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Electrically tunable polarization independent liquid crystal lenses based on orthogonally anisotropic orientations on adjacent micro-domains

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Abstract

Polarization dependency is an intrinsic property of liquid crystals (LC) devices but major problem is optical efficiency. We demonstrated a polarization independent liquid crystal phase modulation based on the orthogonal nematic LC (OLC) mode wherein the optics axes of nematic liquid crystal molecules are set orthogonally to adjacent sub-domains for the first time. Such an OLC mode includes sub-domain with anisotropic orientations but collectively presents a capability of a polarizer-free optical phase modulation. An OLC mode cell provides a tunable optical phase of ∼3.35π radians for unpolarized light and different linearly polarized light. Among the polarizer-free LC mode, the proposed OLC mode is single-layered with large tunable optical phase. We also demonstrated a polarizer-free LC micro-lens. We expect this novel LC mode provide alternatives technology roadmap for upcoming optical applications, such as electrically tunable ophthalmic lenses and optical systems for augmented reality.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Electrically tunable phase-only modulations adopting nematic liquid crystals (LCs) are fundamental electro-optical elements in many applications, such as spatial light modulators, lenses, optical switches, beam steering in antennas for satellite-cellular communication, lidar and 3D sensing [18]. The conventional mechanism is based on an optical phase shift for the extraordinary wave when a linearly polarized light propagates the anisotropic optical medium of nematic liquid crystals [9]. The molecular orientations of nematic LCs are adjustable under applied electric fields which leads to variations of light speed for the incident light and then light is electrically modulated in optical phases (i.e. electrically tunable optical phase modulation). There are other technologies for modulating optical phase of light [1011], and the advantages of LC phase-only modulations are no need mechanical moving part, low power consumption, light weight, and capable of modulating wavefronts continuously or pixel by pixel [12]. However, the requirement of a polarizer results in low light efficiency (∼ 50%) [9]. In order to improve light efficiency, polarization-independent or polarizer-free LC phase modulations are developed, including double-layered structure, residual phase type, a mixed type (i.e., combination of double-layered structure and residual phase type), and the type of Kerr effect [1215]. By means of orthogonally LC molecular orientations in two LC layers resulting in same optical phase shift of two eigen-polarization states, the double-layered structure displays polarizer-free and large optical phase shift (>100π radians) proportional to birefringence of nematic LC and thickness of a LC layer [16]. However, two LC layers have to be identical which is not easy in mass production and the response time is also a trade-off [17]. As to other type of polarization independent LC phase modulators, the optical phase is small (< 1π radians) [1314,18]. Thus, researchers are still developing polarization independent LC phase modulators. In this paper, we proposed a novel LC mode for a polarization independent LC phase modulation by means of orthogonally anisotropic orientations of nematic LC on adjacent micro-domains in a single LC layer. We called it “OLC” mode (i.e. orthogonal nematic LC mode). The two eigen-polarizations of an incident light still experience same optical phase shift after propagating a single LC layer. The corresponding optical mechanism is introduced and followed by experiments. Based on this LC mode, we also fabricate and test a LC micro-lens for proof-of-concept. The impact of this study is to pave a way for ophthalmic lenses using nematic LC.

2. Optical mechanism

To prove optical mechanism of the proposed LC phase modulation, the schematic structure is depicted in Fig. 1. Here we use nematic liquid crystals. At V = 0, the LC molecular orientations are orthogonal in adjacent domains which are either aligned along x-axis or y-axis. Each subdomain in Fig. 1(a) is a homogeneous cell (Fig. 1(b) and 1(c)) which is equivalent to an electrically tunable wave plate with uniaxial optical axis along either x-axis (Fig. 1(b)) or y-axis (Fig. 1(c)). When the applied voltage exceeds the threshold voltage (Vth), the LC molecules in Figs. 1(b) and Fig. 1(c) are re-oriented in x-z plane and y-z plane, respectively. When V >> Vth, all nematic liquid crystal molecules are parallel to z-axis or applied electric fields.

 figure: Fig. 1.

Fig. 1. (a) Schematic structure of the proposed LC phase modulator with 2N+1 by 2N+1 sub-domains. The optics axes (i.e. long axes of nematic LC molecules) at the adjacent domains are orthogonal. Each subdomain in (a) is equivalent to an electrically tunable wave plate with uniaxial optical axis along either (b) x-axis or (c)y-axis.

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Assume the proposed LC phase modulator has 2N+1 by 2N+1 rectangular sub-domains (Fig. 1(a)). The area of each subdomain is Δx times Δy, as depicted in Fig. 1(a). Figures 2(a) and 2(b) illustrate one dimensional refractive index as the function of x-coordinate for x-linearly and y-linearly polarized light. At V = 0, the refractive index as the function of x-coordinate for x-linearly polarized light is:

$$\widetilde n(x,\hat{x}\textrm{ - }{\textrm{polarized}}) = {n_0} + ({n_e} - {n_o}) \cdot \sum\limits_{a ={-} N}^N {rect\left( {\frac{{x - 2 \cdot \Delta x}}{{\Delta x}}} \right)} ,$$
where ne and no are the extraordinary and ordinary refractive indices, respectively; “rect” stands for the rectangular function with a definition of:
$$rect(x) = \left\{ {\begin{array}{cc} 1&{|x |< \frac{1}{2}}\\ {\frac{1}{2}}&{|x |\textrm{ = }\frac{1}{2}}\\ 0&{\textrm{otherwise}} \end{array}} \right..$$

 figure: Fig. 2.

Fig. 2. One dimensional refractive index as the function of x-coordinate when the incident light is (a) x-linearly polarized light and (b) y-linearly polarized light.

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Now let us consider a modulator with the 2D pixels in x and y coordinates, the refractive index for x-linearly polarized light is:

$$\widetilde n(\hat{x} - \textrm{polarized}) = {n_0} + ({n_e} - {n_o}) \cdot \sum\limits_{a ={-} N}^N {rect\left( {\frac{{x - 2a \cdot \Delta x}}{{\Delta x}}} \right)} \cdot \sum\limits_{b ={-} N}^N {rect\left( {\frac{{y - 2b \cdot \Delta y}}{{\Delta y}}} \right)} .$$

Similarly, the refractive index for y-linearly polarized light is:

$$\widetilde n(\hat{y} - \textrm{polarized}) = {n_0} + ({n_e} - {n_o}) \cdot \left[ {1 - \sum\limits_{a ={-} N}^N {rect\left( {\frac{{x - 2a \cdot \Delta x}}{{\Delta x}}} \right) \cdot \sum\limits_{b ={-} N}^N {rect\left( {\frac{{x - 2b \cdot \Delta y}}{{\Delta y}}} \right)} } } \right],$$
where Δx = Δy. When V > Vth, the LC molecules re-orientate and the effective refractive index for e-wave changes respectively. Generally speaking the LC director angle depends on the all three spatial coordinates and the applied voltage, $\theta ({x,y,z,V} )$. To characterize the effective refractive index provided by a given pixel one may average the director angle within the pixel. We shall denote the averaged angle by θz. Then the effective refractive index neff is a function of θz. The refractive indices for x-linearly and y-linearly polarized light are then expressed as:
$$\widetilde n(\hat{x} - \textrm{polarized}) = {n_0} + ({n_{eff}}({\theta _z}) - {n_o}) \cdot \sum\limits_{a ={-} N}^N {rect\left( {\frac{{x - 2a \cdot \Delta x}}{{\Delta x}}} \right)} \cdot \sum\limits_{b ={-} N}^N {rect\left( {\frac{{y - 2b \cdot \Delta y}}{{\Delta y}}} \right)} ,$$
$$\widetilde n(\hat{y} - \textrm{polarized}) = {n_0} + ({n_{eff}}({\theta _z}) - {n_o}) \cdot \left[ {1 - \sum\limits_{a ={-} N}^N {rect\left( {\frac{{x - 2a \cdot \Delta x}}{{\Delta x}}} \right) \cdot \sum\limits_{b ={-} N}^N {rect\left( {\frac{{y - 2b \cdot \Delta y}}{{\Delta y}}} \right)} } } \right].$$

When V >> Vth, the LC molecules are parallel to z-axis which means neffno for both of x-linearly and y-linearly polarized light. Equations (5) and (6) turn out:

$$\widetilde n(\hat{x} - \textrm{polarized}) = \widetilde n(\hat{y} - \textrm{polarized}) = {n_0}.$$

By minimizing the total free energy of the nematic subject to the applied voltage V one can find the LC director spatial profile. For the sake of simplicity, we neglect the neighboring pixel cross talks and the LC director dependence on the x- and y-coordinates. Then it is not difficult to calculate the LC director profile, the averaged value of the cosine squared of the director angle $\left\langle {{{\cos }^2}\theta } \right\rangle = \frac{1}{L}\int\limits_0^L {{{\cos }^2}\theta ({z,V} )dz}$ and phase retardation $\Delta \varphi = \frac{{2\pi }}{\lambda }\int\limits_0^L {[{{n_{eff}}({z,U} )- {n_o}} ]dz} ,$ ${n_{eff}} = ({n_o} \cdot {n_e})/{({n_e^2 \cdot {{\sin }^2}\theta ({z,U} )+ n_o^2 \cdot {{\cos }^2}\theta ({z,U} )} )^{\frac{1}{2}}}$ as the function of the applied voltage. Based on the parameters of nematic LC E7, we plotted dependencies of director angles, phase retardation and $\left\langle {{{\cos }^2}\theta } \right\rangle$, shown in the Figs. 3(a) to 3(c). The threshold voltage magnitude for LC E7 is 0.95 Volts which corresponds to 0.67 Vrms.

 figure: Fig. 3.

Fig. 3. (a) Director angle of LC as a function of z/L at different applied voltage. (b) Phase retardation as a function of electric potential. (c) The averaged value of the cosine squared of the director angle $\left\langle {{{\cos }^2}\theta } \right\rangle$ as a function of electric potential.

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Now let us assume the electric field of an incident plane wave is expressed as:

$${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E} _{in}} = {E_0} \cdot ({A_x} \cdot \hat{x} + {A_y} \cdot \hat{y}) \cdot {{\rm e}^{j(\omega \cdot t - k \cdot z)}},$$
where E0 is the amplitude field, ω is angular frequency of light, t is time, k is wave number of light and z is propagation distance along z-axis. In Eq. (8), the term of $({A_x} \cdot \hat{x} + {A_y} \cdot \hat{y})$ also represents of the polarization of light. Ax and Ay satisfy the relation for normalization: ${|{{A_x}} |^2} + {|{{A_y}} |^2} = 1$. After input light propagates through the proposed LC phase modulator shown in Fig. 1(a), the output light is given by:
$${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E} _{out}}({\boldsymbol r} )= {E_0} \cdot {e^{j(\omega t - k \cdot z)}} \cdot ({A_x} \cdot {e^{ - j \cdot k \cdot {n_x}({\boldsymbol r}) \cdot d}} \cdot \hat{x} + {A_y} \cdot {e^{ - j \cdot k \cdot {n_y}({\boldsymbol r}) \cdot d}} \cdot \hat{y}),$$
where nx and ny are the averaged over the given pixel refractive indices for x- and y-linearly polarized light as the function of position r, and $r = \sqrt {{x^2} + {y^2}}$. We replace nx(r) and ny (r) in Eq. (9) by Eqs. (5) and (6), respectively. Equation (9) can be expressed as:
$$\begin{aligned} &{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E} }_{out}}({\boldsymbol r} )= {E_0} \cdot {e^{j(\omega t - k \cdot z)}} \cdot {e^{ - j \cdot k \cdot {n_0} \cdot d}} \cdot ({A_x} \cdot {e^{ - j \cdot k \cdot ({n_{eff}}(\theta _z^{}) - {n_o}) \cdot d\sum\limits_{a ={-} N}^N {rect\left( {\frac{{x - 2a \cdot \Delta x}}{{\Delta x}}} \right)} \cdot \sum\limits_{b ={-} N}^N {rect\left( {\frac{{y - 2b \cdot \Delta y}}{{\Delta y}}} \right)} }} \cdot \hat{x}\\ &+ {A_y} \cdot {e^{ - j \cdot k \cdot ({n_{eff}}(\theta _z^{}) - {n_o}) \cdot d \cdot \left[ {1 - \sum\limits_{a ={-} N}^N {rect\left( {\frac{{x - 2a \cdot \Delta x}}{{\Delta x}}} \right) \cdot \sum\limits_{b ={-} N}^N {rect\left( {\frac{{y - 2b \cdot \Delta y}}{{\Delta y}}} \right)} } } \right]}} \cdot \hat{y}). \end{aligned}$$

It is convenient to use the Fourier representation to write down the rectangular or square-wave function describing the refractive index profile in 2D pixelized phase modulator. Below we do this for 1D and 2D square-wave functions, and the Fourier representation is an odd function equals to coordinate. The Eqs. (11) and (12) present 1D and 2D square-wave functions in terms of the Fourier series.

$$\eta (x )= 0.5\left[ {1 + \frac{4}{\pi }\sum\limits_{k = 1}^\infty {\frac{{\sin ({2\pi ({2k - 1} )\omega x} )}}{{2k - 1}}} } \right].\quad$$
$$g(x,y) = 0.5\left[ {1 + \left( {\frac{4}{\pi }\sum\limits_{k = 1}^\infty {\frac{{\sin ({2\pi ({2k - 1} )\omega x} )}}{{2k - 1}}} } \right)\left( {\frac{4}{\pi }\sum\limits_{k = 1}^\infty {\frac{{\sin ({2\pi ({2k - 1} )\omega y} )}}{{2k - 1}}} } \right)} \right].$$
here ω is the spatial frequency (ω = 1/T, where T is the spatial period). To illustrate how well this sum describes the square-wave function, we use one hundred harmonics within the non-dimensional spatial interval of [-5:5] at non-dimensional spatial frequency 0.1. Equations (11)–(12) are odd functions. Therefore, below we presented and adopted the Eqs. (11)–(12) with a shifted the coordination by half size of its sub-domain in order to convert square-wave functions become even functions. The result for 5 and 13 pixels is shown in the Fig. 4.

 figure: Fig. 4.

Fig. 4. Function $0.5({1 + \eta (x )\cdot \eta (y )} )$ with coordination shifting by half of the size of its sub-domain. Examples for 5 (left) and 13 pixels (right).

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When on the one hand the pixel size is sufficiently small compared to the light spot and on the other hand the number of pixels is high enough within the characteristic size of the light beam inhomogeneity, one can use the phase retardation averaged over the physically small surface area. Let us check that for both x- and y-polarizations the averaged (effective) phase retardation is the same for both polarizations. For an illustrative example we suppose that the LC director angle θ is spatially modulated and the modulation is described by the Gaussian-type function

$$\theta ({x,y} )= \frac{\pi }{2}\exp \left( { - \frac{{{x^2} + {y^2}}}{{{R^2}}}} \right).$$

Figure 5 shows the effective refractive index profile ${n_{eff}}({x,y} )= ({n_o} \cdot {n_e})/{({n_e^2 \cdot {{\sin }^2}\theta ({x,y} )+ n_o^2 \cdot {{\cos }^2}\theta ({x,y} )} )^{\frac{1}{2}}}$ within the above shown 5 pixels which corresponds to the director profile given by the Eq. (13).

 figure: Fig. 5.

Fig. 5. Effective refractive index profile for 5 pixels case.

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Now we can compute the phase retardation for x- and y-polarized incident beams.

$$\Delta {\Phi _e}({x,y} )= \frac{{2\pi L}}{\lambda }{n_{eff}}g({x,y} );$$
$$\Delta {\Phi _o}({x,y} )= \frac{{2\pi L}}{\lambda }{n_{eff}}[{1 - g({x,y} )} ].$$

In the Fig. 6 we plot the phase retardations for the setting parameters of L = 0.5893 microns, λ = 10 microns.

 figure: Fig. 6.

Fig. 6. (a) and (b) are phase retardation for x- and y-polarized waves, respectively (5 pixels case).

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It is seen that for both x- and y-polarized beams the phase retardation varies within the area occupied by 5 pixels. Assuming that this area is sufficiently small so that the light passed through the area occupied by 5 pixels can be described by the averaged phase retardation we integrate the $\Delta {\Phi _e}({x,y} )$ and $\Delta {\Phi _o}({x,y} )$ over this surface area and get exactly the same averaged phase retardation of ≈2.6 for both polarizations of the incident light. This result can be easily understood from the following simplified 1D example. It is easy to see that

$$\eta (x )= 1 - \eta ({x + T/2} ),$$
then
$$\int\limits_{ - a}^a {\eta (x )} dx = \{{x = x^{\prime} + T/2} \}= \int\limits_{ - a - T/2}^{a - T/2} {\eta ({x^{\prime} + T/2} )} dx^{\prime} = \int\limits_{ - a - T/2}^{a - T/2} {[{1 - \eta ({x^{\prime}} )} ]} dx^{\prime}.$$

When $a \to \infty$

$$\int\limits_{ - \infty }^{ + \infty } {\eta (x )} dx = \mathop{\lim\nolimits _{a \to \infty }}\int\limits_{ - a}^a {\eta (x )} dx = \mathop{\lim\nolimits _{a \to \infty }}\int\limits_{ - a - T/2}^{a - T/2} {[{1 - \eta ({x^{\prime}} )} ]} dx^{\prime} = \int\limits_{ - \infty }^{ + \infty } {[{1 - \eta ({x^{\prime}} )} ]} dx^{\prime}.$$

Therefore, when the pixels are small enough both x- and y-polarized light effectively experience the same phase retardation. This means that the proposed LC phase modulator is polarization independent provided the pixel size is sufficiently small.

At high enough voltage for both polarizations of the incident light the change of phase retardation ($|{\phi (V > {V_{th}}) - \phi (V = 0)} |$) approximately equals to $k \cdot ({n_e} - {n_o}) \cdot d$. The total polarization independent optical phase of the proposed LC phase modulator is proportional to birefringence (ne-no) of nematic liquid crystals times to the thickness which is identical to the optical phase of double-layered LC phase modulators [12]. Therefore, we prove the proposed single-layered design is capable of polarizer-free optical phase modulation which has identical tunable phase of the double-layered LC phase modulator [12]. Based on polarization independent phase modulation of proposed structure, the focal length of a gradient-index (GRIN) LC lens is [3]:

$$f = \frac{{\pi \cdot {r_o}^2}}{{|{\phi (V > {V_{th}}) - \phi (V = 0)} |\cdot \lambda }},$$
where r0 is radius of aperture size. From Eq. (19), the minimum focal length is:
$$f = \frac{{\pi \cdot {r_o}^2}}{{k \cdot ({{n_e} - {n_o}} )\cdot d \cdot \lambda }} = \frac{{{r_o}^2}}{{2 \cdot ({{n_e} - {n_o}} )\cdot d}}.$$

3. Experiments and discussions

To proof-of-concept, we prepared a glass substrate coated with a layer of indium tin oxide (ITO) followed by an alignment layer (Nissan SE-7492). To obtain the orthogonally adjacent micro-domains of the alignment layer, we adopted a tip of the cantilever of Atomic Force Microscope (AC200TS, Olympus AFM-Nanoview 1000AFM) to scratch the alignment layer at the contact mode [1922]. The scanning rate of the tip is 4.34 Hz. We applied 4 different forces: 300 nano Newtons (nN), 500nN, 800nN, and 1000nN to the cantilever. The tip on the cantilever were set to touch alignment layer with a density of 16 lines/μm. In order to determine the condition of the force on the cantilever, we started from area size of 16 μm by 16 μm. After we scratched the alignment layer, we used the tapping mode to scan the surface morphologies with the area size of 1 μm by 1 μm in the scanning direction at 90 degrees with respect to the scanning direction of the contact mode (i.e. scratching direction). The surface morphologies of the alignment at different scratching forces are shown in Figs. 7(a) to 7(d). Larger forces on the cantilever, deeper grooves we observed. Figure 7 (e) plotted the surface roughness (i.e. root mean square roughness) as a function of cantilever forces. Roughness increases from 0.8 nm to 2.7 nm as the cantilever force increases from 300 nN to 1000 nN.

 figure: Fig. 7.

Fig. 7. Surface morphologies of the alignment layers scratched by the cantilever tip of AFM in the contact mode when the force applied to the cantilever was (a) 300 nN, (b) 500 nN, (c) 800 nN, and (d) 1000 nN. The surface morphologies were obtained by scanning at 90 degrees with respect to the grooves at the tapping mode. (e) is the roughness of surface morphologies of (a)-(d) as the function of force at the cantilever. Roughness is defined as the root mean square of the surface height.

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To further test the alignment capability of the scratched alignment layer in Figs. 7(a) to 7(d), we assembled the ITO glass substrate with the scratched alignment layer and a ITO glass substrate with a mechanically rubbed alignment layer (Nissan SE-7492) which is the conventional rubbing method for LC devices. For two glass substrates, one with scratched grooves and the other one with a rubbing surface are assembled together with anti-parallel directions. The Mylar film with a thickness of 10 microns was used to maintain the gap between two glass substrates. Thereafter, nematic LC (E7, Δn = 0.23 at λ = 589.3nm, Merck) was filled between two glass substrates under assistance of a capillary force. The cell was then sealed by UV glue (NOA 81, Norland). Four samples were prepared with different scratched forces applied to the cantilever (300-1000 nN) and then the samples were observed under crossed polarizers. Figures 8(a)-(h) are the photos of samples observed under crossed polarizers with grooves (denoted as R in Fig. 8) either parallel to one of the polarizers or 45 degrees with respect to one of the polarizers. All samples exhibit similar dark states in Figs. 8(a)-(d) and similar bright states in Figs.8 8(e)-(h). This indicates that good alignment of LC molecules. This also means the alignment layers with scratched grooves show good alignment capability to LC molecules. Figs. 8(a)-(d) display similar photos which also means four surface morphologies in Figs. 7(a) to (d) provide similar alignment capability.

 figure: Fig. 8.

Fig. 8. Photos of samples observed under crossed polarizers. One ITO glass substrate is coated with a conventional rubbed alignment layer and the other one is coated with the grooving alignment layer. The nematic LC was sandwiched between two substrates. The grooving alignment layer was scratched by the tip when we applied forces to the cantilever: (a) 300 nN, (b) 500 nN, (c) 800 nN, and (d) 1000 nN. P and A are transmissive axes of two polarizers. R stands for the groove direction with grooves either parallel((a) to (d)) or 45 degrees with respect to one of the polarizers ((e)-(h)).

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To measure anchoring energy and pretilt angle of the alignment layer of 4 samples, we set up experiment to measure the voltage-dependent transmittance (V-T curve) of 4 samples to measure pretilt angle first and then calculated polar anchoring energy based on Rapini-Papoular method [23]. The samples were placed between two crossed polarizers with the rubbing direction at 45 degrees with respect to one of the transmissive axes of a polarizer. He-Ne laser (λ=632.8nm) was used as light source, an objective lens (50x) was used for adjusting the beam size and a photo-detector (Model: 2031, New Focus) was used to measure the transmittance. After obtaining V-T curves, pretilt angles, and threshold voltages of 4 samples, we obtained the phase retardation as a function of applied voltage, and then we calculated polar anchoring energy. The measured and calculated data of samples are listed in Table 1. In Table 1, 4 samples exhibit the phase retardation of 6.68π radians, the threshold voltage of 0.67 Vrms, and a pretilt angle of 1.596 degrees in average under a cell gap of 10 microns. As to the response time, rise time is around 3.98 ms and decay time is around 211 ms. Anchoring energy is around 1.43 × 10−4 J/m2. In comparison, we also prepared a typical LC cell with a similar condition, but both of ITO glass substrates coated with mechanically rubbed polyimide (the 6th column of Table 1). The anchoring energy of the typical LC cell is around 1.28 × 10−4 J/m2. The anchoring energy for 4 samples and typical LC cell are similar ∼10−4 J/m2 in order of magnitude. This means the alignment layer scratched by the cantilever (300-1000 nN) provides similar aligned capability to liquid crystal molecules and then it results in similar electro-optical properties of samples.

Tables Icon

Table 1. Measured and calculated parameters of 4 LC samples in Fig. 3 and a typical LC cell with two mechanically buffered alignment layers.

To further test the proposed polarization-independent phase modulation, we fabricated a liquid crystal micro-lens. The structure, depicted in Fig. 9(a), consists of a LC layer (nematic LC: E7, Δn = 0.23 at λ = 589.3nm, Merck), a top glass substrate coated with hole-patterned aluminum (the diameter of a hole was 100 μm) and a mechanically rubbed vertical alignment layer (N, N-dimethyl-N-octadecyl-3- amonopropyltrimethoxysilyl chloride, DMOAP), and a bottom ITO glass substrate. The bottom glass substrate was coated with mechanically rubbed polyimide (Nissan, 7492) and a small area was scratched by the cantilever tip of AFM (Fig. 7(b)). The small area was full of the orthogonally scratched grooves by means of the cantilever tip of AFM in contact mode (contact force: 500 nN; the scanning density: 16 lines/μm). Each sub-domain is around 4 μm and total domain size is around 112 μm, as depicted in Fig. 9(b). The thickness of LC layer was measured around 46.77 μm. The aperture size of a micro-lens was 100 μm. Figure 9(c) is the side view of Fig. 9(a).

 figure: Fig. 9.

Fig. 9. (a) Schematic configuration for the polarizer-free LC micro-lens (aperture size of a micro-lens: 100 μm). (b) The illustration of scratched PI area in (a). Each sub-domain is 4 μm and total size is 112 μm. Arrows indicate the direction of scratched grooves. The scratched directions (or alignment direction) between adjacent sub-domains are orthogonal. (c) is the side view of (a) when voltage is turned off. (d) and (e) show the focusing capability of the sample at (d) V = 0 and (e) V = 11 Vrms, respectively. (d) and (e) were captured under unpolarized light. The focal length is 1.37 mm for V = 11 Vrms. (f) and (g) show the focusing capability of the sample at V = 0 and V = 11 Vrms for linearly-polarized light where its polarization state is parallel to the rubbing direction. (h) and (i) show the focusing capability sample at V = 0 and V = 11 Vrms for a linearly-polarized light when the polarization state is perpendicular to the rubbing direction. The color bar in (d)-(i) standards for the light intensity in arbitrary unit.

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To measure the focusing properties of the micro-lens (Fig. 9(a)), we setup a system consisting of a unpolarized laser (λ = 543nm), a spatial filter for beam expansion, a polarizer, a lens (focal length = 50 mm) and a camera (Canon 760D) [24]. The sample at V = 0 was placed between the polarizer and the lens. Then we adjusted the sample location till the camera see the image of the hole-pattern electrode. When the sample was applied a voltage (V), we moved the sample location until the camera seeing the focusing spot and then we recorded the distance between the sample at voltage-off state and at voltage-on state. This distance was the focal length of the sample at V [24]. Figures 9(d) and 9(e) show the focusing spot captured by the camera under unpolarized light at the distance of 1.37 mm when V = 0 and V = 11 Vrms, respectively. Similarly, we also captured the focusing spot of LC lens under two linearly-polarized light when V = 0 and V = 11 Vrms, as shown in Figs. 9(f)–9(i). Figures 9(f)–9-(g) were obtained when the polarization state the linearly polarized light is parallel to the rubbing direction, while Figs. 9(h)-(i) were obtained when the polarized state is perpendicular to the rubbing direction. The LC lens present focusing capability for different polarized light and unpolarized light. The focusing spot for Fig. 10(a) showed the measured focal length as a function of applied voltage when the incident light is unpolarized light and linearly polarized light. 0, 45, 90, and 135 degrees stand for the angle of the transmissive axis of the polarizer with respect to the rubbing direction of the top glass substrate coated with DMOAP. For unpolarized light and different linearly polarized light, the averaged focal length of the polarizer-free LC micro-lens decreases from ∼ 2.02 mm (5 Vrms) to ∼1.37 mm (11 Vrms) and then increases to ∼1.60 mm (15 Vrms) with applied voltage. The focal length is independent of polarization of incident light. As a result, the sample of the LC micro-lens is polarization independent and polarizer-free. Figure 10(a) is then converted to the phase shift between the central and peripheral region of the micro-lens according to Eq. (19). Figure 10(b) exhibits the calculated optical phase shift as a function of applied voltage. The averaged optical phase shift increases from ∼2.27π radians (5 Vrms) to ∼3.35π radians (11 Vrms) and the decreases to ∼2.87π radians (15 Vrms) with applied voltage. In comparison, we also prepared a micro-lens with the same structure in Fig. 9(c), except the bottom scratched PI layer. Instead of the scratched PI layer, we used mechanically rubbed PI layer in the bottom ITO glass substrate to construct a conventional polarized LC micro-lens. In the measurement of focal length, we placed a polarizer in front of the conventional polarized LC micro-lens and the transmissive axis of the polarizer was parallel to the rubbing direction. The focal length as a function of the applied voltage is also depicted as yellow triangles in Fig. 10(a). For linearly polarized light, the focal length decreases from ∼2.23 mm (5 Vrms) to ∼1.29 mm (11 Vrms) and then increases to ∼1.53 mm (15 Vrms) with applied voltage. The corresponding optical phase shift increases from ∼2.06 π radians (5 Vrms) to ∼3.57π radians (11 Vrms) and the decreases to ∼3.00π radians (15 Vrms) with applied voltage. From Fig. 10(a) and 10(b), the results of focal lengths and optical phase shifts are similar between the proposed LC micro-lens and the conventional polarized LC micro-lens. Therefore, we can say that liquid crystal micro-lenses based on orthogonally anisotropic orientations on adjacent micro-domains is electrically tunable and polarization independent (polarizer-free). Instead of double-layered structure and residual type LC polarizer-free phase modulation, the proposed design is single layer only. To enlarge the optical phase, the cell gap could be increases and the top vertical alignment layer could be replaced by parallel alignment layer which could enlarge optical phase by 2 times. The method we proposed is also suitable for designing LC lenses with large apertures size. However, the scratched method is going to be time consuming. To obtain LC lenses with orthogonally anisotropic orientations on adjacent micro-domains, different fabrication method, such as photoalignment method and nanoimprinting method, could be good alternative methods. For the size of each mirco-domain, the current design of 4 μm is much smaller than the wavelengths using in 5G communication (millimeter wave) and 6G communication (THz; wavelength ranges from 30 μm to 1 mm) [2527]. Therefore, with a proper ratio of the adjacent domain size, the LC layer could be effectively considered as new medium with tunable optical properties, such as tunable transmittance and reflectance [2830]. By controlling the ratio of adjacent domain size, the effective medium could also present an optic axis.

 figure: Fig. 10.

Fig. 10. (a) Measured focal length as function of voltage of the LC micro-lens when the incident light is unpolarized light (denoted as NO polarizer) and linearly polarized light (0, 45, 90, 135 degree linearly polarized light). Yellow triangles represent the focal length for the conventional polarized LC micro-lens. (b) The corresponding optical phase shift between the central and peripheral region of the micro-lens as a function of applied voltage. Still, yellow triangles represent the focal length for the conventional polarized LC micro-lens. The rest is the polarizer-free LC micro-lens. In (a) and (b), a polarizer was used for the measurement for the conventional polarized LC micro-lens.

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4. Conclusion

We have demonstrated a polarizer-free LC mode for phase modulation based on orthogonal molecular orientations of liquid crystals on adjacent micro-domains. The electrically tunable optical phase is up to 3.35π radians with applied voltage of 11 Vrms, where the tunable range of optical phase is sufficient for LC optics with small apertures or pixelated spatial light modulators. In experiments, we also demonstrated the polarizer-free LC micro-lens. The theoretical mechanism is also proved. The proposed method is also suitable for designing LC lenses with large apertures size. The impact of this study is mainly in applications of beam steering devices, eyeglasses, drones, and augmented realities which require polarizer-free LC optical phase modulations. This concept could be also extended to the applications with wavelengths beyond visible range, such as 5G and 6G communications.

Funding

Ministry of Science and Technology, Taiwan (110-2112-M-A49-024, 110-2218-E-A49-012-MBK).

Acknowledgments

The authors are indebted to Mr. Chia-Hao Kuo for technical assistances.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request. The authors declare no conflicts of interest.

References

1. N. Konforti, E. Marom, and S. T. Wu, “Phase-only modulation with twisted nematic liquid-crystal spatial light modulators,” Opt. Lett. 13(3), 251–253 (1988). [CrossRef]  

2. J. Cho, S. Kim, S. Park, B. Lee, and H. Kim, “DC-free on-axis holographic display using a phase-only spatial light modulator,” Opt. Lett. 43(14), 3397–3400 (2018). [CrossRef]  

3. 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]  

4. M. Decker, C. Kremers, A. Minovich, I. Staude, A. E. Miroshnichenko, D. Chigrin, D. N. Neshev, C. Jagadish, and Y. S. Kivshar, “Electro-optical switching by liquid-crystal controlled metasurfaces,” Opt. Express 21(7), 8879–8885 (2013). [CrossRef]  

5. Y. Kim, K. Won, J. An, J. Y. Hong, Y. Kim, C. S. Choi, H. Song, B. Song, H. S. Kim, K. D. Bae, J. Burm, and H. S. Lee, “Large-area liquid crystal beam deflector with wide steering angle,” Appl. Opt. 59(24), 7462–7468 (2020). [CrossRef]  

6. Q. You, M. Luo, X. Xiao, and S. Yu, “2D optical wireless broadcasting system enabled by a liquid crystal on silicon and rotated-splitting-SLM algorithm,” Opt. Express 28(21), 30851–30860 (2020). [CrossRef]  

7. Z. N. Yuan, Z. B. Sun, H. S. Kwok, and A. K. Srivastava, “Fast LiDAR systems based on ferroelectric liquid crystal Dammann grating,” Liq. Cryst. (2021). Available: https://doi.org/10.1080/02678292.2021.1873437

8. Y. J. Wang, X. Shen, Y. H. Lin, and B. Javidi, “Extended depth-of-field 3D endoscopy with synthetic aperture integral imaging using an electrically tunable focal-length liquid-crystal lens,” Opt. Lett. 40(15), 3564–3567 (2015). [CrossRef]  

9. D. K. Yang and S. T. Wu, Fundamentals of Liquid Crystal Devices (Wiley, 2006).

10. L. Li, L. Xiao, J. Wang, and Q. Wang, “Movable electrowetting optofluidic lens for optical axial scanning in microscopy,” Opto-Electron. Adv. 2(2), 180025 (2019). [CrossRef]  

11. Q. Wang, L. Xiao, C. Liu, and L. Li, “Optofluidic variable optical path modulator,” Sci. Rep. 9(1), 7082 (2019). [CrossRef]  

12. Y. H. Lin, H. Ren, Y. H. Wu, Y. Zhao, J. Fang, Z. Ge, and S. T. Wu, “Polarization-independent liquid crystal phase modulator using a thin polymer-separated double-layered structure,” Opt. Express 13(22), 8746–8752 (2005). [CrossRef]  

13. H. Ren, Y. H. Lin, Y. H. Fan, and S. T. Wu, “Polarization-independent phase modulation using a polymer-dispersed liquid crystal,” Appl. Phys. Lett. 86(14), 141110 (2005). [CrossRef]  

14. Y. Huang, C. H. Wen, and S. T. Wu, “Polarization-independent and submillisecond response phase modulators using a 90° twisted dual-frequency liquid crystal,” Appl. Phys. Lett. 89(2), 021103 (2006). [CrossRef]  

15. 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]  

16. Y. H. Lin and H. S. Chen, “Electrically tunable-focusing and polarizer-free liquid crystal lenses for ophthalmic applications,” Opt. Express 21(8), 9428–9436 (2013). [CrossRef]  

17. H. S. Chen, Y. J. Wang, C. M. Chang, and Y. H. Lin, “A polarizer-free liquid crystal lens exploiting an embedded-multilayered structure,” IEEE Photonics Technol. Lett. 27(8), 899–902 (2015). [CrossRef]  

18. Y. H. Lin, M. S. Chen, W. C. Lin, and Y. S. Tsou, “A polarization-independent liquid crystal phase modulation using polymer-network liquid crystals in a 90 degree twisted cell,” J. Appl. Phys. 112(2), 024505 (2012). [CrossRef]  

19. A. Rastegar, M. Skarabot, B. Blij, and T. Rasing, “Mechanism of liquid crystal alignment on submicron patterned surfaces,” J. Appl. Phys. 89(2), 960–964 (2001). [CrossRef]  

20. J. H. Kim, M. Yoneya, J. Yamamoto, and H. Yokoyama, “Nano-rubbing of a liquid crystal alignment layer by an atomic force microscope: a detailed characterization,” Nanotechnology 13(2), 133–137 (2002). [CrossRef]  

21. J. H. Kim, M. Yoneya, and H. Yokoyama, “Tristable nematic liquid crystal device using micropatterned surface alignment,” Nature 420(6912), 159–162 (2002). [CrossRef]  

22. I. M. Syed, G. Carbone, and C. Rosenblatt, “Planar degenerate substrate for micro- and nanopatterned nematic liquid-crystal cells,” J. Appl. Phys. 98(3), 034403 (2005). [CrossRef]  

23. Yu. A. Nastishin, R. D. Polak, S. V. Shiyanovskii, V. H. Bodnar, and O. D. Lavrentovich, “Nematic polar anchoring strength measured by electric field techniques,” J. Appl. Phys. 86(8), 4199–4213 (1999). [CrossRef]  

24. H. Ren, J. R. Wu, Y. H. Fan, Y. H. Lin, and S. T. Wu, “Hermaphroditic liquid-crystal microlens,” Opt. Lett. 30(4), 376–378 (2005). [CrossRef]  

25. N. Panwar, S. Sharma, and A. K. Singh, “A survey on 5G: The next generation of mobile communication,” Phys. Commun. 18, 64–84 (2016). [CrossRef]  

26. M. Giordani, M. Polese, M. Mezzavilla, S. Rangan, and M. Zorzi, “Toward 6G networks: Use cases and technologies,” IEEE Commun. Mag. 58(3), 55–61 (2020). [CrossRef]  

27. J. Zou, Z. Yang, C. Mao, and S. T. Wu, “Fast-Response Liquid Crystals for 6G Optical Communications,” Crystals 11(7), 797 (2021). [CrossRef]  

28. E. B. Grann, M. G. Moharam, and D. A. Pommet, “Artificial uniaxial and biaxial dielectrics with use of two-dimensional subwavelength binary gratings,” J. Opt. Soc. Am. A 11(10), 2695–2703 (1994). [CrossRef]  

29. Z. W. Xie, J. H. Yang, V. Vashistha, W. Lee, and K. P. Chen, “Liquid-crystal tunable color filters based on aluminum metasurfaces,” Opt. Express 25(24), 30764–30770 (2017). [CrossRef]  

30. T. Sasaki, T. Asano, M. Sakamoto, K. Noda, T. Unuma, K. Goto, K. Tsutsui, N. Kawatsuki, and H. Ono, “Subwavelength liquid crystal gratings for polarization-independent phase shifts in the terahertz spectral range,” Opt. Mater. Express 10(2), 240–248 (2020). [CrossRef]  

References

  • View by:

  1. N. Konforti, E. Marom, and S. T. Wu, “Phase-only modulation with twisted nematic liquid-crystal spatial light modulators,” Opt. Lett. 13(3), 251–253 (1988).
    [Crossref]
  2. J. Cho, S. Kim, S. Park, B. Lee, and H. Kim, “DC-free on-axis holographic display using a phase-only spatial light modulator,” Opt. Lett. 43(14), 3397–3400 (2018).
    [Crossref]
  3. 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]
  4. M. Decker, C. Kremers, A. Minovich, I. Staude, A. E. Miroshnichenko, D. Chigrin, D. N. Neshev, C. Jagadish, and Y. S. Kivshar, “Electro-optical switching by liquid-crystal controlled metasurfaces,” Opt. Express 21(7), 8879–8885 (2013).
    [Crossref]
  5. Y. Kim, K. Won, J. An, J. Y. Hong, Y. Kim, C. S. Choi, H. Song, B. Song, H. S. Kim, K. D. Bae, J. Burm, and H. S. Lee, “Large-area liquid crystal beam deflector with wide steering angle,” Appl. Opt. 59(24), 7462–7468 (2020).
    [Crossref]
  6. Q. You, M. Luo, X. Xiao, and S. Yu, “2D optical wireless broadcasting system enabled by a liquid crystal on silicon and rotated-splitting-SLM algorithm,” Opt. Express 28(21), 30851–30860 (2020).
    [Crossref]
  7. Z. N. Yuan, Z. B. Sun, H. S. Kwok, and A. K. Srivastava, “Fast LiDAR systems based on ferroelectric liquid crystal Dammann grating,” Liq. Cryst. (2021). Available: https://doi.org/10.1080/02678292.2021.1873437
  8. Y. J. Wang, X. Shen, Y. H. Lin, and B. Javidi, “Extended depth-of-field 3D endoscopy with synthetic aperture integral imaging using an electrically tunable focal-length liquid-crystal lens,” Opt. Lett. 40(15), 3564–3567 (2015).
    [Crossref]
  9. D. K. Yang and S. T. Wu, Fundamentals of Liquid Crystal Devices (Wiley, 2006).
  10. L. Li, L. Xiao, J. Wang, and Q. Wang, “Movable electrowetting optofluidic lens for optical axial scanning in microscopy,” Opto-Electron. Adv. 2(2), 180025 (2019).
    [Crossref]
  11. Q. Wang, L. Xiao, C. Liu, and L. Li, “Optofluidic variable optical path modulator,” Sci. Rep. 9(1), 7082 (2019).
    [Crossref]
  12. Y. H. Lin, H. Ren, Y. H. Wu, Y. Zhao, J. Fang, Z. Ge, and S. T. Wu, “Polarization-independent liquid crystal phase modulator using a thin polymer-separated double-layered structure,” Opt. Express 13(22), 8746–8752 (2005).
    [Crossref]
  13. H. Ren, Y. H. Lin, Y. H. Fan, and S. T. Wu, “Polarization-independent phase modulation using a polymer-dispersed liquid crystal,” Appl. Phys. Lett. 86(14), 141110 (2005).
    [Crossref]
  14. Y. Huang, C. H. Wen, and S. T. Wu, “Polarization-independent and submillisecond response phase modulators using a 90° twisted dual-frequency liquid crystal,” Appl. Phys. Lett. 89(2), 021103 (2006).
    [Crossref]
  15. 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]
  16. Y. H. Lin and H. S. Chen, “Electrically tunable-focusing and polarizer-free liquid crystal lenses for ophthalmic applications,” Opt. Express 21(8), 9428–9436 (2013).
    [Crossref]
  17. H. S. Chen, Y. J. Wang, C. M. Chang, and Y. H. Lin, “A polarizer-free liquid crystal lens exploiting an embedded-multilayered structure,” IEEE Photonics Technol. Lett. 27(8), 899–902 (2015).
    [Crossref]
  18. Y. H. Lin, M. S. Chen, W. C. Lin, and Y. S. Tsou, “A polarization-independent liquid crystal phase modulation using polymer-network liquid crystals in a 90 degree twisted cell,” J. Appl. Phys. 112(2), 024505 (2012).
    [Crossref]
  19. A. Rastegar, M. Skarabot, B. Blij, and T. Rasing, “Mechanism of liquid crystal alignment on submicron patterned surfaces,” J. Appl. Phys. 89(2), 960–964 (2001).
    [Crossref]
  20. J. H. Kim, M. Yoneya, J. Yamamoto, and H. Yokoyama, “Nano-rubbing of a liquid crystal alignment layer by an atomic force microscope: a detailed characterization,” Nanotechnology 13(2), 133–137 (2002).
    [Crossref]
  21. J. H. Kim, M. Yoneya, and H. Yokoyama, “Tristable nematic liquid crystal device using micropatterned surface alignment,” Nature 420(6912), 159–162 (2002).
    [Crossref]
  22. I. M. Syed, G. Carbone, and C. Rosenblatt, “Planar degenerate substrate for micro- and nanopatterned nematic liquid-crystal cells,” J. Appl. Phys. 98(3), 034403 (2005).
    [Crossref]
  23. Yu. A. Nastishin, R. D. Polak, S. V. Shiyanovskii, V. H. Bodnar, and O. D. Lavrentovich, “Nematic polar anchoring strength measured by electric field techniques,” J. Appl. Phys. 86(8), 4199–4213 (1999).
    [Crossref]
  24. H. Ren, J. R. Wu, Y. H. Fan, Y. H. Lin, and S. T. Wu, “Hermaphroditic liquid-crystal microlens,” Opt. Lett. 30(4), 376–378 (2005).
    [Crossref]
  25. N. Panwar, S. Sharma, and A. K. Singh, “A survey on 5G: The next generation of mobile communication,” Phys. Commun. 18, 64–84 (2016).
    [Crossref]
  26. M. Giordani, M. Polese, M. Mezzavilla, S. Rangan, and M. Zorzi, “Toward 6G networks: Use cases and technologies,” IEEE Commun. Mag. 58(3), 55–61 (2020).
    [Crossref]
  27. J. Zou, Z. Yang, C. Mao, and S. T. Wu, “Fast-Response Liquid Crystals for 6G Optical Communications,” Crystals 11(7), 797 (2021).
    [Crossref]
  28. E. B. Grann, M. G. Moharam, and D. A. Pommet, “Artificial uniaxial and biaxial dielectrics with use of two-dimensional subwavelength binary gratings,” J. Opt. Soc. Am. A 11(10), 2695–2703 (1994).
    [Crossref]
  29. Z. W. Xie, J. H. Yang, V. Vashistha, W. Lee, and K. P. Chen, “Liquid-crystal tunable color filters based on aluminum metasurfaces,” Opt. Express 25(24), 30764–30770 (2017).
    [Crossref]
  30. T. Sasaki, T. Asano, M. Sakamoto, K. Noda, T. Unuma, K. Goto, K. Tsutsui, N. Kawatsuki, and H. Ono, “Subwavelength liquid crystal gratings for polarization-independent phase shifts in the terahertz spectral range,” Opt. Mater. Express 10(2), 240–248 (2020).
    [Crossref]

2021 (1)

J. Zou, Z. Yang, C. Mao, and S. T. Wu, “Fast-Response Liquid Crystals for 6G Optical Communications,” Crystals 11(7), 797 (2021).
[Crossref]

2020 (4)

2019 (2)

L. Li, L. Xiao, J. Wang, and Q. Wang, “Movable electrowetting optofluidic lens for optical axial scanning in microscopy,” Opto-Electron. Adv. 2(2), 180025 (2019).
[Crossref]

Q. Wang, L. Xiao, C. Liu, and L. Li, “Optofluidic variable optical path modulator,” Sci. Rep. 9(1), 7082 (2019).
[Crossref]

2018 (1)

2017 (2)

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]

Z. W. Xie, J. H. Yang, V. Vashistha, W. Lee, and K. P. Chen, “Liquid-crystal tunable color filters based on aluminum metasurfaces,” Opt. Express 25(24), 30764–30770 (2017).
[Crossref]

2016 (1)

N. Panwar, S. Sharma, and A. K. Singh, “A survey on 5G: The next generation of mobile communication,” Phys. Commun. 18, 64–84 (2016).
[Crossref]

2015 (2)

Y. J. Wang, X. Shen, Y. H. Lin, and B. Javidi, “Extended depth-of-field 3D endoscopy with synthetic aperture integral imaging using an electrically tunable focal-length liquid-crystal lens,” Opt. Lett. 40(15), 3564–3567 (2015).
[Crossref]

H. S. Chen, Y. J. Wang, C. M. Chang, and Y. H. Lin, “A polarizer-free liquid crystal lens exploiting an embedded-multilayered structure,” IEEE Photonics Technol. Lett. 27(8), 899–902 (2015).
[Crossref]

2014 (1)

2013 (2)

2012 (1)

Y. H. Lin, M. S. Chen, W. C. Lin, and Y. S. Tsou, “A polarization-independent liquid crystal phase modulation using polymer-network liquid crystals in a 90 degree twisted cell,” J. Appl. Phys. 112(2), 024505 (2012).
[Crossref]

2006 (1)

Y. Huang, C. H. Wen, and S. T. Wu, “Polarization-independent and submillisecond response phase modulators using a 90° twisted dual-frequency liquid crystal,” Appl. Phys. Lett. 89(2), 021103 (2006).
[Crossref]

2005 (4)

Y. H. Lin, H. Ren, Y. H. Wu, Y. Zhao, J. Fang, Z. Ge, and S. T. Wu, “Polarization-independent liquid crystal phase modulator using a thin polymer-separated double-layered structure,” Opt. Express 13(22), 8746–8752 (2005).
[Crossref]

H. Ren, Y. H. Lin, Y. H. Fan, and S. T. Wu, “Polarization-independent phase modulation using a polymer-dispersed liquid crystal,” Appl. Phys. Lett. 86(14), 141110 (2005).
[Crossref]

H. Ren, J. R. Wu, Y. H. Fan, Y. H. Lin, and S. T. Wu, “Hermaphroditic liquid-crystal microlens,” Opt. Lett. 30(4), 376–378 (2005).
[Crossref]

I. M. Syed, G. Carbone, and C. Rosenblatt, “Planar degenerate substrate for micro- and nanopatterned nematic liquid-crystal cells,” J. Appl. Phys. 98(3), 034403 (2005).
[Crossref]

2002 (2)

J. H. Kim, M. Yoneya, J. Yamamoto, and H. Yokoyama, “Nano-rubbing of a liquid crystal alignment layer by an atomic force microscope: a detailed characterization,” Nanotechnology 13(2), 133–137 (2002).
[Crossref]

J. H. Kim, M. Yoneya, and H. Yokoyama, “Tristable nematic liquid crystal device using micropatterned surface alignment,” Nature 420(6912), 159–162 (2002).
[Crossref]

2001 (1)

A. Rastegar, M. Skarabot, B. Blij, and T. Rasing, “Mechanism of liquid crystal alignment on submicron patterned surfaces,” J. Appl. Phys. 89(2), 960–964 (2001).
[Crossref]

1999 (1)

Yu. A. Nastishin, R. D. Polak, S. V. Shiyanovskii, V. H. Bodnar, and O. D. Lavrentovich, “Nematic polar anchoring strength measured by electric field techniques,” J. Appl. Phys. 86(8), 4199–4213 (1999).
[Crossref]

1994 (1)

1988 (1)

An, J.

Asano, T.

Bae, K. D.

Blij, B.

A. Rastegar, M. Skarabot, B. Blij, and T. Rasing, “Mechanism of liquid crystal alignment on submicron patterned surfaces,” J. Appl. Phys. 89(2), 960–964 (2001).
[Crossref]

Bodnar, V. H.

Yu. A. Nastishin, R. D. Polak, S. V. Shiyanovskii, V. H. Bodnar, and O. D. Lavrentovich, “Nematic polar anchoring strength measured by electric field techniques,” J. Appl. Phys. 86(8), 4199–4213 (1999).
[Crossref]

Burm, J.

Carbone, G.

I. M. Syed, G. Carbone, and C. Rosenblatt, “Planar degenerate substrate for micro- and nanopatterned nematic liquid-crystal cells,” J. Appl. Phys. 98(3), 034403 (2005).
[Crossref]

Chang, C. M.

H. S. Chen, Y. J. Wang, C. M. Chang, and Y. H. Lin, “A polarizer-free liquid crystal lens exploiting an embedded-multilayered structure,” IEEE Photonics Technol. Lett. 27(8), 899–902 (2015).
[Crossref]

Chen, H. S.

H. S. Chen, Y. J. Wang, C. M. Chang, and Y. H. Lin, “A polarizer-free liquid crystal lens exploiting an embedded-multilayered structure,” IEEE Photonics Technol. Lett. 27(8), 899–902 (2015).
[Crossref]

Y. H. Lin and H. S. Chen, “Electrically tunable-focusing and polarizer-free liquid crystal lenses for ophthalmic applications,” Opt. Express 21(8), 9428–9436 (2013).
[Crossref]

Chen, K. P.

Chen, M. S.

Y. H. Lin, M. S. Chen, W. C. Lin, and Y. S. Tsou, “A polarization-independent liquid crystal phase modulation using polymer-network liquid crystals in a 90 degree twisted cell,” J. Appl. Phys. 112(2), 024505 (2012).
[Crossref]

Chigrin, D.

Cho, J.

Choi, C. S.

Decker, M.

Fan, Y. H.

H. Ren, Y. H. Lin, Y. H. Fan, and S. T. Wu, “Polarization-independent phase modulation using a polymer-dispersed liquid crystal,” Appl. Phys. Lett. 86(14), 141110 (2005).
[Crossref]

H. Ren, J. R. Wu, Y. H. Fan, Y. H. Lin, and S. T. Wu, “Hermaphroditic liquid-crystal microlens,” Opt. Lett. 30(4), 376–378 (2005).
[Crossref]

Fang, J.

Ge, Z.

Giordani, M.

M. Giordani, M. Polese, M. Mezzavilla, S. Rangan, and M. Zorzi, “Toward 6G networks: Use cases and technologies,” IEEE Commun. Mag. 58(3), 55–61 (2020).
[Crossref]

Goto, K.

Grann, E. B.

Hong, J. Y.

Huang, Y.

Y. Huang, C. H. Wen, and S. T. Wu, “Polarization-independent and submillisecond response phase modulators using a 90° twisted dual-frequency liquid crystal,” Appl. Phys. Lett. 89(2), 021103 (2006).
[Crossref]

Hyman, R. M.

Jagadish, C.

Javidi, B.

Kawatsuki, N.

Kim, H.

Kim, H. S.

Kim, J. H.

J. H. Kim, M. Yoneya, J. Yamamoto, and H. Yokoyama, “Nano-rubbing of a liquid crystal alignment layer by an atomic force microscope: a detailed characterization,” Nanotechnology 13(2), 133–137 (2002).
[Crossref]

J. H. Kim, M. Yoneya, and H. Yokoyama, “Tristable nematic liquid crystal device using micropatterned surface alignment,” Nature 420(6912), 159–162 (2002).
[Crossref]

Kim, S.

Kim, Y.

Kivshar, Y. S.

Konforti, N.

Kremers, C.

Lavrentovich, O. D.

Yu. A. Nastishin, R. D. Polak, S. V. Shiyanovskii, V. H. Bodnar, and O. D. Lavrentovich, “Nematic polar anchoring strength measured by electric field techniques,” J. Appl. Phys. 86(8), 4199–4213 (1999).
[Crossref]

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Lee, W.

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L. Li, L. Xiao, J. Wang, and Q. Wang, “Movable electrowetting optofluidic lens for optical axial scanning in microscopy,” Opto-Electron. Adv. 2(2), 180025 (2019).
[Crossref]

Q. Wang, L. Xiao, C. Liu, and L. Li, “Optofluidic variable optical path modulator,” Sci. Rep. 9(1), 7082 (2019).
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Y. H. Lin, M. S. Chen, W. C. Lin, and Y. S. Tsou, “A polarization-independent liquid crystal phase modulation using polymer-network liquid crystals in a 90 degree twisted cell,” J. Appl. Phys. 112(2), 024505 (2012).
[Crossref]

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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]

Y. J. Wang, X. Shen, Y. H. Lin, and B. Javidi, “Extended depth-of-field 3D endoscopy with synthetic aperture integral imaging using an electrically tunable focal-length liquid-crystal lens,” Opt. Lett. 40(15), 3564–3567 (2015).
[Crossref]

H. S. Chen, Y. J. Wang, C. M. Chang, and Y. H. Lin, “A polarizer-free liquid crystal lens exploiting an embedded-multilayered structure,” IEEE Photonics Technol. Lett. 27(8), 899–902 (2015).
[Crossref]

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[Crossref]

Y. H. Lin, M. S. Chen, W. C. Lin, and Y. S. Tsou, “A polarization-independent liquid crystal phase modulation using polymer-network liquid crystals in a 90 degree twisted cell,” J. Appl. Phys. 112(2), 024505 (2012).
[Crossref]

H. Ren, J. R. Wu, Y. H. Fan, Y. H. Lin, and S. T. Wu, “Hermaphroditic liquid-crystal microlens,” Opt. Lett. 30(4), 376–378 (2005).
[Crossref]

H. Ren, Y. H. Lin, Y. H. Fan, and S. T. Wu, “Polarization-independent phase modulation using a polymer-dispersed liquid crystal,” Appl. Phys. Lett. 86(14), 141110 (2005).
[Crossref]

Y. H. Lin, H. Ren, Y. H. Wu, Y. Zhao, J. Fang, Z. Ge, and S. T. Wu, “Polarization-independent liquid crystal phase modulator using a thin polymer-separated double-layered structure,” Opt. Express 13(22), 8746–8752 (2005).
[Crossref]

Liu, C.

Q. Wang, L. Xiao, C. Liu, and L. Li, “Optofluidic variable optical path modulator,” Sci. Rep. 9(1), 7082 (2019).
[Crossref]

Lorenz, A.

Luo, M.

Mao, C.

J. Zou, Z. Yang, C. Mao, and S. T. Wu, “Fast-Response Liquid Crystals for 6G Optical Communications,” Crystals 11(7), 797 (2021).
[Crossref]

Marom, E.

Mezzavilla, M.

M. Giordani, M. Polese, M. Mezzavilla, S. Rangan, and M. Zorzi, “Toward 6G networks: Use cases and technologies,” IEEE Commun. Mag. 58(3), 55–61 (2020).
[Crossref]

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Miroshnichenko, A. E.

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Yu. A. Nastishin, R. D. Polak, S. V. Shiyanovskii, V. H. Bodnar, and O. D. Lavrentovich, “Nematic polar anchoring strength measured by electric field techniques,” J. Appl. Phys. 86(8), 4199–4213 (1999).
[Crossref]

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Noda, K.

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N. Panwar, S. Sharma, and A. K. Singh, “A survey on 5G: The next generation of mobile communication,” Phys. Commun. 18, 64–84 (2016).
[Crossref]

Park, S.

Polak, R. D.

Yu. A. Nastishin, R. D. Polak, S. V. Shiyanovskii, V. H. Bodnar, and O. D. Lavrentovich, “Nematic polar anchoring strength measured by electric field techniques,” J. Appl. Phys. 86(8), 4199–4213 (1999).
[Crossref]

Polese, M.

M. Giordani, M. Polese, M. Mezzavilla, S. Rangan, and M. Zorzi, “Toward 6G networks: Use cases and technologies,” IEEE Commun. Mag. 58(3), 55–61 (2020).
[Crossref]

Pommet, D. A.

Rangan, S.

M. Giordani, M. Polese, M. Mezzavilla, S. Rangan, and M. Zorzi, “Toward 6G networks: Use cases and technologies,” IEEE Commun. Mag. 58(3), 55–61 (2020).
[Crossref]

Rasing, T.

A. Rastegar, M. Skarabot, B. Blij, and T. Rasing, “Mechanism of liquid crystal alignment on submicron patterned surfaces,” J. Appl. Phys. 89(2), 960–964 (2001).
[Crossref]

Rastegar, A.

A. Rastegar, M. Skarabot, B. Blij, and T. Rasing, “Mechanism of liquid crystal alignment on submicron patterned surfaces,” J. Appl. Phys. 89(2), 960–964 (2001).
[Crossref]

Ren, H.

Reshetnyak, V.

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]

Rosenblatt, C.

I. M. Syed, G. Carbone, and C. Rosenblatt, “Planar degenerate substrate for micro- and nanopatterned nematic liquid-crystal cells,” J. Appl. Phys. 98(3), 034403 (2005).
[Crossref]

Sakamoto, M.

Sasaki, T.

Sharma, S.

N. Panwar, S. Sharma, and A. K. Singh, “A survey on 5G: The next generation of mobile communication,” Phys. Commun. 18, 64–84 (2016).
[Crossref]

Shen, X.

Shiyanovskii, S. V.

Yu. A. Nastishin, R. D. Polak, S. V. Shiyanovskii, V. H. Bodnar, and O. D. Lavrentovich, “Nematic polar anchoring strength measured by electric field techniques,” J. Appl. Phys. 86(8), 4199–4213 (1999).
[Crossref]

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N. Panwar, S. Sharma, and A. K. Singh, “A survey on 5G: The next generation of mobile communication,” Phys. Commun. 18, 64–84 (2016).
[Crossref]

Skarabot, M.

A. Rastegar, M. Skarabot, B. Blij, and T. Rasing, “Mechanism of liquid crystal alignment on submicron patterned surfaces,” J. Appl. Phys. 89(2), 960–964 (2001).
[Crossref]

Song, B.

Song, H.

Staude, I.

Syed, I. M.

I. M. Syed, G. Carbone, and C. Rosenblatt, “Planar degenerate substrate for micro- and nanopatterned nematic liquid-crystal cells,” J. Appl. Phys. 98(3), 034403 (2005).
[Crossref]

Tsou, Y. S.

Y. H. Lin, M. S. Chen, W. C. Lin, and Y. S. Tsou, “A polarization-independent liquid crystal phase modulation using polymer-network liquid crystals in a 90 degree twisted cell,” J. Appl. Phys. 112(2), 024505 (2012).
[Crossref]

Tsutsui, K.

Unuma, T.

Vashistha, V.

Wang, J.

L. Li, L. Xiao, J. Wang, and Q. Wang, “Movable electrowetting optofluidic lens for optical axial scanning in microscopy,” Opto-Electron. Adv. 2(2), 180025 (2019).
[Crossref]

Wang, Q.

L. Li, L. Xiao, J. Wang, and Q. Wang, “Movable electrowetting optofluidic lens for optical axial scanning in microscopy,” Opto-Electron. Adv. 2(2), 180025 (2019).
[Crossref]

Q. Wang, L. Xiao, C. Liu, and L. Li, “Optofluidic variable optical path modulator,” Sci. Rep. 9(1), 7082 (2019).
[Crossref]

Wang, Y. J.

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]

Y. J. Wang, X. Shen, Y. H. Lin, and B. Javidi, “Extended depth-of-field 3D endoscopy with synthetic aperture integral imaging using an electrically tunable focal-length liquid-crystal lens,” Opt. Lett. 40(15), 3564–3567 (2015).
[Crossref]

H. S. Chen, Y. J. Wang, C. M. Chang, and Y. H. Lin, “A polarizer-free liquid crystal lens exploiting an embedded-multilayered structure,” IEEE Photonics Technol. Lett. 27(8), 899–902 (2015).
[Crossref]

Wen, C. H.

Y. Huang, C. H. Wen, and S. T. Wu, “Polarization-independent and submillisecond response phase modulators using a 90° twisted dual-frequency liquid crystal,” Appl. Phys. Lett. 89(2), 021103 (2006).
[Crossref]

Wilkinson, T. D.

Won, K.

Wu, J. R.

Wu, S. T.

J. Zou, Z. Yang, C. Mao, and S. T. Wu, “Fast-Response Liquid Crystals for 6G Optical Communications,” Crystals 11(7), 797 (2021).
[Crossref]

Y. Huang, C. H. Wen, and S. T. Wu, “Polarization-independent and submillisecond response phase modulators using a 90° twisted dual-frequency liquid crystal,” Appl. Phys. Lett. 89(2), 021103 (2006).
[Crossref]

H. Ren, Y. H. Lin, Y. H. Fan, and S. T. Wu, “Polarization-independent phase modulation using a polymer-dispersed liquid crystal,” Appl. Phys. Lett. 86(14), 141110 (2005).
[Crossref]

Y. H. Lin, H. Ren, Y. H. Wu, Y. Zhao, J. Fang, Z. Ge, and S. T. Wu, “Polarization-independent liquid crystal phase modulator using a thin polymer-separated double-layered structure,” Opt. Express 13(22), 8746–8752 (2005).
[Crossref]

H. Ren, J. R. Wu, Y. H. Fan, Y. H. Lin, and S. T. Wu, “Hermaphroditic liquid-crystal microlens,” Opt. Lett. 30(4), 376–378 (2005).
[Crossref]

N. Konforti, E. Marom, and S. T. Wu, “Phase-only modulation with twisted nematic liquid-crystal spatial light modulators,” Opt. Lett. 13(3), 251–253 (1988).
[Crossref]

D. K. Yang and S. T. Wu, Fundamentals of Liquid Crystal Devices (Wiley, 2006).

Wu, Y. H.

Xiao, L.

L. Li, L. Xiao, J. Wang, and Q. Wang, “Movable electrowetting optofluidic lens for optical axial scanning in microscopy,” Opto-Electron. Adv. 2(2), 180025 (2019).
[Crossref]

Q. Wang, L. Xiao, C. Liu, and L. Li, “Optofluidic variable optical path modulator,” Sci. Rep. 9(1), 7082 (2019).
[Crossref]

Xiao, X.

Xie, Z. W.

Yamamoto, J.

J. H. Kim, M. Yoneya, J. Yamamoto, and H. Yokoyama, “Nano-rubbing of a liquid crystal alignment layer by an atomic force microscope: a detailed characterization,” Nanotechnology 13(2), 133–137 (2002).
[Crossref]

Yang, D. K.

D. K. Yang and S. T. Wu, Fundamentals of Liquid Crystal Devices (Wiley, 2006).

Yang, J. H.

Yang, Z.

J. Zou, Z. Yang, C. Mao, and S. T. Wu, “Fast-Response Liquid Crystals for 6G Optical Communications,” Crystals 11(7), 797 (2021).
[Crossref]

Yokoyama, H.

J. H. Kim, M. Yoneya, J. Yamamoto, and H. Yokoyama, “Nano-rubbing of a liquid crystal alignment layer by an atomic force microscope: a detailed characterization,” Nanotechnology 13(2), 133–137 (2002).
[Crossref]

J. H. Kim, M. Yoneya, and H. Yokoyama, “Tristable nematic liquid crystal device using micropatterned surface alignment,” Nature 420(6912), 159–162 (2002).
[Crossref]

Yoneya, M.

J. H. Kim, M. Yoneya, and H. Yokoyama, “Tristable nematic liquid crystal device using micropatterned surface alignment,” Nature 420(6912), 159–162 (2002).
[Crossref]

J. H. Kim, M. Yoneya, J. Yamamoto, and H. Yokoyama, “Nano-rubbing of a liquid crystal alignment layer by an atomic force microscope: a detailed characterization,” Nanotechnology 13(2), 133–137 (2002).
[Crossref]

You, Q.

Yu, S.

Zhao, Y.

Zorzi, M.

M. Giordani, M. Polese, M. Mezzavilla, S. Rangan, and M. Zorzi, “Toward 6G networks: Use cases and technologies,” IEEE Commun. Mag. 58(3), 55–61 (2020).
[Crossref]

Zou, J.

J. Zou, Z. Yang, C. Mao, and S. T. Wu, “Fast-Response Liquid Crystals for 6G Optical Communications,” Crystals 11(7), 797 (2021).
[Crossref]

Appl. Opt. (2)

Appl. Phys. Lett. (2)

H. Ren, Y. H. Lin, Y. H. Fan, and S. T. Wu, “Polarization-independent phase modulation using a polymer-dispersed liquid crystal,” Appl. Phys. Lett. 86(14), 141110 (2005).
[Crossref]

Y. Huang, C. H. Wen, and S. T. Wu, “Polarization-independent and submillisecond response phase modulators using a 90° twisted dual-frequency liquid crystal,” Appl. Phys. Lett. 89(2), 021103 (2006).
[Crossref]

Crystals (1)

J. Zou, Z. Yang, C. Mao, and S. T. Wu, “Fast-Response Liquid Crystals for 6G Optical Communications,” Crystals 11(7), 797 (2021).
[Crossref]

IEEE Commun. Mag. (1)

M. Giordani, M. Polese, M. Mezzavilla, S. Rangan, and M. Zorzi, “Toward 6G networks: Use cases and technologies,” IEEE Commun. Mag. 58(3), 55–61 (2020).
[Crossref]

IEEE Photonics Technol. Lett. (1)

H. S. Chen, Y. J. Wang, C. M. Chang, and Y. H. Lin, “A polarizer-free liquid crystal lens exploiting an embedded-multilayered structure,” IEEE Photonics Technol. Lett. 27(8), 899–902 (2015).
[Crossref]

J. Appl. Phys. (4)

Y. H. Lin, M. S. Chen, W. C. Lin, and Y. S. Tsou, “A polarization-independent liquid crystal phase modulation using polymer-network liquid crystals in a 90 degree twisted cell,” J. Appl. Phys. 112(2), 024505 (2012).
[Crossref]

A. Rastegar, M. Skarabot, B. Blij, and T. Rasing, “Mechanism of liquid crystal alignment on submicron patterned surfaces,” J. Appl. Phys. 89(2), 960–964 (2001).
[Crossref]

I. M. Syed, G. Carbone, and C. Rosenblatt, “Planar degenerate substrate for micro- and nanopatterned nematic liquid-crystal cells,” J. Appl. Phys. 98(3), 034403 (2005).
[Crossref]

Yu. A. Nastishin, R. D. Polak, S. V. Shiyanovskii, V. H. Bodnar, and O. D. Lavrentovich, “Nematic polar anchoring strength measured by electric field techniques,” J. Appl. Phys. 86(8), 4199–4213 (1999).
[Crossref]

J. Opt. Soc. Am. A (1)

Liq. Cryst. Rev. (1)

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]

Nanotechnology (1)

J. H. Kim, M. Yoneya, J. Yamamoto, and H. Yokoyama, “Nano-rubbing of a liquid crystal alignment layer by an atomic force microscope: a detailed characterization,” Nanotechnology 13(2), 133–137 (2002).
[Crossref]

Nature (1)

J. H. Kim, M. Yoneya, and H. Yokoyama, “Tristable nematic liquid crystal device using micropatterned surface alignment,” Nature 420(6912), 159–162 (2002).
[Crossref]

Opt. Express (5)

Opt. Lett. (4)

Opt. Mater. Express (1)

Opto-Electron. Adv. (1)

L. Li, L. Xiao, J. Wang, and Q. Wang, “Movable electrowetting optofluidic lens for optical axial scanning in microscopy,” Opto-Electron. Adv. 2(2), 180025 (2019).
[Crossref]

Phys. Commun. (1)

N. Panwar, S. Sharma, and A. K. Singh, “A survey on 5G: The next generation of mobile communication,” Phys. Commun. 18, 64–84 (2016).
[Crossref]

Sci. Rep. (1)

Q. Wang, L. Xiao, C. Liu, and L. Li, “Optofluidic variable optical path modulator,” Sci. Rep. 9(1), 7082 (2019).
[Crossref]

Other (2)

D. K. Yang and S. T. Wu, Fundamentals of Liquid Crystal Devices (Wiley, 2006).

Z. N. Yuan, Z. B. Sun, H. S. Kwok, and A. K. Srivastava, “Fast LiDAR systems based on ferroelectric liquid crystal Dammann grating,” Liq. Cryst. (2021). Available: https://doi.org/10.1080/02678292.2021.1873437

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request. The authors declare no conflicts of interest.

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Figures (10)

Fig. 1.
Fig. 1. (a) Schematic structure of the proposed LC phase modulator with 2N+1 by 2N+1 sub-domains. The optics axes (i.e. long axes of nematic LC molecules) at the adjacent domains are orthogonal. Each subdomain in (a) is equivalent to an electrically tunable wave plate with uniaxial optical axis along either (b) x-axis or (c)y-axis.
Fig. 2.
Fig. 2. One dimensional refractive index as the function of x-coordinate when the incident light is (a) x-linearly polarized light and (b) y-linearly polarized light.
Fig. 3.
Fig. 3. (a) Director angle of LC as a function of z/L at different applied voltage. (b) Phase retardation as a function of electric potential. (c) The averaged value of the cosine squared of the director angle $\left\langle {{{\cos }^2}\theta } \right\rangle$ as a function of electric potential.
Fig. 4.
Fig. 4. Function $0.5({1 + \eta (x )\cdot \eta (y )} )$ with coordination shifting by half of the size of its sub-domain. Examples for 5 (left) and 13 pixels (right).
Fig. 5.
Fig. 5. Effective refractive index profile for 5 pixels case.
Fig. 6.
Fig. 6. (a) and (b) are phase retardation for x- and y-polarized waves, respectively (5 pixels case).
Fig. 7.
Fig. 7. Surface morphologies of the alignment layers scratched by the cantilever tip of AFM in the contact mode when the force applied to the cantilever was (a) 300 nN, (b) 500 nN, (c) 800 nN, and (d) 1000 nN. The surface morphologies were obtained by scanning at 90 degrees with respect to the grooves at the tapping mode. (e) is the roughness of surface morphologies of (a)-(d) as the function of force at the cantilever. Roughness is defined as the root mean square of the surface height.
Fig. 8.
Fig. 8. Photos of samples observed under crossed polarizers. One ITO glass substrate is coated with a conventional rubbed alignment layer and the other one is coated with the grooving alignment layer. The nematic LC was sandwiched between two substrates. The grooving alignment layer was scratched by the tip when we applied forces to the cantilever: (a) 300 nN, (b) 500 nN, (c) 800 nN, and (d) 1000 nN. P and A are transmissive axes of two polarizers. R stands for the groove direction with grooves either parallel((a) to (d)) or 45 degrees with respect to one of the polarizers ((e)-(h)).
Fig. 9.
Fig. 9. (a) Schematic configuration for the polarizer-free LC micro-lens (aperture size of a micro-lens: 100 μm). (b) The illustration of scratched PI area in (a). Each sub-domain is 4 μm and total size is 112 μm. Arrows indicate the direction of scratched grooves. The scratched directions (or alignment direction) between adjacent sub-domains are orthogonal. (c) is the side view of (a) when voltage is turned off. (d) and (e) show the focusing capability of the sample at (d) V = 0 and (e) V = 11 Vrms, respectively. (d) and (e) were captured under unpolarized light. The focal length is 1.37 mm for V = 11 Vrms. (f) and (g) show the focusing capability of the sample at V = 0 and V = 11 Vrms for linearly-polarized light where its polarization state is parallel to the rubbing direction. (h) and (i) show the focusing capability sample at V = 0 and V = 11 Vrms for a linearly-polarized light when the polarization state is perpendicular to the rubbing direction. The color bar in (d)-(i) standards for the light intensity in arbitrary unit.
Fig. 10.
Fig. 10. (a) Measured focal length as function of voltage of the LC micro-lens when the incident light is unpolarized light (denoted as NO polarizer) and linearly polarized light (0, 45, 90, 135 degree linearly polarized light). Yellow triangles represent the focal length for the conventional polarized LC micro-lens. (b) The corresponding optical phase shift between the central and peripheral region of the micro-lens as a function of applied voltage. Still, yellow triangles represent the focal length for the conventional polarized LC micro-lens. The rest is the polarizer-free LC micro-lens. In (a) and (b), a polarizer was used for the measurement for the conventional polarized LC micro-lens.

Tables (1)

Tables Icon

Table 1. Measured and calculated parameters of 4 LC samples in Fig. 3 and a typical LC cell with two mechanically buffered alignment layers.

Equations (20)

Equations on this page are rendered with MathJax. Learn more.

n ~ ( x , x ^  -  polarized ) = n 0 + ( n e n o ) a = N N r e c t ( x 2 Δ x Δ x ) ,
r e c t ( x ) = { 1 | x | < 1 2 1 2 | x |  =  1 2 0 otherwise .
n ~ ( x ^ polarized ) = n 0 + ( n e n o ) a = N N r e c t ( x 2 a Δ x Δ x ) b = N N r e c t ( y 2 b Δ y Δ y ) .
n ~ ( y ^ polarized ) = n 0 + ( n e n o ) [ 1 a = N N r e c t ( x 2 a Δ x Δ x ) b = N N r e c t ( x 2 b Δ y Δ y ) ] ,
n ~ ( x ^ polarized ) = n 0 + ( n e f f ( θ z ) n o ) a = N N r e c t ( x 2 a Δ x Δ x ) b = N N r e c t ( y 2 b Δ y Δ y ) ,
n ~ ( y ^ polarized ) = n 0 + ( n e f f ( θ z ) n o ) [ 1 a = N N r e c t ( x 2 a Δ x Δ x ) b = N N r e c t ( y 2 b Δ y Δ y ) ] .
n ~ ( x ^ polarized ) = n ~ ( y ^ polarized ) = n 0 .
E i n = E 0 ( A x x ^ + A y y ^ ) e j ( ω t k z ) ,
E o u t ( r ) = E 0 e j ( ω t k z ) ( A x e j k n x ( r ) d x ^ + A y e j k n y ( r ) d y ^ ) ,
E o u t ( r ) = E 0 e j ( ω t k z ) e j k n 0 d ( A x e j k ( n e f f ( θ z ) n o ) d a = N N r e c t ( x 2 a Δ x Δ x ) b = N N r e c t ( y 2 b Δ y Δ y ) x ^ + A y e j k ( n e f f ( θ z ) n o ) d [ 1 a = N N r e c t ( x 2 a Δ x Δ x ) b = N N r e c t ( y 2 b Δ y Δ y ) ] y ^ ) .
η ( x ) = 0.5 [ 1 + 4 π k = 1 sin ( 2 π ( 2 k 1 ) ω x ) 2 k 1 ] .
g ( x , y ) = 0.5 [ 1 + ( 4 π k = 1 sin ( 2 π ( 2 k 1 ) ω x ) 2 k 1 ) ( 4 π k = 1 sin ( 2 π ( 2 k 1 ) ω y ) 2 k 1 ) ] .
θ ( x , y ) = π 2 exp ( x 2 + y 2 R 2 ) .
Δ Φ e ( x , y ) = 2 π L λ n e f f g ( x , y ) ;
Δ Φ o ( x , y ) = 2 π L λ n e f f [ 1 g ( x , y ) ] .
η ( x ) = 1 η ( x + T / 2 ) ,
a a η ( x ) d x = { x = x + T / 2 } = a T / 2 a T / 2 η ( x + T / 2 ) d x = a T / 2 a T / 2 [ 1 η ( x ) ] d x .
+ η ( x ) d x = lim a a a η ( x ) d x = lim a a T / 2 a T / 2 [ 1 η ( x ) ] d x = + [ 1 η ( x ) ] d x .
f = π r o 2 | ϕ ( V > V t h ) ϕ ( V = 0 ) | λ ,
f = π r o 2 k ( n e n o ) d λ = r o 2 2 ( n e n o ) d .

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