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Abstract
We demonstrate, for the first time, an electrically-tunable and physically-planar freeform optical element made up of nematic liquid crystals (LCs). Continued on numerical study in previous paper (Part I), experimental results here show that it is possible to break the rotational symmetry of the wavefront through the use of uneven tilt angles of the LC molecules even though the electric potential is rotationally symmetric. Our optical element offers the ability to electrically tune the direction of the optical axis, the wavefront deviation, as well as the Zernike polynomials for general descriptions of wavefronts. Corresponding Zernike coefficients of a Zernike polynomial that are related to defocus and spherical aberration, which can be adjusted individually or together. The minimum wavefront deviation is >λ/6. The Zernike coefficients related to coma aberration or the tilt of the optical axis are also electrically tunable. By incorporating our LC phase modulator with tunability of freeform wavefronts into a simple reflective optical system, we demonstrate convincing image performance for off-axis image aberration correction. This approach will inspire further development and design of LC optical elements for applications, such as hyperspectral imagers in aerospace optics, augmented reality, virtual reality, quantum information systems, innovative miniaturized reflective telescopic systems for astrophysics, planetary science, and earth science.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Freeform optics is revolution optical design consisting of at least one element whose optical surface does not have translational or rotational symmetry, also called freeform surface [1–2]. In general, a freeform optical element converts a wavefront into another arbitrary and continuous wavefront by varying the optical path difference (OPD) [2–4]. Merits of freeform optics, compared to traditional optics, are that it provides multiple degrees of freedom for correcting aberration in optical system, more compact system design with extended field of view (FOV), and better image quality [5–9]. Some upcoming applications, such as aerospace optics and augmented reality system, are designed based on freeform optics for the advantages mentioned above [10–14]. Although there are many developments on aberration theories related to freeform optics, it is still challenging for adopting freeform optics widely because the difficulties of manufacture, system assembly, phase error controlling, and even characterizing or measuring the optical component with freeform surface [2–4]. The challenges arise because freeform optics lack a reference center or reference points in nature. Up to date, scientists and engineers are still looking for novel optical devices that can convert one wavefront into another continuous arbitrary wavefront using freeform optical surfaces with the capability of adjusting the wavefront variation (or freeform deviation).
As reported in previous paper (Part I) [15], liquid crystal (LC) phase is a state of matter between a crystalline solid and an amorphous liquid. Rod-like (nematic) LC molecules exhibit the unique optical properties associated with uniaxial crystals and dielectric anisotropy [16–17]. Nematic orientation can be manipulated by applied electric fields. By applying continuously electric field to the nematics, the wavefront should be modulated continuously. In the measurement, the LC optical elements can be made physically planar, and then the wavefront after light passing through a LC optical element can be directly measured by interference method or a wavefront sensor. For the challenges posed by freeform optics, LC devices could be used to arbitrarily modulate the wavefront of incident light wave when LC devices are subject to applied electrical fields. Among current LC devices, phase-only spatial light modulators (SLMs) based on nematics and LC wavefront corrector can be used for dynamic pixel-by-pixel phase correction [18–24]. However, SLMs with a step-like wavefront and limited resolution that depends on its pixel size (∼ 3 µm) [25] is unsuitable for the use of freeform optics because ideal freeform optical element should be capable of a conversion of continuous wavefronts.
Among other LC phase modulators, LC lenses have been developed as planar optical elements since 1979 [26]. Over the past 40 years, researchers have proposed different kinds of LC lenses to convert a plane wave into a converging or a diverging paraboloidal wave for the purpose of approximating an ideal lens or a conventional thin lens [27–29]. Fresnel-type LC lens, which is based on refraction, diffraction or both [29–31], could be polarizer-free with a large aperture size. However, the concentric ring structure of Fresnel-type LC lenses are not suitable in image systems when considering freeform optical element. Gradient-type LC lenses, without concentric ring structure, are recently demonstrated with polarization independency with multi-layered structure and a larger aperture size [32–33]. Other studies have shown that the modulated wavefronts from some LC lenses are continuously tunable [34–37]. For gradient-type LC lens, Begel et al. exploited a divided-electrode pattern for generating asymmetric electric field in order to reduce the aberration of LC lenses [38]. Up to date, an unwanted aberration arises from tilt angle of LC molecules is an issue to achieve a diffractive-limited optical element. However, pursuing diffractive limit here is an optimization in component level but not considering the overall performance of whole optical system, and literatures related to LC lenses have neither discussed nor demonstrated the capability of being a freeform optical element. To our best knowledge, no researchers have studied the possibility of electrically-tunable freeform optical elements made up of nematics. In this paper, we demonstrate such a system for the first time. We demonstrate a polarizer-free LC phase modulator as a freeform optical element via uneven tilt angles of the LC molecules, even though the electric potential is rotationally symmetric. The effect of uneven tilt angles exists but is overlooked by scientists for many years. In the experiments, the polarizer-free LC freeform optical element exhibits electrical tunability of the optical axis, wavefronts, the wavefront deviation and the adjustable Zernike coefficients. We also demonstrate image performance for off-axis aberration correction by using the LC optical element in a reflective optical system. The LC-based freeform optical element which is electrically tunable in wavefront is superior to typical lenses and typical freeform optical elements. The modulated wavefront is freeform-like and continuously tunable. Compared to typical step-like wavefronts in spatial light modulator (SLM), the LC-based freeform optical element indeed is a breakthrough in geometrical optics. We hope that this study will inspire researchers to design tunable optical elements for freeform optical systems.
2. Operating principle and sample preparation
Assume the optical phase transformation of a freeform optical element at the coordinates (x, y) is ${\textrm{t}_{freeform}}(x,y) = {e^{j \cdot \Phi (x,y)}}$, where $\Phi (x,y\textrm{)}$ is the optical phase depending on the optical path during wave propagation [39]. $\Phi (x,y\textrm{)}$ equals ${k_0} \cdot n(x,y) \cdot d(x,y)$, where ${k_0}$ is the wavenumber in free space, n(x, y) is the refractive index, and d(x, y) is the thickness of the freeform optical element. We assume that $\tilde{E}(x,y)$ and $\tilde{E}^{\prime}(x,y)$ are the incident and transmitted complex fields of the freeform optical element, respectively. The relation between $\tilde{E}(x,y)$ and $\tilde{E}^{\prime}(x,y)$ is $\tilde{E}^{\prime}\textrm{(x,y) = }{\textrm{t}_{freeform}}(x,y) \cdot \tilde{E}(x,y)$. When a wavefront error or aberration exists, $\tilde{E}^{\prime}(x,y)$ is the result of a combination of a perfect spherical wave (the reference wavefront) and an extra phase-shifting plate over an aperture [40]. As a result, the optical phase transformation can be written as ${\textrm{t}_{freeform}}(x,y) = p(x,y) \cdot {{\mathop{\rm e}\nolimits} ^{j \cdot {k_0} \cdot W(x,y)}}$, where p(x,y) is the pupil function, W(x,y) is the path length error at (x,y), and k is still the wavenumber in free space. W(x, y) is a so-called aberration function, wavefront, or wave aberration polynomial, and can be decomposed as a linear superposition of Zernike polynomials (${Z_i}(\rho ,\theta )$ ): $W(\rho ,\theta ) = \sum\limits_i {{C_i} \cdot {Z_i}} (\rho ,\theta )$, where ${C_i}$ is the Zernike coefficient corresponding to each polynomial, i is the index of the Zernike polynomial, $\rho$ is the normalized radial position ($\rho = r/{r_0}$ and $r \equiv \sqrt {{x^2} + {y^2}}$), ${r_0}$ is the aperture radius, and $\theta$ is the azimuthal angle in cylindrical coordinates. Zernike polynomials, a complete basis for circular domain, are commonly used in aberration theory in geometrical optics. Each polynomial represents an aberration, and it is a product of radial polynomials $R(\rho )$ and angular functions $\Theta (\theta )$ [40]:
Fig. 1. (a) Schematic showing the structure of the nematic-based freeform optical element. (b)-(d) are side views of (a). (b) A spherical lens with a positive focal length. (c) A spherical lens with a negative focal length. (d) A freeform optical element with a non-rotationally-symmetric distribution of the LC tilt angles (in other words, an LC device with a tilted optical axis). The red lines in (b)-(d) represent electrodes. (e) is an equivalent circuit of (a). Rs is the sheet resistance of the high resistivity layer, Cd is specific capacitance of the insulating layer, Ceff is the effective specific capacitance of the LC layer and the glass layer, and Reff is the effective sheet resistive of the LC layer and the glass layer.
The structure consists of three layers of glass substrates, two layers of indium tin oxide (ITO) sheet electrodes, a layer of ITO electrode patterned with 10-mm-diameter hole (that is, aperture size = 10 mm), an insulting layer (a UV glue, NOA81, Norland), one high-resistivity layer with a resistance of 106 ohms per square or ohm/square [a mixture of polyvinyl alcohol (PVA, Merck) and a conductive polymer (S300, AGFA)], a layer of polyimide (PI, model: SE-7492, Nissan Chemical), two alignment layers (PVA from Merck and PI from Nissan Chemical), a polymeric layer with alignment ability on both sides, and two LC layers each with a thickness of 50 µm (LN3, also named as NJU-LDn-2, Δɛ = 5.98, K11= 9, Δn = 0.369 for λ = 589.3 nm at 20°C, synthetized and provided by Xiao Liang in Tsinghua University) [44–46]. The orientations of the two LC layers are set by the alignment layers and polymeric layers. The transmittance of the structure is larger than 80% without Fresnel reflection and absorption from two ITO substrates and the polymeric layer. The structure here was developed and extended from early stage LC lens structure with three electrodes [47], high-resistive layer [34], and polymeric layer for separating LC layers [32].
The fabrication process for a polymeric layer is as follows. Each polymeric layer consists of a mixture of nematic LC (Merck, MLC 2144, Δn = 0.2493 for λ = 589.3 nm at 20°C), reactive mesogen (Merck, RM-257), and a photoinitiator (Merck, IRG-184) with a 20:79:1 wt% ratio. The mixture was filled into an empty cell at 90°C, where the empty cell consists of two ITO glass substrates coated with alignment layers which were mechanically buffered in orthogonal directions. The pretilt angle of two mechanically buffered alignment layers is 3-5 degrees. The thickness of the empty cell was 35 µm sustained by a mylar film with thickness of 35 µm and fixed by UV glue (NOA81, Norland). At 90°C, the mixture is in nematic phase, and the molecule are well aligned by two alignments. Thereafter, an alternating current (AC) voltage of 300 Vrms with a frequency of 1 kHz was applied to the cell, and the cell was then exposed to ultraviolet (UV) light with an intensity of 3 mW/cm2 at 90°C for an hour to enable polymerization. After polymerization, a polymeric layer with thickness of 35 µm was obtained. The pretilt angle on the polymeric layer is also 3-5 degrees because the mixture is polymerized with pretilt angle of 3-5 degrees. The keys of controlling cell-gap and maintaining the flatness of LC polymeric layer are as follow. After polymerization, the cell was heated up to 70°C to soften the sealing (i.e., UV glue), and a knife was used to separate the glass substrates. Next, the LC polymeric layer which was still attached on one glass substrate was assembled with bottom glass substrate coated with ITO and polyimide. The gap between the bottom glass substrate and the LC polymeric layer was controlled by the mylar film with thickness of 50 µm. Since the LC polymer layer was attached on glass substrate and fixed by a mylar film, the surface flatness was maintained. After removing the rest of glass substrate, the polymeric layer is fixed by the bottom glass with the mylar film. We deposited conductive layer on the buffering layer with the surface coated with 10mm-diameter ITO pattern [see Fig. 1(a)], and then the conductive polymer was sealed with flat ITO sheet using UV glue. Finally, the buffering layer was spin-coated with alignment, and then it was attached with polymeric layer with 50µm-thick mylar film. The structure was completed after filling the LC mixture using capillary effect.
An equivalent circuit of the structure shown in Fig. 1(a) is displayed in Fig. 1(e), where we assume that U is the electric potential at top of the buffering layer and that U is a periodic function of time. The homogeneous equation $\nabla _{x,y}^2U - {\kappa ^2}U = 0$ for a disk-like high-resistivity electrode is also the equation for the zeroth-order Bessel function. The simplified solution of U is
The operating principle has been illustrated in detail in previous article [15], and we summary it here briefly. For V1>V2, the LC molecules near the edge of the aperture are more perpendicular to the substrate than the ones at the center of the aperture, as illustrated in Fig. 1(b). Therefore, the LC device functions as a positive lens with a positive focal length. Two orthogonal LC layers lead to the same phase shift for two eigenpolarizations (i.e., the extraordinary wave and ordinary wave), and then the LC optical element acts as a polarization-independent lens. The similarly, the LC molecules near the edge of the aperture are more parallel to the substrate than the ones at the center of the aperture for V2>V1, as illustrated in Fig. 1(c). The LC device now functions as a polarization-independent negative lens with a negative focal length. For oblique incident light, the structure remains polarization independence when incident angle of light less than 5 degree; the wavefront error between two eigen-polarized light is below λ/4. In optometry, devices such as those depicted in Figs. 1(b)–2(c) are called single vision lenses. In Fig. 1(d), the LC optical element possesses an asymmetric distribution of the LC molecular tilt angles (the angle between the long axis of the LC molecule and the glass substrate). The rotational axis is no longer along the center of the aperture. Therefore, the LC device exhibits a phase profile with an oblique optical axis and can be considered as a freeform optical element. Under the condition that the structure and electric field are rotational symmetry, the mechanism to break the rotational symmetry is the initial angle of LC molecules. When the voltages are turned on, the dipole moments of the LC molecules are induced by the applied electric field (E). The dielectric torque (τ) exerted on an individual LC molecule makes the LC molecule to rotate (creating a dynamic state). Next, the LC molecules stop rotating and reach a steady state when there is balance between the dielectric torque, the elastic recoil force of the LC molecules and the anchoring force from the alignment layers. Since the initial tilt angle of LC molecules is 3-5 degree in the proposed LC device, the dielectric torque is non-rotationally symmetric. This indicates that we get the asymmetric LC profile in the steady even with a rotationally symmetric electric field as illustrated in previous article [15].
3. Methods
3.1 Measurement of wavefront and focal length
To test the optical properties of the LC optical element, the fabricated LC optical element was studied using a Shack-Hartmann wavefront sensor (Thorlabs, WFS150-7AR). The experimental setup consists of a light source (He-Ne laser, Model: 30990, Research Electro-Optics, Inc., λ = 633 nm), a single mode fiber, a solid lens for collimation with a focal length of 75.6 cm, a pair of solid lenses as relay optics (focal length: 19.0 cm and 6.25 cm), and a wavefront sensor. The light source was coupled into the single mode fiber; the light coming out from another end of the fiber can be regarded as a point light source. The point source located at the focal plane of the lens was coupled to a lens to generate a collimated light. The LC sample, relay optics, and the wavefront sensor are placed behind the collimating lens, accordingly. The LC sample was placed in front of the relay optics at the distance of the front focal length of the first solid lens. The wavefront sensor was placed behind the relay optics at the distance of the back focal length of the second lens of replay optics. The wavefront generated by the sample was relayed to the wavefront sensor. In the measurement, the wavefront sensor provides the fitting data in the form of 5th order Zernike coefficients [i = 0-20 in Eq. (1)]. The first three coefficients (i = 0-2) are neglected in further calculation because piston is an uniform phase and the tilt is generated by deviation from the center of the sample aperture and the center of the wavefront sensor. Based on the measured Zernike coefficients, we can reconstruct the wavefronts and calculate the lens power P (i.e. an inverse of focal length) according to a formula [40]: $P ={-} {{4\sqrt {3 \cdot } {c_4}} \mathord{\left/ {\vphantom {{4\sqrt {3 \cdot } {c_4}} {{r_0}^2}}} \right. } {{r_0}^2}}$, where ${r_0}$ is the radius of the aperture (${r_0}$ = 5 mm) and c4 is defocus coefficient.
3.2 Measurement of difference in OPD of wavefront vs applied voltages
To measure the minimum departure of wavefront, we applied a fixed value of one of the voltages, either V1 or V2, at a fixed frequency to the LC optical element and then varied the other voltage by means of an incremental change of 2Vrms. We plotted the distribution of aberration function of wavefront based on the following equation of $W({V_1},{V_2}) = \sum\limits_{i = 3}^{20} {{C_i}({V_1},{V_2}) \cdot } {Z_i}$. Then, we calculated the difference between two aberration function which is denoted as $\Delta W$=|W(V1+2, V2)-W(V1, V2)| or $\Delta W$=|W(V1, V2+2)-W(V1, V2)|. We have done the measurements three times and then averaged the results. $\Delta W$, the difference in OPD, was plotted in Fig. 2(a).
Fig. 2. (a) Lens power and difference in the OPD as a function of applied voltage pairs V1 and V2. With an incremental voltage of 2 Vrms (V1 or V2), the wavefront deviation between the two adjacent voltages was measured as the difference in the OPD. (b) OPD of the nematic-based freeform optical element with 48 Vrms, 10Vrms at f = 18.5 kHz. (c) OPD of the nematic-based freeform optical element with 10 Vrms, 60Vrms at f = 5 kHz. (d), (e): OPD as a function of position in the x-pupil coordinate for different voltage pairs. (f), (g): Lens power as a function of position in the x-pupil coordinate for different voltage pairs. (h) The difference in the OPD is less than 0.3 µm when V2 increases from 30 Vrms to 32 Vrms for V1 = 48 Vrms at f = 18.5 kHz. (i) The difference in the OPD is less than 0.2 µm when V1 increases from 30 Vrms to 32 Vrms for V2 = 60 Vrms at f = 5 kHz. In (d), (e), (h) and (i), the reference wave is a plane wave with λ = 633 nm.
3.3 Calculation of lens power distribution
The aperture center of the LC optical element was set as the origin of the coordinate (i.e. x = 0, y = 0, z = 0). We calculate the lens power distribution P(r) according to: $\textrm{P}(r)\textrm{ = }\frac{{{{\partial W(r)} \mathord{\left/ {\vphantom {{\partial W(r)} {\partial r}}} \right.} {\partial r}}}}{r}$ [40,48]. Since the aberration function W can be measured and reconstructed using the equation: $W(\rho ,\theta ) = \sum\limits_{i = 3}^{20} {{C_i} \cdot {Z_i}(\rho ,\theta )}$, then the lens power distribution can be expressed as: $\textrm{P}(r)\textrm{ = }\sum\limits_{i = 3}^{20} {{C_i} \cdot \frac{{{{\partial {Z_i}(\rho ,\theta )} \mathord{\left/ {\vphantom {{\partial {Z_i}(\rho ,\theta )} {\partial r}}} \right.} {\partial r}}}}{r}}$. A singularity of lens power appears near the center of the LC optical element. The region near the singularity is within r = ± 0.5 mm and it is usually ignored in Optometry [49].
3.4 Image performance in a reflective optical system
The system consists of a broadband light source (Axen L-150W), a pin-hole aperture (< 1 mm in diameter) for confining the light as a point source, a resolution chart (USAF 1951; negative type), a cylindrical lens with a directional lens power of -1.5D (the axis of cylindrical lens is parallel to y-axis), a tilted concave mirror with a focal length of 50 mm, the LC optical element with its x-z plane set on horizontal plane, and a camera (Canon 760D) for mimicking our eyes. The pinhole was attached to the resolution chart and the images of resolution chart were recorded by the camera after the light beam reflected by the mirror with a tilt angle of 25 degrees with respect to the horizontal plane (x-z plane). The camera was placed at the distance of 17 cm away from the concave mirror. The distance between resolution chart and the concave mirror was 4 cm. At first, we took a photo before adding any optical component in order to correct the aberration. The settings of the camera were: shutter speed = 1/250, aperture = f/5.6, and ISO sensitivity = 100. Next, we took a photo after we attached a cylindrical lens whose axis is along y-axis for correcting the astigmatism. Thereafter, we placed the LC optical element 6 cm away from the concave mirror. We took two photos when we applied no voltages and applied the voltage pair of (48Vrms, 10Vrms) at f = 18.5 kHz to the LC optical element. In addition, the images at three wavelengths were recorded when color filters were placed between the LC optical element and the camera. The color filters were: λ = 650 nm (Oriel Model: 5308), λ = 550 nm, (Edmund Model: 14802), and λ = 450 nm, (Lambda Research Optics, Inc. Model: 3037). Because the transmittance of the color filters are different, the shutter speed of camera was set as 1/125 (λ = 650 nm), 1/50 (λ = 550 nm), and 2” (λ = 450 nm).
4. Experimental results and discussion
The lens power obtained as a function of applied voltage pairs (V1 and V2) is shown in Fig. 2(a). The LC device is a positive lens when V1 > V2 for V1 = 48 Vrms at 18.5 kHz [red dotted line in Fig. 2(a)]. The power of the positive lens can be tuned between + 0.13 and + 0.76 D. The LC device is a negative lens when V1 < V2 for V2 = 60 Vrms at 5 kHz [blue dotted line in Fig. 2(a)]. The power of the negative lens can be tuned up to -0.63 D. When V1 >> 0 and V2 >> 0, the lens powers are + 0.13 D for the positive lens and + 0.10 D for the negative lens, which are nonzero because non-uniformity of the homemade polymeric layer (35 ${\pm}$1.2 µm) results in small lensing effects. Figures 2(b)–2(c) show the OPD of the nematic-based freeform optical element when acting either as a positive and negative lens under different applied voltage pairs. The applicable range of optical lens power of the LC device is from -0.63D to + 0.76D (∼1.39D in total) while the response is less than 2 seconds. The response time could be further reduced by adopting driving scheme (e.g., over-driving).
Figure 2(d) shows the OPD as a function of position in the x-pupil coordinate for different voltage pairs, plotted from Figs. 2(b)–2(c). We see that both OPD and lens power decrease as V2 increases for a fixed V1. From Fig. 2(d), we see that the OPD at an x-pupil coordinate of 0 mm is larger (up to 9 micron) than the OPD at ${\pm} $5 mm, which indicates that the nematic-based freeform optical element functions as a positive lens. Based on similar analysis in Figs. 2(a) and 2(d), the nematic-based freeform optical element functions as a negative lens with the data in Figs. 2(a) and 2(e). No matter whether the nematic-based freeform optical element is operated as a positive lens or negative lens in Figs. 2(d)–2(e), the wavefronts and OPDs are not only electrically tunable but also continuous through the aperture and not step-like.
Figures 2(f)–2(g) show the lens power as a function of position in the x-pupil coordinate, obtained by converting the wavefronts or OPD in Figs. 2(d)–2(e). Typically, only the region outside ${\pm} $0.5 mm in the x-pupil coordinate is meaningful in optometry, because an unreliable singularity appears at the radial center when the local lens power is calculated from the wavefront [48–49]. The constant lens powers shown in Figs. 2(f)–2(g) represent single vision lenses because of the uniform lens power through the entire aperture. Non-constant lens powers represent progressive lenses whose lens powers vary spatially. In addition, we also observe that the lens power through the aperture can be asymmetric under certain voltage conditions, which also indicates an asymmetric OPD or wavefront. To further measure the minimum deviation of the wavefront with the applied voltages, we plotted [in Fig. 2(h)] the cross-sections of the OPD for the voltage pairs of (48Vrms, 30Vrms) and (48Vrms, 32Vrms). The difference between the two OPDs is less than 0.3 µm (< λ/2) when V2 increases from 30 Vrms to 32 Vrms for a fixed V1.
Figure 2(i) shows the cross-sections of the OPD corresponding to two voltage pairs. The change in the OPD is less than 0.2 µm (< λ/3) when V1 increases from 30 Vrms to 32 Vrms for a fixed V2. Similarly, we plotted the difference in OPD between the two adjacent voltages as we applied an incremental voltage of 2 Vrms for either V1 or V2, as shown in Fig. 2(a). Most of the OPD difference is less than λ/2. This also means the wavefront deviation is smaller than λ/2 when the applied voltage difference is smaller. In experiments, the smallest wavefront difference is around∼λ/6, which is the smallest value that the wavefront sensor can measure in our experiments. This is an implication of measurement limitation. For practical device production, for instance typical lenses, the fabricating errors of curved surfaces exist during mass production. When many lenses are assembled together in an optical system, the fabricating errors are amplified. Thus, the feature of electrically tunable wavefront deviation of the LC optical element is great benefit from the point of view of manufacturing.
To test the ability of our device for tuning aberrations, we used Zernike polynomials to fit the wavefronts under the conditions in Fig. 2(a). Then we plotted the Zernike coefficients (Ci, i = 0-20) as a function of the voltage pairs in Figs. 3(a)–3(e). In Fig. 3(a), the defocus (characterized by C4 Zernike coefficient) is positive and decreases from 2.61 micron to 0.47 micron as V2 increases from 0 to 48 Vrms. In Fig. 3(b), the defocus (C4) is negative and increases from -2.27 micron to 0.34 micron as V1 increases from 0 to 60 Vrms. Comparing the results in Figs. 2(a), 2(d), 2(f), 3(a) and 3(b), a positive C4 indicates a positive lens power, and the magnitude of C4 is proportional to the lens power. Moreover, in Fig. 3(a), the spherical aberration (characterized by C12 Zernike coefficient) decreases from a positive (+0.27 micron) to a negative value (-0.08 micron), and then C12 increases to zero when the voltage increases. Figure 3(b) shows a similar trend for C12. Comparing the results in Figs. 3(a), 3(b), 2(f), and 2(g), we see that the lens powers are constants (or uniform) at x-pupil coordinate in Figs. 2(f)–2(g) when there is no spherical aberration (that is, when C12 is zero). In contrast, spherical aberration (a non-zero C12) indicates non-uniform lens power at x-pupil coordinate in Figs. 2(f)–2(g). When C12 >0, the lens power in the center of the aperture is smaller than that around the edge of the aperture; when C12 <0, the lens power in the center of the aperture is larger than that around the edge of the aperture. Although the x-coma (C8 Zernike coefficient) and y-coma (C7 Zernike coefficient) in Figs. 3(a)–3(b) are relatively smaller than C4 and C12, the non-zero x-coma (C8) and y-coma (C7) values indicate that the oblique optical axis of the LC optical element is not parallel to the z-axis (Fig. 1). Such an oblique optical axis results in the asymmetric lens power distribution (or rotationally asymmetric wavefront) displayed in Figs. 2(f)–2(g) and is indicative of a freeform optical element. In addition, the remaining Zernike coefficients are independent of the applied voltage, as shown in Figs. 3(c)–3(e). The results indicate that the Zernike coefficients C4, C12, C7 and C8 of the LC optical element are indeed electrically tunable.
Fig. 3. Zernike coefficients of the nematic-based freeform optical element as a function of the applied voltages. (a) For V1 = 48 Vrms, and f = 18.5 kHz, the Zernike coefficients for defocus (C4), spherical aberration (C12), x-coma (C8) and y-coma (C7) vary as a function of V2. (b) For V2 = 60 Vrms and f = 5 kHz, C4, C12, C8 and C7 vary as a function of V1. (c), (d) and (e) show that the Zernike coefficients C10, C11, C13 and C14 do not depend on voltage, regardless of the voltage conditions of (a) or (b). The wavelength is λ = 633 nm.
It is seen in Fig. 3, that several Zernike coefficients of the LC optical element can be electrically tuned at the same time. For some optical systems, an electrical adjustment of single Zernike coefficient for specific aberration correction is required. To test whether our device also enables a single Zernike coefficient to be tuned while the other Zernike coefficients remain constant, we chose to look at the Zernike coefficient for defocus (C4) as an example. Four different voltage pairs and frequencies – S1, S2, S3, and S4 – were selected. The results are shown in Figs. 4(a)–4(e). From Figs. 4(a)–4(b), we can see that the lens powers are positive for S1 and S2 and negative for S3 and S4. In Fig. 4(b), the lens powers are uniform through the aperture. In Figs. 4(c)–4(e), only C4 changes almost linearly when different voltage pairs and frequencies are applied while the other Zernike coefficients remain almost constant. To further test the tunability of the spherical aberration C12, we selected three additional conditions (denoted as S5, S6, and S7). The results are shown in Figs. 5(a)–5(f). In Fig. 5(a), three conditions exhibit similar OPDs with small variations depending on the specific voltage/frequency combination. When we look at the difference in OPD between the two conditions, as depicted in Fig. 5(b), the optical axis of the LC optical element is along the center of the aperture, but the variation in OPD is electrically adjustable. The lens power distribution is more uniform (∼0.50 D) for S6 [Fig. 5(c)]. The lens power distributions of S5 and S7 are not constants. The lens power distribution of S5 displays the opposite pattern in S7. That is the lens power at the center of aperture is ∼0.26D smaller than that at the edge of the aperture ∼1.06D for S5. In S7, the lens power at the center of aperture is ∼0.63D larger than that at the edge of the aperture ∼0.37D This indicates a positive spherical aberration C12 for S5 and a negative spherical aberration for S7. In Figs. 5(d)–5(f), the spherical aberration changes, while the other Zernike coefficients remain almost constant in S5, S6, and S7. In details, for selected conditions of S1, S2, S3, and S4, the tunable coefficients of defocus (C4) is from -3.034 µm to + 2.809 µm (5.843 µm in total) while the largest change of non-targeted coefficient is 0.049 µm in total (i.e., 0.8% error). For selected condition of S5, S6, and S7, the tunable coefficient of spherical aberration (C12) is from -0.267 µm to 0.318 µm (0.585 µm in total) while the maximum change of non-targeted coefficient is 0.140 µm (i.e., 24% error). It is therefore possible to electrically tune a single Zernike coefficient of the LC optical element by adjusting the voltage pairs and frequencies. Here we just use Zernike spherical aberration as example. In fact, other Zernike coefficient corresponding to different aberrations could also be manipulated alone for requirement of optical systems.
Fig. 4. Electrically tunable Zernike coefficient for defocus (C4) of the freeform optical element. Four different voltage pairs and frequencies (S1, S2, S3, and S4) were applied. (a) OPD and (b) lens power as a function of the x-pupil coordinate. (c), (d) and (e): Zernike coefficients for S1, S2, S3 and S4. The wavelength is λ=633 nm.
Fig. 5. Electrically tunable spherical aberration (C12) of the freeform optical element. Three different voltage pairs and frequencies were applied, denoted as S5, S6, and S7. (a) OPD and lens power as a function of the x-pupil coordinate. (b) Difference in the OPD between the two conditions. (c) Lens power as a function of the x-pupil coordinate. (d), (e) and (f): Zernike coefficients for S5, S6, and S7. The wavelength is λ = 633 nm.
To measure the inclination of the equivalent optical axis of the LC optical element [Fig. 6(a)], we analyzed the OPD distributions [Figs. 6(b)–6(c)] and used them to calculate the angles Ψ and θ [shown in Fig. 6(a)]. Figures 6(b)–6(c) display the OPD in x- and y-pupil coordinates, from which the location of the local maximum or minimum was found. The shifts of the local maximum/minimum in x- and y-pupil coordinates are denoted as Δx and Δy, respectively. This analysis would help us to understand the inclination of the freeform optical element. According to Fig. 6(a), Ψ and θ were calculated using the formulae $\Psi = {\tan ^{ - 1}}({{\sqrt {\Delta {x^2} + \Delta {y^2}} } \mathord{\left/ {\vphantom {{\sqrt {\Delta {x^2} + \Delta {y^2}} } {\Delta z}}} \right. } {\Delta z}})$ and $\theta = {\tan ^{ - 1}}({{\Delta y} \mathord{\left/ {\vphantom {{\Delta y} {\Delta x}}} \right.} {\Delta x}})$, where $\Delta z$ is the distance between the center of the LC optical element and the top substrate ($\Delta z$ = 1.35 mm in the experiments). In Fig. 6(b), Ψ1 and θ1 are 5.71° and -68.6° when the lens power of the LC optical element is positive [corresponding to Fig. 2(a)]. In Fig. 6(c), Ψ2 and θ2 are 8.00° and -83.9° when the lens power of the LC optical element is negative [corresponding to Fig. 2(a)]. Figure 6(d) depicts the calculated values of Ψ and θ as a function of V2 for V1 = 48 Vrms (f = 18.5 kHz). We can see that Ψ changes between 2° and 6° with V2 and rotates around the z-axis (θ changes from + 30° to -75°) when the lens power of the LC optical element is positive [corresponding to Fig. 2(a)]. Similarly, Ψ changes between 6° and 8° with V1, but $\theta$ remains constant at approximately -85°. This means that the tilt of the optical axis Ψ changes but does not rotate around the z-axis when the lens power of the LC optical element is negative. Since there is an asymmetric electric field along the z-axis, the two LC layers should have equivalent optical axes with two different tilt angles Ψ and θ angles for x-polarized light and y-polarized light. Thus, the equivalent optical axis of the LC optical element rotates and tilts along the z-axis as a function of voltage, as can be seen in Fig. 6(d) and Fig. 6(e).
Fig. 6. Inclination of the equivalent optical axis. (a) Schematic showing the LC optical element, its optical axis and related coordinates. OPDs of the optical element and the shifts in the maximum or minimum OPD for (b) 48 Vrms, 20Vrms and f = 18.5 kHz and (c) 24 Vrms, 60 Vrms and f = 5 kHz. (d) Ψ and θ as a function of voltage V2 with V1 = 48 Vrms (f = 18.5 kHz). (e) Ψ and θ as a function of voltage V1 with V2 = 60 Vrms (f = 5 kHz). λ = 633 nm.
Up to this point, we have demonstrated the ability of a nematic-based optical element to act as a freeform optical element. To further demonstrate its potential use, we set up a simple reflective imaging system, as illustrated in Fig. 7(a). The system consists of a broadband light source (Axen L-150W), a pinhole aperture (< 1 mm in diameter) for confining the light as a point source, a resolution chart (USAF 1951, negative type), a cylindrical lens with a directional lens power of -1.5D (the axis of the cylindrical lens is parallel to the y-axis), a tilted concave mirror with a focal length of 50 mm, the LC optical element with its x-z plane as the horizontal plane, and a camera (Canon, 760D) to mimic human vision. The pinhole was attached to the resolution chart, and images of the resolution chart were recorded by the camera after the light reflected off the mirror with a tilt angle of 25° with respect to the horizontal plane (x-z plane). The distance between the resolution chart and the mirror is 4 cm while it is 6 cm between the mirror and the LC optical element. It is well-known that a spherical mirror generates cylindrical aberration and coma aberration under oblique angle of incidence, and the amount of coma aberration increases with angle of incidence. According to the results of wavefront measurement in terms of Zernike polynomials, we found operating condition of the LC optical element in order to correct the non-rotationally symmetric aberration (i.e., coma) generated by the titled spherical mirror.
Fig. 7. Freeform LC optical element in a simple reflective imaging system. (a) Illustration of the experimental setup with the concave mirror tilted 25° with respect to the x-z plane. (b) Image formed with no cylindrical lens and LC optical element removed from the setup shown in (a). (c) Image formed with cylindrical lens but no LC optical element in the setup. (d) Image formed with cylindrical lens and an LC optical element in the voltage-off state. (e) Image formed with cylindrical lens and an LC optical element in the voltage-on state. V1 = 48Vrms, V2 = 16Vrms and f = 18.5kHz. (c), (d), and (e) are images of the area highlighted by the blue square in (b).
Figure 7(b) shows the image obtained when the LC optical element and cylindrical lens are not present in the system. We magnified the small area of Fig. 7(b) and corrected this area using a cylindrical lens and the LC optical element. Typically, to improve the image quality in Fig. 7(b), the solution is to correct the tilt-induced astigmatism and coma [38]. Since the LC optical element here is mainly able to correct coma, we used a cylindrical lens in the system to correct astigmatism. The recorded image is shown in Fig. 7(c), and it can be seen that the image is still blurred. The LC optical element was then placed in the system in the voltage-off state. The resulting recorded image is shown in Fig. 7(d). Since the LC optical element features a voltage-independent cylindrical aberration [Fig. 3(c)], the recorded image in Fig. 7(d) is slightly corrected compared to Fig. 7(c). Figure 7(e) shows the recorded image when we applied voltage pair of (48Vrms, 16Vrms) at f = 18.5 kHz to the LC optical element. By comparing the images in Figs. 7(b)–7(e), we can see that the LC optical element can indeed help correct the off-axis blurred image, which also improves the FOV. The magnification of Figs. 7(c)–7(e) changes slightly because the LC optical element also provide focusing function (defocus Zernike coefficient) as well. By rotating the freeform LC device in the xy-plane, we effectively rotate the equivalent optical axis and alter the wavefront correction, which is the basic approach underpinning freeform optics. In addition, when LC optical elements is not present in the system, the monochromatic image at λ=550 nm is focused but with aberration while monochromatic image at λ=450 nm and λ=650 nm are not focused. This indicates the existence of chromatic aberration in our system, which mainly results from the dielectric cylindrical lens. Despite the large dispersion of LC material, the thin LC layer make less chromatic issue than a thick dielectric cylindrical lens. For a quantitively analysis, we measured the wavefront of the optical system shown in Fig. 7(a), and then we converted the wavefront into MTF curve as shown Fig. 8. The cylindrical lens improved MTF of the optical system (from red line to grey line) by correcting cylindrical aberration. When the LC freeform optical element was placed in the optical system but was turned off (green line), the MTF curves are similar to the one without LC optical element (grey line). After the LC freeform optical element was turned on, the MTF (blue line) was further improved because of the compensation of coma aberration. The MTF results show that we still have space the improve the optical quality by further designing the LC optical elements. A transmissive type of LC freeform optical element was demonstrated for proof-of-concept, and it could be further designed as a mirror-based LC freeform optical element. The optical system with a concave mirror is generally extended and used in folded optical systems, such as space telescopes, cubesats, and optical system of augmented reality. Similar to any optical elements for versatile applications, we should specially design and optimize the LC freeform optical elements. The LC optical elements can generate other aberration for optical system as long as we design the profile distribution of LC directors.
Fig. 8. Modulation transfer function (MTF) of the optical system. Based on the experimental set up in Fig. 7. Red line stands for MTF when the light is reflected from a mirror. Grey line stands for MTF when the light is reflected from a mirror and the cylindrical lens. Green line stands for MTF when the light is reflected from a mirror, the cylindrical lens, and a LC optical element at voltage-off. Blue line stands for MTF when the light is reflected from a mirror, the cylindrical lens, and a LC optical element at voltage-on.
Up to this point, we have demonstrated a proof-of-concept simple freeform LC optical element. In practice, to realize a useful device with new functionality or practical utility, the optical phase and lens power should be large, so we need to increase the tunable optical phase or defocus term of the Zernike coefficients (C4). This in turn involves overcoming the constraints posed by some of the chemical and physical properties of the nematics, specifically, the birefringence (Δn < 0.8) which is limited in a chemical sense, and the thickness of the LC layer (< 50 µm) which is limited by the anchoring strength of the alignment layer and LC director thermal fluctuations. A multi-layered LC phase modulator with a built-in polymeric layer could offer a way to increase the thickness of the LC layers without slowing the response time [32–33]. Nevertheless, such a modulator still requires an increase in the OPD to increase the aperture size, because the aperture size is proportional to $\sqrt {\textrm{OPD}}$ [29]. A cascaded LC structure, with many of the LC optical elements of Fig. 1(a) stacked together, could be an alternative option. A general expression for the transfer function of a cascaded LC structure is:
Regarding the wavefront deviation, the difference in OPD obtained with voltage conditions spaced by 2Vrms is less than 0.3 µm (< λ/2), and the smallest wavefront deviation < λ/6, which is similar to the results obtained from a computer numerical control (CNC) process. The wavefront deviation could be decreased (∼λ/40) by improving the accuracy of the applied voltage (0.01 V). This suggests that the LC optical element could compensate for the wavefront error of the optical system due to manufacturing processes by adjusting the applied voltages and frequencies.
Although the modulated wavefront is continuous in our experiments, the spatial resolution of the LC optical element still needs to be evaluated. In isotropic liquids the orientation coherence length is on a scale of few nanometers [50]. In nematic liquid crystal the electric coherence length is given by $\xi (E )= {\left( {\frac{K}{{{\varepsilon_0}{\varepsilon_a}}}} \right)^{1/2}}\frac{1}{E}$ [15]. The electric field coherence length says that the electric field of a given magnitude is able to induce the elastic deformation over a length scale $\xi $. Typical value of LC elastic constant $K\sim {10^{ - 12}}N$, dielectric anisotropy ${\varepsilon _a}\sim 10$, then for usual value of the electric field of 1 V per micron, one gets an estimate for the electric field coherence length $\xi $∼ 0.1 um. It should be noted that there are blue phase LC with rather large dielectric anisotropy (>100) [51], in which case the electric correlation length may be significantly lower than 100 nm. In addition, according to the swarm theory of nematics, a LC medium constitutes an aggregation of mutually-oriented LC molecules that are gathered into a large group (swarm) by intermolecular interactions [52]. Spatially, a swarm can be considered as the smallest area in which the modulated wavefront is uniform in optical phase. A swarm comprises approximately 104-106 molecules, and the size of a swarm in which molecular orientations are correlated can be up to 10-20 times larger than the dimension of the LC molecule (∼10−10 m) [52–53]. With a molecular length of 10 Å, the size of the swarm is approximately 10-20 nm (below λ/20 = 0.05 λ in the visible range). Compared to commercial diamond-turning technology, the LC optical element could provide a more continuous spatially-modulated wavefront, which is required for freeform optics. Actually, the modulated wavefront should be continuous because the energy required for many LC molecules (104∼106) to remain in one region of a certain dimension (10∼20 nm) and then shift abruptly to another orientation is prohibitively large [52–53]. Compared to an SLM with a pixel size of >3 µm, the modulated wavefront obtained using the LC optical element demonstrated in this paper can be regarded as continuously electrically tunable.
5. Conclusion
We experimentally present, for the first time, an electrically-tunable, polarizer-free, freeform optical element that is based on nematic LCs. Even though the electric potential is rotationally symmetric, the rotational symmetry of the wavefront can be broken by generating uneven tilt angles of the LC molecules. This allows us to realize electrically tunable free-form optical elements, where numerical study for the mechanism is studied in previous paper (Part I) [15]. The minimum wavefront deviation is >λ/6. The Zernike coefficients related to coma aberration or the tilt of the optical axis are also electrically tunable. Our LC optical element is incorporated into a simple reflective optical system as a proof-of-concept demonstration of off-axis aberration correction. The LC phase modulator as freeform optical element puts forward here could overcome the challenges faced in typical freeform optics: (1) the capability of adjusting the wavefront variations of LC freeform optical elements could compensate of phase errors arising from fabrication and assembly. (2) Manufacturing nematic-based freeform optical elements is relatively simple due to the maturity of LC technology and the fact that there is no need for a CNC process. (3) Furthermore, our nematic-based optical element is easy to manufacture and overcomes the difficulty encountered in characterizing a conventional optical component with a freeform surface, because LC freeform optical elements are physically planar and measurement of the wavefront is simple. These findings pave the way towards the development and design of a variety of nematic-based optical elements in applications of hyperspectral imagers in aerospace optics, augmented reality, virtual reality, quantum information systems, and integrated optics devices.
Funding
Ministry of Science and Technology, Taiwan (107-2112-M-009-019-MY3); Google.
Acknowledgments
The authors are indebted to Prof. Liang Xiao in Tsinghua University in China for providing LC materials.
Disclosures
YJW: Google LLC (R), YHL: Google LLC (R), OC: Google LLC (E).
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