Abstract

We investigate numerically the optical properties of a hexagonal half-Skyrmion lattice exhibited by a highly chiral liquid crystal confined between two parallel plates. Our study focuses on the near and far-field reflection for normally incident light with different polarizations. We show that, when the wavelength of the incident light is longer than a threshold value, the reflectivity is almost insensitive to the polarization of the incident light, although the intensity profiles of the reflected light, in particular in the near-field regime, depend significantly on the polarization. The former property is attributable to the quasi two-dimensional nature of the half-Skyrmion lattice, that is, almost uniform orientational order along the direction normal to the confining plates. Our results for the intensity of reflected light generated by evanescent as well as propagating contributions suggest that direct evidence of the formation and structure of half-Skyrmions could be provided by near-field optics with resolutions higher than that of conventional optical microscopy.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
  31. P. J. Ackerman, Z. Qi, and I. I. Smalyukh, “Optical generation of crystalline, quasicrystalline, and arbitrary arrays of torons in confined cholesteric liquid crystals for patterning of optical vortices in laser beams,” Phys. Rev. E 86, 021703 (2012).
    [Crossref]
  32. P. J. Ackerman, R. P. Trivedi, B. Senyuk, J. van de Lagemaat, and I. I. Smalyukh, “Two-dimensional skyrmions and other solitonic structures in confinement-frustrated chiral nematics,” Phys. Rev. E 90, 012505 (2014).
    [Crossref]
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    [Crossref]
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  49. One might feel it strange that the intensity is negative at some points in the profiles. However, the intensity of the evanescent contributions, when averaged over a two-dimensional plane, is zero, which means that its intensity can be positive and negative unless it is zero throughout the plane. Therefore, local “negative intensity” does not contradict any physics. Note also that the intensity averaged over the plane for given handedness of polarization is independent of the distance, and positive, as it should be.
  50. K. B. Dossou and L. C. Botten, “A combined three-dimensional finite element and scattering matrix method for the analysis of plane wave diffraction by bi-periodic, multilayered structures,” J. Comp. Phys. 231, 6969–6989 (2012).
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    [Crossref]

2017 (2)

S. Afghah and J. V. Selinger, “Theory of helicoids and skyrmions in confined cholesteric liquid crystals,” Phys. Rev. E 96, 012708 (2017).
[Crossref]

A. Nych, J.-i. Fukuda, U. Ognysta, S. Žumer, and I. Muševič, “Spontaneous formation and dynamics of half-skyrmions in a chiral liquid-crystal film,” Nature Physics 13, 1215–1220 (2017).
[Crossref]

2016 (2)

A. Tan, J. Li, A. Scholl, E. Arenholz, A. T. Young, Q. Li, C. Hwang, and Z. Q. Qiu, “Topology of spin meron pairs in coupled Ni/Fe/Co/Cu(001) disks,” Phys. Rev. B 94, 014433 (2016).
[Crossref]

J.-i. Fukuda, Y. Okumura, and H. Kikuchi, “Calculation of confocal microscope images of cholesteric blue phases,” Proc. SPIE 9769, 976906 (2016).
[Crossref]

2015 (2)

P. J. Ackerman, J. van de Lagemaat, and I. I. Smalyukh, “Self-assembly and electrostriction of arrays and chains of hopfion particles in chiral liquid crystals,” Nature Communications 6, 6012 (2015).
[Crossref] [PubMed]

W. Jiang, P. Upadhyaya, W. Zhang, G. Yu, M. B. Jungfleisch, F. Y. Fradin, J. E. Pearson, Y. Tserkovnyak, K. L. Wang, O. Heinonen, S. G. E. te Velthuis, and A. Hoffmann, “Blowing magnetic skyrmion bubbles,” Science 349, 283–286 (2015).
[Crossref] [PubMed]

2014 (4)

A. O. Leonov, I. E. Dragunov, U. K. Rößler, and A. N. Bogdanov, “Theory of skyrmion states in liquid crystals,” Phys. Rev. E 90, 042502 (2014).
[Crossref]

M. Pereiro, D. Yudin, J. Chico, C. Etz, O. Eriksson, and A. Bergman, “Topological excitations in a kagome magnet,” Nature Communications 5, 4815 (2014).
[Crossref] [PubMed]

J. Fukuda and S. Žumer, “Exotic Defect Structures and Their Optical Properties in a Strongly Confined Chiral Liquid Crystal,” Mol. Cryst. Liq. Cryst. 594, 70–77 (2014).
[Crossref]

P. J. Ackerman, R. P. Trivedi, B. Senyuk, J. van de Lagemaat, and I. I. Smalyukh, “Two-dimensional skyrmions and other solitonic structures in confinement-frustrated chiral nematics,” Phys. Rev. E 90, 012505 (2014).
[Crossref]

2013 (4)

V. C. Venugopal, “Three-dimensional periodic chiral sculptured thin films,” J. Nanophotonics 7, 073502 (2013).
[Crossref]

A. Fert, V. Cros, and J. Sampaio, “Skyrmions on the track,” Nature Nanotechnology 8, 152–156 (2013).
[Crossref] [PubMed]

N. Nagaosa and Y. Tokura, “Topological properties and dynamics of magnetic skyrmions,” Nature Nanotechnology 8, 899–911 (2013).
[Crossref] [PubMed]

N. Romming, C. Hanneken, M. Menzel, J. E. Bickel, B. Wolter, K. von Bergmann, A. Kubetzka, and R. Wiesendanger, “Writing and Deleting Single Magnetic Skyrmions,” Science 341, 636–639 (2013).
[Crossref] [PubMed]

2012 (2)

P. J. Ackerman, Z. Qi, and I. I. Smalyukh, “Optical generation of crystalline, quasicrystalline, and arbitrary arrays of torons in confined cholesteric liquid crystals for patterning of optical vortices in laser beams,” Phys. Rev. E 86, 021703 (2012).
[Crossref]

K. B. Dossou and L. C. Botten, “A combined three-dimensional finite element and scattering matrix method for the analysis of plane wave diffraction by bi-periodic, multilayered structures,” J. Comp. Phys. 231, 6969–6989 (2012).
[Crossref]

2011 (2)

M. Ezawa, “Compact merons and skyrmions in thin chiral magnetic films,” Phys. Rev. B 83, 100408 (2011).
[Crossref]

J. Fukuda and S. Žumer, “Quasi-two-dimensional Skyrmion lattices in a chiral nematic liquid crystal,” Nature Communications 2, 246 (2011).
[Crossref] [PubMed]

2010 (4)

I. I. Smalyukh, Y. Lansac, N. A. Clark, and R. P. Trivedi, “Three-dimensional structure and multistable optical switching of triple-twisted particle-like excitations in anisotropic fluids,” Nature Materials 9, 139–145 (2010).
[Crossref]

X. Yu, Y. Onose, N. Kanazawa, J. Park, J. Han, Y. Matsui, N. Nagaosa, and Y. Tokura, “Real-space observation of a two-dimensional skyrmion crystal,” Nature 465, 901–904 (2010).
[Crossref] [PubMed]

C. Pfleiderer and A. Rosch, “Condensed-matter physics: Single skyrmions spotted,” Nature 465, 880–881 (2010).
[Crossref] [PubMed]

O. Henrich, D. Marenduzzo, K. Stratford, and M. E. Cates, “Thermodynamics of blue phases in electric fields,” Phys. Rev. E 81, 031706 (2010).
[Crossref]

2009 (2)

S. Mühlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A. Rosch, A. Neubauer, R. Georgii, and P. Böni, “Skyrmion Lattice in a Chiral Magnet,” Science 323, 915–919 (2009).
[Crossref] [PubMed]

L. S. Leslie, A. Hansen, K. C. Wright, B. M. Deutsch, and N. P. Bigelow, “Creation and Detection of Skyrmions in a Bose-Einstein Condensate,” Phys. Rev. Lett. 103, 250401 (2009).
[Crossref]

2006 (1)

U. Rößler, A. Bogdanov, and C. Pfleiderer, “Spontaneous skyrmion ground states in magnetic metals,” Nature 442, 797–801 (2006).
[Crossref]

2004 (1)

T. A. Davis, “Algorithm 832: Umfpack v4.3—an unsymmetric-pattern multifrontal method,” ACM Trans. Math. Softw. 30, 196–199 (2004).
[Crossref]

2003 (2)

A. E. Leanhardt, Y. Shin, D. Kielpinski, D. E. Pritchard, and W. Ketterle, “Coreless vortex formation in a spinor bose-einstein condensate,” Phys. Rev. Lett. 90, 140403 (2003).
[Crossref] [PubMed]

A. N. Bogdanov, U. K. Rößler, and A. A. Shestakov, “Skyrmions in nematic liquid crystals,” Phys. Rev. E 67, 016602 (2003).
[Crossref]

2001 (1)

U. Al Khawaja and H. Stoof, “Skyrmions in a ferromagnetic bose-einstein condensate,” Nature 411, 918–920 (2001).
[Crossref] [PubMed]

1998 (1)

A. Bogdanov and A. Shestakov, “Inhomogeneous two-dimensional structures in liquid crystals,” J. Exp. Theo. Phys. 86, 911–923 (1998).
[Crossref]

1997 (1)

V. M. H. Ruutu, J. Kopu, M. Krusius, U. Parts, B. Plaçais, E. V. Thuneberg, and W. Xu, “Critical Velocity of Vortex Nucleation in Rotating Superfluid 3He- A,” Phys. Rev. Lett. 79, 5058–5061 (1997).
[Crossref]

1995 (2)

S. E. Barrett, G. Dabbagh, L. N. Pfeiffer, K. W. West, and R. Tycko, “Optically Pumped NMR Evidence for Finite-Size Skyrmions in GaAs Quantum Wells near Landau Level Filling ν = 1,” Phys. Rev. Lett. 74, 5112–5115 (1995).
[Crossref] [PubMed]

A. Schmeller, J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, “Evidence for Skyrmions and Single Spin Flips in the Integer Quantized Hall Effect,” Phys. Rev. Lett. 75, 4290–4293 (1995).
[Crossref] [PubMed]

1994 (1)

M. J. Bowick, L. Chandar, E. A. Schiff, and A. M. Srivastava, “The Cosmological Kibble Mechanism in the Laboratory: String Formation in Liquid Crystals,” Science 263, 943–945 (1994).
[Crossref] [PubMed]

1993 (1)

S. L. Sondhi, A. Karlhede, S. A. Kivelson, and E. H. Rezayi, “Skyrmions and the crossover from the integer to fractional quantum Hall effect at small Zeeman energies,” Phys. Rev. B 47, 16419–16426 (1993).
[Crossref]

1991 (1)

I. Chuang, R. Durrer, N. Turok, and B. Yurke, “Cosmology in the laboratory: defect dynamics in liquid crystals,” Science 251, 1336–1342 (1991).
[Crossref] [PubMed]

1990 (1)

1989 (3)

J. W. Goodby, M. Waugh, S. Stein, E. Chin, R. Pindak, and J. Patel, “Characterization of a new helical smectic liquid crystal,” Nature 337, 449–452 (1989).
[Crossref]

G. Heppke, B. Jéròme, H.-S. Kitzerow, and P. Pieranski, “Observation of a hexagonal blue phase in systems with negative dielectric anisotropy,” Liq. Cryst. 5, 813–828 (1989).
[Crossref]

R. M. Hornreich and S. Shtrikman, “Invited lecture. Hexagonal cholesteric blue phases,” Liq. Cryst. 5, 777–789 (1989).
[Crossref]

1988 (1)

S. R. Renn and T. C. Lubensky, “Abrikosov dislocation lattice in a model of the cholesteric–to–smectic- A transition,” Phys. Rev. A 38, 2132–2147 (1988).
[Crossref]

1978 (1)

C. G. Callan, R. Dashen, and D. J. Gross, “Toward a theory of the strong interactions,” Phys. Rev. D 17, 2717–2763 (1978).
[Crossref]

1977 (1)

P. W. Anderson and G. Toulouse, “Phase Slippage without Vortex Cores: Vortex Textures in Superfluid 3He,” Phys. Rev. Lett. 38, 508–511 (1977).
[Crossref]

1976 (1)

N. D. Mermin and T.-L. Ho, “Circulation and Angular Momentum in the A Phase of Superfluid Helium-3,” Phys. Rev. Lett. 36, 594–597 (1976).
[Crossref]

1961 (1)

T. H. R. Skyrme, “A Non-Linear Field Theory,” Proc. Royal Soc. London A 260, 127–138 (1961).
[Crossref]

1957 (1)

A. Abrikosov, “Magnetic properties of superconductors of the second group,” Sov. Phys. JETP 5, 1774–1182 (1957).

Abrikosov, A.

A. Abrikosov, “Magnetic properties of superconductors of the second group,” Sov. Phys. JETP 5, 1774–1182 (1957).

Ackerman, P. J.

P. J. Ackerman, J. van de Lagemaat, and I. I. Smalyukh, “Self-assembly and electrostriction of arrays and chains of hopfion particles in chiral liquid crystals,” Nature Communications 6, 6012 (2015).
[Crossref] [PubMed]

P. J. Ackerman, R. P. Trivedi, B. Senyuk, J. van de Lagemaat, and I. I. Smalyukh, “Two-dimensional skyrmions and other solitonic structures in confinement-frustrated chiral nematics,” Phys. Rev. E 90, 012505 (2014).
[Crossref]

P. J. Ackerman, Z. Qi, and I. I. Smalyukh, “Optical generation of crystalline, quasicrystalline, and arbitrary arrays of torons in confined cholesteric liquid crystals for patterning of optical vortices in laser beams,” Phys. Rev. E 86, 021703 (2012).
[Crossref]

Afghah, S.

S. Afghah and J. V. Selinger, “Theory of helicoids and skyrmions in confined cholesteric liquid crystals,” Phys. Rev. E 96, 012708 (2017).
[Crossref]

Al Khawaja, U.

U. Al Khawaja and H. Stoof, “Skyrmions in a ferromagnetic bose-einstein condensate,” Nature 411, 918–920 (2001).
[Crossref] [PubMed]

Anderson, P. W.

P. W. Anderson and G. Toulouse, “Phase Slippage without Vortex Cores: Vortex Textures in Superfluid 3He,” Phys. Rev. Lett. 38, 508–511 (1977).
[Crossref]

Arenholz, E.

A. Tan, J. Li, A. Scholl, E. Arenholz, A. T. Young, Q. Li, C. Hwang, and Z. Q. Qiu, “Topology of spin meron pairs in coupled Ni/Fe/Co/Cu(001) disks,” Phys. Rev. B 94, 014433 (2016).
[Crossref]

Bagby, J. S.

Barrett, S. E.

S. E. Barrett, G. Dabbagh, L. N. Pfeiffer, K. W. West, and R. Tycko, “Optically Pumped NMR Evidence for Finite-Size Skyrmions in GaAs Quantum Wells near Landau Level Filling ν = 1,” Phys. Rev. Lett. 74, 5112–5115 (1995).
[Crossref] [PubMed]

Bergman, A.

M. Pereiro, D. Yudin, J. Chico, C. Etz, O. Eriksson, and A. Bergman, “Topological excitations in a kagome magnet,” Nature Communications 5, 4815 (2014).
[Crossref] [PubMed]

Bickel, J. E.

N. Romming, C. Hanneken, M. Menzel, J. E. Bickel, B. Wolter, K. von Bergmann, A. Kubetzka, and R. Wiesendanger, “Writing and Deleting Single Magnetic Skyrmions,” Science 341, 636–639 (2013).
[Crossref] [PubMed]

Bigelow, N. P.

L. S. Leslie, A. Hansen, K. C. Wright, B. M. Deutsch, and N. P. Bigelow, “Creation and Detection of Skyrmions in a Bose-Einstein Condensate,” Phys. Rev. Lett. 103, 250401 (2009).
[Crossref]

Binz, B.

S. Mühlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A. Rosch, A. Neubauer, R. Georgii, and P. Böni, “Skyrmion Lattice in a Chiral Magnet,” Science 323, 915–919 (2009).
[Crossref] [PubMed]

Bogdanov, A.

U. Rößler, A. Bogdanov, and C. Pfleiderer, “Spontaneous skyrmion ground states in magnetic metals,” Nature 442, 797–801 (2006).
[Crossref]

A. Bogdanov and A. Shestakov, “Inhomogeneous two-dimensional structures in liquid crystals,” J. Exp. Theo. Phys. 86, 911–923 (1998).
[Crossref]

Bogdanov, A. N.

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S. Mühlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A. Rosch, A. Neubauer, R. Georgii, and P. Böni, “Skyrmion Lattice in a Chiral Magnet,” Science 323, 915–919 (2009).
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U. Rößler, A. Bogdanov, and C. Pfleiderer, “Spontaneous skyrmion ground states in magnetic metals,” Nature 442, 797–801 (2006).
[Crossref]

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A. O. Leonov, I. E. Dragunov, U. K. Rößler, and A. N. Bogdanov, “Theory of skyrmion states in liquid crystals,” Phys. Rev. E 90, 042502 (2014).
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A. N. Bogdanov, U. K. Rößler, and A. A. Shestakov, “Skyrmions in nematic liquid crystals,” Phys. Rev. E 67, 016602 (2003).
[Crossref]

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V. M. H. Ruutu, J. Kopu, M. Krusius, U. Parts, B. Plaçais, E. V. Thuneberg, and W. Xu, “Critical Velocity of Vortex Nucleation in Rotating Superfluid 3He- A,” Phys. Rev. Lett. 79, 5058–5061 (1997).
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A. Fert, V. Cros, and J. Sampaio, “Skyrmions on the track,” Nature Nanotechnology 8, 152–156 (2013).
[Crossref] [PubMed]

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M. J. Bowick, L. Chandar, E. A. Schiff, and A. M. Srivastava, “The Cosmological Kibble Mechanism in the Laboratory: String Formation in Liquid Crystals,” Science 263, 943–945 (1994).
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A. Schmeller, J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, “Evidence for Skyrmions and Single Spin Flips in the Integer Quantized Hall Effect,” Phys. Rev. Lett. 75, 4290–4293 (1995).
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A. Tan, J. Li, A. Scholl, E. Arenholz, A. T. Young, Q. Li, C. Hwang, and Z. Q. Qiu, “Topology of spin meron pairs in coupled Ni/Fe/Co/Cu(001) disks,” Phys. Rev. B 94, 014433 (2016).
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S. Afghah and J. V. Selinger, “Theory of helicoids and skyrmions in confined cholesteric liquid crystals,” Phys. Rev. E 96, 012708 (2017).
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P. J. Ackerman, R. P. Trivedi, B. Senyuk, J. van de Lagemaat, and I. I. Smalyukh, “Two-dimensional skyrmions and other solitonic structures in confinement-frustrated chiral nematics,” Phys. Rev. E 90, 012505 (2014).
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A. Bogdanov and A. Shestakov, “Inhomogeneous two-dimensional structures in liquid crystals,” J. Exp. Theo. Phys. 86, 911–923 (1998).
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A. N. Bogdanov, U. K. Rößler, and A. A. Shestakov, “Skyrmions in nematic liquid crystals,” Phys. Rev. E 67, 016602 (2003).
[Crossref]

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A. E. Leanhardt, Y. Shin, D. Kielpinski, D. E. Pritchard, and W. Ketterle, “Coreless vortex formation in a spinor bose-einstein condensate,” Phys. Rev. Lett. 90, 140403 (2003).
[Crossref] [PubMed]

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R. M. Hornreich and S. Shtrikman, “Invited lecture. Hexagonal cholesteric blue phases,” Liq. Cryst. 5, 777–789 (1989).
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T. H. R. Skyrme, “A Non-Linear Field Theory,” Proc. Royal Soc. London A 260, 127–138 (1961).
[Crossref]

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P. J. Ackerman, J. van de Lagemaat, and I. I. Smalyukh, “Self-assembly and electrostriction of arrays and chains of hopfion particles in chiral liquid crystals,” Nature Communications 6, 6012 (2015).
[Crossref] [PubMed]

P. J. Ackerman, R. P. Trivedi, B. Senyuk, J. van de Lagemaat, and I. I. Smalyukh, “Two-dimensional skyrmions and other solitonic structures in confinement-frustrated chiral nematics,” Phys. Rev. E 90, 012505 (2014).
[Crossref]

P. J. Ackerman, Z. Qi, and I. I. Smalyukh, “Optical generation of crystalline, quasicrystalline, and arbitrary arrays of torons in confined cholesteric liquid crystals for patterning of optical vortices in laser beams,” Phys. Rev. E 86, 021703 (2012).
[Crossref]

I. I. Smalyukh, Y. Lansac, N. A. Clark, and R. P. Trivedi, “Three-dimensional structure and multistable optical switching of triple-twisted particle-like excitations in anisotropic fluids,” Nature Materials 9, 139–145 (2010).
[Crossref]

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S. L. Sondhi, A. Karlhede, S. A. Kivelson, and E. H. Rezayi, “Skyrmions and the crossover from the integer to fractional quantum Hall effect at small Zeeman energies,” Phys. Rev. B 47, 16419–16426 (1993).
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M. J. Bowick, L. Chandar, E. A. Schiff, and A. M. Srivastava, “The Cosmological Kibble Mechanism in the Laboratory: String Formation in Liquid Crystals,” Science 263, 943–945 (1994).
[Crossref] [PubMed]

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J. W. Goodby, M. Waugh, S. Stein, E. Chin, R. Pindak, and J. Patel, “Characterization of a new helical smectic liquid crystal,” Nature 337, 449–452 (1989).
[Crossref]

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U. Al Khawaja and H. Stoof, “Skyrmions in a ferromagnetic bose-einstein condensate,” Nature 411, 918–920 (2001).
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O. Henrich, D. Marenduzzo, K. Stratford, and M. E. Cates, “Thermodynamics of blue phases in electric fields,” Phys. Rev. E 81, 031706 (2010).
[Crossref]

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A. Tan, J. Li, A. Scholl, E. Arenholz, A. T. Young, Q. Li, C. Hwang, and Z. Q. Qiu, “Topology of spin meron pairs in coupled Ni/Fe/Co/Cu(001) disks,” Phys. Rev. B 94, 014433 (2016).
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W. Jiang, P. Upadhyaya, W. Zhang, G. Yu, M. B. Jungfleisch, F. Y. Fradin, J. E. Pearson, Y. Tserkovnyak, K. L. Wang, O. Heinonen, S. G. E. te Velthuis, and A. Hoffmann, “Blowing magnetic skyrmion bubbles,” Science 349, 283–286 (2015).
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V. M. H. Ruutu, J. Kopu, M. Krusius, U. Parts, B. Plaçais, E. V. Thuneberg, and W. Xu, “Critical Velocity of Vortex Nucleation in Rotating Superfluid 3He- A,” Phys. Rev. Lett. 79, 5058–5061 (1997).
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N. Nagaosa and Y. Tokura, “Topological properties and dynamics of magnetic skyrmions,” Nature Nanotechnology 8, 899–911 (2013).
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X. Yu, Y. Onose, N. Kanazawa, J. Park, J. Han, Y. Matsui, N. Nagaosa, and Y. Tokura, “Real-space observation of a two-dimensional skyrmion crystal,” Nature 465, 901–904 (2010).
[Crossref] [PubMed]

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P. W. Anderson and G. Toulouse, “Phase Slippage without Vortex Cores: Vortex Textures in Superfluid 3He,” Phys. Rev. Lett. 38, 508–511 (1977).
[Crossref]

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P. J. Ackerman, R. P. Trivedi, B. Senyuk, J. van de Lagemaat, and I. I. Smalyukh, “Two-dimensional skyrmions and other solitonic structures in confinement-frustrated chiral nematics,” Phys. Rev. E 90, 012505 (2014).
[Crossref]

I. I. Smalyukh, Y. Lansac, N. A. Clark, and R. P. Trivedi, “Three-dimensional structure and multistable optical switching of triple-twisted particle-like excitations in anisotropic fluids,” Nature Materials 9, 139–145 (2010).
[Crossref]

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W. Jiang, P. Upadhyaya, W. Zhang, G. Yu, M. B. Jungfleisch, F. Y. Fradin, J. E. Pearson, Y. Tserkovnyak, K. L. Wang, O. Heinonen, S. G. E. te Velthuis, and A. Hoffmann, “Blowing magnetic skyrmion bubbles,” Science 349, 283–286 (2015).
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S. E. Barrett, G. Dabbagh, L. N. Pfeiffer, K. W. West, and R. Tycko, “Optically Pumped NMR Evidence for Finite-Size Skyrmions in GaAs Quantum Wells near Landau Level Filling ν = 1,” Phys. Rev. Lett. 74, 5112–5115 (1995).
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W. Jiang, P. Upadhyaya, W. Zhang, G. Yu, M. B. Jungfleisch, F. Y. Fradin, J. E. Pearson, Y. Tserkovnyak, K. L. Wang, O. Heinonen, S. G. E. te Velthuis, and A. Hoffmann, “Blowing magnetic skyrmion bubbles,” Science 349, 283–286 (2015).
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P. J. Ackerman, J. van de Lagemaat, and I. I. Smalyukh, “Self-assembly and electrostriction of arrays and chains of hopfion particles in chiral liquid crystals,” Nature Communications 6, 6012 (2015).
[Crossref] [PubMed]

P. J. Ackerman, R. P. Trivedi, B. Senyuk, J. van de Lagemaat, and I. I. Smalyukh, “Two-dimensional skyrmions and other solitonic structures in confinement-frustrated chiral nematics,” Phys. Rev. E 90, 012505 (2014).
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W. Jiang, P. Upadhyaya, W. Zhang, G. Yu, M. B. Jungfleisch, F. Y. Fradin, J. E. Pearson, Y. Tserkovnyak, K. L. Wang, O. Heinonen, S. G. E. te Velthuis, and A. Hoffmann, “Blowing magnetic skyrmion bubbles,” Science 349, 283–286 (2015).
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Waugh, M.

J. W. Goodby, M. Waugh, S. Stein, E. Chin, R. Pindak, and J. Patel, “Characterization of a new helical smectic liquid crystal,” Nature 337, 449–452 (1989).
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A. Schmeller, J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, “Evidence for Skyrmions and Single Spin Flips in the Integer Quantized Hall Effect,” Phys. Rev. Lett. 75, 4290–4293 (1995).
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N. Romming, C. Hanneken, M. Menzel, J. E. Bickel, B. Wolter, K. von Bergmann, A. Kubetzka, and R. Wiesendanger, “Writing and Deleting Single Magnetic Skyrmions,” Science 341, 636–639 (2013).
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N. Romming, C. Hanneken, M. Menzel, J. E. Bickel, B. Wolter, K. von Bergmann, A. Kubetzka, and R. Wiesendanger, “Writing and Deleting Single Magnetic Skyrmions,” Science 341, 636–639 (2013).
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V. M. H. Ruutu, J. Kopu, M. Krusius, U. Parts, B. Plaçais, E. V. Thuneberg, and W. Xu, “Critical Velocity of Vortex Nucleation in Rotating Superfluid 3He- A,” Phys. Rev. Lett. 79, 5058–5061 (1997).
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A. Tan, J. Li, A. Scholl, E. Arenholz, A. T. Young, Q. Li, C. Hwang, and Z. Q. Qiu, “Topology of spin meron pairs in coupled Ni/Fe/Co/Cu(001) disks,” Phys. Rev. B 94, 014433 (2016).
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W. Jiang, P. Upadhyaya, W. Zhang, G. Yu, M. B. Jungfleisch, F. Y. Fradin, J. E. Pearson, Y. Tserkovnyak, K. L. Wang, O. Heinonen, S. G. E. te Velthuis, and A. Hoffmann, “Blowing magnetic skyrmion bubbles,” Science 349, 283–286 (2015).
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X. Yu, Y. Onose, N. Kanazawa, J. Park, J. Han, Y. Matsui, N. Nagaosa, and Y. Tokura, “Real-space observation of a two-dimensional skyrmion crystal,” Nature 465, 901–904 (2010).
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I. Chuang, R. Durrer, N. Turok, and B. Yurke, “Cosmology in the laboratory: defect dynamics in liquid crystals,” Science 251, 1336–1342 (1991).
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W. Jiang, P. Upadhyaya, W. Zhang, G. Yu, M. B. Jungfleisch, F. Y. Fradin, J. E. Pearson, Y. Tserkovnyak, K. L. Wang, O. Heinonen, S. G. E. te Velthuis, and A. Hoffmann, “Blowing magnetic skyrmion bubbles,” Science 349, 283–286 (2015).
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A. Nych, J.-i. Fukuda, U. Ognysta, S. Žumer, and I. Muševič, “Spontaneous formation and dynamics of half-skyrmions in a chiral liquid-crystal film,” Nature Physics 13, 1215–1220 (2017).
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J. Fukuda and S. Žumer, “Quasi-two-dimensional Skyrmion lattices in a chiral nematic liquid crystal,” Nature Communications 2, 246 (2011).
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Nature (5)

X. Yu, Y. Onose, N. Kanazawa, J. Park, J. Han, Y. Matsui, N. Nagaosa, and Y. Tokura, “Real-space observation of a two-dimensional skyrmion crystal,” Nature 465, 901–904 (2010).
[Crossref] [PubMed]

C. Pfleiderer and A. Rosch, “Condensed-matter physics: Single skyrmions spotted,” Nature 465, 880–881 (2010).
[Crossref] [PubMed]

J. W. Goodby, M. Waugh, S. Stein, E. Chin, R. Pindak, and J. Patel, “Characterization of a new helical smectic liquid crystal,” Nature 337, 449–452 (1989).
[Crossref]

U. Al Khawaja and H. Stoof, “Skyrmions in a ferromagnetic bose-einstein condensate,” Nature 411, 918–920 (2001).
[Crossref] [PubMed]

U. Rößler, A. Bogdanov, and C. Pfleiderer, “Spontaneous skyrmion ground states in magnetic metals,” Nature 442, 797–801 (2006).
[Crossref]

Nature Communications (3)

P. J. Ackerman, J. van de Lagemaat, and I. I. Smalyukh, “Self-assembly and electrostriction of arrays and chains of hopfion particles in chiral liquid crystals,” Nature Communications 6, 6012 (2015).
[Crossref] [PubMed]

J. Fukuda and S. Žumer, “Quasi-two-dimensional Skyrmion lattices in a chiral nematic liquid crystal,” Nature Communications 2, 246 (2011).
[Crossref] [PubMed]

M. Pereiro, D. Yudin, J. Chico, C. Etz, O. Eriksson, and A. Bergman, “Topological excitations in a kagome magnet,” Nature Communications 5, 4815 (2014).
[Crossref] [PubMed]

Nature Materials (1)

I. I. Smalyukh, Y. Lansac, N. A. Clark, and R. P. Trivedi, “Three-dimensional structure and multistable optical switching of triple-twisted particle-like excitations in anisotropic fluids,” Nature Materials 9, 139–145 (2010).
[Crossref]

Nature Nanotechnology (2)

A. Fert, V. Cros, and J. Sampaio, “Skyrmions on the track,” Nature Nanotechnology 8, 152–156 (2013).
[Crossref] [PubMed]

N. Nagaosa and Y. Tokura, “Topological properties and dynamics of magnetic skyrmions,” Nature Nanotechnology 8, 899–911 (2013).
[Crossref] [PubMed]

Nature Physics (1)

A. Nych, J.-i. Fukuda, U. Ognysta, S. Žumer, and I. Muševič, “Spontaneous formation and dynamics of half-skyrmions in a chiral liquid-crystal film,” Nature Physics 13, 1215–1220 (2017).
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O. Henrich, D. Marenduzzo, K. Stratford, and M. E. Cates, “Thermodynamics of blue phases in electric fields,” Phys. Rev. E 81, 031706 (2010).
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P. J. Ackerman, Z. Qi, and I. I. Smalyukh, “Optical generation of crystalline, quasicrystalline, and arbitrary arrays of torons in confined cholesteric liquid crystals for patterning of optical vortices in laser beams,” Phys. Rev. E 86, 021703 (2012).
[Crossref]

P. J. Ackerman, R. P. Trivedi, B. Senyuk, J. van de Lagemaat, and I. I. Smalyukh, “Two-dimensional skyrmions and other solitonic structures in confinement-frustrated chiral nematics,” Phys. Rev. E 90, 012505 (2014).
[Crossref]

A. N. Bogdanov, U. K. Rößler, and A. A. Shestakov, “Skyrmions in nematic liquid crystals,” Phys. Rev. E 67, 016602 (2003).
[Crossref]

A. O. Leonov, I. E. Dragunov, U. K. Rößler, and A. N. Bogdanov, “Theory of skyrmion states in liquid crystals,” Phys. Rev. E 90, 042502 (2014).
[Crossref]

S. Afghah and J. V. Selinger, “Theory of helicoids and skyrmions in confined cholesteric liquid crystals,” Phys. Rev. E 96, 012708 (2017).
[Crossref]

Phys. Rev. Lett. (7)

S. E. Barrett, G. Dabbagh, L. N. Pfeiffer, K. W. West, and R. Tycko, “Optically Pumped NMR Evidence for Finite-Size Skyrmions in GaAs Quantum Wells near Landau Level Filling ν = 1,” Phys. Rev. Lett. 74, 5112–5115 (1995).
[Crossref] [PubMed]

A. Schmeller, J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, “Evidence for Skyrmions and Single Spin Flips in the Integer Quantized Hall Effect,” Phys. Rev. Lett. 75, 4290–4293 (1995).
[Crossref] [PubMed]

A. E. Leanhardt, Y. Shin, D. Kielpinski, D. E. Pritchard, and W. Ketterle, “Coreless vortex formation in a spinor bose-einstein condensate,” Phys. Rev. Lett. 90, 140403 (2003).
[Crossref] [PubMed]

L. S. Leslie, A. Hansen, K. C. Wright, B. M. Deutsch, and N. P. Bigelow, “Creation and Detection of Skyrmions in a Bose-Einstein Condensate,” Phys. Rev. Lett. 103, 250401 (2009).
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V. M. H. Ruutu, J. Kopu, M. Krusius, U. Parts, B. Plaçais, E. V. Thuneberg, and W. Xu, “Critical Velocity of Vortex Nucleation in Rotating Superfluid 3He- A,” Phys. Rev. Lett. 79, 5058–5061 (1997).
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M. J. Bowick, L. Chandar, E. A. Schiff, and A. M. Srivastava, “The Cosmological Kibble Mechanism in the Laboratory: String Formation in Liquid Crystals,” Science 263, 943–945 (1994).
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S. Mühlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A. Rosch, A. Neubauer, R. Georgii, and P. Böni, “Skyrmion Lattice in a Chiral Magnet,” Science 323, 915–919 (2009).
[Crossref] [PubMed]

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[Crossref] [PubMed]

W. Jiang, P. Upadhyaya, W. Zhang, G. Yu, M. B. Jungfleisch, F. Y. Fradin, J. E. Pearson, Y. Tserkovnyak, K. L. Wang, O. Heinonen, S. G. E. te Velthuis, and A. Hoffmann, “Blowing magnetic skyrmion bubbles,” Science 349, 283–286 (2015).
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One might feel it strange that the intensity is negative at some points in the profiles. However, the intensity of the evanescent contributions, when averaged over a two-dimensional plane, is zero, which means that its intensity can be positive and negative unless it is zero throughout the plane. Therefore, local “negative intensity” does not contradict any physics. Note also that the intensity averaged over the plane for given handedness of polarization is independent of the distance, and positive, as it should be.

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Figures (5)

Fig. 1
Fig. 1 Top (top row) and side (bottom row) view of (a) a hexagonal half-Skyrmion lattice, and (b) a different structure with three-fold rotational symmetry. Red surfaces represent the location of topological defects of orientational order (Topological defects near the confining surfaces are not shown for the clarity of the top views). Short gray rods show the orientational order of the liquid crystal.
Fig. 2
Fig. 2 Reflectivity of our hexagonal half-Skyrmion lattice [Fig. 1(a)] as a function of λ for different polarizations of the incident light.
Fig. 3
Fig. 3 (a–f) Intensity profiles of reflected light for incident light with λ = 0.953p and (a–c) left-circular polarization and (d–f) right-circular polarization, for our hexagonal lattice of half-Skyrmions [Fig. 1(a)]. The planes are located at z = (a,d) −0.0796p, (b,e) −0.796p and (c,f) −7.96p. These intensity profiles are viewed from the light source (z = −∞).
Fig. 4
Fig. 4 (a–f) Intensity profiles of reflected light for incident light with λ = 0.896p and (a–c) left-circular polarization and (d–f) right-circular polarization, for our hexagonal lattice of half-Skyrmions [Fig. 1(a)]. The planes are located at (a,d) −0.0796p, (b,e) −0.796p and (c,f) −7.96p.
Fig. 5
Fig. 5 (a–f) Intensity profiles of reflected light for the structure shown in Fig. 1(b), for incident light with λ = 0.944p and (a–c) left-circular polarization and (d–f) right-circular polarization. The planes are located at z = (a,d) −0.0796p, (b,e) −0.796p and (c,f) −7.96p. (g) Reflectivity as a function of λ for different polarizations of the incident light.

Equations (6)

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f local { Q α β } = c ( T T * ) Q α β Q α β 6 b Q α β Q β γ Q γ α + a ( Q α β Q α β ) 2 ,
f el { Q α β , γ } = 1 4 K 1 [ α γ δ γ Q δ β + 2 q 0 Q α β ] 2 + 1 4 K 0 [ β Q α β ] 2 ,
f s { Q α β } = 1 2 W ( Q Q s ) α β 2 ,
× × E ( ω / c ) 2 E = 0 ,
E ( r ) = { E i exp ( i k i r ) + m , n E r ( m , n ) exp ( i k r ( m , n ) r ) with k r ( m , n ) k + G ( m , n ) + k rz ( m , n ) z ^ ( z < 0 ) , m , n E ( m , n ) ( z ) exp ( i [ k + G ( m , n ) ] r ) ( 0 z d ) , m , n E t ( m , n ) exp ( i k t ( m , n ) r ) with k t ( m , n ) k + G ( m , n ) + k r t ( m , n ) z ^ ( z > d ) .
tan κ d = 2 κ γ κ 2 γ 2 ( for the transverse electric ( TE ) mode ) tan κ d = 2 n LC 2 glass κ γ glass 2 κ 2 n LC 4 γ 2 ( for the transverse magnetic ( TM ) mode )

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