ω 2 2 f 1 z 0 k 2 c 2 | A ω | 2 exp [ i 2 φ ω ( x , y ) ] ,
φ 2 ω ( x , y ) = 2 φ ω ( x , y ) ,
t 1 2 π C φ ( r ) d r ,
φ ω ( x , y )  =  α [ ϕ ( x , y ) 2 π m M ] ,
t ω = M f l o o r ( α / M + 1 / 2 ) t 2 ω = M f l o o r ( 2 α / M + 1 / 2 ) ,
h l = { M sin ( α π / M ) π ( α l ) e i α π M , l / M N 0 , l / M N ,
L z = [ ( r r 0 ) × p ] z d x = L z x 0 P y y 0 P x ,

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