Open Access
Add to BibSonomyAbstract
Inspired by crystalline lenses in human eyes, liquid lenses have a simple yet elegant working principle, and result in compact optical systems. Recent numerical studies showed that membranes with variable thicknesses could affect the lens profile. However, fabrication and assembly of a liquid lens with an inhomogeneous membrane is difficult. There is also a lack of experimental studies about the changes of a lens profile during deformation. In this paper, we provided a new experimental approach for characterizing the performance of a liquid lens with an inhomogeneous membrane. A 2D axisymmetric lens model was built in finite element analysis software to theoretically study the non-linear deformation behavior of the inhomogeneous membrane. Then we provided a new approach to fabricate inhomogeneous membranes using a pre-machined aluminum mold. An optical coherence tomography (OCT) system was used to dynamically measure the changes of a lens profile without contact. Both simulation and the experiments indicated that the variation of the thickness of the membrane could affect the lens profile in a predictable manner. A negative conic constant was achieved when a plano-concave membrane was adopted in a liquid lens. Larger increments of the thickness of the membrane in the radial direction resulted in a larger contribution of a conic constant to the lens profile. The presented study offers guidance for image-quality analysis and optimization of a liquid-lens-based optical system.
© 2017 Optical Society of America
1. Introduction
Liquid lens have been developed in recent years for being used in miniaturized optical systems because of the compact format, low cost, and optical tuning capability over a large dynamic range [1–3]. Because of these advantages, liquid lens can be integrated in a miniaturized optical system and holds the potential to be used in various industrial applications, such as intraocular lens [4, 5], mobile electronics [6], minimally invasive surgery [7–9], microscopy [10, 11]. An important approach to implement focus-tunable liquid lens is using flexible elastomer membrane to encapsulate the optical liquid [12–14]. As the pressure inside the fluid cavity increases, the elastomer membrane deforms and a liquid lens is formed [15, 16].
The influence of the membrane structure on the optical performance of the liquid lens has brought great attention to researchers. D. Shaw group studied variable-focus liquid-filled optical lens with different membrane shapes, which suggested that s membrane with arc cross section has better optical quality than a flat one [17]. S. T. Choi group studied a microlens with a uniform membrane. Their simulation suggested that the diameter and thickness of the membrane were the key parameters to control the effect of gravity [12]. K. Wei group studied a liquid lens with an aspherical membrane at high diopters [18]. Their device showed that the optical resolving power of a thickness varied liquid lens had improved at + 100 dpt, when compared to a liquid lens with an uniform membrane. P. Zhao designed a liquid-tunable lens with a plano-convex membrane. They numerically demonstrated that the optical performance of the aspherical liquid lens improved over a focal length range from 6 mm to 12 mm for a 3 mm clear aperture [13].
Although many researchers have noticed the importance of a thickness variable elastomer membrane on the optical performance of a liquid lens, there is little experimental study on the dynamic changes of lens profile when the focal length varies. It is because of the difficulty in fabricating a flexible membrane with desirable structure and non-contact measuring of the lens profile during deformation. In this paper, we provided a new experimental approach for characterizing the performance of liquid lens with inhomogeneous membranes. A 2D axisymmetric lens model was built in finite element analysis software to theoretically study the non-linear deformation behavior of the inhomogeneous membrane. Then we provided a new approach to fabricate inhomogeneous membranes using a pre-machined aluminum mold. The measurement of a lens profile without contact is difficult. Here we used Optical Coherence Tomography (OCT) to experimentally measure the changes of lens profile when the liquid lens deforms. Finally, the measured lens profiles were imported into an optical design software (ZEMAX) to analysis the optical performance of the liquid lens. Both the simulation and the experiments indicated that the conic constant of the aspherical liquid lens varied as the lens deformed. So the aberration should be optimized across the tuning range of a lens system. A negative conic constant was achieved when a plano-concave membrane was adopted in a liquid lens. Larger increment of the thickness of the membrane in the radial direction resulted in a larger contribution of conic constant to the lens profile. The presented study offered a guidance for image-quality analysis and optimization of liquid lens based optical system.
2. Liquid lens structure and fabrication method
2.1 Lens structure
As shown in Fig. 1, the membrane based fluidic lens consists of a glass substrate, a liquid chamber, a supporting ring, and a polydimethylsiloxane (PDMS) membrane. Optical liquid with a refractive index different from the environment is sealed inside the liquid chamber to provide optical power. The supporting ring works as a clear aperture (6 mm) of the lens. On the sidewall of the liquid chamber, there is an inlet for liquid injection. A syringe pump (not shown in the figure) is connected with the liquid inlet and works as an actuator for lens deformation.
When operating, extra optical fluid is pumped into or out of the liquid chamber. As a result, the curvature of the lens membrane and subsequently optical power is changed. Moreover, the optical resolving power is changed while the surface profile of the PDMS membrane deforms. In order to study the relationship between the membrane structure and surface profile during deformation, PDMS membranes with inhomogeneous thickness are studied, including uniform membrane, plano-concave membrane, and plano-convex membrane. The thickness of a uniform membrane is set to be 0.1 mm. For the plano-convex membranes, the radii of the curve surfaces are 45 mm, and 15 mm, respectively. The boundary thickness is kept to be 0.1 mm. For plano-concave membranes, the radii of the curve surfaces are 15 mm, 25 mm and 45 mm, respectively. Yet the central thickness of the membranes is kept to be 0.1 mm. Figure 2 showed the details of these inhomogeneous membrane.
Fig. 2 Inhomogeneity membrane structure. (a) Isometric view of membrane. (b) Across-section view ofplano-concave membrane (c) Cross-section view of plano-convex membrane. (d) and (e) Details of two typical membranes under investigation.
2.2 Device fabrication
The PDMS membrane is the key component in the lens structure. To define the spherical surfaces of the inhomogeneous membranes, several molds were carefully machined. The membranes were all made by spin coating transparent PDMS (Sylgard 184) layer on top of a plane silicon wafer or a spherical mold.
To make the lens, a glass slide with a thickness of 1 mm was first sonicated in acetone, methanol, and isopropanol for 5 min, successively. After solvent cleaning, the glass slide was rinsed by deionized (DI) water and blown dry by nitrogen. The glass slide was further dehydrated on a hot plate at 200 °C for 10 min. In the next step, the liquid chamber is made by Sylgard 170, a black Polydimethylsiloxane (PDMS) from Dow Corning. The Sylgard 170 curing agent and prepolymer was mixed with 1: 10 ratio and cured in oven at 60 °C for 4 hours. After heat curing, the black PDMS was diced to device size. A circular hole was punched through the black PDMS to form the liquid chamber. Since Sylgard 170 is opaque, scattering light is effectively reduced. To make the supporting ring, a carefully machined aluminum ring with an inner diameter of 6 mm was coated with a thin layer of silicon oxide (100 nm). Finally, the glass slide, the liquid chamber, the supporting ring, as well as the PDMS membrane made beforehand was bonded together. All of the bonding process was performed with UV/ozone surface treatment by placing the PDMS in an UV/ozone cleaner for 3 mins and in an oven at 100 °C overnight. The bonding process proved to be universally applicable to all the sealing steps involved in this work. Finally, the device was filled with optical fluid with a refractive index of 1.67 and connected with a syringe pump for actuation [19, 20].
3. Numerical analysis of lens deformation
The change of surface profile of liquid lens during deformation was numerically studied using finite element analysis software (COMSOL Multiphysics). Since the boundary condition and the exerted force is axial symmetric, we built a two-dimensional axisymmetric model of the liquid lens in solid mechanics module. Under large strains, the deformation of the elastic membrane could be as large as 300%. So Hooke’s law for linear elastic deformation expressed by Young’s modulus is no longer valid. Assuming PDMS as incompressive hyperelastic material, we used Mooney-Rivlin model to examine the nonlinear behavior of the flexible membrane during deformation [14, 21–23]. The strain energy density function for an incompressible material is:
where W represents strain energy density, C10 and C01 are empirically determined material coefficients, I1 and I2 are the first and second invariant of the unimodular component of deformation tensor. For Sylgard 184, C10 is 0.219 MPa, and C01 is 0.0951 MPa, according to prior study by S. J. Lee [24].Figure 3 showed an example of the surface profile of a liquid lens with a uniform membrane under the pressure of 8000 Pa. Figure 3(a) represented the relationship between curvature and conic constants for different membrane structure. To give a detailed analysis of the surface profile, the simulated profile of the membrane is mathematically fitted to Eq. (2) using curvature and conic constant as fitting parameters:
where C is the spherical curvature of the surface, K is the conic constant, R is the radius of a liquid lens, and the αk are the high order aspherical coefficients. When αk are all zeros, the surface of a liquid lens has a form determined by K, namely, hyperbola (K< −1), parabola (K = −1), ellipse (−1 < K < 0), sphere (K = 0), or ellipse (K> 0). The curve fitting was performed over 80% of the diameter of the clear aperture. As showed in Fig. 3(b), the fitting curve fit the simulated data well and the R-square error was within 0.999. Image-quality analyses based on ray-tracing simulations suggest that such deviations in lens profile have insignificant effects on lens design, hence demonstrating that Eq. (2) with two fitting parameters describes the profiles of molded liquid lenses closely. Figure 3(c) showed the deformation of a liquid lens with a plano-concave (r: 25 mm) membrane under different pressure.Fig. 3 (a) A 3D graph of uniform membrane profile. (b) A fitting result of uniform membrane profile. (c) The deformation of plano-concave(r: 25 mm) membrane profile.
The influence of membrane thickness variation along the axial direction was carefully studied in Fig. 4. Figure 4(a) showed the lens profile of various liquid lens with inhomogeneous membrane under the same center deflection (Z = 0.7 mm). Figure 4(b) represented various combinations of curvature and conic constants of thickness variable membrane. From the figures, the liquid lens had an aspherical surface, and the conic constant of the lens varied as the optical power changed. According to Eq. (2), conic constant was of little significance when the lens power was small. As the lens power increased, the contribution of conic constant became important. It should be noted that the conic constant became stable when the optical power of a liquid lens increased. Moreover, the membrane structure significantly affected the conic constant of a deformed surface. Positive conic constants were obtained for liquid lens with a uniform membrane or plano-convex membranes that had the same periphery thickness as the uniform membrane. It was interesting to note that we have achieved negative conic constants for several liquid lens with plano-concave membranes. Larger increment of the thickness in the radial direction resulted in a larger contribution of conic constant. It was because more deformation happened at the central area of the membrane when the periphery was thicker. Same effect can be anticipated by changing elastic constants in the radial direction.
Fig. 4 (a) The surface profiles of PDMS membranes at the same center deflection. (b) Curvature versus conic constant of liquid lens with variable thickness in the radial direction.
4. Experiment and study of lens deformation
4.1 Setup for measuring liquid lens
The liquid lens is deformable with a tunable optical power. As a result, the measurement of the surface profile during deformation becomes tricky. The conventional method of measuring aspherical surfaces is either touch based or need to know the radius of the lens beforehand. Interferometer offers a method to measure the liquid lens without contact, but it can only measure a small area each time. Here we used OCT to measure the dynamic changes of liquid lens during deformation. OCT is a non-invasive, non-contact optical imaging method that can provide micrometer-scale, cross-sectional images of liquid lens [25–27]. The schematic of a spectral domain OCT is shown in Fig. 5. Light emanates from SLD and was divided into two parts, with one part directed towards mirror and another part towards the liquid lens. The reflected light beams from mirror and liquid lens interference, and a reflectivity profile containing spatial structure information in the depth axis was acquired. By laterally scanning a series of these axial depth profiles, a cross-sectional (B-scan) image was created.
4.2 Experimental results
Before measuring the surface profile of a deformed liquid lens, we evaluated the structure of the fabricated membranes using interferometer. Figure 6 showed an example of the measurement. The measured thickness of the membrane agreed with the designed curve, with a deviation smaller than 0.95 μm. It indicated that the fabrication process of the inhomogeneous membranes was reliable.
The OCT B-scan image of a deformed liquid lens was shown in Fig. 7(a). The cross-sectional photograph of the lens was then binarized and the data of the deformed liquid lens was obtained. The curve was fitted according to Eq. (2) (αk are all zero) with two parameters using 80% of the experimental data. As shown in Fig. 7(b), the fitting curve fit the experimental data well with a R-square error of 0.996. Liquid lens with inhomogeneous membrane were fabricated and their profile were measured during deformation. The conic constants and curvature of these four liquid lens during deformation were plotted in Fig. 8(a). The solid curve showed the simulation result and the dots represented the experimental data. A good agreement between the measurement and the simulation indicated that the variation of thickness of the PDMS membrane affected the lens profile in a predictable manner. The data further suggested that liquid lens with plano-concave membrane did produce negative conic constant for certain range of curvature. Plano-concave membrane with smaller radius produced lens profile with smaller conic constant. When analysis the image quality of a liquid lens based optical system, the variation of conic constant of the liquid lens should be taken into consideration, especially when the lens power is large. The membrane structure of the liquid lens provides a tunable parameter to achieve desirable lens profile, as well as optimized image quality and reduced aberration. The small discrepancy between the simulation and the measurement might be attributed to the slightly mismatch of the fitting parameter of Sylgard 184. The measured surface profiles in Fig. 8(a) were then imported into an optical design software (ZEMAX) to simulate the optical performance of the liquid lens. Figure 8(b) showed the obtained RMS spot radius at the vertex deflection of 0.7 mm. From the figure, the RMS spot radius of liquid lens with plano-concave (r = 25 mm) membrane was smaller than that of liquid lens with uniform or plano-convex membranes. However, it should be noted that the optical analysis in Fig. 8(b) was based on a single lens. In a liquid lens tuning/zooming system, several pieces of solid lens and liquid lens should be involved to further minimize aberration over the whole tuning range. In this case, different types of asphericity, as well as membrane structure, will be desirable to achieve appreciated optical performance depending on the applications.
Fig. 7 Experimental results. (a) A OCT image of a liquid lens. The scale bar is 100 μm. (b) A fitting result of a deformed lens profile.
Fig. 8 (a) The relationship between conic constant and curvature of deformed aspherical liquid lens. (b) RMS spot radius at the vertex deflection of 0.7 mm
5. Discussion and conclusion
In this paper, we designed an aspherical liquid lens with predictable surface profile by engineering the structure of PDMS membrane. Plano-convex, plano-concave, as well as uniform membranes were fabricated and tested experimentally in liquid lens. Both simulation and the experiments indicated that the conic constant of the aspherical liquid lens varied as the lens deformed. Liquid lens with uniform membrane and plano-convex membrane produced an elliptical surface, while a negative conic constant was achieved when a plano-concave membrane was adopted in a liquid lens. Moreover, larger increment of the thickness of the membrane in the radial direction resulted in a larger contribution of conic constant. It was because more deformation happened at the central area of the membrane when the periphery was thicker.
Ray tracing analysis suggested that liquid lens with plano-concave membrane exhibit improved optical performance. However, it should be noted that the optical analysis was based on a single lens. The image-quality analysis of a liquid lens tuning/zooming system should be done over the whole tuning range. Just like the traditional lens system, a good liquid lens system will be comprised of several pieces of solid lens and liquid lens. Depending on the system architecture, different types of asphericity will be desirable to achieve appreciated optical performance. Therefore, the membrane structure of the liquid lens provide a tunable parameter to achieve desirable lens profile, as well as optimized image quality and reduced aberration. We believe the study of the lens profile for deformable liquid lens paves the way for the optical design of liquid lens based lens system. Same effect can be anticipated by changing the young’s modulus of the membrane in the radial direction.
Funding
National Natural Science Foundation of China (NSFC) (Grant No. 61401292); the Natural Science Foundation of Jiangsu Province, China (Grant No. BK20140350); the Key University Science Research Project of Jiangsu Province (16KJA510002); the National Key Research and Development Program of China (2016YFF0201005); and the Priority Academic Program Development (PAPD) of Jiangsu Higher Education Institutions.
References and links
1. N. Chronis, G. Liu, K.-H. Jeong, and L. Lee, “Tunable liquid-filled microlens array integrated with microfluidic network,” Opt. Express 11(19), 2370–2378 (2003). [CrossRef] [PubMed]
2. J. Draheim, F. Schneider, R. Kamberger, C. Mueller, and U. Wallrabe, “Fabrication of a fluidic membrane lens system,” J. Micromech. Microeng. 19(9), 095013 (2009). [CrossRef]
3. F. Schneider, J. Draheim, R. Kamberger, P. Waibel, and U. Wallrabe, “Optical characterization of adaptive fluidic silicone-membrane lenses,” Opt. Express 17(14), 11813–11821 (2009). [CrossRef] [PubMed]
4. W. Qiao, D. Johnson, F. S. Tsai, S. H. Cho, and Y.-H. Lo, “Bio-inspired accommodating fluidic intraocular lens,” Opt. Lett. 34(20), 3214–3216 (2009). [CrossRef] [PubMed]
5. W. Qiao, F. S. Tsai, S. H. Cho, H. Yan, and Y.-H. Lo, “Fluidic intraocular lens with a large accommodation range,” IEEE Photonic. Tech. L. 21(5), 304–306 (2009). [CrossRef]
6. I. Park, J. Yang, S. Oh, and S. Chung, “Multifuctional liquid lens for high-performance miniature cameras,” in 2016 IEEE 29th International Conference on Micro Electro Mechanical Systems (MEMS) (IEEE, 2016), pp. 776–779. [CrossRef]
7. F. S. Tsai, D. Johnson, C. S. Francis, S. H. Cho, W. Qiao, A. Arianpour, Y. Mintz, S. Horgan, M. Talamini, and Y. H. Lo, “Fluidic lens laparoscopic zoom camera for minimally invasive surgery,” J. Biomed. Opt. 15(3), 030504 (2010). [CrossRef] [PubMed]
8. A. Miks and J. Novak, “Analysis of two-element zoom systems based on variable power lenses,” Opt. Express 18(7), 6797–6810 (2010). [CrossRef] [PubMed]
9. N. Savidis, G. Peyman, N. Peyghambarian, and J. Schwiegerling, “Nonmechanical zoom system through pressure-controlled tunable fluidic lenses,” Appl. Opt. 52(12), 2858–2865 (2013). [CrossRef] [PubMed]
10. S. Murali, K. P. Thompson, and J. P. Rolland, “Three-dimensional adaptive microscopy using embedded liquid lens,” Opt. Lett. 34(2), 145–147 (2009). [CrossRef] [PubMed]
11. J. M. Jabbour, B. H. Malik, C. Olsovsky, R. Cuenca, S. Cheng, J. A. Jo, Y. S. Cheng, J. M. Wright, and K. C. Maitland, “Optical axial scanning in confocal microscopy using an electrically tunable lens,” Biomed. Opt. Express 5(2), 645–652 (2014). [CrossRef] [PubMed]
12. S. T. Choi, B. S. Son, G. W. Seo, S.-Y. Park, and K.-S. Lee, “Opto-mechanical analysis of nonlinear elastomer membrane deformation under hydraulic pressure for variable-focus liquid-filled microlenses,” Opt. Express 22(5), 6133–6146 (2014). [CrossRef] [PubMed]
13. P. Zhao, Ç. Ataman, and H. Zappe, “Spherical aberration free liquid-filled tunable lens with variable thickness membrane,” Opt. Express 23(16), 21264–21278 (2015). [CrossRef] [PubMed]
14. M. Flores-Bustamante, M. Rosete-Aguilar, and S. Calixto-Carrera, “Mechanical and optical behavior of a tunable liquid lens using a variable cross section membrane: modeling results,” Proc. SPIE 9699, 969908 (2016). [CrossRef]
15. H. Ren and S.-T. Wu, “Variable-focus liquid lens by changing aperture,” Appl. Phys. Lett. 86(21), 211107 (2005). [CrossRef]
16. H. Ren, D. Fox, P. A. Anderson, B. Wu, and S. T. Wu, “Tunable-focus liquid lens controlled using a servo motor,” Opt. Express 14(18), 8031–8036 (2006). [CrossRef] [PubMed]
17. D. Shaw, “Optical properties of variable-focus liquid-filled optical lenses with different membrane shapes,” Opt. Eng. 46(2), 024002 (2007). [CrossRef]
18. K. Wei, H. Huang, Q. Wang, and Y. Zhao, “Focus-tunable liquid lens with an aspherical membrane for improved central and peripheral resolutions at high diopters,” Opt. Express 24(4), 3929–3939 (2016). [CrossRef] [PubMed]
19. W. H. Grover, A. M. Skelley, C. N. Liu, E. T. Lagally, and R. A. Mathies, “Monolithic membrane valves and diaphragm pumps for practical large-scale integration into glass microfluidic devices,” Sens. Actuators B Chem. 89(3), 315–323 (2003). [CrossRef]
20. W. Zhang, S. Lin, C. Wang, J. Hu, C. Li, Z. Zhuang, Y. Zhou, R. A. Mathies, and C. J. Yang, “PMMA/PDMS valves and pumps for disposable microfluidics,” Lab Chip 9(21), 3088–3094 (2009). [CrossRef] [PubMed]
21. F. Schneider, T. Fellner, J. Wilde, and U. Wallrabe, “Mechanical properties of silicones for MEMS,” J. Micromech. Microeng. 18(6), 065008 (2008). [CrossRef]
22. I. Johnston, D. McCluskey, C. Tan, and M. Tracey, “Mechanical characterization of bulk Sylgard 184 for microfluidics and microengineering,” J. Micromech. Microeng. 24(3), 035017 (2014). [CrossRef]
23. T. K. Kim, J. K. Kim, and O. C. Jeong, “Measurement of nonlinear mechanical properties of PDMS elastomer,” Microelectron. Eng. 88(8), 1982–1985 (2011). [CrossRef]
24. S. Lee, J. C. Chan, K. Maung, E. Rezler, and N. Sundararajan, “Characterization of laterally deformable elastomer membranes for microfluidics,” J. Micromech. Microeng. 17(5), 843–851 (2007). [CrossRef]
25. J. G. Fujimoto, M. E. Brezinski, G. J. Tearney, S. A. Boppart, B. Bouma, M. R. Hee, J. F. Southern, and E. A. Swanson, “Optical biopsy and imaging using optical coherence tomography,” Nat. Med. 1(9), 970–972 (1995). [CrossRef] [PubMed]
26. J. P. Dunkers, F. R. Phelan, D. P. Sanders, M. J. Everett, W. H. Green, D. L. Hunston, and R. S. Parnas, “The application of optical coherence tomography to problems in polymer matrix composites,” Opt. Lasers Eng. 35(3), 135–147 (2001). [CrossRef]
27. P. H. Tomlins and R. K. Wang, “Theory, developments and applications of optical coherence tomography,” J. Phys. D Appl. Phys. 38(15), 2519–2535 (2005). [CrossRef]
