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Abstract
A novel membrane-based tunable liquid aspherical lens design capable of dynamically reducing spherical aberration over certain focal length tuning range is presented. Different from conventional treatment using elastic membrane with uniform thickness, in current case, the membrane’s cross-section is designed to be a plano-convex shape with well-defined aspherical contour. A brand new design flow combining optical ray-tracing and mechanical finite element analysis is especially developed, in which through properly selecting the initial focal length and the center membrane thickness with respect to particularly designed figure of merit (FOM), the membrane can be flexibly optimized to achieve dynamic spherical aberration correction capability over the desired focal length range. For proof of concept demonstration, a tunable aspherical lens with clear aperture of 4 mm is designed. Compared with conventional counterpart, after being individually optimized using current design strategy, lenses with distinctly reduced spherical aberration have been successfully achieved over different focal length tuning ranges. Besides, no special refractive index matching between the filling liquid and the membrane is required, making current design more practical in real applications.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
As for optical society, tunable liquid lenses, consisting of a liquid chamber sandwiched between an elastic membrane based deformable refractive surface and a fixed flat surface, have attracted a lot of research interest due to the fact that it can realize particular tuning functions as autofocus and optical zoom without mechanically moving multiple optical components along axial direction, thus largely simplifying system configuration and improving long-term stability [1,2]. Given its compact size, low cost, fast and large range focal length tuning capability, liquid lenses have inspired next-generation miniaturized autofocus and zoom lens design [3–5], and offered a great improvement in microscopy [6,7], endoscopy [8], beam steering [9], wavefront sensing [10]and spectral imaging [11] etc.
Despite of their distinct advantages and extensive study, the widespread application of tunable liquid lenses is still largely limited due to their inherent optical aberrations associated with the membrane deformation based focal length tuning process. As for conventional tunable liquid lenses, a membrane with uniform thickness is the most commonly used structure. From mechanical analysis it can be seen that the membrane will be deformed into a spherical like shape under actuation with uniform hydraulic pressure, thus introducing significant spherical aberration (SA). To address this issue, various approaches have been proposed. The first attempt is to replace the flat surface at the fixed end of the tunable liquid lens with an aspherical one possessing certain phase contour, through which the SA improvement for the lens has been experimentally demonstrated [12]. However, due to the fixed configuration, only limited correction capability within relatively small focal length tuning range can be obtained. In order to pursue better dynamic performance, more works have been focused on the deformable end of the lens. One way is to use additional electrostatic actuator to reshape the hydraulic pressure induced spherical membrane contour into aspherical one during operation, so as to compensate the SA [13]. Despite of successful dynamic correction, the device structure as well as its control is a bit complicated. More recently, an non-uniform membrane strategy is proposed [14–17]. In this case, the membrane with variable meniscus thickness, which is determined by the deformed contour of the uniform membrane, is adopted. Due to the modified mechanical property, the resultant membrane deformation helps to improve the SA within a certain focal length tuning range. Nevertheless, the capability to design and optimize the membrane targeting for user-defined operation range has never been demonstrated. Moreover, refractive index match between the membrane and the filling liquid is used in the design and no discussion about the effect of the filling liquid especially its refractive index on the lens optical performance can be found, which increase the implementation difficulty further.
In this paper, a new membrane design strategy is proposed. Based on the exact surface contour fabrication capability associated with our previously developed fabrication technology combining single point diamond turning with soft lithography [18–20], the membrane’s cross-section is designed to be a plano-convex shape with well-defined aspherical contour, which can be optimized using specially developed design flow together with particularly designed FOM based on the combination of optical ray-tracing and mechanical finite element analysis. With this strategy, dynamic SA correction over arbitrarily selected focal length operation range has been achieved, showing remarkable flexibility and versatility. Furthermore, the optical performance of the proposed design demonstrates less sensitivity to the refractive index of the filling liquid, thus facilitating its practical applications.
2. Structure design and optimization flow
Figure 1 shows the schematic of the proposed tunable liquid aspherical lens as well as its operation comparison with conventional counterpart. Similarly, the lens consists of a cylinder liquid chamber sealed by an elastic polydimethylsiloxane (PDMS) membrane. During operation, with the filling of optical fluid into the liquid chamber, an equivalent refractive lens will be formed. Through changing the hydraulic pressure, the membrane will be deformed upward or downward accordingly, resulting in the focal length shift. Different from conventional design using uniform membrane, in current case, a non-uniform membrane with plano-convex cross-section is adopted instead. Considering its mechanical deformation characteristics, an equivalent aspherical lens rather than spherical one will be obtained under lens actuation. Given proper membrane design, dynamic SA correction over certain focal length tuning range can be achieved.
In order to find the optimized structure parameters for the membrane, a strategy combining optical ray-tracing and mechanical simulation is developed, detailed design flow chart of which is provided in Fig. 2. Firstly, given certain lens diameter and effective aperture size with respect to different application requirements, the initial spherical profile can be easily obtained to meet the predetermined focal length using ZEMAX. Based on which, an aspherical profile can be further obtained through performing SA optimization, constituting the convex end of the membrane. After presetting the center membrane thickness, the initial membrane structure can be determined. Considering the fact that the focal length tuning is realized based on the membrane deformation and the surface profile of the deformed membrane will directly affect the lens optical performance. As a result, the membrane deformation characteristic should be studied. According to mechanical analysis, the deformation of a membrane with non-uniform thickness under the effect of uniform hydrostatic pressure can be described by a pair of nonlinear differential equations [21,22],
These equations can be solved numerically using the boundary conditions as listed below,
For analysis simplification, the membrane deformation is commonly studied using commercial finite element analysis (FEA) software, such as COMSOL and ANSYS. Currently, the membrane deformation characteristic during the focal length tuning is studied using COMSOL, in which hyperelastic material based model is built for the membrane. Its circumference will be set to be clamped boundary and uniform pressure will be applied onto its surface. The deformed profiles under different applied pressures will be extracted out and a polynomial fitting procedure will be conducted using MATLAB. The lens model will be subsequently refreshed with the fitted surface profile and optical analysis will be performed again in ZEMAX, thus revealing the resultant change in lens optical performance such as focal length and spherical aberration associated with the membrane deformation. Upon these treatments, a curve reflecting the change of the spherical aberration with respect to the focal length tuning can be achieved. Then, a membrane thickness sweep will be performed, in which the initial aspherical profile will be kept constant whilst only changing the center membrane thickness. During which, the same processes including the membrane deformation simulation followed by the optical analysis will be repeated with the updated membrane structure so as to get a series of spherical aberration curves. By evaluating these curves with particularly designed figure of merit (FOM), detailed information about which is provided in the following section, the locally optimized membrane structure at this preset focal length can be determined. Subsequently, another parameter sweep about the preset focal length will also be conducted and a group of optimized membrane designs at individual preset focal lengths can be obtained by using similar iterative loop processing. Through comparing all the curves with respect to FOM, the globally optimized membrane structure parameters can be eventually determined. Since the calculation of FOM is highly correlated to the selected focal length operation range, the membrane can be exclusively optimized with respect to any user-defined lens aperture size as well as its focal length tuning range to meet various application requirements, making the proposed method more flexible, adaptive and universal.
3. Simulation results
For proof of concept demonstration, lens with 6mm physical diameter and 4mm effective aperture size is used. Its effective focal length of 55mm at operation wavelength of 587nm is firstly chosen as the predetermined parameter, with which the initial spherical profile can be determined as shown in Fig. 3(a). Based on which, an aspherical lens profile can be obtained after performing SA optimization procedure (see Fig. 3(b)). From the ZEMAX simulation results, it can be seen that distinct decrease in both of the PV and RMS of the wavefront error at focus, that are mainly caused by SA, from original values of 0.3683waves and 0.1085waves to optimized 0.0051waves and 0.0015waves, respectively, can be achieved. Then, the initial membrane structure will be determined by choosing the center membrane thickness to be 0.375 mm and its physical model will be built in COMSOL as mentioned above. The deformed surface profiles of the membrane two ends under actuation pressures ranging from −400 Pa to 1400 Pa with a step of 200 Pa can be extracted out from the FEA mechanical analysis as shown in Figs. 4(a) and 4(b), in which the Young’s modulus, Poisson’s ratio and density of PDMS are set as 1.86MPa, 0.49 and 970 kg/m3, respectively. Considering the axial rotational symmetry of the membrane structure, the deformed profile can be further fitted with a 6th order polynomial involving only the even order items:
where A, B, C and D represent the corresponding polynomial coefficients.
Fig. 4. Simulation results about the deformed membrane profile under different actuation pressures (a) The upper surface (b) The bottom surface of the membrane.
All of the fitted profiles are individually imported into ZEMAX ray-tracing software for optical performance analysis so as to acquire the corresponding information about the focal length and SA values. In this case, the refractive index of PDMS is set to be 1.4 and the water with refractive index of 1.334 is selected as the lens filling liquid. From simulation results, it can be seen that with the change of the applied pressure from −400 Pa to 1400 Pa, the resultant focal length tuning range from 140 mm to 25 mm can be achieved. Furthermore, the curve disclosing the SA variation with respect to the focal length can also be obtained. To explore the locally optimized membrane design at this predetermined focal length, a parametric sweep about the center membrane thickness from 0.25 mm to 0.45 mm with a step of 0.025 mm is performed. Using the same treatments, a series of the SA change curves can be obtained (see Fig. 5(a)). Obviously, all of these curves reach zero SA at the focal length of 55 mm, agreeing well with the predetermined focal length for SA optimization. On the other hand, distinct difference between these curves can be observed, which further validates the fact that the lens performance is sensitive to the membrane design. In order to quantitatively evaluate these curves for optimization purpose, particular FOM should be designed. Considering the real application requirement, within a given focal length tuning range, the SA with small amplitude and less change (namely flat response) is much more desired. As for the SA curve shape, the SA amplitude is positively correlated to the area enclosed between the SA curve and the axis of zero SA within the interested focal length region (the larger the SA amplitude, the bigger the area), whilst the standard deviation concept, commonly being used for data fluctuation evaluation in probability and statistics, can reveal the change degree of the SA with respect to the focal length. As a result, these two items are individually calculated for all the curves and subsequently used as FOM for comparison. Obviously, the smaller the better for both cases. From Fig. 5(b), it can be seen that as for the area enclosed by the curve, with the increasing center membrane thickness from 0.25 mm, a monotonic decrease tendency in initial status can be observed. After reaching the minimum around the center membrane thickness of 0.375 mm, the area value tends to increase instead with the further increase of the membrane thickness. Similar tendency can also be found in the case of standard deviation item except that the minimum appears at a bit larger thickness of 0.4 mm. By taking these two items both into consideration, the best center membrane thickness for the case of the predetermined focal length of 55 mm is selected to be at the cross point of these two curves, namely 0.39 mm.
Fig. 5. Simulation results of (a) the change of SA within focal length tuning range from 25 mm to 140 mm (b) the normalized standard deviation and the normalized area of the SA curve under different membrane thickness designs.
In order to find the global optimization for the membrane design, another parametric sweep about the predetermined focal length from 30 mm to 100 mm is performed. By using similar treatments, the membrane designs optimized at corresponding focal lengths can be obtained and parallel comparison between them is conducted using the same FOM as shown in Figs. 6(a) and 6(b). It can be found that with the increase of the predetermined focal length value, the optimal center membrane thickness is gradually reduced. Moreover, the membrane with 0.425mm center thickness that optimized at 50mm focal length undoubtedly can be treated as the global best design based on the FOM when considering the current focal length range from 25mm to 140mm.
Fig. 6. Comparison between different membrane designs optimized at different focal lengths for searching global optimization for focal length tuning range from 25 mm to 140 mm with respect to adopted FOM of (a) normalized absolute area and (b) normalized standard deviation.
For evaluating the lens performance improvement, the case of the conventional lens design with uniform membrane thickness is also provided for comparison in Fig. 7, in which three different membrane thicknesses, namely 0.25 mm, 0.3 mm and 0.4 mm are used. As for conventional cases, negative Zernike coefficient with relatively large amplitude can be always found within the whole interested focal length range despite of a bit difference between different designs within shorter focal length region. In contrast, through adopting current design, not only the Zernike coefficient amplitude, but also its change with respect to the focal length both can be largely decreased, demonstrating excellent dynamic SA correction capability.
Fig. 7. Comparison of the SA change curve between the optimized lens and the conventional lens design with different membrane thicknesses.
To further explore the inherent cause of this improvement, the surface profiles of the optimized lens operating at several different focal lengths of 25mm, 40mm, 50mm, 60mm, 100mm and 140mm are intentionally extracted out. They are compared to the ideal aspherical profiles exclusively optimized at these individual focal length points, and the corresponding surface profile differences are provided in Fig. 8(a). As expected, within the effective aperture region of 4mm diameter, negligible deviation from the ideal profile can be found at the focal length of 50mm, which is the selected focal length for membrane optimization. As for the cases of the focal length longer than 50mm, negative deviation with relatively small amplitude can be found and a slightly increased deviation with the increasing focal length can also be observed. In contrast, with respect to the cases in the shorter focal length region, positive profile difference with a bit larger amplitude can be observed instead and it will also increase with the deviation of the focal length further away from the optimization point. Obviously, the change tendency agrees well with that of the SA curve. For comparison purpose, the corresponding surface profile deviation condition for the case of conventional lens with 0.3mm membrane thickness is also studied. From the results shown in Fig. 8(b), more distinct deviation especially at shorter focal length can be observed within the same effective aperture region. Undoubtedly, the SA is directly related to the residual profile deviation from the ideal aspherical one. With current design, the resultant lens surface profiles are closer to ideal case during the focal length tuning process, therefore demonstrating excellent dynamic SA correction capability.
Fig. 8. The deviation of the membrane profile from the ideal aspherical profile for SA correction at different focal lengths (a) proposed lens (b) conventional lens.
4. Experimental results
For proof of concept demonstration, an injection-molding process as shown in Fig. 9 is developed for prototype fabrication. First, two molds involving the above designed membrane contour and the lens chamber, respectively, are fabricated into polymethylmethacrylate (PMMA) substrates using diamond turning method. These two molds are subsequently assembled together and degassed liquid PDMS prepolymer (a mixture of PDMS base and curing agent (Sylgard 184, Dow Corning, USA) with weight ratio of 10:1) is injected into the hollow cavity of the assembly. After complete curing and demolding process, the lens body can be obtained, and it will be bonded to a glass sheet for cavity sealing purpose so as to form the final device.
Figures 10(a) and 10(b) show the pictures of the as-fabricated molds and the lens boy, respectively. Although currently we have no capability to quantitatively characterize the lens performance, a shearing interference based method as reported in [15,21] can also be used for qualitative demonstration purpose and detailed experiment setup is provided in Fig. 10(c). During experiment, the collimation lens 2 will be moved along the optical axis to capture the interferograms at focus and defocus regions of the lens under test, through which its SA condition can be evaluated.
Fig. 10. Pictures of (a) the adopted mold (b) the as-fabricated lens body and (c) lens performance characterization setup.
Figure 11 shows the captured interferograms from the same lens under test operating at three different focal lengths, namely 24mm, 33mm and 41mm. For each status, three interferograms are captured at positions corresponding to focus, within focus and out of focus (both are 2mm away from the focus) regions. It can be seen that with the lens being adjusted toward longer focal length, the diameter of the laser beam will be monotonously decreaseddue to the reduced amplification ratio of the combined optical system consisting of the tunable lens and the collimation lens 2 with fixed focal length. Most importantly, the interferograms tend to change from curved pattern to straight one accompanying with focal length tuning. Considering theoretical analysis, the interferogram patterns are highly dependent on the SA. If there exists larger SA, more curved pattern will be induced. On the contrary, with the decreasing SA, the pattern will become more and more straight. As a result, reduced SA in the tunable lens can be concluded with the increasing operation focal length, agreeing well with the simulation expectation.
Fig. 11. Captured interferograms from the proposed tunable liquid lens operating at different focal lengths.
5. Discussions
From above mentioned optimization process, it is obvious that the FOM, including the area enclosed by the SA change curve and the standard deviation between all of the data points on the curve, are the most important since they are used as the assessment criteria for optimization. Considering the fact that the calculation of the FOM is highly dependent on the selected focal length range, the membrane can be specifically optimized with respect to certain focal range to meet various application requirements, making current optimization method more flexible and universal. For proof of concept demonstration, the same lens is used except that the targeted operation range is changed to shorter end (25-50 mm) rather than covering a relatively large range from 25 mm to 140 mm. Figures 12(a) and 12(b) show the updated results, it can be clearly seen that if only considering the dynamic SA correction within shorter focal range, the current best membrane design will change to 0.75 mm center membrane thickness with aspherical profile being optimized at 30 mm focal length. For visualized comparison, the corrected SA change curves within the focal length range from 25 mm to 50 mm provided by current (optimized focal length at 30 mm) and above optimized membrane (optimized focal length at 50 mm) designs both are shown in Fig. 13. Obviously, more flat response with smaller amplitude can be achieved with current design, representing better dynamic SA correction performance as expectation.
Fig. 12. Comparison between different membrane designs optimized at different focal lengths for searching global optimization for focal length tuning range from 25 mm to 50 mm with respect to adopted FOM of (a) normalized absolute area and (b) normalized standard deviation.
Fig. 13. Comparison of SA correction performance within focal length range from 25 mm to 50 mm between different membrane designs optimized according to different operation range.
Considering the tunable lens configuration, certain liquid should be filled into the lens body acting as the working medium for both of the mechanical actuation and optical refraction. It’s well known that given the same membrane profile, any variation of the liquid refractive index will change the focal length accordingly. On the other hand, if keeping the focus unchanged with different liquids, the surface profile of the deformed membrane must be changed instead. Since the membrane must be optimized at particular focal length so as to achieve the best dynamic SA correction performance and the residual SA is still focal length dependent, the effect of the liquid refractive index on the lens performance should be also studied, in which the above optimized lens structure is used whilst the refractive index of the filling liquid is changed from 1.334 to 1.4. After performing similar mechanical and optical analysis, similar SA change curve within the focal length tuning range from 25mm to 140mm can be obtained a shown in Fig. 14. For comparison, the original curve as shown in Fig. 7 is also provided. Although a relatively large refractive index change about 5% is introduced, smaller SA within the whole operation range still can be achieved when compared with conventional design, thus keeping the excellent dynamic SA correction capability. The less sensitivity of lens performance to the refractive index change of the filled liquid is thought to be attributed to current membrane arrangement, in which its convex and planar ends are faced to environment air and working liquid in the lens chamber, respectively, constructing a double agglutination lens like structure (one is a equivalent refractive lens consisting of the non-uniform PDMS membrane, the other is a refractive lens made of liquid). In this case, the convex end of membrane will contribute relatively more dioptric power especially in operation region with shorter focal length than the liquid one. Since the membrane convex end is not directly contacted with the filled liquid, the weight of the effect of the liquid refractive index change on optical performance of the whole lens will be largely reduced. Obviously, this insensitivity to the liquid refractive index change will largely simplify the lens fabrication as well as process control, and improve the lens stability in practical application. Besides, it will also become a remarkable advantage for our future achromatic lens design.
Fig. 14. Simulation result about the effect of the liquid refractive index on the dynamic SA correction.
From application point of view, the adoption of even larger lens aperture is much more desired for high efficient light collection and high resolution imaging. As a result, the effectiveness of the proposed design strategy for treating such cases is also studied, in which lens with the same physical diameter of 6 mm but larger effective aperture size of 5 mm is used. Based on the proposed design procedure, the best dynamic spherical aberration correction performance within the focal length range from 25 mm to 140 mm can be obtained through using the membrane with center thickness of 0.44 mm and aspherical surface contour that being optimized at 40 mm focal length. For visualized demonstration, the performance comparison similar to Fig. 7 is also provided in Fig. 15. It can be seen that although lens with larger aperture size is used, which will give rise to a bit larger SA value, significant SA correction capability still can be achieved through using current lens optimization method, demonstrating good availability for various applications.
Fig. 15. Comparison of the SA change curve between the optimized lens and the conventional lens design with different membrane thicknesses in the case of 5 mm effective aperture size.
6. Conclusions
In summary, a new tunable liquid aspherical lens design is proposed for achieving dynamic SA correction capability. Different from existing methods, the membrane’s cross-section is designed to be a plano-convex shape with well-defined aspherical contour, equivalent to a solid elastic lens. It will be deformed during the focal length tuning process. A design flow combining optical ray-tracing and mechanical finite element analysis for the membrane optimization is developed, in which two parametric sweeps about the predetermined focal length and the center membrane thickness are used. Through introducing particularly designed FOMs that can evaluate the SA amplitude and SA change within certain focal length range, the membrane can be exclusively optimized for the interested operation range, with which excellent dynamic SA correction performance has been successfully demonstrated. Besides, good insensitivity of the SA correction performance to the refractive index of the filling liquid has also been disclosed, providing additional advantage for real application.
Funding
National Natural Science Foundation of China (61875244); Ministry of Education of the People's Republic of China (2016JCTD112); Shenzhen Science and Technology Innovation Commission (JCYJ20160414102014801).
Acknowledgments
Mold fabrication and lens characterization were supported by Chongqing Institute of Green and Intelligent Technology, Chinese Academy of Sciences.
Disclosures
For illustrations, authors Hang Zhou, Xinfeng Zhang, Zijian Xu, Peng Wu and Hongbin Yu are represented below as HZ, XZ, ZX, PW and HY.
HZ, XZ, ZX and HB: School of Optical and electronic information, Huazhong University of Science and Technology, PW: Chongqing Institute of Green and Intelligent Technology, Chinese Academy of Sciences.
The authors declare no conflicts of interest.
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