Low f ‐Number Diffraction‐Limited Pancharatnam–Berry

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Low f-Number Diffraction-Limited Pancharatnam–Berry Microlenses Enabled by Plasmonic Photopatterning of Liquid Crystal Polymers

Miao Jiang, Yubing Guo, Hao Yu, Ziyuan Zhou, Taras Turiv, Oleg D. Lavrentovich,

and Qi-Huo Wei*

Dr. M. Jiang, Dr. Y. Guo, H. Yu, Z. Zhou, T. Turiv, Prof. O. D. Lavrentovich, Prof. Q.-H. WeiAdvanced Materials and Liquid Crystal InstituteKent State UniversityKent, OH 44242, USAE-mail: [email protected]

Prof. O. D. Lavrentovich, Prof. Q.-H. WeiDepartment of PhysicsKent State UniversityKent, OH 44242, USA

The ORCID identification number(s) for the author(s) of this article can be found under https://doi.org/10.1002/adma.201808028.

DOI: 10.1002/adma.201808028

For example, phase profiles required for lensing can be realized by sculpturing dielectric substrates with discretized steps or sloped ridges through photo lithography and reactive ion etching.[18–20] These lithography processes make diffractive microlenses more tailorable than refrac-tive microlenses in terms of filling factor and low f-number. However, to achieve high performances, very complex fabrica-tion processes are needed. For example, up to 8 lithography steps and sometimes electron-beam lithography are needed to achieve low f-numbers,[21–23] diffrac-tion-limited performance, and over 90% efficiency.[19,24–26]

The phase profiles desired for lensing can also be realized with the Pancharatnam–Berry (PB) phases[27,28] by using bire-fringent materials or metasurfaces with spatially variant optical axes.[13,14,21,29–34] The PB microlenses made of liquid crystals are particularly attractive because of their close to 100% efficiency and switchable focal lengths.[31,33,35,36] Large-sized PB lenses with low f-numbers can be made by holography photopat-terning with a refractive master lens,[31] while liquid crystal PB microlenses are still limited to large f-number (>10)[32,33] and no work has been able to show diffraction-limited quality.

A range of techniques have been developed in recent years to align liquid crystal molecules into arbitrary designer orien-tation patterns, which are either based on photoalignments using digital micromirror device (DMD),[37,38] pixel-to-pixel direct laser writing,[30] holography interference[31] and plas-monic photo patterning,[39–41] or based on nanostructured sur-faces by using nanoimprinting[33] or atomic force microscopy scribing.[42] These techniques have enabled various applications and research ranging from optical devices to programmable origami.[30,33,37,38,43–47]

Here, we show that high-quality microlenses based on PB phases can be designed and made with liquid crystal poly-mers by using a plasmonic photopatterning technique. The plasmonic photopatterning technique allows for arbitrary molecular orientations encoded in designs of so-called plas-monic metamasks. We designed and fabricated microlenses with a set of focal lengths and f-numbers to test the quality and resolution limit of the plasmonic photopatterning tech-nique. As discussed later, the microlenses with f-number down to 2 requires 1.5 µm of smallest molecular pitches.

Microlenses are desired by a wide range of industrial applications while it

is always challenging to make them with diffraction-limited quality. Here,

it is shown that high-quality microlenses based on Pancharatnam–Berry

(PB) phases can be made with liquid crystal polymers by using a plasmonic

photopatterning technique. Based on the generalized Snell’s law for the PB

phases, PB microlenses with a range of focal lengths and f-numbers are

designed and fabricated and their point-spread functions and ability to image

micrometer-sized particles are carefully characterized. The results show that

these PB microlenses with f-number down to 2 are all diffraction-limited. The

capability of arraying these PB microlenses with 100% filling factor with a

step-and-flash approach is further demonstrated.

Microlenses

Microlenses with sub-millimeter diameters are indispensable optical components in a wide variety of miniaturized systems for applications,[1,2] covering optical interconnects,[3,4] Hartman wave front sensing,[5–7] beam homogenization,[8,9] visual system of microrobots,[10,11] light emitting diode display,[12] and vir-tual/augmented reality.[13,14] They can be either refractive or diffractive, depending on how the wavefronts are altered. Refractive microlenses refracting light according to the Snell’s law are miniaturized versions of traditional optical lenses,[1,2] and are mass-produced by industry through processes such as photoresists thermal reflow, gray scale photolithography, and direct laser writing.[2,15–17] However, the filling factor and corrections for spherical aberrations of these refractive micro-lenses are often limited by the spherical lens shapes.[16]

Diffractive microlenses rely on phase gradients generated by micro/nanostructured surfaces to alter optical wavefronts and can be engineered in a wide variety of geometrical shapes.[18]

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We also carefully characterized the point-spread functions (PSFs) and imaging capability of these PB microlenses. The results show that these microlenses are all diffraction-limited and of high quality. Furthermore, these PB microlenses can be arrayed with a 100% filling factor in a step-and-flash approach.

When passing through a birefringent material, light acquires a phase difference, named PB phase which is primarily dependent on the orientation of the optical axis.[44,48] This PB phase can be readily seen using the Jones calculus. For a thin film of birefringent materials with optical axes in the film plane, its Jones matrix can be generally expressed as T = R(−θ)T′R(θ). Here the optical axis is assumed to be oriented at an angle θ with respect to the x-axis, R(θ) is the rotation matrix, and

T10

0

e i′ =

− Γ is the Jones matrix of the film when the

slow optical axis is along the x-axis. The phase retardation Γ between the ordinary and extraordinary beams is defined

as n n tπ

λΓ = −

2( )

0

e o , where t is the film thickness, λ0 is the

wavelength of light in air and ne and no are the refractive indices for the extraordinary and ordinary beams, respec-tively. When the birefringent film is a half-wave retarder, the

Jones matrix can be simplified as cos2 sin 2sin 2 cos2

T θ θθ θ

=−

.

With circularly polarized incident light represented by the

Jones vector 1

2

1i

J = ±

± , the output light beam, represented by

the Jones vector J T1

2

1i

1

2

1i

e 2i′ = ⋅ ±

=

θ±

± , is circularly polar-

ized with the opposite handedness and an additional phase Φ = ± 2θ, i.e., the PB phase.

In comparison with crystalline birefringent materials such as quartz, the liquid crystals can be advantageously aligned with spatially variant orientations, leading to a spatially non-uniform, PB phase profile. The gradients of the PB phase cause the transmitted light to change its propagation direction. It can be shown that when the thickness of the liquid crystal film is smaller than the characteristic length scale over which the PB phase varies by π, the incident (θi) and transmitted (θt) angle can be related to the PB phase Φ by the so-called generalized Snell’s law[44]

sin sin2

d

dt i

0

xθ θ

λ

π( ) ( )− =

Φ

(1)

where the medium outside the liquid crystal film is assumed to be air. To focus a parallel incident beam with θi = 0 onto the focal point, the bending angle of the transmitted light should satisfy sin( ) /t

2 2r r fθ = + , where f is the focal length and r is the radial coordinate from the lens center. Combining this condi-tion with Equation (1) yields the required PB phase profile Φ of a PB lens

22

0

2 2f r fθπ

λ( )Φ = = + −

(2)

When θ = mπ, we can define 2202

0r m m fm λ λ= + and the grating pitch p = rm + 1 − rm (over which the director rotates by π). For large m and f ≫ mλ, it can be shown that p ≈ λ0f/rm.

For a square microlens with width l and focal length f, the f-number Nf is defined as Nf = f/l. Considering that the maximum of rm is 2 / 2r lm = , the f-number of the PB micro-lens can be related to the smallest pitch and the working wavelength as

2 2 0

Nf

r

pf

m λ= ≈

(3)

For a given working wavelength, the smallest achievable f-number for a PB microlens is dictated by the smallest grating pitch p.

As testbeds for the fabrication techniques, we designed square-shaped PB microlenses with two different widths and a range of focal lengths working at wavelength λ0 = 532 nm. The design parameters are summarized in Table1. The corre-sponding f-number is varied between 2 and 30, meaning that the desired smallest pitch for the molecular orientation is 1.5 µm.

To fabricate these designed PB microlenses, we combined a plasmonic photopatterning technique with photopoly-merization.[39,40] The plasmonic photopatterning technique uses engineered plasmonic metamasks (PMMs) to pattern molecular orientations in a way similar to projection photolithography.[39,40] Unlike the traditional photomasks used in photolithography which generate spatial modulations of light intensity, the PMMs generate light with spatially variant polarization orientations.[39] By projecting such structured light onto films of photoalignment materials such as the azo-dye brilliant yellow, the rod-shaped photoactive molecules are induced to reorient themselves with long axes perpendicular to the local polarization direction of the exposing light.[49,50]

The PMMs used in the device fabrication are composed of rectangular nanoaperture arrays carved in Al films.[39] The key steps in fabricating the PMMs include successive deposition of Al and SiO2, electron-beam lithography, and sequential reac-tive ion etching of SiO2 and Al. The rectangular nanoapertures are positioned in a triangular lattice and oriented with their long axes following the target orientations of the liquid crystal molecules. Further details of the processes can be found in two previous publications.[39,40]

Figure 1a presents an optical microscopy image of a PMM observed under a polarization analyzer, showing that the trans-mitted light is polarized with spatially variant polarization direc-tions. The scanning electron microscopy (SEM) image shows that the fabricated PMM is composed of rectangular holes in an Al film positioned in a triangular lattice with a 270 nm perio-dicity (Figure 1b). The smallest orientational pitch of the hole arrays is ≈1.5 µm. To determine the polarization orientation of the transmitted light, the PMM is illuminated with nonpolar-ized light and imaged with a polarization analyzer at different orientation angle θ. By fitting the intensity I of each pixel with

Adv. Mater. 2019, 1808028

Table 1. Parameters of the designed PB microlenses.

N = 2 N = 5 N = 15 N = 30

l = 150 µm f = 300 µm f = 750 µm f = 2250 µm f = 4500 µm

l = 300 µm f = 1500 µm f = 4500 µm f = 9000 µm



a function I = I0 cos2(θ + θ0), we can determine the polarization direction θ0. Figure 1c shows the measured spatially varying polarization orientation corresponding to the area shown in Figure 1b. It can be observed that the local light polarization is perpendicular to the long axis of the nanoapertures, which is in agreement with the design expectation.

To print the predesigned molecular orientations in the photo-alignment films, a home-built projection photopatterning system was used to project the polarized light generated by the PMMs onto thin films of brilliant yellow spin-coated on glass substrates. In addition, a 1:1 magnification was used in the projection to maximize the spatial resolution of the photopatterning. On the substrates with patterned photoalignment films, a toluene solution of RM257 (Wilshire Technology) and photo initiator I651 (CIBA-GEIGY Corp.) at 400:20:1 weight ratio was spin-coated at a speed of 3000 RPM for 30 s and then polymerized under vacuum at room temperature by illumination of UV light (365, 80 nm bandwidth) for 15 min. Three additional layers of the RM257 were then coated sequentially in a similar fashion. The concentration of the RM257 and spin speed were fine-tuned to obtain the designed phase retardation of 266 nm. The phase retardation and molecular orientation after coating each addi-tional layer of RM257 were examined by using a PolScope.[51]

A representative PB microlens fabricated by the metamask in Figure 1a is shown in Figure 1d. The molecular orientation in this PB microlens measured by the PolScope (Figure 1e,f) shows good agreements with the designed molecular orien-tation pattern, i.e., that of the rectangular holes in the PMM. It can also be observed that the PB microlens with 1.5 µm smallest pitch is well produced with almost no defects. Two

objectives used in our system have identical numerical aperture NA = 0.42, so the diffraction limited resolution based on the Rayleigh criteria can be estimated as R = 0.61λ/NA ≈ 0.7 µm. Since one pitch corresponds to the distance over which the LC molecules rotate by π, the smallest pitch of 1.5 µm achieved in PB microlens fabrication indicates that the resolution of plas-monic photopatterning goes beyond the Rayleigh criteria. This resolution is the highest among the existing photoalignment techniques. Based on the literature, the smallest pixel sizes achieved are ≈2 µm in the DMD-based photoalignment[37] and 4 µm in laser direct writing,[30] meaning 10 µm pitches.

The physical mechanisms behind this high resolution in photopatterning may be related to the superoscillatory focusing, as the plasmonic metamasks can be considered as phase masks.[52,53] According to Huang et al., the resolution of sup-eroscillatory focusing can go below 0.38λ/NA ≈ 0.4 µm, which seems to be in agreement with our experiments.[53] Further studies are necessary in the future to elucidate the dependence of the photopatterning resolution on metamask patterns and the physics behind.

Figure 2a presents the optical setup used to measure the PSFs of the PB microlenses. A laser beam from a solid-state laser with wavelength λ0 = 532 nm passes through a polarizer and a quarter wave plate to convert into circularly polarized light. The beam size is then expanded and collimated by using two convex lenses with 20 and 200 mm focal lengths, respec-tively. The expanded collimated laser beam is then focused by a PB microlens. An Olympus 40× objective with numerical aper-ture NA = 0.6 is used together with a tube lens to image the focal spot and couple it onto a CCD camera.

Adv. Mater. 2019, 1808028

Figure 1. a) A polarized optical microscopy image of the fabricated PMM for the microlens with f-number Nf = 2. b) SEM image of the blue square area in (a). c) Measured polarization directions (red bars) of light transmitted through the area in (b). d) Cross-polarized microscopy image of the PB microlens fabricated using the PMM in (a). e,f) Molecular orientations in the blue and green boxes in (d) measured using the PolScope.



Figure 2b,c present experimental results for two exemplary PB microlenses: one for Nf = 2 and l = 150 µm (Figure 2b) and the other for Nf = 5 and size l = 300 µm (Figure 2c). The insets at the top left are images of the focused laser spots, and the data points are the intensity profiles along horizontal axis through the spot center. As the image of the focal spot has gone through two steps of diffractions sequentially by the PB micro-lens and the objective, the measured intensity profile should be the square of the convolution between the amplitude PSFs of the microlens and the objective, which can be expressed math-ematically as I(r) = I0 [f*g]2, where f(r) is the amplitude PSF of the PB microlens and g(r) is the amplitude PSF of the objective.

High-quality objectives are considered as diffraction-limited and their amplitude PSFs due to diffraction by circular aper-tures are the Airy patterns, i.e.,[54]

/1g r J kr kr( )( ) = (4)

where J1 is the first-order Bessel function and k = π · NA/λ0. With NA = 0.6, λ0 = 0.532 µm in our experiments, the diam-eter D of the Airy disc, defined as the first root to the zero of the Bessel function, can be obtained from J1 (kD/2) = 0 or D = 1.22λ0/NA = 1.08 µm.

For a square aperture with width l, the far-field amplitude due to Fraunhofer diffraction can be written as[54]

sin sinf r

x

x

y

y

α

α

α

α

( ) =

(5)

where α = πl/dλ0 with d being the distance from the aperture. The intensity distribution measured by the CCD camera can thus be expressed as

rr rr rr rrI x y I f g, d0

2

∫∫( ) ( ) ( )= ′ − ′ ′

−∞

∞

(6)

To extract the PSFs of the PB microlenses, we use this con-voluted intensity distribution I(x, y) to fit the experimental data by using α and I0 as the fitting parameters. Figure 2b,c shows that the experimental data can be well fitted with Equation (6). Based on the microlens PSFs obtained from the fitting (e.g., red lines in Figure 2b,c), we can obtain their spot diameters D. The results are plotted as a function of the f-number Nf in Figure 2d.

For an aberration-free microlens with a square shape and focal length f, the amplitude PSF is just Equation (5) with d replaced by the focal length f. The first root of the PSF is αD/2 = π, so the focal spot diameter can be expressed as

2 / 20 0D f l N fλ λ= = (7)

The experimental data are in perfect agreements with this expression (red line in Figure 2d). This shows that these fabricated PB microlenses are diffraction-limited and of high quality.

To further illustrate the diffraction-limited quality of the fab-ricated PB microlenses, we also used a single PB microlens to image colloidal particles of 5 µm in diameter. Because the small focal lengths of the microlenses make it hard to couple such images onto a CCD camera, we use a microscope to observe the colloidal images formed by the microlens. The setup is shown schematically in Figure 3a. Circularly polarized white light is used to illuminate the colloidal sample from the bottom, and the PB microlens used in the experiment has a width l = 150 µm. The sign of focal length is determined by the hand-edness of the circularly polarized incident light. In this case, the focal length f = −750 µm. The separation between the inner sur-face of colloidal sample and the microlens was set at ≈300 µm (smaller than f). Based on the lens imaging formula, the col-loidal image formed by this microlens is a virtual image located at ≈200 µm from the microlens. Figure 3b–d shows the images obtained when the objective is focused on the particle plane,

Adv. Mater. 2019, 1808028

Figure 2. a) Schematic optical setup used to measure the PSFs. b,c) Measured (data points) and fitted (solid lines) intensity distributions of the focal spots for two representative microlenses with Nf = 2, l = 150 µm (b) and Nf = 5, l = 300 µm (c). Insets show the CCD images (left corner) and fitted images (right corner) of the focused spots. The blue lines represent the fittings with Equation (6). For clarity, the baselines are shifted vertically. d) Measured focal spot diameter D as a function of the f-number Nf. The red solid line shows the theoretical curve: D = 2λ0Nf.



the microlens plane and the colloidal image plane respectively. It can be seen that around the image center (Figure 3d), indi-vidual colloidal particles are clearly visible, indicating the capa-bility of this single PB microlens in imaging microsized objects. It is also visible that particles become blurry and colorful at the sample edges due to chromatic dispersion and finite field of view (Figure 3d). A schematic field of view angle <5° is marked with a circle in the image as a reference (Figure 3d).

This experiment represents a proof of concept that these PB microlenses can be used to image microstructures. Since the plasmonic photopatterning allows for fabricating such PB microlenses on almost all flat substrates with designable size and focal length, these microlenses can easily be integrated with various kinds of microsystems and photodetectors for imaging, sensing and optoelectronic applications.

Microlens arrays are also demanded in numerous applica-tions, for example, in light field imaging and displays.[55–58] These liquid crystal PB microlenses can easily be made in arrays by using a step-and-flash approach. As a proof of the capability, we fabricated a 10 × 10 array of the PB microlenses which show almost no defects under a cross-polarizing micro-scope (part of the array shown in Figure 4).

It is worth noting that the shapes of the PB microlenses can be designed to fill the whole 2D space, 100% filling factor can easily be achieved, for example, with square microlenses in a square lattice array shown here. This is important for various applications like beam shaping. For traditional refractive micro-lenses, it is often impossible to fill the whole 2D space with spherical microlens arrays. As a result, light passing through the empty space between microlenses is not focused, leading to zero-order leakage, stray light, and unoptimized performances.

To conclude, we show that diffraction limited Pan-charatnam–Berry microlenses with f-number as low as 2 can be designed and fabricated with the plasmonic photopatterning of liquid crystal polymers. We also demonstrated that the meas-ured point-spread functions of these PB microlenses are in excellent agreements with the diffraction limits and these PB microlenses can be used to image micrometer-sized particles. These PB microlenses can also be fabricated in arrays through a step-and-flash approach. In comparison with the refractive and other diffractive microlenses, the liquid crystal PB micro-lenses exhibit several advantages including diffraction-limited quality, high efficiency, potentially switchable functions and low costs, and are expected to find various unique applications.

Adv. Mater. 2019, 1808028

Figure 3. a) Schematic optical setup used to image 5 µm diameter colloid particles with one PB microlens. b–d) Images taken when the microscope objective is focused respectively on the colloid particles (b), on the microlens (c), and on the particle image formed by the microlens (d).



Adv. Mater. 2019, 1808028

Acknowledgements

M.J. and Y.G. contributed equally to this work. Financial support by the NSF through CMMI-1663394 and CMMI-1436565 is acknowledged. The author Q.-H.W. also acknowledges the financial support by the AvH foundation and Clemens Bechinger for his hospitality during his sabbatical in University of Konstanz when this manuscript was finished.

Conflict of Interest

The authors declare no conflict of interest.

Keywords

liquid crystal devices, microlenses, optical design and fabrication, Pancharatnam–Berry phase, photoalignments

Received: December 12, 2018

Revised: March 6, 2019

Published online:

[1] H. P. Herzig, Micro-Optics: Elements, Systems, and Applications,

Taylor & Francis, London 1997.

[2] P. Nussbaum, R. Völkel, H. P. Herzig, M. Eisner, S. Haselbeck,

Pure Appl. Opt. 1997, 6, 617.

[3] J. S. Leggatt, M. C. Hutley, Electron. Lett. 1991, 27, 238.

[4] Y. Yaacob, O. Mikami, C. Fujikawa, S. Nakajima, S. Ambran,

IOP Conf. Ser.: Mater. Sci. Eng. 2017, 210, 012029.

[5] G. E. Artzner, Opt. Eng. 1992, 31, 1311.

[6] M. Ares, S. Royo, J. Caum, Opt. Lett. 2007, 32, 769.

[7] M. M. Vekshin, A. S. Levchenko, A. V. Nikitin, V. A. Nikitin,

N. A. Yacovenko, Meas. Sci. Technol. 2010, 21, 054010.

[8] M. Zimmermann, N. Lindlein, R. Voelkel, K. J. Weible, Proc. SPIE

2007, 6663, 666302.

[9] F. Wippermann, U.-D. Zeitner, P. Dannberg, A. Bräuer, S. Sinzinger,

Opt. Express 2007, 15, 6218.

[10] Wook Choi, M. Akbarian, V. Rubtsov, Chang-Jin Kim, IEEE Trans.

Ind. Electron. 2009, 56, 1005.

[11] J. W. Duparré, F. C. Wippermann, Bioinspir. Biomim. 2006, 1, R1.

[12] J.-H. Lee, Y.-H. Ho, K.-Y. Chen, H.-Y. Lin, J.-H. Fang, S.-C. Hsu,

J.-R. Lin, M.-K. Wei, Opt. Express 2008, 16, 21184.

[13] Y.-H. Lee, G. Tan, T. Zhan, Y. Weng, G. Liu, F. Gou, F. Peng,

N. V. Tabiryan, S. Gauza, S.-T. Wu, Opt. Data Process. Storage 2017,

3, 79.

[14] T. Zhan, Y.-H. Lee, S.-T. Wu, Opt. Express 2018, 26, 4863.

[15] Z. D. Popovic, R. A. Sprague, G. A. Neville Connell, Appl. Opt. 1988,

27, 1281.

[16] H. Ottevaere, R. Cox, H. P. Herzig, T. Miyashita, K. Naessens,

M. Taghizadeh, R. Völkel, H. J. Woo, H. Thienpont, J. Opt. A: Pure

Appl. Opt. 2006, 8, S407.

[17] T. J. Suleski, D. C. O’Shea, Appl. Opt. 1995, 34, 7507.

[18] E. Noponen, J. Turunen, A. Vasara, J. Opt. Soc. Am. A 1993, 10, 434.

[19] M. Rossi, R. E. Kunz, H. P. Herzig, Appl. Opt. 1995, 34, 5996.

[20] M. R. Taghizadeh, P. Blair, B. Layet, I. M. Barton, A. J. Waddie,

N. Ross, Microelectron. Eng. 1997, 34, 219.

[21] N. Yu, F. Capasso, Nat. Mater. 2014, 13, 139.

[22] N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso,

Z. Gaburro, Science 2011, 334, 333.

[23] J. Scheuer, Nanophotonics 2017, 6, 137.

[24] J. Jahns, S. J. Walker, Appl. Opt. 1990, 29, 931.

Figure 4. A cross-polarized optical microscopy image of an exemplary PB microlens array.



Adv. Mater. 2019, 1808028

[25] T. Fujita, H. Nishihara, J. Koyama, Opt. Lett. 1982, 7, 578.

[26] T. Shiono, T. Hamamoto, K. Takahara, Appl. Opt. 2002, 41, 2390.

[27] S. Pancharatnam, Proc. – Indian Acad. Sci., Sect. A 1956, 44, 398.

[28] M. V. Berry, J. Mod. Opt. 1987, 34, 1401.

[29] N. V. Tabiryan, S. V. Serak, S. R. Nersisyan, D. E. Roberts,

B. Y. Zeldovich, D. M. Steeves, B. R. Kimball, Opt. Express 2016, 24,

7091.

[30] J. Kim, Y. Li, M. N. Miskiewicz, C. Oh, M. W. Kudenov, M. J. Escuti,

Optica 2015, 2, 958.

[31] K. Gao, H. Cheng, A. K. Bhowmik, P. J. Bos, Opt. Express 2015, 23,

26086.

[32] U. Ruiz, P. Pagliusi, C. Provenzano, E. Lepera, G. Cipparrone, Appl.

Opt. 2015, 54, 3303.

[33] Z. He, Y. Lee, R. Chen, D. Chanda, S. Wu, Opt. Lett. 2018, 43, 5062.

[34] H.-T. Chen, A. J. Taylor, N. Yu, Rep. Prog. Phys. 2016, 79, 076401.

[35] S. R. Nersisyan, N. V Tabiryan, D. M. Steeves, B. R. Kimball, Opt.

Photonics News 2010, 21, 40.

[36] N. V. Tabiryan, S. V. Serak, D. E. Roberts, D. M. Steeves,

B. R. Kimball, Opt. Express 2015, 23, 25783.

[37] C. Culbreath, N. Glazar, H. Yokoyama, Rev. Sci. Instrum. 2011, 82,

126107.

[38] H. Wu, W. Hu, H. Hu, X. Lin, G. Zhu, J.-W. Choi, V. Chigrinov, Y. Lu,

Opt. Express 2012, 20, 16684.

[39] Y. Guo, M. Jiang, C. Peng, K. Sun, O. Yaroshchuk, O. Lavrentovich,

Q.-H. Wei, Adv. Mater. 2016, 28, 2353.

[40] Y. Guo, M. Jiang, C. Peng, K. Sun, O. Yaroshchuk, O. Lavrentovich,

Q.-H. Wei, Crystals 2016, 7, 8.

[41] C. Peng, Y. Guo, T. Turiv, M. Jiang, Q.-H. Wei, O. D. Lavrentovich,

Adv. Mater. 2017, 29, 1606112.

[42] B. S. Murray, R. A. Pelcovits, C. Rosenblatt, Phys. Rev. E 2014, 90,

52501.

[43] P. Chen, Y.-Q. Lu, W. Hu, Liq. Cryst. 2016, 43, 2051.

[44] M. Jiang, H. Yu, X. Feng, Y. Guo, I. Chaganava, T. Turiv,

O. D. Lavrentovich, Q.-H. Wei, Adv. Opt. Mater. 2018, 6,

1800961.

[45] L. De Sio, D. E. Roberts, Z. Liao, J. Hwang, N. Tabiryan,

D. M. Steeves, B. R. Kimball, Appl. Opt. 2018, 57, A118.

[46] T. H. Ware, M. E. McConney, J. J. Wie, V. P. Tondiglia, T. J. White,

Science 2015, 347, 982.

[47] T. J. White, D. J. Broer, Nat. Mater. 2015, 14, 1087.

[48] L. Marrucci, C. Manzo, D. Paparo, Appl. Phys. Lett. 2006, 88, 2.

[49] Vladimir G. Chigrinov, Vladimir M. Kozenkov, H.-S. Kwok,

Vladimir G. Chigrinov, Vladimir M. Kozenkov, H.-S. Kwok,

V. G. Chigrinov, V. M. Kozenkov, H.-S. Kwok, Photoalignment of

Liquid Crystalline Materials: Physics and Applications, John Wiley &

Sons Ltd., West Sussex, UK 2008.

[50] O. Yaroshchuk, Y. Reznikov, J. Mater. Chem. 2012, 22, 286.

[51] M. Shribak, R. Oldenbourg, Appl. Opt. 2003, 42, 3009.

[52] E. T. F. Rogers, J. Lindberg, T. Roy, S. Savo, J. E. Chad, M. R. Dennis,

N. I. Zheludev, Nat. Mater. 2012, 11, 432.

[53] K. Huang, H. Ye, J. Teng, S. P. Yeo, B. Luk’yanchuk, C.-W. Qiu,

Laser Photonics Rev. 2014, 8, 152.

[54] M. Born, E. Wolf, Principles of Optics, Cambridge University Press,

Cambridge 1999.

[55] T. Hou, C. Zheng, S. Bai, Q. Ma, D. Bridges, A. Hu, W. W. Duley,

Appl. Opt. 2015, 54, 7366.

[56] Y. Lee, J. Baek, Y. Kim, J. U. Heo, Y. Moon, J. S. Gwag, C. Yu, J. Kim,

Opt. Express 2013, 21, 129.

[57] Z. He, Y.-H. Lee, D. Chanda, S.-T. Wu, Opt. Express 2018, 26,

21184.

[58] X.-W. Cao, Q.-D. Chen, L. Zhang, Z.-N. Tian, Q.-K. Li, L. Wang,

S. Juodkazis, H.-B. Sun, Opt. Lett. 2018, 43, 831.

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