Achromatic polymer-dispersed liquid crystal lens with diffractive–refractive hybrid structure

The focal length of liquid crystal Fresnel lens is electrically controllable, but the disadvantage is the chromatic aberration is obvious. In this paper, the electrically controlled zoom characteristics of liquid crystal lens and the basic theory of Fresnel lens achromatic are analyzed, and a diffractive–refractive hybrid lens model made of polymer-dispersed liquid-crystal (PDLC) material is proposed. Fresnel liquid crystal lenses and PDLC hybrid lenses are tested with light at R (700.0 nm), G (546.1 nm) and B (435.8 nm) wavelengths, respectively. The simulation results show that the hybrid lens provides a greater range of adjustment of the focus position. After voltage modulation, the measured light with three different wavelengths have the same focal length, which means that the axial chromatic aberration is eliminated. In addition, with the elimination of axial chromatic aberration, the zoom range of the hybrid lens is doubled by further accurately controlling the electrode, which breaks through the limitation that the traditional liquid crystal lens can only work in monochromatic light spectrum.


Introduction
Nematic Liquid Crystals (NLCs) have anisotropic and birefringent properties [1,2]. The orientation of LC molecules can be controlled by the voltage applied to the pixelated electrodes, which can generate gradient refractive index profiles induced by an electric field to achieve an electronically controlled zoom function [3,4]. Due to the advantages of strong light transmission, low power consumption, high reliability, simple control and small size, LC microlenses are widely used in daily life and industrial production. For example, it can be applied to 3D stereo display technology in cinemas or monitors [5][6][7][8], to constitute fast non-mechanical lens sets in cameras [9], and it can also be seen in biomedicine, microfluidic sensing, and optical communication.
The LC lens with higher light control ability will have a larger adjustable focal length range. In order to enhance the optical control ability, thicker LC layers are usually used, but the thicker LC layer will lead to higher driving voltage and slower response time. In response to these problems, some researchers have proposed ideas on electrode structures: e.g., hole-patterned-electrodes, concave-convex electrodes, and trapezoidal electrodes [10][11][12], expecting to reduce the driving voltage of LC lenses. However, some of these lenses are detrimental to image quality due to poor refractive index profiles, or some have complicated manufacturing processes. On the other hand, there have been many reports of research on the application of large-aperture flattened Fresnel-type LC lenses [13][14][15][16]. LC Fresnel lens is a new type of lens combining Fresnel lens and liquid crystal material. It can accurately control the voltage of adjacent zones through phase difference to realize the lens function. In 2016, Shih-Hung Lin et al developed an optically controllable LC Fresnel lens with different focal lengths using an interferometric method [17]. In 2018, Jamali et al designed a large-aperture Fresnel LC lens for applications in stereoscopic displays [18]. In 2020, Hu Dou et al used holographic exposure technology to fabricate a switchable blue-phase LC/polymer Fresnel lens [19]. These studies focus on the electro-optic characteristics of Fresnel LC lens when monochromatic light is incident. In the wide spectrum range of visible light, there is still a lack of relevant research on the chromatic aberration of incident light with different wavelengths. The imaging quality of the visual optical system is very sensitive to the change of wavelength, so the chromatic aberration of Fresnel lens as a binary diffraction system is well worth studying. The diffractive-refractive hybrid achromatic design method has received a lot of attention from researchers due to its high performance and small size characteristics [20][21][22][23]. However, because of the limitation of materials and complex fabrication processes, these studies can only fabricate lenses with fixed focal length.
In this paper, combining the advantages of LC electronically controlled zoom and lightweight Fresnel lens, a method of refraction-diffraction hybrid lens achromatization in the visible light band is proposed. The hybrid lens modulates R, G, and B three-color light to achieve the same focal length under electrical switching. PDLC is used as the filling material of the lens. PDLC has the characteristics of electronically controlled zooming [24], and has a very fast regulation speed under the condition of applying voltage, with electrical switching speed reaching the microsecond level [25]. Through the synchronous rapid change of the three primary colors timesequence pulse light sources, the color display can be realized by the time-sequence color mixing method by utilizing the persistence of vision characteristics of the human eye. And it realizes multi-focal plane imaging when working with polychromatic light. It is a hybrid optical system suitable for wide-spectrum band application equipment. It can be applied to 3D display, AR/VR enhanced display, liquid crystal shutter glasses and other scenarios.
2. Design structure and principle of hybrid PDLC lens 2.1. Overview of refractive liquid crystal lenses PDLC is a mixture of polymer monomer and liquid crystal. Under illumination, this mixture will undergo polymerization to form PDLC droplets, and the size of the droplets can reach nanoscale. The photocurable PDLC is filled between two glass substrates coated with ITO conductive film, and the refractive index forms a parabolic distribution under the action of an electric field, as shown in figure 1(a), which finally produces the characteristics of lens focusing. The physical basis of liquid crystal focusing lens is the optical anisotropy of liquid crystal, i.e., the birefringence property of liquid crystal. The birefringence property of LC is usually described by the difference between the extraordinary refractive index n e and the ordinary refractive index n o :Δn = n e −n o . When the voltage is applied, the LC molecules rotate, and the included angle θ between the incident light and the long axis of the LC molecules changes. Under the action of voltage, the functional relationship between the tilt angle of LC molecules and voltage is: where U is the driving voltage applied to both ends of the LC cell, and Uc is the threshold voltage when the LC molecular axis starts to rotate under the voltage action, which is determined by experiment. U 0 is a constant. However, the ordinary refractive index n o of LC does not change with the voltage, and only n e will change. At this time, the effective refractive index of LC is n eff (θ), where  [26]. The study shows that the birefringence changes little within the threshold voltage of 0.8 V; Between 0.8 V and 4 V, Δn decreases rapidly and changes continuously with the voltage. When the voltage is greater than 4 V, Δn → 0. However, the interaction between LC molecules and electromagnetic waves is complex, and the liquid crystal birefringence will also be affected by voltage frequency, LC concentration, temperature and other factors.

Overview of Fresnel liquid crystal lenses
To achieve the desired zoom effect of the refractive paraboloidal LC lens described above, the LC layer has to be made very thick, so the driving voltage can reach tens or even hundreds of volts. In contrast, the LC Fresnel lens can make the effective refractive index continuously variable control from unusual to unusual light with only a few volts of low voltage, and its structure is shown in figure 2. The lower ITO film is an aluminum-coated glass substrate with Fresnel zone pattern etched by laser. Through the electrodes on the lens, the relative phase difference between the zones could be accurately controlled. Where the phase difference is Δδ (Δδ = 2π(m even −m odd )d/λ), and m is the number of wave bands. When the phase difference between adjacent electrodes is the same, a parabolic phase can be formed, thus obtaining a phase-type focusable LC Fresnel diffraction type lens. According to Fresnel zone theory, the main focal length of Fresnel zone lens is: where r j is the radius of the jth waveband of the Fresnel lens, j is the number of wavebands contained in the lens, λ is the wavelength of incident light , and n represents the focal order. Fresnel lens imaging will produce multiple imaging focuses due to multi-order diffraction. Figure 3 shows the energy distribution of the center three diffraction points of a 700 nm electromagnetic wave incident on a LC Fresnel lens with an aperture of 3.5 mm.
The middle diffraction point FWHM = 9.7 μm (full width at half maximum, FWHM) is the smallest, which means that the diffraction point is the main focus position. In this paper, only the focal length value of the main focus is discussed. If the number of bands continues to increase, thus increasing the lens aperture, the imaging FWHM will be further reduced.  As can be seen from equation (3), the imaging focal length of a Fresnel diffractive optical lens is inversely proportional to the wavelength. For LC lens with Fresnel structure, when the polychromatic light is incident, it will inevitably produce obvious dispersion effect, resulting in obvious chromatic aberration of this type of lens affected by the wavelength.

Design of achromatic refractive-diffractive hybrid lens
Aiming at the above problems, this paper designed an LC refractive-diffractive hybrid lens system filled with a PDLC material( figure 4). The LC cell substrate is a polymer monomer with a refractive index np = 1.5. The nematic liquid crystal is a mixed liquid crystal of 99%TEB50 + 0.1%CB15. At room temperature, n o = 1.50, n e = 1.70, and the range of effective refractive index changes is 1.5 ∼ 1.7. The hybrid lens is made of three transparent indium tin oxides (ITO) conductive glasses, a plano-convex parabolic lens and a Fresnel lens. The upper and lower ITO electrodes are connected to the positive electrode, and the middle ITO electrode is grounded.
The refractive indices of the two combined LC lenses can be adjusted separately by electrodes, and it is stipulated that the effective refractive index of the lower Fresnel lens is n 1 , the effective refractive index of the upper LC plano-convex lens is n 2 , and the refractive index of the base is n p . Set the origin of coordinates in the middle of the LC Fresnel lens and plano-convex lens, with the Z axis being the thickness direction of the LC layer, and the center of the Z axis coinciding with the center of the lens. The innermost radius of the Fresnel lens, r 1 = 0.135 mm, and the radius of the nth zone is based on the equation r jr j 2 1 2 = (j is the number of zones). The Fresnel lens thickness is d 1 and has 21 LC filled ring bands. The flat-convex LC lens has an aperture size of 2.1 mm and a thickness of d 2 . The distance of the electromagnetic wave from the center of the Fresnel lens is 0.55 mm, and it is incident vertically into the hybrid lens from one side, with the central focus falling in the positive direction of the z-axis.
In this paper, the phase distribution of the refractive index profile of the upper lens and the lower LC Fresnel lens can be changed by controlling the ITO electrodes added to the lens separately, so that the reproduced images of the three colors of light can be presented in the same position. The visual retention time of the human eye is about 1/10th of a second, and the response time of PDLC material is in the microsecond range. When the switching frequency of the trichromatic hologram is high enough, the color reproduction image of the object can be observed by using this visual temporary effect, and this technology has been widely used in imaging systems, zoom systems and holographic projection.

Achromatic simulation verification
According to Formula (3), the focusing position of the Fresnel lens depends on the wavelength. In order to verify the chromatic aberration theory of Fresnel LC lenses, the propagation of electromagnetic waves in PDLC Fresnel lenses is simulated by BeamPROP with the commercial software RSOFT. To prevent the boundary reflection from affecting the calculation results, the perfect matching layer is set to 0.5 μm. Meanwhile, the grid is set to 0.01 μm to ensure the accuracy of the calculation. The wavelengths of 700 nm (R), 546.1 nm (G), and 435.8 nm (B) were selected to be incident into a 10-μm-thick single-layer PDLC Fresnel lens. The optical transmittance of PDLC film changes with the change of the applied electrical signal. Let L 0 be the transmittance and L be the maximum transmittance, then where η denotes the density of LC molecules; s denotes the cross-sectional area of average scattering; Epdlc denotes the applied voltage; and d denotes the thickness of the LC optical element, i.e., the path of light propagation. The transmission of PDLC films is enhanced when PDLC is loaded with the applied voltage. By making the voltage change from 0 to 15 V, the light transmittance of each pixel of the liquid crystal is changed and the refractive index of the LC is changed accordingly, so that the LC molecules are arranged into Fresnel zones with the number of ring bands N varying in a certain range, thus a Fresnel zone lens with dynamically changing main focal length is obtained. The correspondence between the focal length of the main focus imaging on the z-axis center and the refractive index n1 was recorded, and the focal length curves at different wavelengths were plotted( figure 5).
As can be seen in figure 5, the focal length of red light ranges from 40.3 mm to 46.6 mm; The focal length of green light is 52.6 mm ∼ 61.3 mm; The focal length of blue light is 68.8 mm ∼80.0 mm. Their maximum relative change in focal length was 12.4%,13.5%,13.1%, respectively. But apparently the focal length values of the three curves have no overlapping area, and the axial chromatic aberration between the three color lights cannot be eliminated by changing the refractive index through voltage modulation. The imaging diagram of the focal position of the single-layer Fresnel LC lens for RGB light operation with electronically controlled n 1 = 1.6 is given in figure 6, where the electromagnetic wave is incident below Z = 0 and focused in the positive direction of  the Z-axis. Comparing the focal position in figure 6(a-c), it can be seen that the focal distance difference is larger for different wavelengths, so the dispersion effect is more obvious.
In optical imaging, in order to obtain rich image information, multi-color light imaging system is often used [27], so the chromatic aberration problem must be considered in lens structure design. To verify the function of the hybrid lens to eliminate axial chromatic aberration, the RGB trichromatic light is vertically incident on the hybrid lens structure, and the structural parameters of the hybrid lens are: the thickness of the LC layer of the plano-convex lens d 2 is 77 μm, the effective refractive index of the lens n 1 is 1.6, and the effective refractive index of the plano-convex LC lens n 2 is 1.5 ∼ 1.7. The voltage variation range is 0 ∼ 28 V. The focal length curves of the samples were measured at different n 2 ( figure 7(a)). Figure 7(a) shows that the change trend of the focal length curve of RGB tricolor light tends to be consistent with the increase of n 2 , and they all decreased with the increase of n 2 .
In order to compare with the focal length of the single-layer Fresnel lens in figure 6, the BeamPROP simulation plots of the focused imaging of R, G, and B light under specific parameters are given in figure 7(b ∼ d). From figure 7(b ∼ d), it can be visualized that the focal lengths of R, G, and B are the same , which is 42.9 mm. Under the conditions of λ = 435.8 nm, n 1 = 1.6 and n 2 = 1.7, the focal length of B light reaches the minimum value of 39.2 mm, which overlaps with R and G light in the focal length range of 39.2 ∼ 42.9 mm, and defines this common focal length overlap range as d f . In the gray shadow area with focal length of 39.2∼42.9 mm, the RGB tri-color light can achieve the electronically controlled zoom with d f = 3.6 mm under the condition of eliminating chromatic aberration.

Zooming of achromatic hybrid lenses
In this paper, two factors affecting d f are discussed, which are the thickness of the plano-convex lens liquid crystal layer d 2 and the effective refractive index n 1 of the Fresnel LC layer.
Different thicknesses of the LC layer of the plano-convex lens with different refractive index paraboloidal curvature will affect the magnitude of d f . Figure 7(e) shows the change of d f with d 2 under the conditions that n 1 is the same and n 1 is different.
From the figure, it can be seen that the d f value gradually decreases with d 2 . When the n 1 of RGB is 1.6(n 1 is the same), at d 2 = 64 μm, d f = 0, the focal lengths of the three wavelengths are just equal, reaching the achromatic threshold, and the total thickness of the LC layer is d = 74 μm (ignoring the electrode thickness).
Combined with the curve in figure 5, it can be seen that the variation of imaging focal length with the effective refractive index n 1 of Fresnel LC lens is: the focal length increases with n 1 . In order to increase the d f , the system parameters of modulating n 1 are set as follows: λ = 700 nm for red light, n 1 = 1.7; λ = 546.1 nm for green light, n 1 = 1.6; λ = 435.8 nm for red light, n 1 = 1.51. Thus, the relationship between the variation of d f and d 2 (n 1 is different) is derived in figure 7(e). It can be seen from the figure that at d 2 = 54 μm, d f = 0, the focal lengths of the three wavelengths are just equal, reaching the threshold of achromatic aberration, when the total thickness of the LC cell d = 64 μm.
Comparing the curves with the same n 1 and different n 1 under the incidence of electromagnetic waves with different wavelengths, longitudinally, it can be seen that the lens structure with the same liquid crystal layer thickness can obtain a larger zoom range of polychromatic light by modulating n 1 by voltage. Figure 7(f) gives the focal length variation of the LC Fresnel layer lens after voltage modulation of the refractive index (operating parameters are as follows: R light,n 1 = 1.7; G light,n 1 = 1.6; B light,n 1 = 1.51), and it can be seen from figure 7(f) that after voltage modulation, the overlapping range d f of R, G and B focal lengths increases to 7.8 mm at the position of 38.8 ∼ 46.6 mm, and the zoom range is doubled compared with d f = 3.7 mm in figure 7(a).
The FWHM is used as the standard to characterize the spot size. Under voltage modulation n 1 = 1.6, the relationship between effective refractive index n 2 and FWHM at different wavelengths is discussed, as shown in figure 8. From the figure, it can be seen that the spot shows a shrinking trend with n 2 . We know that the n o of the liquid crystal lens is constant, and the n e will change due to the applied voltage. Therefore, if the FWHM of the light spot is to be reduced, the birefringence Δn of the liquid crystal lens modulated to be as large as possible by voltage modulation.

Conclusion
In this paper, we propose a hybrid lens system based on PDLC material with plano-convex and Fresnel lenses, which adopts R, G, and B trichromatic electromagnetic waves as the working wavelengths and is used for solving the chromatic aberration problem when a 3D display system works in a wide spectrum. Through simulation, a hybrid lens structure with common focal length of R, G, and B light is obtained to overcome the axial chromatic aberration problem. The effect of the thickness of the liquid crystal layer on the zoom range of the achromatic lens is discussed at a certain aperture of the lens. The results show that when the effective refractive index n 1 of the Fresnel liquid crystal layer is not modulated, the axial chromatic aberration can be eliminated and the zoom can be realized when the thickness of the liquid crystal layer of the hybrid lens is greater than 74 μm. When modulating the variation of n 1 , the axial chromatic aberration can be eliminated when the thickness of the liquid crystal layer of the hybrid lens is greater than 64 μm, and then zoom can be realized. Therefore, the hybrid lens can reduce the thickness of the liquid crystal layer with achromatic function by modulating the refractive index, thus reducing the driving voltage and the response time. We also find that increasing the liquid crystal birefringence difference will reduce the size of the diffracted spot. This model can provide guidance for the design of liquid crystal achromatic microlenses. The fly in the ointment is that in order to reduce the complexity of the process, the electrode used in this paper is a flat electrode structure, and if lower driving voltage is desired, it can be combined with a round-hole type electrode, a trapezoidal electrode or other more refined electrode models.