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Abstract
Focus-tunable lenses are indispensable to optical systems. This paper proposes an electrically modulated varifocal metalens combined with twisted nematic liquid crystals. In our design, a metalens is employed to focus on different points depending on the polarization state of incident light. We demonstrated that the varifocal metalens has a sub-millisecond response time. Furthermore, the numerical aperture of both the first and second focal points can be customized to achieve a wide range of 0.2–0.7. Moreover, the full width at half maximum approached the diffraction limit at multiple focal points. Because of the advantages of our proposed electrically modulated metalens, it has the potential for application in optical technology and biomedical science, both of which require high image quality and a rapid response time.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
In recent years, devices have increasingly required lenses that are lightweight and have a tunable focal length with no moving parts. Focus-tunable lenses are indispensable to optical systems. The traditional design of a focus-tunable lens is two lenses positioned along the optical axis and separated by an adjustable distance. This design is straightforward but requires considerable space. Many focus-tunable lenses without moving parts are available, including liquid lenses [1–4], electro-wetting lenses [5], and liquid crystal (LC) lenses. An LC lens is a useful optical component owing to its thinness and electrical tunability [6–9]. The key principle of a LC lens is an incident plane wave that propagates and experiences a lens-like phase change based on the orientation of each LC director. The wavefront of the incident plane wave then bends to form a converging or diverging spherical wave. However, LC lenses have two crucial problems; they require several hundred milliseconds to change focus, and their focal length is primarily determined by three physical factors, namely aperture size, cell gap thickness, and refractive index difference of the LCs. Blue phase LC lenses have a relatively rapid response time at the microsecond level but also have a limited operating temperature range and require a high voltage [10–12]. Therefore, this study used the function of LCs to alter the polarization state of incident light rather than produce a lens-like phase difference. Twisted nematic (TN) LCs have been discussed extensively in previous studies following the technological breakthrough of liquid-crystal displays [13]. The main principle of TN LCs is the precisely controlled realignment of LC molecules between differently ordered molecular configurations under the influence of an applied electrical field. TN LCs with a low operating voltage can be developed to attain a high operation speed at the microsecond level but a slower relaxing speed at the sub-millisecond level [14–16].
Metasurfaces have been researched extensively because of their nanoscale size and versatile functionalities [17–22]. A metasurface is a two-dimensional arrangement of subwavelength scatters that manipulates the wavefront, polarization, and light intensity. Rather than relying on gradual phase accumulation, each subwavelength scatter causes an abrupt change in the phase of incident light. Through an abrupt phase change at the interface, light can be arbitrarily deflected in any direction. Metasurfaces are composed of dielectric materials with lower loss and higher transmission than metals [23]. A metalens is a type of metasurface that controls the wavefront of light for focus through the application of a specialized phase profile. The flat structure of a metalens enables it to overcome the spherical aberration that can occur in conventional lenses. Compared with traditional lenses, metalenses are flatter and thinner. However, a metalens has a fixed focal length without ability to change focal point once its structure has been determined.
In recent years, bifocal metalenses have been researched by dividing metalenses into multi-ring areas, with each area corresponding to one focal point [24,25]. However, this method can result in low focusing efficiency, a low signal-to-noise ratio, and inconvenience in controlling the relative intensity between focal points. Some bifocal metalenses have achieved relatively high focusing efficiencies after alteration of the polarization of incident light [26–28]. TN LCs combined with nanostructure has been researched in previous, such as using TN LCs to design electro-optical control [29–31]. The present paper proposes an electrically controlled method that combines a metasurface with TN LCs. In our proposed multifocus metalens, we take advantage of the metalens and TN LCs separately. TN LCs are employed to adjust the polarization of incident light at the sub-millisecond level according to whether the voltage is in the on-state or off-state, and the metalens focuses on multiple points with high image quality. Compared with conventional tunable lenses, our proposed electrically modulated metalens has improved image quality and a considerably reduced the response time. The simulation results revealed that the proposed metalens can be customized, so that the numerical aperture (NA) of each bifocal point falls between 0.7 and 0.21. Moreover, the full width at half maximum (FWHM) of each point is close to the diffraction limit, and the focusing efficiency is approximately 40% at the first focal point and 70% at the second focal point.
2. Principle and design of the metalens
The fundamental goal of the proposed metalens is to achieve a hyperbolic phase profile to focus incident light [32], as expressed by the following equation:
Fig. 1. Schematic of the unit cell structure of the electrically modulated metalens combined with TN LCs. (a) Unit cell structure; (b) side view of the electrically modulated metalens combined with TN LCs.
For each unit cell of the aperiodic structure, previous research [33–35] assumed that the metasurface is locally periodic: the scattering in any small region is almost the same as the scattering from a periodic surface. Thus, we calculated the phase delay and transmission passing through the unit cell structure by using the commercial finite-difference time-domain (FDTD) software from Lumerical Inc. To cover the entire 0 to 2π phase and to achieve high transmission of unit cell, the length and width of our nanostructures are between 0.05 and 0.27 µm; the height is fixed at 0.6 µm. Both length and width unit cell are 0.4 µm. Considering computing resources and time consuming, we used 101 steps to establish our data library. The x-polarized and y-polarized light refer to a linearly polarized plane wave whose polarized direction is along the x- and y-axes, corresponding to width and length in Fig. 1 respectively. Because nanostructures can be considered truncated waveguides [36], those with different dimensions can generate different effective refractive indices by changing their sizes in the x and y directions to provide different phase distributions in these directions. We calculated each phase of each unit cell structure by sweeping their lengths and widths. The phase results for linear polarization are shown in Figs. 2(a) and 2(b); the corresponding transmission results for polarization are shown in Figs. 2(c) and 2(d). The operation wavelength is 650 nm, and the length and width have 101 steps each.
Fig. 2. Sweep results of unit cells over 101 steps. (a, b) Simulated sweep phase delay of unit cell structures; (c, d) simulated transmission of unit cell structures. The operation wavelength was 650 nm, and the height was 600 nm.
By using Eq. (1), we calculated the target phases of the first focal length and second focal length for a metalens with a diameter of 20 µm. We designed the first focal length of approximate 10-µm and the second focal length of approximate 45-µm. We then subtracted multiples of 2π, as shown in Fig. 3(a). Designed focal length can be more accurate by using more steps of our data library but it will cost more time. The number of points we use in Fig. 3. can be calculated as lens diameter divided by unit cell. Subsequently, we used the least squares method to find the most fitting parameters for length, width, and high transmission with respect to the previously calculated target phase. According to the results shown in Fig. 3(b), the unit cell structure and the phase have different optimal parameters for linear polarization.
Fig. 3. Metalens parameters. (a) Phase versus radial distance; (b) Length and width versus radial distance.
3. Simulation of the electrically modulated multifocus metalens
Our metalens has a diameter of 20 µm. The thickness of the TN LC layer is 5 µm, and the length and width of the nanostructure at each location are shown in Fig. 3. A schematic of the metalens generated using Lumerical and detailing the top of the structure is presented in Figs. 4(a)–4(b).
Fig. 4. (a) Schematic of the proposed electrically modulated multi-focus metalens; (b) top view of the metalens.
Figures 5(a) and 5(b) show the tilt angle cross profile of the TN LCs when the voltage was in the off-state and on-state, respectively. When no voltage is applied across the TN LCs, they are oriented in a twisted configuration, and the LC director makes a homogeneous 90° turn away from the bottom surface of the LC layer and toward the top surface. As the light enters the LCs with polarization parallel to the bottom director, the linear polarization of the light approximately follows the rotation of the director; consequently, the transmitted light is polarized in the y direction. When voltage is applied, the external electrical field forces the LC director to become homogeneous and parallel to the propagation direction; no change in the polarization state of the light occurs. We simulated the TN LCs by using TechWiz LCD 3D and then imported the obtained TN LC data into Lumerical to calculate the x-polarized fraction and y-polarized fraction, the results of which are shown in Figs. 5(c) and 5(d), respectively. The thickenss of TN LCs is 5 µm with a distance between 2 µm and 7 µm and the fraction of loss due to TN LCs is around 5%.
Fig. 5. Schematics of the TN LC tilt angle and the electrical field fraction. (a) TN LCs in the off-state; (b) TN LCs in the on-state; (c) x-polarized and y-polarized fractions in the off-state; (d) x-polarized and y-polarized fractions in the on-state.
The response time depends on the operating voltage, as shown in Fig. 6. We simulated the response times of the TN LCs at voltages between 10 and 200 V; the results revealed that the optimal response time of the TN LCs is at the microsecond level at a voltage of approximately 25 V. At a higher voltage, the response time of the TN LCs is considerably quicker, but the change is not significant. Our results revealed that the TN LCs can alter the polarization at the sub-millisecond level.
4. Results and discussion
To verify the validity of the proposed design, electromagnetic propagation through the dielectric metalens was analyzed using the FDTD method. We employed near-to-far-field transformation [37] to calculate the image quality of the bifocal point. The intensity distribution results for the first focal points on the x-z, y-z, and x-y planes are shown in Figs. 7(a)–7(c), respectively; the focus was around 10 µm. The intensity distribution results for the second focal points on the x-z, y-z, and x-y planes are shown in Figs. 7(d)–7(f), respectively; the focus was around 45 µm. Figures 7(c) and Fig. 7(f) were visualized by Mayavi [38].
Fig. 7. Schematics of far fields on the x-z, y-z, and x-y planes. (a) First focal point on the x-z plane; (b) first focal point on the y-z plane; (c) first focal point on the x-y plane; (d) second focal point on the x-z plane; (d) second focal point on the y-z plane; (d) second focal point on the x-y plane (Visualization 1).
We analyzed multiple NAs to verify the image quality of our proposed metalens. The intensity of the Airy pattern followed the Fraunhofer diffraction pattern of a circular aperture, which is expressed as the squared modulus of the Fourier transform of said aperture:
where ${I_0}$ is the maximum intensity of the pattern at the Airy disk center, ${J_1}$ is the Bessel function of the first kind of order one, k is the wavenumber, a is the radius of the aperture, and θ is the angle of observation. The focusing efficiency can be defined as the fraction of incident light that passes through the TN LCs divided by the area of a circular iris on the focal plane with a radius equal to three times the FWHM spot size [39,40]. The simulation results of two focal points are shown in Table 1; notably, the FWHM that we employed for comparison with the diffraction limit is the average of the normalized intensity scores in the x- and y-directions. Table 1 shows that the FWHM of our proposed metalens approaches the diffraction limit in every design. In design 4, the focusing efficiencies are 41% at the first focal point and 76% at the second focal point. In addition, we compared the FWHM at each focal point with the diffraction limit; a corresponding schematic is shown in Fig. 8. The horizontal ordinate is radial distance, and the vertical ordinate is normalized intensity.Fig. 8. Schematics for comparison between the FWHM and the diffraction limit in design 4. (a) First focal point at around 10 µm; (b) second focal point at around 45 µm.
Table 1. Comparison between four designs of the proposed metalens with respect to the diffraction limit.
Furthermore, if multiple incident light wavelengths are applied, our proposed metalens can hold a more consistent focal point around the designed bifocal point. In future research, we intend to focus on this aspect to develop a design with a more continuous focal point with high focusing efficiency and high image quality. The advantages of our proposed electrically modulated metalens, namely its rapid response time and high image quality, should enable it to play a vital role in advancing the field of tunable lenses.
5. Conclusions
This paper proposes an electrically modulated multifocus metalens combined with TN LCs. By combining the metalens with TN LCs, we were able to develop a high-quality multifocus lens and a rapid response time with 0.8 millisecond at 25 volts. The results revealed that the FWHM of our metalens approaches the diffraction limit in multiple simulations, and the metalens has a quick operation speed at the microsecond level and a slower relaxing speed at the sub-millisecond level. Moreover, the NA of the first and second focal points can be customized to achieve a wide bifocal point range of 0.2–0.7. The focusing efficiencies of the first and second focal points are approximately 40% and 70%, respectively. The results revealed the high quality of our metalens, which is attributed to its ability to approach the diffraction limit through adjustment of the focal point. Compared with traditional tunable lenses, our proposed metalens offers high image quality without optical or mechanical compensation; furthermore, it is flatter and thinner and can shift its focal point at the sub-millisecond level. Crucially, our metalens has the potential for multiple applications, including those in optical technology, biomedical science, display technology, augmented reality, and virtual reality.
Funding
Ministry of Science and Technology, Taiwan (105-2221-E-002 -158 -MY3, 108-2218-E-002-056-, 108-2221-E-002-159-); National Taiwan University.
Disclosures
The authors declare no conflicts of interest.
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