Results and Discussions The relationship between the phase change and the applied voltage is shown in Fig. 5. It shows the phase of liquid crystal layer is proportional to the applied voltage in the range from 1.56V_rms to 2.50V_rms. When V₁<V₂, the liquid crystal lens operates in the positive lens state; conversely, it operates in the negative-lens state. Thus, the proposed liquid crystal lens is positive-negative tunable by adjusting the driving voltages V₁ and V₂. From Fig. 7, the number of interference fringes increases with the increase of |V₁-V₂|, which indicates that the optical power of the lens is electrically adjustable, and it is positive-negative tunable. The concentric interference fringes (Fig. 7) also indicate the high performance of the liquid crystal lens. The phase profiles extracted from Fig. 7 are shown in Fig. 8(a). The scatter points are measurements, and the curves are fittings. It can be seen that the phase keeps a parabolic phase distribution in all states. In addition, the phase shift from the edge to the center of the lens is 22.4π for the positive lens and 21.4π for the negative lens, which basically agrees with the 22π measured in Fig. 5. The results imply that the optical power or focal length of the lens can be accurately designed as long as the phase response curve is obtained, which can provide effective guidance for the design of liquid crystal lenses. The measured optical power [Fig.8(b)] of positive and negative lens states is proportional to the voltage difference |V₁-V₂|, which is in good agreement with Eq. (7).

The electrode patterns of liquid crystal lenses are used to generate an inhomogeneous electric field by controlling the rotation of the liquid crystal molecules, thus producing a lens-like phase distribution. In the last few decades of development of liquid crystal lens, many different structures have been proposed, such as hole-patterned electrodes, concentric electrodes, modal lenses, and some other variations of these structures. With the development of liquid crystal lenses, the performance has been significantly improved, and many associated problems, such as small aperture, high driving voltage, slow response, and disclination line, have been solved. Despite of this, the traditional liquid crystal lenses still face some problems that hinder the practical application of liquid crystal lenses. For traditional liquid crystal lenses, the voltage distribution formed by electrode is affected by many parameters, such as the voltage frequency, voltage phase, size of aperture, and thickness of liquid crystal layer. Therefore, it is difficult to obtain a parabolic voltage profile of an ideal liquid crystal lens for the traditional liquid crystal lenses, which increases the aberrations. To improve the performance of a liquid crystal lens, an electrode design that generates a parabolic voltage profile is desired. On the other hand, traditional liquid crystal lenses need high-resistance layers to enlarge aperture size. However, the properties of high-resistance layers usually change over time, resulting in changes in the properties of the liquid crystal lenses. The faced problems by traditional liquid crystal lenses have become an obstacle to the mass production of liquid crystal lenses. The primary objective of this study is to design a high-performance liquid crystal lens with ideal phase profile, which also overcomes the associated drawbacks of traditional liquid crystal lenses mentioned above.

The proposed liquid crystal lens combines the electrode structure design and the linear response range of liquid crystal materials to improve the performance. The designed electrode structure is used to generate a parabolic voltage profile, and the parabolic phase profile can be achieved when the driving voltage is controlled within the linear response range. To measure the linear response range of the LC material, a liquid crystal cell with plane electrodes (not patterned) on the inner faces of two substrates is fabricated. One plane electrode is grounded, and the other is applied on a voltage. Increase voltage and record normalized intensity captured by complementary metal oxide semiconductor camera. Then the phase can be extracted from record normalized intensity, and the linear response range can be obtained. The designed electrode is processed by photolithography, and the polyimide layer is spun and rubbed on electrodes to align nematic director parallel to the substrate surfaces. Then two substrates are separated by 50 μm spacers and optically aligned facing each other's interior surface with an opposite rubbing direction. Finally, the liquid crystal material is injected into the gap between the two substrates and the liquid crystal cell is sealed using the UV curing adhesive. The phase profiles are extracted from interference fringes obtained by use of polarization interferometry.

A design method of a high-performance liquid crystal lens based on the linear response range of liquid crystal materials is proposed, and the performance of the lens is verified by experiments. The driving method of the liquid crystal lens is simple, the structure is simple, the driving voltage is low, and the phase follows the parabolic distribution. In theory, an equation is established according to the requirement of parabolic voltage distribution, and the corresponding analytical expression of the electrode structure is obtained by solving the equation. Through the analysis, it can be seen that the optical power of the liquid crystal lens is positive-negative tunable, and the optical power is proportional to the difference between the two driving voltages. Experimentally, the electrode is developed by lithography. A liquid crystal lens with an aperture of 4 mm and a liquid crystal layer of 50 μm is fabricated, and the interference fringes are obtained by polarization interference principle. The experimental results show that the phase of the liquid crystal lens keeps the ideal parabolic distribution during the zoom process, which verifies the high performance of the liquid crystal lens and the accuracy of the design method. In addition, the experimental results show the optical power of the liquid crystal lens is proportional to the difference of two driving voltages, which is consistent with the theoretical analysis.