Abstract

In addition to displays, liquid crystals (LCs) have also found widespread applications in photonic devices, such as adaptive lens, adaptive optics, and sensors, because of their responses to electric field, temperature, and light. As the fabrication technique advances, more sophisticated devices can be designed and created. In this review, we report recent advances of two-photon polymerization-based direct-laser writing enabled LC devices. Firstly, we describe the basic working principle of two-photon polymerization. With this powerful fabrication technique, we can generate anchoring energy by surface morphology to align LC directors on different form factors. To prove this concept, we demonstrate LC alignment on planar, curvilinear surfaces as well as in three-dimensional volumes. Based on the results, we further propose a novel, ultra-broadband, twisted-nematic diffractive waveplate that can potentially be fulfilled by this technique. Next, we briefly discuss the current status of direct-laser writing on LC reactive mesogens and its potential applications. Finally, we present two design challenges: fabrication yield and polymer relaxation/deformation, remaining to be overcome.

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

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2019 (2)

R. Chen, Y. H. Lee, T. Zhan, K. Yin, Z. An, and S. T. Wu, “Multi-stimuli-responsive self-organized liquid crystal Bragg gratings,” Adv. Opt. Mater. 7, 1900101 (2019).
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K. Yin, Y. H. Lee, Z. He, and S. T. Wu, “Stretchable, flexible, rollable, and adherable polarization volume grating film,” Opt. Express 27(4), 5814–5823 (2019).
[Crossref] [PubMed]

2018 (14)

H. W. Chen, J. H. Lee, B. Y. Lin, S. Chen, and S. T. Wu, “Liquid crystal display and organic light-emitting diode display: present status and future perspectives,” Light Sci. Appl. 7(3), 17168 (2018).
[Crossref] [PubMed]

Z. He, H. Chen, Y. H. Lee, and S. T. Wu, “Tuning the correlated color temperature of white light-emitting diodes resembling Planckian locus,” Opt. Express 26(2), A136–A143 (2018).
[Crossref] [PubMed]

T. Zhan, Y. H. Lee, and S. T. Wu, “High-resolution additive light field near-eye display by switchable Pancharatnam-Berry phase lenses,” Opt. Express 26(4), 4863–4872 (2018).
[Crossref] [PubMed]

Y. Weng, Y. Zhang, J. Cui, A. Liu, Z. Shen, X. Li, and B. Wang, “Liquid-crystal-based polarization volume grating applied for full-color waveguide displays,” Opt. Lett. 43(23), 5773–5776 (2018).
[Crossref] [PubMed]

S. Nocentini, C. Parmeggiani, D. Martella, and D. S. Wiersma, “Optically driven soft micro robotics,” Adv. Opt. Mater. 6(14), 1800207 (2018).
[Crossref]

D. Martella, S. Nocentini, C. Parmeggiani, and D. S. Wiersma, “Self-regulating capabilities in photonic robotics,” Adv. Mater. Technol. 4(2), 1800571 (2018).
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P. I. Dietrich, M. Blaicher, I. Reuter, M. Billah, T. Hoose, A. Hofmann, C. Caer, R. Dangel, B. Offrein, U. Troppenz, M. Moehrle, W. Freude, and C. Koos, “In situ 3D nanoprinting of free-form coupling elements for hybrid photonic integration,” Nat. Photonics 12(4), 241–247 (2018).
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J. D. Lin, Y. L. Daniel Ho, L. Chen, M. Lopez-Garcia, S. A. Jiang, M. P. C. Taverne, C. R. Lee, and J. G. Rarity, “Microstructure-stabilized blue phase liquid crystals,” ACS Omega 3(11), 15435–15441 (2018).
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Z. He, Y. H. Lee, R. Chen, D. Chanda, and S. T. Wu, “Switchable Pancharatnam-Berry microlens array with nano-imprinted liquid crystal alignment,” Opt. Lett. 43(20), 5062–5065 (2018).
[Crossref] [PubMed]

Z. He, Y. H. Lee, D. Chanda, and S. T. Wu, “Adaptive liquid crystal microlens array enabled by two-photon polymerization,” Opt. Express 26(16), 21184–21193 (2018).
[Crossref] [PubMed]

E. Descrovi, F. Pirani, V. P. Rajamanickam, S. Licheri, and C. Liberale, “Photo-responsive suspended micro-membranes,” J. Mater. Chem. C Mater. Opt. Electron. Devices 6(39), 10428–10434 (2018).
[Crossref]

S. Nocentini, D. Martella, C. Parmeggiani, S. Zanotto, and D. S. Wiersma, “Structured optical materials controlled by light,” Adv. Opt. Mater. 6(15), 1800167 (2018).
[Crossref]

S. Nocentini, F. Riboli, M. Burresi, D. Martella, C. Parmeggiani, and D. S. Wiersma, “Three-dimensional photonic circuits in rigid and soft polymers tunable by light,” ACS Photonics 5(8), 3222–3230 (2018).
[Crossref]

C. C. Tartan, J. J. Sandford O’Neill, P. S. Salter, J. Aplinc, M. J. Booth, M. Ravnik, S. M. Morris, and S. J. Elston, “Read on demand images in laser-written polymerizable liquid crystal devices,” Adv. Opt. Mater. 6(20), 1800515 (2018).
[Crossref]

2017 (16)

A. A. Bauhofer, S. Krödel, J. Rys, O. R. Bilal, A. Constantinescu, and C. Daraio, “Harnessing photochemical shrinkage in direct laser writing for shape morphing of polymer sheets,” Adv. Mater. 29(42), 1703024 (2017).
[Crossref] [PubMed]

B. A. Kowalski, T. C. Guin, A. D. Auguste, N. P. Godman, and T. J. White, “Pixelated polymers: directed self assembly of liquid crystalline polymer networks,” ACS Macro Lett. 6(4), 436–441 (2017).
[Crossref]

D. Martella, S. Nocentini, D. Nuzhdin, C. Parmeggiani, and D. S. Wiersma, “Photonic microhand with autonomous action,” Adv. Mater. 29(42), 1704047 (2017).
[Crossref] [PubMed]

D. Martella, D. Antonioli, S. Nocentini, D. S. Wiersma, G. Galli, M. Laus, and C. Parmeggiani, “Light activated non-reciprocal motion in liquid crystalline networks by designed microactuator architecture,” RSC Advances 7(32), 19940–19947 (2017).
[Crossref]

C. C. Tartan, P. S. Salter, T. D. Wilkinson, M. J. Booth, S. M. Morris, and S. J. Elston, “Generation of 3-dimensional polymer structures in liquid crystalline devices using direct laser writing,” RSC Advances 7(1), 507–511 (2017).
[Crossref]

Y. H. Lee, D. Franklin, F. Gou, G. Liu, F. Peng, D. Chanda, and S. T. Wu, “Two-photon polymerization enabled multi-layer liquid crystal phase modulator,” Sci. Rep. 7(1), 16260 (2017).
[Crossref] [PubMed]

Z. He, Y. H. Lee, F. Gou, D. Franklin, D. Chanda, and S. T. Wu, “Polarization-independent phase modulators enabled by two-photon polymerization,” Opt. Express 25(26), 33688–33694 (2017).
[Crossref]

K. Gao, C. McGinty, H. Payson, S. Berry, J. Vornehm, V. Finnemeyer, B. Roberts, and P. Bos, “High-efficiency large-angle Pancharatnam phase deflector based on dual-twist design,” Opt. Express 25(6), 6283–6293 (2017).
[Crossref] [PubMed]

F. Gou, F. Peng, Q. Ru, Y.-H. Lee, H. Chen, Z. He, T. Zhan, K. L. Vodopyanov, and S.-T. Wu, “Mid-wave infrared beam steering based on high-efficiency liquid crystal diffractive waveplates,” Opt. Express 25(19), 22404–22410 (2017).
[Crossref] [PubMed]

D. Franklin, R. Frank, S. T. Wu, and D. Chanda, “Actively addressed single pixel full-colour plasmonic display,” Nat. Commun. 8, 15209 (2017).
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Y. H. Lin, Y. J. Wang, and V. Reshetnyak, “Liquid crystal lenses with tunable focal length,” Liq. Cryst. Rev. 5(2), 111–143 (2017).
[Crossref]

H. W. Chen, R. D. Zhu, J. He, W. Duan, W. Hu, Y. Q. Lu, M. C. Li, S. L. Lee, Y. J. Dong, and S. T. Wu, “Going beyond the limit of an LCD’s color gamut,” Light Sci. Appl. 6(9), e17043 (2017).
[Crossref] [PubMed]

A. H. Gelebart, D. Jan Mulder, M. Varga, A. Konya, G. Vantomme, E. W. Meijer, R. L. B. Selinger, and D. J. Broer, “Making waves in a photoactive polymer film,” Nature 546(7660), 632–636 (2017).
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H. Shahsavan, L. Yu, A. Jákli, and B. Zhao, “Smart biomimetic micro/nanostructures based on liquid crystal elastomers and networks,” Soft Matter 13(44), 8006–8022 (2017).
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A. Komar, Z. Fang, J. Bohn, J. Sautter, M. Decker, A. Miroshnichenko, T. Pertsch, I. Brener, Y. Kivshar, I. Staude, and D. Neshev, “Electrically tunable all-dielectric optical metasurfaces based on liquid crystals,” Appl. Phys. Lett. 110(7), 071109 (2017).
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Y. H. Lee, G. Tan, T. Zhan, Y. Weng, G. Liu, F. Gou, F. Peng, N. V. Tabiryan, S. Gauza, and S. T. Wu, “Recent progress in Pancharatnam-Berry phase optical elements and the applications for virtual/augmented realities,” Opt. Data Process. Storage 3(1), 79–88 (2017).
[Crossref]

2016 (6)

H. K. Bisoyi and Q. Li, “Light-driven liquid crystalline materials: from photo-induced phase transitions and property modulations to applications,” Chem. Rev. 116(24), 15089–15166 (2016).
[Crossref] [PubMed]

Z. Ji, X. Zhang, B. Shi, W. Li, W. Luo, I. Drevensek-Olenik, Q. Wu, and J. Xu, “Compartmentalized liquid crystal alignment induced by sparse polymer ribbons with surface relief gratings,” Opt. Lett. 41(2), 336–339 (2016).
[Crossref] [PubMed]

W. Ji, C. H. Lee, P. Chen, W. Hu, Y. Ming, L. Zhang, T. H. Lin, V. Chigrinov, and Y. Q. Lu, “Meta-q-plate for complex beam shaping,” Sci. Rep. 6(1), 25528 (2016).
[Crossref] [PubMed]

H. Chen, Y. Weng, D. Xu, N. V. Tabiryan, and S. T. Wu, “Beam steering for virtual/augmented reality displays with a cycloidal diffractive waveplate,” Opt. Express 24(7), 7287–7298 (2016).
[Crossref] [PubMed]

C. C. Tartan, P. S. Salter, M. J. Booth, S. M. Morris, and S. J. Elston, “Localised polymer networks in chiral nematic liquid crystals for high speed photonic switching,” J. Appl. Phys. 119(18), 183106 (2016).
[Crossref]

S. Nocentini, D. Martella, C. Parmeggiani, and D. S. Wiersma, “Photoresist design for elastomeric light tunable photonic devices,” Materials (Basel) 9(7), 525 (2016).
[Crossref] [PubMed]

2015 (12)

H. Zeng, P. Wasylczyk, C. Parmeggiani, D. Martella, M. Burresi, and D. S. Wiersma, “Light-fueled microscopic walkers,” Adv. Mater. 27(26), 3883–3887 (2015).
[Crossref] [PubMed]

C. H. Ho, Y. C. Cheng, L. Maigyte, H. Zeng, J. Trull, C. Cojocaru, D. S. Wiersma, and K. Staliunas, “Controllable light diffraction in woodpile photonic crystals filled with liquid crystal,” Appl. Phys. Lett. 106(2), 021113 (2015).
[Crossref]

A. M. Flatae, M. Burresi, H. Zeng, S. Nocentini, S. Wiegele, C. Parmeggiani, H. Kalt, and D. Wiersma, “Optically controlled elastic microcavities,” Light Sci. Appl. 4(4), e282 (2015).
[Crossref]

D. Franklin, Y. Chen, A. Vázquez-Guardado, S. Modak, J. Boroumand, D. Xu, S. T. Wu, and D. Chanda, “Polarization-independent actively tunable colour generation on imprinted plasmonic surfaces,” Nat. Commun. 6(1), 7337 (2015).
[Crossref] [PubMed]

J. Kim, Y. Li, M. N. Miskiewicz, C. Oh, M. W. Kudenov, and M. J. Escuti, “Fabrication of ideal geometric-phase holograms with arbitrary wavefronts,” Optica 2(11), 958–964 (2015).
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J. K. Hohmann, M. Renner, E. H. Waller, and G. von Freymann, “Three-Dimensional μ-Printing: an enabling technology,” Adv. Opt. Mater. 3(11), 1488–1507 (2015).
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H. Zeng, P. Wasylczyk, G. Cerretti, D. Martella, C. Parmeggiani, and D. S. Wiersma, “Alignment engineering in liquid crystalline elastomers: free-form microstructures with multiple functionalities,” Appl. Phys. Lett. 106(11), 111902 (2015).
[Crossref]

H. Chen, Z. Luo, R. Zhu, Q. Hong, and S. T. Wu, “Tuning the correlated color temperature of white LED with a guest-host liquid crystal,” Opt. Express 23(10), 13060–13068 (2015).
[Crossref] [PubMed]

C.-C. Huang, Y.-Y. Kuo, S.-H. Chen, W.-T. Chen, and C.-Y. Chao, “Liquid-crystal-modulated correlated color temperature tunable light-emitting diode with highly accurate regulation,” Opt. Express 23(3), A149–A156 (2015).
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M. Kim, K. J. Park, S. Seok, J. M. Ok, H.-T. Jung, J. Choe, D. H. Oh, and D. H. Kim, “Fabrication of microcapsules for dye-doped polymer-dispersed liquid crystal-based smart windows,” ACS Appl. Mater. Interfaces 7(32), 17904–17909 (2015).
[Crossref] [PubMed]

J. Sautter, I. Staude, M. Decker, E. Rusak, D. N. Neshev, I. Brener, and Y. S. Kivshar, “Active tuning of all-dielectric metasurfaces,” ACS Nano 9(4), 4308–4315 (2015).
[Crossref] [PubMed]

T. J. White and D. J. Broer, “Programmable and adaptive mechanics with liquid crystal polymer networks and elastomers,” Nat. Mater. 14(11), 1087–1098 (2015).
[Crossref] [PubMed]

2014 (4)

J. L. Digaum, J. J. Pazos, J. Chiles, J. D’Archangel, G. Padilla, A. Tatulian, R. C. Rumpf, S. Fathpour, G. D. Boreman, and S. M. Kuebler, “Tight control of light beams in photonic crystals with spatially-variant lattice orientation,” Opt. Express 22(21), 25788–25804 (2014).
[Crossref] [PubMed]

R. M. Hyman, A. Lorenz, S. M. Morris, and T. D. Wilkinson, “Polarization-independent phase modulation using a blue-phase liquid crystal over silicon device,” Appl. Opt. 53(29), 6925–6929 (2014).
[Crossref] [PubMed]

H. Zeng, D. Martella, P. Wasylczyk, G. Cerretti, J. C. G. Lavocat, C. H. Ho, C. Parmeggiani, and D. S. Wiersma, “High-resolution 3D direct laser writing for liquid-crystalline elastomer microstructures,” Adv. Mater. 26(15), 2319–2322 (2014).
[Crossref] [PubMed]

L. Yang, J. Li, Y. Hu, C. Zhang, Z. Lao, W. Huang, and J. Chu, “Projection two-photon polymerization using a spatial light modulator,” Opt. Commun. 331, 82–86 (2014).
[Crossref]

2013 (3)

F. Serra, S. M. Eaton, R. Cerbino, M. Buscaglia, G. Cerullo, R. Osellame, and T. Bellini, “Nematic liquid crystals embedded in cubic microlattices: memory effects and bistable pixels,” Adv. Funct. Mater. 23(32), 3990–3994 (2013).
[Crossref]

R. K. Komanduri, K. F. Lawler, and M. J. Escuti, “Multi-twist retarders: broadband retardation control using self-aligning reactive liquid crystal layers,” Opt. Express 21(1), 404–420 (2013).
[Crossref] [PubMed]

K. Obata, A. El-Tamer, L. Koch, U. Hinze, and B. N. Chichkov, “High-aspect 3D two-photon polymerization structuring with widened objective working range (WOW-2PP),” Light Sci. Appl. 2(12), e116 (2013).
[Crossref]

2012 (9)

W. Hu, A. Kumar Srivastava, X. W. Lin, X. Liang, Z. J. Wu, J. T. Sun, G. Zhu, V. Chigrinov, and Y. Q. Lu, “Polarization independent liquid crystal gratings based on orthogonal photoalignments,” Appl. Phys. Lett. 100(11), 111116 (2012).
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W. Hu, A. Srivastava, F. Xu, J. T. Sun, X. W. Lin, H. Q. Cui, V. Chigrinov, and Y. Q. Lu, “Liquid crystal gratings based on alternate TN and PA photoalignment,” Opt. Express 20(5), 5384–5391 (2012).
[Crossref] [PubMed]

T. Bückmann, N. Stenger, M. Kadic, J. Kaschke, A. Frölich, T. Kennerknecht, C. Eberl, M. Thiel, and M. Wegener, “Tailored 3D mechanical metamaterials made by dip-in direct-laser-writing optical lithography,” Adv. Mater. 24(20), 2710–2714 (2012).
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Y. J. Liu, G. Y. Si, E. S. P. Leong, N. Xiang, A. J. Danner, and J. H. Teng, “Light-driven plasmonic color filters by overlaying photoresponsive liquid crystals on gold annular aperture arrays,” Adv. Mater. 24(23), OP131–OP135 (2012).
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O. Yaroshchuk and Y. Reznikov, “Photoalignment of liquid crystals: basics and current trends,” J. Mater. Chem. 22(2), 286–300 (2012).
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J.-H. Na, S.-C. Park, S.-U. Kim, Y. Choi, and S.-D. Lee, “Physical mechanism for flat-to-lenticular lens conversion in homogeneous liquid crystal cell with periodically undulated electrode,” Opt. Express 20(2), 864–869 (2012).
[Crossref] [PubMed]

A. Orth and K. Crozier, “Microscopy with microlens arrays: high throughput, high resolution and light-field imaging,” Opt. Express 20(12), 13522–13531 (2012).
[Crossref] [PubMed]

H. Yoshida, “Functionalisation of cholesteric liquid crystals by direct laser writing,” Liq. Cryst. Today 21(1), 3–19 (2012).
[Crossref]

C. P. Jisha, K.-C. Hsu, Y. Lin, J.-H. Lin, C.-C. Jeng, and R.-K. Lee, “Tunable pattern transitions in a liquid-crystal-monomer mixture using two-photon polymerization,” Opt. Lett. 37(23), 4931–4933 (2012).
[Crossref] [PubMed]

2011 (3)

H. G. Park, J. J. Lee, K. Y. Dong, B. Y. Oh, Y. H. Kim, H. Y. Jeong, B. K. Ju, and D. S. Seo, “Homeotropic alignment of liquid crystals on a nano-patterned polyimide surface using nanoimprint lithography,” Soft Matter 7(12), 5610–5614 (2011).
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F. Klein, B. Richter, T. Striebel, C. M. Franz, G. von Freymann, M. Wegener, and M. Bastmeyer, “Two-component polymer scaffolds for controlled three-dimensional cell culture,” Adv. Mater. 23(11), 1341–1345 (2011).
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Figures (16)

Fig. 1
Fig. 1 Working principles of (a) traditional UV photolithography and (b) TPP. The dashed rectangles highlight the polymerized volume. In traditional photolithography, almost all the exposed volume is polymerized due to the low threshold of single-photon absorption. While for TPP, only a localized focal volume is polymerized.
Fig. 2
Fig. 2 Three working configurations of TPP: (a) air mode, (b) oil-immersion mode, and (c) dip-in mode. Among them, dip-in mode suffers from least aberration and thus offers highest resolution.
Fig. 3
Fig. 3 The main feature of (a) 1D and (b) 2D binary LC gratings on one substrate. The green lines denote the direction of microgroove alignment and the yellow ellipsoids show the LC directors near the alignment surface.
Fig. 4
Fig. 4 SEM images of (a, b) 1D and (c, d) 2D grating alignment on one substrate where the nanogrooves have 300-nm period and the red arrows highlight the local LC alignment directions. Scale bar: 20 μm (a), 2 μm (b), 20 μm (c), and 2 μm (d).
Fig. 5
Fig. 5 (a) Schematic illustration of the patterned alignment layer on one substrate for a PBMLA. The highlight shows the desired LC alignment for a single microlens. In experiment, P = 148 μm. (b, c) Working principle of the PBMLA. When the PBMLA serves as converging lenses for RCP, it serves as diverging lenses for LCP. PBMLA: Pancharatnam-Berry microlens array.
Fig. 6
Fig. 6 SEM images of the imprinted alignment for a PB microlens array. (a) Top view where (b, c) are zoom-in views showing the nanogrooves orienting at different directions. (d) Angled view. Scale bar: 20 μm (a), 2 μm (b), 2 μm (c), and 10 μm (d).
Fig. 7
Fig. 7 Schematic plot of two-state degenerate alignment achieved by TPP. Polymerizing the lines (green grids) in both orthogonal directions in plane, the LC directors (yellow ellipsoids) are degenerate in both diagonal directions.
Fig. 8
Fig. 8 (a) Schematic illustration of the composite-lens type LC microlens array. (b, c) An example showing the working principle of RMLA. As a bi-focal lens, the focal length is tunable only for one linear polarization of light (parallel to the alignment direction). By applying different voltages, the RMLA can either diverge or converge input light. RMLA: refractive microlens array.
Fig. 9
Fig. 9 Slanted SEM images of the passive refractive microlens array with nanogroove alignment fabricated by TPP. (a) Top view where (b) is a zoom-in view showing the nanogrooves orienting uniformly toward one direction as the red arrow denotes. Scale bar: 20 μm (a), 2 μm (b).
Fig. 10
Fig. 10 Schematic illustration of three dual-layer LC devices: (a) phase modulator, (b) polarization-independent phase modulator and (c) TN. The green structures are 3D scaffolds fabricated by TPP and the yellow ellipsoids are LC directors.
Fig. 11
Fig. 11 Step-by-step fabrication process of the 3D scaffold for dual-layer TN. After dropping photoresist on the substrate, a laser lithography system is applied to expose the photoresist line-by-line. During the exposure, the entire structure will be divided into many writing fields, determined by the objective and the laser scanning system. The polymerization process happens at one writing field at a time. (a) Line-by-line exposure to form a uniform groove alignment covering one writing field. (b) After the bottom alignment layer is exposed, the laser will then polymerize pillars that are used to support the floating layer. (c) Line-by-line exposure to form the floating layer which offers alignment at 45° relative to the bottom alignment. (d) After finishing this writing field, the laser system will move to the next writing field and keep scanning until the whole structure is accomplished.
Fig. 12
Fig. 12 A dual-layer TN sample compared to a conventional single-layer TN sample with the same total effective LC thickness of 3.3 µm. (a) The relaxation time (intensity change from 100% to 10% when releasing from 10 Vrms) of dual- and single-layer TN samples are 3.3 ms and 11.8 ms, respectively. Inset: the SEM image of a two-layer scaffold, scale bar: 2 µm. The white arrows denote the alignment direction at each layer. (b) The measured voltage-dependent transmittance changes of the dual-layer and single-layer TN samples at λ = 633 nm.
Fig. 13
Fig. 13 SEM images of a four-layer scaffold for a phase modulator: (a) Top view and (b) side view. From the side view, it can be clearly distinguished that there are three polymer layers. Scale bar: 20 µm (a) and 2 µm (b).
Fig. 14
Fig. 14 The schematic diagram of the proposed twisted-nematic diffractive waveplate with an additional floating alignment layer enabled by two photon polymerization.
Fig. 15
Fig. 15 Simulated diffraction efficiency for a TN diffractive waveplate with circularly polarized incident light.
Fig. 16
Fig. 16 SEM images of a peeled-off four-layer scaffold for a phase modulator: (a) Top view and (b) side view. Scale bar: 20 µm (a) and 2 µm (b).

Equations (8)

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dΔn=( 3 /2)λ,
J ± = 1 2 [ cos2φ sin2φ sin2φ cos2φ ][ 1 ±i ]= 1 2 [ 1 i ] e ±2iφ .
δ=2πdΔn/λ.
τ= γ 1 d s 2 /(K π 2 ),
ϕ twist << Γ 2 = πΔnd λ ,
ϕ LC =2π z d x p ,
[ E x E y ]= 1 2 cos( Γ 2 )[ 1 i ] e i πx/p 1 2 sin( Γ 2 )[ i 1 ] e i πx/p .
[ E x E y ]= 1 2 [ e iΓ/2 i e iΓ/2 ] e i πx/p .

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