Abstract

A continuously deformative space possesses trivial or nontrivial topological characteristics depending on the associated homotopy groups associated with spaces describing the physical processes. Moreover, the interaction of spatial warping and structural symmetry always presents fantastic phenomena, especially in the systems with unique symmetrical properties such as quasicrystals. Here, we propose a quasi-periodic structure (QPS) with topological defects. The analytical expression of the corresponding Fourier spectrum is derived, which reflects the combined effects of topological structure and quasi-translational symmetry. Light-matter interaction therein brings unusual diffraction characteristics with exotic evolution of orbital angular momentum (OAM). Long-range correlation of QPS resulted in multi-fractal and pairwise distribution of optical singularities. A general conservation law of OAM is revealed. A liquid crystal photopatterned QPS is fabricated to demonstrate the above characteristics. Dynamic reconfigurable manipulation of optical singularities is achieved. Our approach offers the opportunity to manipulate OAM with multiple degrees of freedom, which has promising applications in multi-channel quantum information processing and high-dimensional quantum state generation.

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

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References

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  22. T. Janssen, “Crystallography of quasi-crystals,” Acta Crystallogr. A 42(4), 261–271 (1986).
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    [Crossref] [PubMed]
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    [Crossref]
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2018 (1)

P. Chen, L.-L. Ma, W. Duan, J. Chen, S.-J. Ge, Z.-H. Zhu, M.-J. Tang, R. Xu, W. Gao, T. Li, W. Hu, and Y.-Q. Lu, “Digitalizing Self-Assembled Chiral Superstructures for Optical Vortex Processing,” Adv. Mater. 30(10), 1705865 (2018).
[Crossref] [PubMed]

2016 (4)

J. Kobashi, H. Yoshida, and M. Ozaki, “Polychromatic optical vortex generation from patterned cholesteric liquid crystals,” Phys. Rev. Lett. 116(25), 253903 (2016).
[Crossref] [PubMed]

Z.-Y. Zhou, S.-L. Liu, Y. Li, D.-S. Ding, W. Zhang, S. Shi, M.-X. Dong, B.-S. Shi, and G.-C. Guo, “Orbital angular momentum-entanglement frequency transducer,” Phys. Rev. Lett. 117(10), 103601 (2016).
[Crossref] [PubMed]

P. Chen, S.-J. Ge, L.-L. Ma, W. Hu, V. Chigrinov, and Y.-Q. Lu, “Generation of equal-energy orbital angular momentum beams via photopatterned liquid crystals,” Phys. Rev. Appl. 5(4), 044009 (2016).
[Crossref]

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

2015 (1)

Y. Ming, J. Tang, Z. Chen, F. Xu, L. Zhang, and Y. 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)

R. Fickler, R. Lapkiewicz, M. Huber, M. P. Lavery, M. J. Padgett, and A. Zeilinger, “Interface between path and orbital angular momentum entanglement for high-dimensional photonic quantum information,” Nat. Commun. 5(1), 4502 (2014).
[Crossref] [PubMed]

D. Tanese, E. Gurevich, F. Baboux, T. Jacqmin, A. Lemaître, E. Galopin, I. Sagnes, A. Amo, J. Bloch, and E. Akkermans, “Fractal energy spectrum of a polariton gas in a Fibonacci quasiperiodic potential,” Phys. Rev. Lett. 112(14), 146404 (2014).
[Crossref] [PubMed]

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

2013 (1)

R. Barboza, U. Bortolozzo, G. Assanto, E. Vidal-Henriquez, M. G. Clerc, and S. Residori, “Harnessing optical vortex lattices in nematic liquid crystals,” Phys. Rev. Lett. 111(9), 093902 (2013).
[Crossref] [PubMed]

2012 (1)

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] [PubMed]

2011 (2)

A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photonics 3(2), 161–204 (2011).
[Crossref]

A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, and E. Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys. 7(9), 677–680 (2011).
[Crossref]

2010 (1)

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

2007 (1)

W. Steurer and D. Sutter-Widmer, “Photonic and phononic quasicrystals,” J. Phys. D 40(13), R229–R247 (2007).
[Crossref]

2006 (1)

E. Maciá, “The role of aperiodic order in science and technology,” Rep. Prog. Phys. 69(2), 397–441 (2006).
[Crossref]

2005 (1)

M. A. Levin and X.-G. Wen, “String-net condensation: A physical mechanism for topological phases,” Phys. Rev. B Condens. Matter Mater. Phys. 71(4), 045110 (2005).
[Crossref]

2004 (1)

V. Chigrinov, S. Pikin, A. Verevochnikov, V. Kozenkov, M. Khazimullin, J. Ho, D. D. Huang, and H.-S. Kwok, “Diffusion model of photoaligning in azo-dye layers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69(6 Pt 1), 061713 (2004).
[Crossref] [PubMed]

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] [PubMed]

2001 (1)

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292(5518), 912–914 (2001).
[Crossref] [PubMed]

1997 (1)

S. Zhu, Y. Y. Zhu, and N. B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science 278(5339), 843–846 (1997).
[Crossref]

1994 (1)

W. Gellermann, M. Kohmoto, B. Sutherland, and P. C. Taylor, “Localization of light waves in Fibonacci dielectric multilayers,” Phys. Rev. Lett. 72(5), 633–636 (1994).
[Crossref] [PubMed]

1992 (3)

V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39(5), 985–990 (1992).
[Crossref]

S. Martin, S. Klaus, K. Vladimir, and C. Vladimir, “Surface-induced parallel alignment of liquid crystals by linearly polymerized photopolymers,” Jpn. J. Appl. Phys. 31(1), 2155–2164 (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] [PubMed]

1990 (1)

C. Godreche and J. M. Luck, “Multifractal analysis in reciprocal space and the nature of the Fourier transform of self-similar structures,” J. Phys. A 23(16), 3769–3797 (1990).
[Crossref]

1989 (1)

A. Chhabra and R. V. Jensen, “Direct determination of the f( α ) singularity spectrum,” Phys. Rev. Lett. 62(12), 1327–1330 (1989).
[Crossref] [PubMed]

1987 (1)

M. Kohmoto, B. Sutherland, and K. Iguchi, “Localization of optics: Quasiperiodic media,” Phys. Rev. Lett. 58(23), 2436–2438 (1987).
[Crossref] [PubMed]

1986 (2)

T. C. Halsey, M. H. Jensen, L. P. Kadanoff, I. Procaccia, and B. I. Shraiman, “Fractal measures and their singularities: The characterization of strange sets,” Phys. Rev. A Gen. Phys. 33(2), 1141–1151 (1986).
[Crossref] [PubMed]

T. Janssen, “Crystallography of quasi-crystals,” Acta Crystallogr. A 42(4), 261–271 (1986).
[Crossref]

1984 (2)

D. Levine and P. J. Steinhardt, “Quasicrystals: a new class of ordered structures,” Phys. Rev. Lett. 53(26), 2477–2480 (1984).
[Crossref]

D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, “Metallic phase with long-range orientational order and no translational symmetry,” Phys. Rev. Lett. 53(20), 1951–1953 (1984).
[Crossref]

1979 (1)

N. D. Mermin, “The topological theory of defects in ordered media,” Rev. Mod. Phys. 51(3), 591–648 (1979).
[Crossref]

1971 (1)

E. Noether, “Invariant variation problems,” Transp. Theory Stat. Phys. 1(3), 186–207 (1971).
[Crossref]

Akkermans, E.

D. Tanese, E. Gurevich, F. Baboux, T. Jacqmin, A. Lemaître, E. Galopin, I. Sagnes, A. Amo, J. Bloch, and E. Akkermans, “Fractal energy spectrum of a polariton gas in a Fibonacci quasiperiodic potential,” Phys. Rev. Lett. 112(14), 146404 (2014).
[Crossref] [PubMed]

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] [PubMed]

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]

Amo, A.

D. Tanese, E. Gurevich, F. Baboux, T. Jacqmin, A. Lemaître, E. Galopin, I. Sagnes, A. Amo, J. Bloch, and E. Akkermans, “Fractal energy spectrum of a polariton gas in a Fibonacci quasiperiodic potential,” Phys. Rev. Lett. 112(14), 146404 (2014).
[Crossref] [PubMed]

Andersson, E.

A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, and E. Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys. 7(9), 677–680 (2011).
[Crossref]

Arie, A.

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] [PubMed]

Arlt, J.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292(5518), 912–914 (2001).
[Crossref] [PubMed]

Assanto, G.

R. Barboza, U. Bortolozzo, G. Assanto, E. Vidal-Henriquez, M. G. Clerc, and S. Residori, “Harnessing optical vortex lattices in nematic liquid crystals,” Phys. Rev. Lett. 111(9), 093902 (2013).
[Crossref] [PubMed]

Baboux, F.

D. Tanese, E. Gurevich, F. Baboux, T. Jacqmin, A. Lemaître, E. Galopin, I. Sagnes, A. Amo, J. Bloch, and E. Akkermans, “Fractal energy spectrum of a polariton gas in a Fibonacci quasiperiodic potential,” Phys. Rev. Lett. 112(14), 146404 (2014).
[Crossref] [PubMed]

Barboza, R.

R. Barboza, U. Bortolozzo, G. Assanto, E. Vidal-Henriquez, M. G. Clerc, and S. Residori, “Harnessing optical vortex lattices in nematic liquid crystals,” Phys. Rev. Lett. 111(9), 093902 (2013).
[Crossref] [PubMed]

Bazhenov, V. Y.

V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39(5), 985–990 (1992).
[Crossref]

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]

Blech, I.

D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, “Metallic phase with long-range orientational order and no translational symmetry,” Phys. Rev. Lett. 53(20), 1951–1953 (1984).
[Crossref]

Bloch, J.

D. Tanese, E. Gurevich, F. Baboux, T. Jacqmin, A. Lemaître, E. Galopin, I. Sagnes, A. Amo, J. Bloch, and E. Akkermans, “Fractal energy spectrum of a polariton gas in a Fibonacci quasiperiodic potential,” Phys. Rev. Lett. 112(14), 146404 (2014).
[Crossref] [PubMed]

Bloch, N. V.

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] [PubMed]

Bortolozzo, U.

R. Barboza, U. Bortolozzo, G. Assanto, E. Vidal-Henriquez, M. G. Clerc, and S. Residori, “Harnessing optical vortex lattices in nematic liquid crystals,” Phys. Rev. Lett. 111(9), 093902 (2013).
[Crossref] [PubMed]

Bryant, P. E.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292(5518), 912–914 (2001).
[Crossref] [PubMed]

Buller, G. S.

A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, and E. Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys. 7(9), 677–680 (2011).
[Crossref]

Cahn, J. W.

D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, “Metallic phase with long-range orientational order and no translational symmetry,” Phys. Rev. Lett. 53(20), 1951–1953 (1984).
[Crossref]

Chen, J.

P. Chen, L.-L. Ma, W. Duan, J. Chen, S.-J. Ge, Z.-H. Zhu, M.-J. Tang, R. Xu, W. Gao, T. Li, W. Hu, and Y.-Q. Lu, “Digitalizing Self-Assembled Chiral Superstructures for Optical Vortex Processing,” Adv. Mater. 30(10), 1705865 (2018).
[Crossref] [PubMed]

Chen, P.

P. Chen, L.-L. Ma, W. Duan, J. Chen, S.-J. Ge, Z.-H. Zhu, M.-J. Tang, R. Xu, W. Gao, T. Li, W. Hu, and Y.-Q. Lu, “Digitalizing Self-Assembled Chiral Superstructures for Optical Vortex Processing,” Adv. Mater. 30(10), 1705865 (2018).
[Crossref] [PubMed]

P. Chen, S.-J. Ge, L.-L. Ma, W. Hu, V. Chigrinov, and Y.-Q. Lu, “Generation of equal-energy orbital angular momentum beams via photopatterned liquid crystals,” Phys. Rev. Appl. 5(4), 044009 (2016).
[Crossref]

Chen, Z.

Y. Ming, J. Tang, Z. Chen, F. Xu, L. Zhang, and Y. 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]

Chhabra, A.

A. Chhabra and R. V. Jensen, “Direct determination of the f( α ) singularity spectrum,” Phys. Rev. Lett. 62(12), 1327–1330 (1989).
[Crossref] [PubMed]

Chigrinov, V.

P. Chen, S.-J. Ge, L.-L. Ma, W. Hu, V. Chigrinov, and Y.-Q. Lu, “Generation of equal-energy orbital angular momentum beams via photopatterned liquid crystals,” Phys. Rev. Appl. 5(4), 044009 (2016).
[Crossref]

V. Chigrinov, S. Pikin, A. Verevochnikov, V. Kozenkov, M. Khazimullin, J. Ho, D. D. Huang, and H.-S. Kwok, “Diffusion model of photoaligning in azo-dye layers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69(6 Pt 1), 061713 (2004).
[Crossref] [PubMed]

Clerc, M. G.

R. Barboza, U. Bortolozzo, G. Assanto, E. Vidal-Henriquez, M. G. Clerc, and S. Residori, “Harnessing optical vortex lattices in nematic liquid crystals,” Phys. Rev. Lett. 111(9), 093902 (2013).
[Crossref] [PubMed]

Dada, A. C.

A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, and E. Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys. 7(9), 677–680 (2011).
[Crossref]

Dholakia, K.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292(5518), 912–914 (2001).
[Crossref] [PubMed]

Ding, D.-S.

Z.-Y. Zhou, S.-L. Liu, Y. Li, D.-S. Ding, W. Zhang, S. Shi, M.-X. Dong, B.-S. Shi, and G.-C. Guo, “Orbital angular momentum-entanglement frequency transducer,” Phys. Rev. Lett. 117(10), 103601 (2016).
[Crossref] [PubMed]

Dong, M.-X.

Z.-Y. Zhou, S.-L. Liu, Y. Li, D.-S. Ding, W. Zhang, S. Shi, M.-X. Dong, B.-S. Shi, and G.-C. Guo, “Orbital angular momentum-entanglement frequency transducer,” Phys. Rev. Lett. 117(10), 103601 (2016).
[Crossref] [PubMed]

Duan, W.

P. Chen, L.-L. Ma, W. Duan, J. Chen, S.-J. Ge, Z.-H. Zhu, M.-J. Tang, R. Xu, W. Gao, T. Li, W. Hu, and Y.-Q. Lu, “Digitalizing Self-Assembled Chiral Superstructures for Optical Vortex Processing,” Adv. Mater. 30(10), 1705865 (2018).
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Figures (5)

Fig. 1
Fig. 1 Structure and Fourier spectra for QPS with τ = ( 1 + 5 ) / 2 and l 1 = l 2 = 1 . (a) Structure of the QPS; (b) Self-similar Fourier spectrum of QPS for max ( k m n ) = 15 μ m 1 ; (c) Fourier spectrum of the low-order diffraction part, with some diffraction orders labeled; (d) Two-dimensional distribution of the Fourier spectrum; the magnitude of the Fourier coefficient is represented by a color bar.
Fig. 2
Fig. 2 Multifractal analysis of the Fourier spectrum for QPS. (a) Generalized dimension Dq; (b) Associated singular spectrum f(α). Maximum values of the Fourier coefficient subscript m (or n) and the order q for numerical calculation are 40 and 50, respectively.
Fig. 3
Fig. 3 Schematic illustration of experiment. A Gaussian beam emitted by a He-Ne laser incident on a reconfigurable photo-patterned liquid crystal (LC) cell. Optical singularities carrying OAM are generated and detected. Here we show the result of topological defects modulated with a QPS of l1 = l2 = 1.
Fig. 4
Fig. 4 Experimental and numerical distributions of diffracted optical singularities in QPS with various topological defects. The golden ratio ( 1 + 5 ) / 2 is picked as the irrational number in all QPS. (a, b) Numerical and experimental results for l1 = l2 = 1; (c, d) Numerical and experimental results for l1 = 1 and l2 = 2. Diffraction orders (m, n) are labeled by brackets with two indices; the carried OAM of optical singularities are indicated correspondingly.
Fig. 5
Fig. 5 Experimental and numerical distributions of diffracted optical singularities in CPS with various topological defects. (a, b) Fourier spectra for various CPS with τ = 2 and τ = 3/2, respectively; the degenerate axis is indicated by a green arrow; (c, e) Numerical and experimental results for an arbitrary topological defects doping case in CPS with τ = 2; (g, i) Numerical and experimental results under the specific condition l2 = τl1 in CPS with τ = 2; (d, f, h, j) Analogy results of CPS with τ = 3/2.

Equations (5)

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g ( x , ϕ ) = s i g n { cos ( 2 π Λ x + l 1 ϕ ) + cos ( 2 π τ Λ x + l 2 ϕ ) } ,
F x { g ( x , ϕ ) } = m , n sin c ( m + n 2 ) sin c ( n m 2 ) e i ( m l 1 + n l 2 ) ϕ δ ( k 2 π Λ ( m + n τ ) ) = m , n f m n e i l m n ϕ δ ( k k m n ) ,
l m n = m l 1 + n l 2 .
( m , m 1 ) and ( m 1 , m ) ,
γ = n n m m = 1 τ ,

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