Abstract

Periodic birefringence is today extensively explored as an interesting route for controlling the flow of light. Distinctly, complex fluids with periodic modulations of birefringence can perform as photonic crystals, with the main examples being cholesteric and blue phases birefringent profiles. Here we demonstrate the characteristics of light propagation in heliconical liquid crystal and demonstrate their tunable optical and photonic properties, specifically as one-dimensional photonic crystals, in the regime of heliconical pitch comparable to the wavelength of light. Using a combination of frequency- and time-domain simulations, we show the existence and properties of the photonic band gap, as determined by the relative handedness of the polarization of light and the heliconical structure. We calculate photonic eigenmodes of the light and find the emergence of electric field component along the propagation axis of light, for both left- and right-handed polarization of light, which in turn results in strongly spatially varying Poynting vector that exhibits circular-like and four-leaf-clover-like patterns. As this variation of the Poynting vector is tunable with various material parameters and external (electric) fields, heliconical birefringence photonic crystals show interesting potential for use in tunable photonic applications, such as complex modulation of light beams.

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

Full Article  |  PDF Article
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2018 (2)

2017 (4)

A. Bregar, T. J. White, and M. Ravnik, “Refraction of light on flat boundary of liquid crystals or anisotropic metamaterials,” Liq. Cryst. Rev. 5, 53–68 (2017).
[Crossref]

J. Kobashi, H. Yoshida, and M. Ozaki, “Circularly-polarized, semitransparent and double-sided holograms based on helical photonic structures,” Sci. Rep. 7, 016470 (2017).
[Crossref]

G. Tan, Y.-H. Lee, F. Gou, H. Chen, Y. Huang, Y.-F. Lan, C.-Y. Tsai, and S.-T. Wu, “Review on polymer-stabilized short-pitch cholesteric liquid crystal displays,” J. Phys. D: Appl. Phys. 50, 493001 (2017).
[Crossref]

Y. Huang, H. Chen, G. Tan, H. Tobata, S. ichi Yamamoto, E. Okabe, Y.-F. Lan, C.-Y. Tsai, and S.-T. Wu, “Optimized blue-phase liquid crystal for field-sequential-color displays,” Opt. Mater. Express 7, 641–650 (2017).
[Crossref]

2016 (8)

A. Jullien, U. Bortolozzo, S. Grabielle, J.-P. Huignard, N. Forget, and S. Residori, “Continuously tunable femtosecond delay-line based on liquid crystal cells,” Opt. Express 24, 14483–14493 (2016).
[Crossref] [PubMed]

M. Wahle, J. Ebel, D. Wilkes, and H.-S. Kitzerow, “Asymmetric band gap shift in electrically addressed blue phase photonic crystal fibers,” Opt. Express 24, 22718–22729 (2016).
[Crossref] [PubMed]

I. Muševič, “Liquid-crystal micro-photonics,” Liq. Cryst. Rev. 4, 1–34 (2016).
[Crossref]

J. Xiang, A. Varanytsia, F. Minkowski, D. A. Paterson, J. M. D. Storey, C. T. Imrie, O. D. Lavrentovich, and P. Palffy-Muhoray, “Electrically tunable laser based on oblique heliconical cholesteric liquid crystal,” Proc. Natl. Acad. Sci. 113, 12925–12928 (2016).
[Crossref] [PubMed]

J. Kobashi, H. Yoshida, and M. Ozaki, “Planar optics with patterned chiral liquid crystals,” Nat. Photon. 10, 389–392 (2016).
[Crossref]

R. Barboza, U. Bortolozzo, M. G. Clerc, and S. Residori, “Berry phase of light under bragg reflection by chiral liquid-crystal media,” Phys. Rev. Lett. 117, 053903 (2016).
[Crossref] [PubMed]

J. Aplinc, M. Štimulak, S. Čopar, and M. Ravnik, “Nematic liquid crystal gyroids as photonic crystals,” Liq. Cryst. 43, 2320–2331 (2016).
[Crossref]

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

2015 (3)

J. Xiang, Y. Li, Q. Li, D. A. Paterson, J. M. D. Storey, C. T. Imrie, and O. D. Lavrentovich, “Electrically tunable selective reflection of light from ultraviolet to visible and infrared by heliconical cholesterics,” Adv. Mater. 27, 3014–3018 (2015).
[Crossref] [PubMed]

M. Rybin, I. Shishkin, K. Samusev, P. Belov, Y. Kivshar, R. Kiyan, B. Chichkov, and M. Limonov, “Band structure of photonic crystals fabricated by two-photon polymerization,” Crystals 5, 61–73 (2015).
[Crossref]

N. Wang, J. S. Evans, Q. Liu, S. Wang, I.-C. Khoo, and S. He, “Electrically controllable self-assembly for radial alignment of gold nanorods in liquid crystal droplets,” Opt. Mater. Express 5, 1065–1070 (2015).
[Crossref]

2014 (5)

J. Xiang, S. V. Shiyanovskii, C. Imrie, and O. D. Lavrentovich, “Electrooptic response of chiral nematic liquid crystals with oblique helicoidal director,” Phys. Rev. Lett. 112, 217801 (2014).
[Crossref]

M. Čančula, M. Ravnik, and S. Žumer, “Generation of vector beams with liquid crystal disclination lines,” Phys. Rev. E 90, 022503 (2014).
[Crossref]

D. Chen, M. Nakata, R. Shao, M. R. Tuchband, M. Shuai, U. Baumeister, W. Weissflog, D. M. Walba, M. A. Glaser, J. E. Maclennan, and N. A. Clark, “Twist-bend heliconical chiral nematic liquid crystal phase of an achiral rigid bent-core mesogen,” Phys. Rev. E 89, 022506 (2014).
[Crossref]

I. C. Khoo, “Nonlinear optics, active plasmonics and metamaterials with liquid crystals,” Prog. Quantum Electron. 38, 77–117 (2014).
[Crossref]

M. Štimulak and M. Ravnik, “Tunable photonic crystals with partial bandgaps from blue phase colloidal crystals and dielectric-doped blue phases,” Soft Matter 10, 6339–6346 (2014).
[Crossref] [PubMed]

2013 (2)

D. Chen, J. H. Porada, J. B. Hooper, A. Klittnick, Y. Shen, M. R. Tuchband, E. Korblova, D. Bedrov, D. M. Walba, M. A. Glaser, J. E. Maclennan, and N. A. Clark, “Chiral heliconical ground state of nanoscale pitch in a nematic liquid crystal of achiral molecular dimers,” Proc. Natl. Acad. Sci. 110, 15931–15936 (2013).
[Crossref] [PubMed]

V. Borshch, Y.-K. Kim, J. Xiang, M. Gao, A. Jákli, V. P. Panov, J. K. Vij, C. T. Imrie, M. G. Tamba, G. H. Mehl, and O. D. Lavrentovich, “Nematic twist-bend phase with nanoscale modulation of molecular orientation,” Nat. Commun. 4, 2635 (2013).
[Crossref] [PubMed]

2012 (2)

T. Kosa, L. Sukhomlinova, L. Su, B. Taheri, T. J. White, and T. J. Bunning, “Light-induced liquid crystallinity,” Nature 485, 347–349 (2012).
[Crossref] [PubMed]

N. I. Zheludev and Y. S. Kivshar, “From metamaterials to metadevices,” Nat. Mater. 11, 917–924 (2012).
[Crossref] [PubMed]

2011 (1)

M. D. Leistikow, A. P. Mosk, E. Yeganegi, S. R. Huisman, A. Lagendijk, and W. L. Vos, “Inhibited spontaneous emission of quantum dots observed in a 3d photonic band gap,” Phys. Rev. Lett. 107, 193903 (2011).
[Crossref] [PubMed]

2010 (1)

H. Coles and S. Morris, “Liquid-crystal lasers,” Nat. Photon. 4, 676–685 (2010).
[Crossref]

2007 (1)

S. Noda, M. Fujita, and T. Asano, “Spontaneous-emission control by photonic crystals and nanocavities,” Nat. Photon. 1, 449–458 (2007).
[Crossref]

2006 (1)

S. Gottardo, M. Burresi, F. Geobaldo, L. Pallavidino, F. Giorgis, and D. Wiersma, “Self-alignment of liquid crystals in three-dimensional photonic crystals,” Phys. Rev. E 74, 040702 (2006).
[Crossref]

2005 (1)

H. J. Coles and M. N. Pivnenko, “Liquid crystal blue phases with a wide temperature range,” Nature 436, 997–1000 (2005).
[Crossref] [PubMed]

2003 (1)

S. Kutter and M. Warner, “Reflectivity of cholesteric liquid crystals with spatially varying pitch,” Eur. Phys. J. E 12, 515–521 (2003).
[Crossref]

2001 (1)

1997 (1)

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature 386, 143–149 (1997).
[Crossref]

1991 (1)

E. Yablonovitch, T. Gmitter, and K. Leung, “Photonic band structure: The face-centered-cubic case employing nonspherical atoms,” Phys. Rev. Lett. 67, 2295–2298 (1991).
[Crossref] [PubMed]

1968 (1)

R. B. Meyer, “Effects of electric and magnetic fields on the structure of cholesteric liquid crystals,” Appl. Phys. Lett. 12, 281–282 (1968).
[Crossref]

Aplinc, J.

J. Aplinc, M. Štimulak, S. Čopar, and M. Ravnik, “Nematic liquid crystal gyroids as photonic crystals,” Liq. Cryst. 43, 2320–2331 (2016).
[Crossref]

Asano, T.

S. Noda, M. Fujita, and T. Asano, “Spontaneous-emission control by photonic crystals and nanocavities,” Nat. Photon. 1, 449–458 (2007).
[Crossref]

Babakhanova, G.

Barboza, R.

R. Barboza, U. Bortolozzo, M. G. Clerc, and S. Residori, “Berry phase of light under bragg reflection by chiral liquid-crystal media,” Phys. Rev. Lett. 117, 053903 (2016).
[Crossref] [PubMed]

Baumeister, U.

D. Chen, M. Nakata, R. Shao, M. R. Tuchband, M. Shuai, U. Baumeister, W. Weissflog, D. M. Walba, M. A. Glaser, J. E. Maclennan, and N. A. Clark, “Twist-bend heliconical chiral nematic liquid crystal phase of an achiral rigid bent-core mesogen,” Phys. Rev. E 89, 022506 (2014).
[Crossref]

Bedrov, D.

D. Chen, J. H. Porada, J. B. Hooper, A. Klittnick, Y. Shen, M. R. Tuchband, E. Korblova, D. Bedrov, D. M. Walba, M. A. Glaser, J. E. Maclennan, and N. A. Clark, “Chiral heliconical ground state of nanoscale pitch in a nematic liquid crystal of achiral molecular dimers,” Proc. Natl. Acad. Sci. 110, 15931–15936 (2013).
[Crossref] [PubMed]

Belov, P.

M. Rybin, I. Shishkin, K. Samusev, P. Belov, Y. Kivshar, R. Kiyan, B. Chichkov, and M. Limonov, “Band structure of photonic crystals fabricated by two-photon polymerization,” Crystals 5, 61–73 (2015).
[Crossref]

Borshch, V.

V. Borshch, Y.-K. Kim, J. Xiang, M. Gao, A. Jákli, V. P. Panov, J. K. Vij, C. T. Imrie, M. G. Tamba, G. H. Mehl, and O. D. Lavrentovich, “Nematic twist-bend phase with nanoscale modulation of molecular orientation,” Nat. Commun. 4, 2635 (2013).
[Crossref] [PubMed]

Bortolozzo, U.

R. Barboza, U. Bortolozzo, M. G. Clerc, and S. Residori, “Berry phase of light under bragg reflection by chiral liquid-crystal media,” Phys. Rev. Lett. 117, 053903 (2016).
[Crossref] [PubMed]

A. Jullien, U. Bortolozzo, S. Grabielle, J.-P. Huignard, N. Forget, and S. Residori, “Continuously tunable femtosecond delay-line based on liquid crystal cells,” Opt. Express 24, 14483–14493 (2016).
[Crossref] [PubMed]

Bregar, A.

A. Bregar, T. J. White, and M. Ravnik, “Refraction of light on flat boundary of liquid crystals or anisotropic metamaterials,” Liq. Cryst. Rev. 5, 53–68 (2017).
[Crossref]

Bunning, T. J.

T. Kosa, L. Sukhomlinova, L. Su, B. Taheri, T. J. White, and T. J. Bunning, “Light-induced liquid crystallinity,” Nature 485, 347–349 (2012).
[Crossref] [PubMed]

Burresi, M.

S. Gottardo, M. Burresi, F. Geobaldo, L. Pallavidino, F. Giorgis, and D. Wiersma, “Self-alignment of liquid crystals in three-dimensional photonic crystals,” Phys. Rev. E 74, 040702 (2006).
[Crossref]

Cancula, M.

M. Čančula, M. Ravnik, and S. Žumer, “Generation of vector beams with liquid crystal disclination lines,” Phys. Rev. E 90, 022503 (2014).
[Crossref]

Chandrasekhar, S.

S. Chandrasekhar, Liquid crystals (Cambridge University, 1992), 2nd ed.
[Crossref]

Chen, D.

D. Chen, M. Nakata, R. Shao, M. R. Tuchband, M. Shuai, U. Baumeister, W. Weissflog, D. M. Walba, M. A. Glaser, J. E. Maclennan, and N. A. Clark, “Twist-bend heliconical chiral nematic liquid crystal phase of an achiral rigid bent-core mesogen,” Phys. Rev. E 89, 022506 (2014).
[Crossref]

D. Chen, J. H. Porada, J. B. Hooper, A. Klittnick, Y. Shen, M. R. Tuchband, E. Korblova, D. Bedrov, D. M. Walba, M. A. Glaser, J. E. Maclennan, and N. A. Clark, “Chiral heliconical ground state of nanoscale pitch in a nematic liquid crystal of achiral molecular dimers,” Proc. Natl. Acad. Sci. 110, 15931–15936 (2013).
[Crossref] [PubMed]

Chen, H.

G. Tan, Y.-H. Lee, F. Gou, H. Chen, Y. Huang, Y.-F. Lan, C.-Y. Tsai, and S.-T. Wu, “Review on polymer-stabilized short-pitch cholesteric liquid crystal displays,” J. Phys. D: Appl. Phys. 50, 493001 (2017).
[Crossref]

Y. Huang, H. Chen, G. Tan, H. Tobata, S. ichi Yamamoto, E. Okabe, Y.-F. Lan, C.-Y. Tsai, and S.-T. Wu, “Optimized blue-phase liquid crystal for field-sequential-color displays,” Opt. Mater. Express 7, 641–650 (2017).
[Crossref]

Chichkov, B.

M. Rybin, I. Shishkin, K. Samusev, P. Belov, Y. Kivshar, R. Kiyan, B. Chichkov, and M. Limonov, “Band structure of photonic crystals fabricated by two-photon polymerization,” Crystals 5, 61–73 (2015).
[Crossref]

Clark, N. A.

D. Chen, M. Nakata, R. Shao, M. R. Tuchband, M. Shuai, U. Baumeister, W. Weissflog, D. M. Walba, M. A. Glaser, J. E. Maclennan, and N. A. Clark, “Twist-bend heliconical chiral nematic liquid crystal phase of an achiral rigid bent-core mesogen,” Phys. Rev. E 89, 022506 (2014).
[Crossref]

D. Chen, J. H. Porada, J. B. Hooper, A. Klittnick, Y. Shen, M. R. Tuchband, E. Korblova, D. Bedrov, D. M. Walba, M. A. Glaser, J. E. Maclennan, and N. A. Clark, “Chiral heliconical ground state of nanoscale pitch in a nematic liquid crystal of achiral molecular dimers,” Proc. Natl. Acad. Sci. 110, 15931–15936 (2013).
[Crossref] [PubMed]

Clerc, M. G.

R. Barboza, U. Bortolozzo, M. G. Clerc, and S. Residori, “Berry phase of light under bragg reflection by chiral liquid-crystal media,” Phys. Rev. Lett. 117, 053903 (2016).
[Crossref] [PubMed]

Coles, H.

H. Coles and S. Morris, “Liquid-crystal lasers,” Nat. Photon. 4, 676–685 (2010).
[Crossref]

Coles, H. J.

H. J. Coles and M. N. Pivnenko, “Liquid crystal blue phases with a wide temperature range,” Nature 436, 997–1000 (2005).
[Crossref] [PubMed]

Copar, S.

J. Aplinc, M. Štimulak, S. Čopar, and M. Ravnik, “Nematic liquid crystal gyroids as photonic crystals,” Liq. Cryst. 43, 2320–2331 (2016).
[Crossref]

Cucinotta, A.

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J. Xiang, A. Varanytsia, F. Minkowski, D. A. Paterson, J. M. D. Storey, C. T. Imrie, O. D. Lavrentovich, and P. Palffy-Muhoray, “Electrically tunable laser based on oblique heliconical cholesteric liquid crystal,” Proc. Natl. Acad. Sci. 113, 12925–12928 (2016).
[Crossref] [PubMed]

J. Xiang, Y. Li, Q. Li, D. A. Paterson, J. M. D. Storey, C. T. Imrie, and O. D. Lavrentovich, “Electrically tunable selective reflection of light from ultraviolet to visible and infrared by heliconical cholesterics,” Adv. Mater. 27, 3014–3018 (2015).
[Crossref] [PubMed]

Su, L.

T. Kosa, L. Sukhomlinova, L. Su, B. Taheri, T. J. White, and T. J. Bunning, “Light-induced liquid crystallinity,” Nature 485, 347–349 (2012).
[Crossref] [PubMed]

Sukhomlinova, L.

T. Kosa, L. Sukhomlinova, L. Su, B. Taheri, T. J. White, and T. J. Bunning, “Light-induced liquid crystallinity,” Nature 485, 347–349 (2012).
[Crossref] [PubMed]

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A. Taflove, S. G. Johnson, and A. Oskooi, Advances in FDTD Computational Electrodynamics: Photonics and Nanotechnology (Artech House, 2013).

A. Taflove and S. C. Hagness, Computational electrodynamics: the finite difference time-domain method (Artech House, Boston, 2000), 2nd ed.

Taheri, B.

T. Kosa, L. Sukhomlinova, L. Su, B. Taheri, T. J. White, and T. J. Bunning, “Light-induced liquid crystallinity,” Nature 485, 347–349 (2012).
[Crossref] [PubMed]

Tamba, M. G.

V. Borshch, Y.-K. Kim, J. Xiang, M. Gao, A. Jákli, V. P. Panov, J. K. Vij, C. T. Imrie, M. G. Tamba, G. H. Mehl, and O. D. Lavrentovich, “Nematic twist-bend phase with nanoscale modulation of molecular orientation,” Nat. Commun. 4, 2635 (2013).
[Crossref] [PubMed]

Tan, G.

G. Tan, Y.-H. Lee, F. Gou, H. Chen, Y. Huang, Y.-F. Lan, C.-Y. Tsai, and S.-T. Wu, “Review on polymer-stabilized short-pitch cholesteric liquid crystal displays,” J. Phys. D: Appl. Phys. 50, 493001 (2017).
[Crossref]

Y. Huang, H. Chen, G. Tan, H. Tobata, S. ichi Yamamoto, E. Okabe, Y.-F. Lan, C.-Y. Tsai, and S.-T. Wu, “Optimized blue-phase liquid crystal for field-sequential-color displays,” Opt. Mater. Express 7, 641–650 (2017).
[Crossref]

Tobata, H.

Tsai, C.-Y.

Y. Huang, H. Chen, G. Tan, H. Tobata, S. ichi Yamamoto, E. Okabe, Y.-F. Lan, C.-Y. Tsai, and S.-T. Wu, “Optimized blue-phase liquid crystal for field-sequential-color displays,” Opt. Mater. Express 7, 641–650 (2017).
[Crossref]

G. Tan, Y.-H. Lee, F. Gou, H. Chen, Y. Huang, Y.-F. Lan, C.-Y. Tsai, and S.-T. Wu, “Review on polymer-stabilized short-pitch cholesteric liquid crystal displays,” J. Phys. D: Appl. Phys. 50, 493001 (2017).
[Crossref]

Tuchband, M. R.

D. Chen, M. Nakata, R. Shao, M. R. Tuchband, M. Shuai, U. Baumeister, W. Weissflog, D. M. Walba, M. A. Glaser, J. E. Maclennan, and N. A. Clark, “Twist-bend heliconical chiral nematic liquid crystal phase of an achiral rigid bent-core mesogen,” Phys. Rev. E 89, 022506 (2014).
[Crossref]

D. Chen, J. H. Porada, J. B. Hooper, A. Klittnick, Y. Shen, M. R. Tuchband, E. Korblova, D. Bedrov, D. M. Walba, M. A. Glaser, J. E. Maclennan, and N. A. Clark, “Chiral heliconical ground state of nanoscale pitch in a nematic liquid crystal of achiral molecular dimers,” Proc. Natl. Acad. Sci. 110, 15931–15936 (2013).
[Crossref] [PubMed]

Varanytsia, A.

J. Xiang, A. Varanytsia, F. Minkowski, D. A. Paterson, J. M. D. Storey, C. T. Imrie, O. D. Lavrentovich, and P. Palffy-Muhoray, “Electrically tunable laser based on oblique heliconical cholesteric liquid crystal,” Proc. Natl. Acad. Sci. 113, 12925–12928 (2016).
[Crossref] [PubMed]

Vij, J. K.

V. Borshch, Y.-K. Kim, J. Xiang, M. Gao, A. Jákli, V. P. Panov, J. K. Vij, C. T. Imrie, M. G. Tamba, G. H. Mehl, and O. D. Lavrentovich, “Nematic twist-bend phase with nanoscale modulation of molecular orientation,” Nat. Commun. 4, 2635 (2013).
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J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature 386, 143–149 (1997).
[Crossref]

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M. D. Leistikow, A. P. Mosk, E. Yeganegi, S. R. Huisman, A. Lagendijk, and W. L. Vos, “Inhibited spontaneous emission of quantum dots observed in a 3d photonic band gap,” Phys. Rev. Lett. 107, 193903 (2011).
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Walba, D. M.

D. Chen, M. Nakata, R. Shao, M. R. Tuchband, M. Shuai, U. Baumeister, W. Weissflog, D. M. Walba, M. A. Glaser, J. E. Maclennan, and N. A. Clark, “Twist-bend heliconical chiral nematic liquid crystal phase of an achiral rigid bent-core mesogen,” Phys. Rev. E 89, 022506 (2014).
[Crossref]

D. Chen, J. H. Porada, J. B. Hooper, A. Klittnick, Y. Shen, M. R. Tuchband, E. Korblova, D. Bedrov, D. M. Walba, M. A. Glaser, J. E. Maclennan, and N. A. Clark, “Chiral heliconical ground state of nanoscale pitch in a nematic liquid crystal of achiral molecular dimers,” Proc. Natl. Acad. Sci. 110, 15931–15936 (2013).
[Crossref] [PubMed]

Wang, N.

Wang, S.

Warner, M.

S. Kutter and M. Warner, “Reflectivity of cholesteric liquid crystals with spatially varying pitch,” Eur. Phys. J. E 12, 515–521 (2003).
[Crossref]

Weissflog, W.

D. Chen, M. Nakata, R. Shao, M. R. Tuchband, M. Shuai, U. Baumeister, W. Weissflog, D. M. Walba, M. A. Glaser, J. E. Maclennan, and N. A. Clark, “Twist-bend heliconical chiral nematic liquid crystal phase of an achiral rigid bent-core mesogen,” Phys. Rev. E 89, 022506 (2014).
[Crossref]

Welch, C.

White, T. J.

A. Bregar, T. J. White, and M. Ravnik, “Refraction of light on flat boundary of liquid crystals or anisotropic metamaterials,” Liq. Cryst. Rev. 5, 53–68 (2017).
[Crossref]

T. Kosa, L. Sukhomlinova, L. Su, B. Taheri, T. J. White, and T. J. Bunning, “Light-induced liquid crystallinity,” Nature 485, 347–349 (2012).
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S. Gottardo, M. Burresi, F. Geobaldo, L. Pallavidino, F. Giorgis, and D. Wiersma, “Self-alignment of liquid crystals in three-dimensional photonic crystals,” Phys. Rev. E 74, 040702 (2006).
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G. Tan, Y.-H. Lee, F. Gou, H. Chen, Y. Huang, Y.-F. Lan, C.-Y. Tsai, and S.-T. Wu, “Review on polymer-stabilized short-pitch cholesteric liquid crystal displays,” J. Phys. D: Appl. Phys. 50, 493001 (2017).
[Crossref]

Y. Huang, H. Chen, G. Tan, H. Tobata, S. ichi Yamamoto, E. Okabe, Y.-F. Lan, C.-Y. Tsai, and S.-T. Wu, “Optimized blue-phase liquid crystal for field-sequential-color displays,” Opt. Mater. Express 7, 641–650 (2017).
[Crossref]

Xiang, J.

J. Xiang, A. Varanytsia, F. Minkowski, D. A. Paterson, J. M. D. Storey, C. T. Imrie, O. D. Lavrentovich, and P. Palffy-Muhoray, “Electrically tunable laser based on oblique heliconical cholesteric liquid crystal,” Proc. Natl. Acad. Sci. 113, 12925–12928 (2016).
[Crossref] [PubMed]

J. Xiang, Y. Li, Q. Li, D. A. Paterson, J. M. D. Storey, C. T. Imrie, and O. D. Lavrentovich, “Electrically tunable selective reflection of light from ultraviolet to visible and infrared by heliconical cholesterics,” Adv. Mater. 27, 3014–3018 (2015).
[Crossref] [PubMed]

J. Xiang, S. V. Shiyanovskii, C. Imrie, and O. D. Lavrentovich, “Electrooptic response of chiral nematic liquid crystals with oblique helicoidal director,” Phys. Rev. Lett. 112, 217801 (2014).
[Crossref]

V. Borshch, Y.-K. Kim, J. Xiang, M. Gao, A. Jákli, V. P. Panov, J. K. Vij, C. T. Imrie, M. G. Tamba, G. H. Mehl, and O. D. Lavrentovich, “Nematic twist-bend phase with nanoscale modulation of molecular orientation,” Nat. Commun. 4, 2635 (2013).
[Crossref] [PubMed]

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E. Yablonovitch, T. Gmitter, and K. Leung, “Photonic band structure: The face-centered-cubic case employing nonspherical atoms,” Phys. Rev. Lett. 67, 2295–2298 (1991).
[Crossref] [PubMed]

E. Yablonovitch, “Optical antennas; led’s faster than lasers,” in 2017 IEEE Photonics Society Summer Topical Meeting Series (SUM), (2017), pp. 107.
[Crossref]

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M. D. Leistikow, A. P. Mosk, E. Yeganegi, S. R. Huisman, A. Lagendijk, and W. L. Vos, “Inhibited spontaneous emission of quantum dots observed in a 3d photonic band gap,” Phys. Rev. Lett. 107, 193903 (2011).
[Crossref] [PubMed]

Yoshida, H.

J. Kobashi, H. Yoshida, and M. Ozaki, “Circularly-polarized, semitransparent and double-sided holograms based on helical photonic structures,” Sci. Rep. 7, 016470 (2017).
[Crossref]

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

J. Kobashi, H. Yoshida, and M. Ozaki, “Planar optics with patterned chiral liquid crystals,” Nat. Photon. 10, 389–392 (2016).
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N. I. Zheludev and Y. S. Kivshar, “From metamaterials to metadevices,” Nat. Mater. 11, 917–924 (2012).
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M. Čančula, M. Ravnik, and S. Žumer, “Generation of vector beams with liquid crystal disclination lines,” Phys. Rev. E 90, 022503 (2014).
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Adv. Mater. (1)

J. Xiang, Y. Li, Q. Li, D. A. Paterson, J. M. D. Storey, C. T. Imrie, and O. D. Lavrentovich, “Electrically tunable selective reflection of light from ultraviolet to visible and infrared by heliconical cholesterics,” Adv. Mater. 27, 3014–3018 (2015).
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R. B. Meyer, “Effects of electric and magnetic fields on the structure of cholesteric liquid crystals,” Appl. Phys. Lett. 12, 281–282 (1968).
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Crystals (1)

M. Rybin, I. Shishkin, K. Samusev, P. Belov, Y. Kivshar, R. Kiyan, B. Chichkov, and M. Limonov, “Band structure of photonic crystals fabricated by two-photon polymerization,” Crystals 5, 61–73 (2015).
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Eur. Phys. J. E (1)

S. Kutter and M. Warner, “Reflectivity of cholesteric liquid crystals with spatially varying pitch,” Eur. Phys. J. E 12, 515–521 (2003).
[Crossref]

J. Phys. D: Appl. Phys. (1)

G. Tan, Y.-H. Lee, F. Gou, H. Chen, Y. Huang, Y.-F. Lan, C.-Y. Tsai, and S.-T. Wu, “Review on polymer-stabilized short-pitch cholesteric liquid crystal displays,” J. Phys. D: Appl. Phys. 50, 493001 (2017).
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Liq. Cryst. (1)

J. Aplinc, M. Štimulak, S. Čopar, and M. Ravnik, “Nematic liquid crystal gyroids as photonic crystals,” Liq. Cryst. 43, 2320–2331 (2016).
[Crossref]

Liq. Cryst. Rev. (2)

A. Bregar, T. J. White, and M. Ravnik, “Refraction of light on flat boundary of liquid crystals or anisotropic metamaterials,” Liq. Cryst. Rev. 5, 53–68 (2017).
[Crossref]

I. Muševič, “Liquid-crystal micro-photonics,” Liq. Cryst. Rev. 4, 1–34 (2016).
[Crossref]

Nat. Commun. (1)

V. Borshch, Y.-K. Kim, J. Xiang, M. Gao, A. Jákli, V. P. Panov, J. K. Vij, C. T. Imrie, M. G. Tamba, G. H. Mehl, and O. D. Lavrentovich, “Nematic twist-bend phase with nanoscale modulation of molecular orientation,” Nat. Commun. 4, 2635 (2013).
[Crossref] [PubMed]

Nat. Mater. (1)

N. I. Zheludev and Y. S. Kivshar, “From metamaterials to metadevices,” Nat. Mater. 11, 917–924 (2012).
[Crossref] [PubMed]

Nat. Photon. (3)

J. Kobashi, H. Yoshida, and M. Ozaki, “Planar optics with patterned chiral liquid crystals,” Nat. Photon. 10, 389–392 (2016).
[Crossref]

H. Coles and S. Morris, “Liquid-crystal lasers,” Nat. Photon. 4, 676–685 (2010).
[Crossref]

S. Noda, M. Fujita, and T. Asano, “Spontaneous-emission control by photonic crystals and nanocavities,” Nat. Photon. 1, 449–458 (2007).
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H. J. Coles and M. N. Pivnenko, “Liquid crystal blue phases with a wide temperature range,” Nature 436, 997–1000 (2005).
[Crossref] [PubMed]

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature 386, 143–149 (1997).
[Crossref]

T. Kosa, L. Sukhomlinova, L. Su, B. Taheri, T. J. White, and T. J. Bunning, “Light-induced liquid crystallinity,” Nature 485, 347–349 (2012).
[Crossref] [PubMed]

Opt. Express (4)

Opt. Lett. (1)

Opt. Mater. Express (2)

Phys. Rev. E (3)

S. Gottardo, M. Burresi, F. Geobaldo, L. Pallavidino, F. Giorgis, and D. Wiersma, “Self-alignment of liquid crystals in three-dimensional photonic crystals,” Phys. Rev. E 74, 040702 (2006).
[Crossref]

D. Chen, M. Nakata, R. Shao, M. R. Tuchband, M. Shuai, U. Baumeister, W. Weissflog, D. M. Walba, M. A. Glaser, J. E. Maclennan, and N. A. Clark, “Twist-bend heliconical chiral nematic liquid crystal phase of an achiral rigid bent-core mesogen,” Phys. Rev. E 89, 022506 (2014).
[Crossref]

M. Čančula, M. Ravnik, and S. Žumer, “Generation of vector beams with liquid crystal disclination lines,” Phys. Rev. E 90, 022503 (2014).
[Crossref]

Phys. Rev. Lett. (5)

J. Xiang, S. V. Shiyanovskii, C. Imrie, and O. D. Lavrentovich, “Electrooptic response of chiral nematic liquid crystals with oblique helicoidal director,” Phys. Rev. Lett. 112, 217801 (2014).
[Crossref]

M. D. Leistikow, A. P. Mosk, E. Yeganegi, S. R. Huisman, A. Lagendijk, and W. L. Vos, “Inhibited spontaneous emission of quantum dots observed in a 3d photonic band gap,” Phys. Rev. Lett. 107, 193903 (2011).
[Crossref] [PubMed]

E. Yablonovitch, T. Gmitter, and K. Leung, “Photonic band structure: The face-centered-cubic case employing nonspherical atoms,” Phys. Rev. Lett. 67, 2295–2298 (1991).
[Crossref] [PubMed]

R. Barboza, U. Bortolozzo, M. G. Clerc, and S. Residori, “Berry phase of light under bragg reflection by chiral liquid-crystal media,” Phys. Rev. Lett. 117, 053903 (2016).
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J. Kobashi, H. Yoshida, and M. Ozaki, “Polychromatic optical vortex generation from patterned cholesteric liquid crystals,” Phys. Rev. Lett. 116, 253903 (2016).
[Crossref] [PubMed]

Proc. Natl. Acad. Sci. (2)

D. Chen, J. H. Porada, J. B. Hooper, A. Klittnick, Y. Shen, M. R. Tuchband, E. Korblova, D. Bedrov, D. M. Walba, M. A. Glaser, J. E. Maclennan, and N. A. Clark, “Chiral heliconical ground state of nanoscale pitch in a nematic liquid crystal of achiral molecular dimers,” Proc. Natl. Acad. Sci. 110, 15931–15936 (2013).
[Crossref] [PubMed]

J. Xiang, A. Varanytsia, F. Minkowski, D. A. Paterson, J. M. D. Storey, C. T. Imrie, O. D. Lavrentovich, and P. Palffy-Muhoray, “Electrically tunable laser based on oblique heliconical cholesteric liquid crystal,” Proc. Natl. Acad. Sci. 113, 12925–12928 (2016).
[Crossref] [PubMed]

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I. C. Khoo, “Nonlinear optics, active plasmonics and metamaterials with liquid crystals,” Prog. Quantum Electron. 38, 77–117 (2014).
[Crossref]

Sci. Rep. (1)

J. Kobashi, H. Yoshida, and M. Ozaki, “Circularly-polarized, semitransparent and double-sided holograms based on helical photonic structures,” Sci. Rep. 7, 016470 (2017).
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Soft Matter (1)

M. Štimulak and M. Ravnik, “Tunable photonic crystals with partial bandgaps from blue phase colloidal crystals and dielectric-doped blue phases,” Soft Matter 10, 6339–6346 (2014).
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Other (8)

S. S. F. Poli and A. Cucinotta, Photonic Crystal Fibers: Properties and Applications (Springer, 2007).

E. Yablonovitch, “Optical antennas; led’s faster than lasers,” in 2017 IEEE Photonics Society Summer Topical Meeting Series (SUM), (2017), pp. 107.
[Crossref]

P. G. de Gennes and J. Prost, The Physics of Liquid Crystals (Oxford University, 1995), 2nd ed.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic crystals - Molding the flow of light (Princeton University, Princeton, New Jersey, 2008), 2nd ed.

A. Taflove and S. C. Hagness, Computational electrodynamics: the finite difference time-domain method (Artech House, Boston, 2000), 2nd ed.

M. Kleman and O. Lavrentovich, Soft Matter Physics: An Introduction (Springer, New York, 2003).

S. Chandrasekhar, Liquid crystals (Cambridge University, 1992), 2nd ed.
[Crossref]

A. Taflove, S. G. Johnson, and A. Oskooi, Advances in FDTD Computational Electrodynamics: Photonics and Nanotechnology (Artech House, 2013).

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Figures (6)

Fig. 1
Fig. 1 A sketch of the heliconical director structure. a) Representation of heliconical director profile in the region of one pitch p with z-direction as illustrated. b) A sketch of the index ellipsoid at an arbitrary z coordinate. Inscribed is the coordinate system along with the tilt angle θ and rotation angle φ. The ordinary no and extraordinary refractive index ne are the semi-axes of the ellipsoid. Depicted are also the circular cross section of the ellipsoid with the plane orthogonal to its optical axis, and the elliptical cross section of the ellipsoid with the xy-plane, which is the incoming plane for the plane wave, used in all the simulations. The semi-axes of the xy-ellipse are the ordinary refractive index no and the effective extraordinary refractive index n e eff ( θ ) (see Eq. (11)), which determine the optical response of the structure. c) Scheme of refraction between two birefringent layers.
Fig. 2
Fig. 2 a) Band-gap diagram for a heliconical structure with heliconical tilt angle θ of 45°. At the edge of the first Brillouin zone, a band gap for RCP light emerges. The bands are numbered 1 − 4 as marked on the figure. b) Opening of the band gap with increasing θ. Values for the band-gap frequencies, obtained from simulations (dots, crosses and diamonds), are compared to the theoretical values, corresponding to ordinary refractive index no (violet curve) and effective extraordinary index n e eff (blue curve). The green curve describes the band gap center and thus the dispersion point of the LCP wave: it shows the average between the dispersion points of ordinary and extraordinary index.
Fig. 3
Fig. 3 Transmittivity of heliconical liquid crystals for different heliconical tilt angle and pitch. a) Intensity transmittivity as function of vacuum wavelength of incoming light λ0 for different θ. Pitch is chosen to be p = 300 nm, which makes the optical periodicity of the heliconical structure equal to a = p/2 = 150 nm. The lower wavelength band gap edge is at λ0 = nop, and the upper wavelength band gap edge at λ 0 = n e eff ( θ ) p. b) Intensity transmittance as function of vacuum wavelength for different pitches. Angle θ is chosen to be 60°.
Fig. 4
Fig. 4 Diagram of electric field E eigenmodes at the edge of irreducible Brillouin zone (k = π/a) for heliconical θ = π/4 (full line) and helical cholesteric θ = π/2 (dashed line) structure. Figs. (a) and (d) represent RCP light eigenmodes, while (b) and (c) represent LCP light eigenmodes. Band 2 and Band 3 belong to degenerate eigenstates, whereas Band 1 and Band 4 are nondegenerate, as shown in Fig. 2. Compared to cholesterics, heliconical structure keeps the same symmetry for x and y components, but an additional longitudinal Ez component emerges.
Fig. 5
Fig. 5 The emergent periodicity of the Ez component around the band gap for RCP wave. a) Modulation of Ez component for different vacuum wavelengths λ0 (with their corresponding wavelengths λ inside the heliconical medium). The wavelengths λ0 are approaching the band gap wavelength, which is for given parameters equal to λ0 = 450 nm or, equally, λ = 300nm. The Ez modulation wavelength λ z RCP is much larger than the pitch and is increasing when nearing the band gap. When approaching the exact lower band gap edge, λ z RCP becomes infinite, and Ez component is constant. b) Analysis of the λz around the band gap. Values of λ z RCP with respect to λ0, obtained from full FDTD simulations for θ = 30° and θ = 60°, and from Eq. (17) are shown, as in full quantitative agreement.
Fig. 6
Fig. 6 Representation of the rotation of E, H, P for four different wavelengths of the incoming light λ0. a) One pitch p region of the heliconical director structure. b) Tilt and rotation of the RCP polarized light for four different wavelengths: λ = p, λ = p/2, λ = 2p and λ = 1.42p. The direction of the electric field is shown in dark yellow, of magnetic field in dark blue, and of the Poynting vector in red. The rotation of the three vectors depends on λ relative to p, whereas the tilting with respect to the z-axis has a period of λ z RCP. c) The z-component of the RCP electric field. The Ez component is constant for λ = p (at the edge of the first Brillouin zone), and has a periodicity λz of p, 2p and 3.38p for λ = p/2, 2p, 1.42p, respectively. d) Visualization of the Poynting vector rotation with respect to z for RCP waves around the band gap wavelengths. On the band-gap edge (λ0 = 445nm, λ = 298nm), the Poynting vector rotates in an effective circular helical pattern with respect to z. e) For LCP light for λ = 300nm (λ0 = 482nm), the Poynting vector inscribes a four-leaf clover like pattern.

Equations (17)

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n = ( cos φ ( z ) sin θ , sin φ ( z ) sin θ , cos θ ) , where φ ( z ) = 2 π z / p and θ = const .
f = 1 2 K 1 ( n ) 2 + 1 2 K 2 ( n ( × n ) 2 π p * ) 2 + 1 2 K 3 ( n × ( × n ) ) 2 1 2 a 0 ( n E ext ) 2 ,
E N * C 2 π p * K 2 0 a K 3 κ ( 2 + 2 ( 1 κ ) ) ( 1 + κ ) ,
p = 2 π E ext K 3 0 a , sin 2 θ = κ 1 κ ( E NC E ext 1 ) .
E NC = 2 π p * ( K 2 0 a K 3 ) .
∊̳ ( z ) = o + a n ( z ) n T ( z )
× { ∊̳ 1 ( × H ) } = ( ω c ) 2 H
( D t D i ) n it = 0 ,
E = ε 0 ε̳ 1 ( z ) D ,
k t = k i n t eff / n i eff .
n e eff = n o n e n o 2 sin 2 θ + n e 2 cos 2 θ ,
p n o < λ 0 < p n e eff , n o < n e
p n o > λ 0 > p n e eff , n e > n o
φ + φ 0 = 2 π
λ z LCP = λ p p + λ
φ 0 φ = ± 2 π
λ z RCP = ± λ p p λ ,

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